Multiple Correct Answers MCQs for Sub-Topics of Topic 10: Calculus

Question 1. Which of the following statements accurately describe the concept of a limit $\lim\limits_{x \to a} f(x) = L$? (Multiple Correct Answers)

(A) For the limit to exist, $f(x)$ must be defined at $x=a$.

(B) The value $L$ is the value that $f(x)$ approaches as $x$ gets arbitrarily close to $a$ from both sides.

(C) If the limit exists, the left-hand limit and the right-hand limit must be equal to $L$ and be finite.

(D) The limit describes the behavior of the function *near* the point $a$, not necessarily *at* $a$.

(E) The limit can exist even if the function is undefined at $x=a$.

Question 2. Consider the function $f(x) = \frac{|x-3|}{x-3}$. Which of the following statements are true regarding the point $x=3$? (Multiple Correct Answers)

(A) $f(3)$ is defined.

(B) The left-hand limit $\lim\limits_{x \to 3^-} f(x) = -1$.

(C) The right-hand limit $\lim\limits_{x \to 3^+} f(x) = 1$.

(D) The limit $\lim\limits_{x \to 3} f(x)$ exists.

(E) The function has a jump discontinuity at $x=3$.

Question 3. Which of the following yield an indeterminate form upon direct substitution? (Multiple Correct Answers)

(A) $\lim\limits_{x \to 0} \frac{\sin x}{x}$

(B) $\lim\limits_{x \to 1} \frac{x^2 - 1}{x - 1}$

(C) $\lim\limits_{x \to 0} \frac{e^x - 1}{x}$

(D) $\lim\limits_{x \to 2} \frac{\sqrt{x+2} - 2}{x-2}$

(E) $\lim\limits_{x \to 0} \frac{1}{x}$

Question 4. To evaluate $\lim\limits_{x \to a} f(x)$ when direct substitution yields an indeterminate form, which of the following techniques might be applicable? (Multiple Correct Answers)

(A) Factorizing and cancelling common factors.

(B) Rationalizing the numerator or the denominator.

(C) Using algebraic manipulation to simplify the expression.

(D) Applying standard limit formulas (if applicable).

(E) Checking if the left-hand limit equals the right-hand limit (often done after simplification).

Question 5. Consider the function $g(x) = \begin{cases} 5-2x & , & x < 2 \\ 1 & , & x = 2 \\ x^2-3 & , & x > 2 \end{cases}$. Which of the following statements are true? (Multiple Correct Answers)

(A) The left-hand limit of $g(x)$ as $x \to 2$ is 1.

(B) The right-hand limit of $g(x)$ as $x \to 2$ is 1.

(C) $\lim\limits_{x \to 2} g(x)$ exists and is equal to 1.

(D) $g(2) = 1$.

(E) The function $g(x)$ is continuous at $x=2$.

Question 6. For $\lim\limits_{x \to a} f(x)$ to exist, which properties must hold true for the left-hand limit ($\lim\limits_{x \to a^-} f(x)$) and the right-hand limit ($\lim\limits_{x \to a^+} f(x)$)? (Multiple Correct Answers)

(A) Both limits must exist.

(B) Both limits must be finite.

(C) The two limits must be equal.

(D) The limits must be non-zero.

(E) The function value $f(a)$ must be equal to the limits.

Question 7. Evaluate $\lim\limits_{x \to 9} \frac{3 - \sqrt{x}}{x - 9}$. Which evaluation technique(s) are appropriate? (Multiple Correct Answers)

(A) Direct Substitution.

(B) Factorization.

(C) Rationalization of the numerator.

(D) Using the standard form $\lim\limits_{x \to a} \frac{x^n - a^n}{x - a} = n a^{n-1}$ by rewriting the expression.

(E) Applying the Squeeze Theorem.

Question 8. If $\lim\limits_{x \to a} f(x) = L$, where $L$ is a finite number, which of the following must be true? (Multiple Correct Answers)

(A) $f(x)$ is continuous at $x=a$.

(B) The graph of $f(x)$ has no hole or jump at $x=a$.

(C) For any sequence $x_n \to a$ (with $x_n \neq a$), the sequence $f(x_n) \to L$.

(D) The values of $f(x)$ are arbitrarily close to $L$ for $x$ sufficiently close to $a$ (but $x \neq a$).

(E) $f(a) = L$.

Question 9. Identify the limits that can be evaluated using direct substitution. (Multiple Correct Answers)

(A) $\lim\limits_{x \to 5} (x^2 - 3x + 1)$

(B) $\lim\limits_{x \to \pi/2} \cos x$

(C) $\lim\limits_{x \to 0} \frac{x^2+1}{x+1}$

(D) $\lim\limits_{x \to -1} \sqrt{x^2 + 3}$

(E) $\lim\limits_{x \to 0} \frac{|x|}{x}$

Question 10. If $\lim\limits_{x \to a} f(x) = L$ and $\lim\limits_{x \to a} g(x) = M$, where $L$ and $M$ are finite real numbers, which of the following limit properties are valid? (Multiple Correct Answers)

(A) $\lim\limits_{x \to a} (f(x) + g(x)) = L+M$

(B) $\lim\limits_{x \to a} (f(x) \cdot g(x)) = L \cdot M$

(C) $\lim\limits_{x \to a} \frac{f(x)}{g(x)} = \frac{L}{M}$, provided $M \neq 0$.

(D) $\lim\limits_{x \to a} (f(x))^n = L^n$, for any positive integer $n$.

(E) $\lim\limits_{x \to a} \sqrt{f(x)} = \sqrt{L}$, provided $L \geq 0$.

Question 11. Consider the function $f(x) = \begin{cases} ax+3 & , & x < 2 \\ b & , & x = 2 \\ x^2+a & , & x > 2 \end{cases}$. For $\lim\limits_{x \to 2} f(x)$ to exist, which conditions must be satisfied? (Multiple Correct Answers)

(A) The left-hand limit must exist.

(B) The right-hand limit must exist.

(C) $\lim\limits_{x \to 2^-} f(x) = \lim\limits_{x \to 2^+} f(x)$.

(D) The value of $b$ must be equal to the limit.

(E) The limit must be a specific value independent of $b$, but possibly depending on $a$.

Question 1. Which of the following are standard algebraic limits? (Multiple Correct Answers)

(A) $\lim\limits_{x \to a} \frac{x^n - a^n}{x - a} = n a^{n-1}$ (for rational $n$)

(B) $\lim\limits_{x \to 0} \frac{\sin x}{x} = 1$

(C) $\lim\limits_{x \to 0} (1+x)^{1/x} = e$

(D) $\lim\limits_{x \to \infty} \frac{1}{x^n} = 0$ (for $n > 0$)

(E) $\lim\limits_{x \to 0} \frac{e^x - 1}{x} = 1$

Question 2. According to the Algebra of Limits, if $\lim\limits_{x \to a} f(x) = L$ and $\lim\limits_{x \to a} g(x) = M$, where $L, M \in \mathbb{R}$, which of the following are true? (Multiple Correct Answers)

(A) $\lim\limits_{x \to a} (f(x) - g(x)) = L - M$

(B) $\lim\limits_{x \to a} (f(x) \cdot g(x)) = L \cdot M$

(C) $\lim\limits_{x \to a} \frac{f(x)}{g(x)} = \frac{L}{M}$, provided $M \neq 0$.

(D) $\lim\limits_{x \to a} c f(x) = c L$ for any constant $c$.

(E) $\lim\limits_{x \to a} (f(x))^p = L^p$ for any real number $p$, provided $L^p$ is defined.

Question 3. The Squeeze Play Theorem can be used to evaluate $\lim\limits_{x \to a} f(x)$ if which conditions are met? (Multiple Correct Answers)

(A) There exist functions $g(x)$ and $h(x)$ such that $g(x) \leq f(x) \leq h(x)$ for all $x$ in some open interval containing $a$, except possibly at $a$.

(B) $\lim\limits_{x \to a} g(x)$ exists.

(C) $\lim\limits_{x \to a} h(x)$ exists.

(D) $\lim\limits_{x \to a} g(x) = \lim\limits_{x \to a} h(x) = L$ for some limit $L$.

(E) $f(x)$ must be continuous at $x=a$.

Question 4. Which of the following are standard trigonometric limits? (Multiple Correct Answers)

(A) $\lim\limits_{x \to 0} \frac{\tan x}{x} = 1$

(B) $\lim\limits_{x \to 0} \frac{\sin x}{x} = 1$

(C) $\lim\limits_{x \to 0} \frac{1 - \cos x}{x} = 0$

(D) $\lim\limits_{x \to 0} \frac{\sin ax}{bx} = a/b$ (for $b \neq 0$)

(E) $\lim\limits_{x \to 0} \frac{\tan ax}{\tan bx} = a/b$ (for $b \neq 0$)

Question 5. Which of the following are standard limits involving exponential or logarithmic functions? (Multiple Correct Answers)

(A) $\lim\limits_{x \to 0} \frac{e^x - 1}{x} = 1$

(B) $\lim\limits_{x \to 0} \frac{\log_e(1+x)}{x} = 1$

(C) $\lim\limits_{x \to \infty} (1 + \frac{1}{x})^x = e$

(D) $\lim\limits_{x \to 0} \frac{a^x - 1}{x} = \log_e a$ (for $a > 0$)

(E) $\lim\limits_{x \to e} \frac{\log_e x - 1}{x - e} = 1/e$

Question 6. Evaluate $\lim\limits_{x \to 0} \frac{\sin(3x)}{5x}$. Which of the following steps or results are correct? (Multiple Correct Answers)

(A) This is an indeterminate form of type $0/0$.

(B) We can rewrite this as $\lim\limits_{x \to 0} \frac{\sin(3x)}{3x} \cdot \frac{3x}{5x}$.

(C) Using the standard limit, $\lim\limits_{x \to 0} \frac{\sin(3x)}{3x} = 1$.

(D) The limit is equal to $3/5$.

(E) Direct substitution works for this limit.

Question 7. Consider the limit $\lim\limits_{x \to 1} \frac{\log_e x}{x-1}$. Which of the following are true? (Multiple Correct Answers)

(A) Direct substitution yields the indeterminate form $0/0$.

(B) We can use the substitution $y = x-1$, so as $x \to 1$, $y \to 0$.

(C) The limit becomes $\lim\limits_{y \to 0} \frac{\log_e(1+y)}{y}$.

(D) Using a standard limit, the value is 1.

(E) This limit is related to the definition of the derivative of $\log_e x$ at $x=1$.

Question 8. Which of the following functions are continuous at $x=0$, allowing for direct substitution of the limit? (Multiple Correct Answers)

(A) $f(x) = x^2 + 5$

(B) $f(x) = \sin x$

(C) $f(x) = e^x$

(D) $f(x) = |x|$

(E) $f(x) = \frac{1}{x}$

Question 9. If $\lim\limits_{x \to a} f(x) = L$ and $\lim\limits_{x \to a} g(x) = M$, where $L$ is finite and $M=0$, which of the following statements about $\lim\limits_{x \to a} \frac{f(x)}{g(x)}$ can be true? (Multiple Correct Answers)

(A) The limit is undefined.

(B) The limit is $L/M$ if $M \neq 0$.

(C) The limit can be $\infty$ or $-\infty$ if $L \neq 0$.

(D) The limit can be an indeterminate form if $L = 0$.

(E) The limit does not exist in $\mathbb{R}$.

Question 10. Using the standard limit $\lim\limits_{x \to 0} \frac{a^x - 1}{x} = \log_e a$, evaluate $\lim\limits_{x \to 0} \frac{2^{x+1} - 2}{x}$. Which steps or results are correct? (Multiple Correct Answers)

(A) The expression can be written as $\frac{2 \cdot 2^x - 2}{x}$.

(B) The expression can be written as $2 \cdot \frac{2^x - 1}{x}$.

(C) Using the standard limit, $\lim\limits_{x \to 0} \frac{2^x - 1}{x} = \log_e 2$.

(D) The value of the limit is $2 \log_e 2$.

(E) The value of the limit is $\log_e 4$.

Question 11. The limit $\lim\limits_{x \to \infty} (1 + \frac{k}{x})^x$ for a constant $k$ is equal to $e^k$. Which of the following limits can be evaluated using this result? (Multiple Correct Answers)

(A) $\lim\limits_{x \to \infty} (1 + \frac{5}{x})^x$

(B) $\lim\limits_{x \to \infty} (1 + \frac{1}{2x})^x$

(C) $\lim\limits_{x \to \infty} (1 - \frac{3}{x})^x$

(D) $\lim\limits_{x \to 0} (1 + x)^{1/x}$ (by substitution)

(E) $\lim\limits_{n \to \infty} (1 + \frac{1}{n})^n$ (where n is an integer)

Question 1. For a function $f(x)$ to be continuous at a point $x=a$, which conditions must be satisfied? (Multiple Correct Answers)

(A) $\lim\limits_{x \to a} f(x)$ must exist.

(B) $f(a)$ must be defined.

(C) $\lim\limits_{x \to a} f(x) = f(a)$.

(D) The left-hand limit and the right-hand limit must be equal and finite.

(E) The function must be differentiable at $x=a$.

Question 2. Identify the points of discontinuity for the function $f(x) = \frac{x^2 - 1}{x - 1}$. (Multiple Correct Answers)

(A) $x=1$

(B) $x=-1$

(C) $x=0$

(D) No points of discontinuity.

(E) The function is discontinuous at $x=1$ because it is undefined there, even though the limit exists.

Question 3. If $f(x)$ and $g(x)$ are continuous at $x=a$, which of the following functions are also necessarily continuous at $x=a$? (Assume denominators are non-zero where applicable). (Multiple Correct Answers)

(A) $f(x) + g(x)$

(B) $f(x) - g(x)$

(C) $f(x) \cdot g(x)$

(D) $\frac{f(x)}{g(x)}$ (provided $g(a) \neq 0$)

(E) $c \cdot f(x)$ (for any constant $c$)

Question 4. Consider the function $f(x) = \begin{cases} 2x+1 & , & x \leq 0 \\ x^2+1 & , & x > 0 \end{cases}$. Which of the following are true? (Multiple Correct Answers)

(A) $\lim\limits_{x \to 0^-} f(x) = 1$.

(B) $\lim\limits_{x \to 0^+} f(x) = 1$.

(C) $\lim\limits_{x \to 0} f(x)$ exists and is equal to 1.

(D) $f(0) = 1$.

(E) The function is continuous at $x=0$.

Question 5. Which of the following describe types of discontinuity? (Multiple Correct Answers)

(A) Removable Discontinuity (where the limit exists but is not equal to the function value or the function is undefined).

(B) Jump Discontinuity (where left and right limits exist but are not equal).

(C) Infinite Discontinuity (where one or both limits are infinite).

(D) Point Discontinuity (can refer to removable discontinuity).

(E) Oscillating Discontinuity (where the function oscillates wildly near the point, and the limit does not exist).

Question 6. A function is continuous on a closed interval $[a, b]$ if it is continuous on the open interval $(a, b)$ and which additional conditions? (Multiple Correct Answers)

(A) $\lim\limits_{x \to a^+} f(x)$ exists.

(B) $\lim\limits_{x \to b^-} f(x)$ exists.

(C) $\lim\limits_{x \to a^+} f(x) = f(a)$.

(D) $\lim\limits_{x \to b^-} f(x) = f(b)$.

(E) $f(x)$ is differentiable on $(a, b)$.

Question 7. If $g$ is continuous at $a$ and $f$ is continuous at $g(a)$, then the composite function $(f \circ g)(x) = f(g(x))$ is continuous at $x=a$. Which of the following are consequences of this property? (Multiple Correct Answers)

(A) If $f(x)$ is continuous, then $|f(x)|$ is continuous.

(B) If $f(x)$ is continuous and $f(x) \geq 0$, then $\sqrt{f(x)}$ is continuous.

(C) If $f(x)$ is a polynomial, it is continuous everywhere.

(D) If $f(x) = \sin x$, then $f(x^2) = \sin(x^2)$ is continuous.

(E) If $f(x)$ is continuous, then $1/f(x)$ is continuous (assuming $f(x) \neq 0$).

Question 8. For what values of the constants $a$ and $b$ is the function $f(x) = \begin{cases} x+a\sqrt{2}\sin x & , & 0 \leq x < \pi/4 \\ 2x \cot x + b & , & \pi/4 \leq x \leq \pi/2 \\ a \cos 2x - b \sin x & , & \pi/2 < x \leq \pi \end{cases}$ continuous on $[0, \pi]$? (Multiple Correct Answers)

(A) The function must be continuous at $x=\pi/4$ and $x=\pi/2$.

(B) At $x=\pi/4$: $\lim\limits_{x \to \pi/4^-} f(x) = \frac{\pi}{4} + a\sqrt{2} \sin(\pi/4) = \frac{\pi}{4} + a$. $f(\pi/4) = 2(\pi/4)\cot(\pi/4) + b = \pi/2 + b$. So $\frac{\pi}{4} + a = \frac{\pi}{2} + b$.

(C) At $x=\pi/2$: $f(\pi/2) = 2(\pi/2)\cot(\pi/2) + b = b$. $\lim\limits_{x \to \pi/2^+} f(x) = a \cos(\pi) - b \sin(\pi/2) = -a - b$. So $b = -a - b$.

(D) From the equations, $a-b = \pi/4$ and $a+2b=0$.

(E) Solving the system gives $3b = -\pi/4$, $b = -\pi/12$, and $a = \pi/4 + b = \pi/4 - \pi/12 = 2\pi/12 = \pi/6$.

Question 9. Which of the following functions are continuous on their respective natural domains in $\mathbb{R}$? (Multiple Correct Answers)

(A) Polynomial functions.

(B) Rational functions (except at points where the denominator is zero).

(C) Trigonometric functions $\sin x$ and $\cos x$ (on $\mathbb{R}$).

(D) Exponential functions $a^x$ ($a>0$) (on $\mathbb{R}$).

(E) Logarithmic functions $\log_a x$ ($a>0, a\neq 1$) on their domain $(0, \infty)$.

Question 10. If $f(x)$ is continuous on $[a, b]$ and $f(a) \cdot f(b) < 0$, then according to the Intermediate Value Theorem, which of the following are true? (Multiple Correct Answers)

(A) There exists at least one $c \in (a, b)$ such that $f(c) = 0$.

(B) The function must be monotonic on $[a, b]$.

(C) The function must cross the x-axis at least once in $(a, b)$.

(D) There exists exactly one $c \in (a, b)$ such that $f(c) = 0$.

(E) For any value $k$ between $f(a)$ and $f(b)$, there exists $c \in (a, b)$ such that $f(c) = k$.

Question 11. Consider the function $f(x) = \lfloor x \rfloor$. Which of the following are true? (Multiple Correct Answers)

(A) The function is continuous at any non-integer point.

(B) The function has a jump discontinuity at every integer point.

(C) $\lim\limits_{x \to n^-} \lfloor x \rfloor = n-1$ for any integer $n$.

(D) $\lim\limits_{x \to n^+} \lfloor x \rfloor = n$ for any integer $n$.

(E) The function is continuous on $[0, 1)$.

Question 12. If a function $f(x)$ is continuous on $[a, b]$, which properties are guaranteed by theorems? (Multiple Correct Answers)

(A) $f(x)$ attains its maximum value in $[a, b]$ (Extreme Value Theorem).

(B) $f(x)$ attains its minimum value in $[a, b]$ (Extreme Value Theorem).

(C) $f(x)$ is differentiable on $(a, b)$.

(D) If $f(a) < 0$ and $f(b) > 0$, there is a root in $(a, b)$ (Intermediate Value Theorem).

(E) $\int_a^b f(x) dx$ exists.

Question 1. The derivative of a function $f(x)$ at a point $x=a$, denoted by $f'(a)$, is defined using limits. Which expressions correctly represent this definition? (Multiple Correct Answers)

(A) $\lim\limits_{h \to 0} \frac{f(a+h) - f(a)}{h}$, provided the limit exists and is finite.

(B) $\lim\limits_{x \to a} \frac{f(x) - f(a)}{x - a}$, provided the limit exists and is finite.

(C) $\lim\limits_{h \to 0} \frac{f(a) - f(a-h)}{h}$, provided the limit exists and is finite.

(D) $\lim\limits_{h \to 0^+} \frac{f(a+h) - f(a)}{h}$ (Right-hand derivative)

(E) $\lim\limits_{h \to 0^-} \frac{f(a+h) - f(a)}{h}$ (Left-hand derivative)

Question 2. For a function $f(x)$ to be differentiable at a point $x=a$, which conditions must be met? (Multiple Correct Answers)

(A) $f(x)$ must be continuous at $x=a$.

(B) The left-hand derivative at $x=a$ must exist.

(C) The right-hand derivative at $x=a$ must exist.

(D) The left-hand derivative must be equal to the right-hand derivative at $x=a$ and be finite.

(E) The derivative $f'(a)$ must be non-zero.

Question 3. If a function is differentiable at a point $x=a$, which of the following statements are consequences? (Multiple Correct Answers)

(A) The function is continuous at $x=a$.

(B) The graph of the function has a unique tangent line at $(a, f(a))$ that is not vertical.

(C) The derivative $f'(a)$ exists and is finite.

(D) The left-hand derivative equals the right-hand derivative at $x=a$.

(E) The function $f(x)$ must be monotonic in a neighborhood of $a$.

Question 4. Which of the following functions are continuous at $x=0$ but not differentiable at $x=0$? (Multiple Correct Answers)

(A) $f(x) = |x|$

(B) $f(x) = x |x|$

(C) $f(x) = \lfloor x \rfloor$

(D) $f(x) = x^{1/3}$

(E) $f(x) = \begin{cases} x \sin(1/x) & , & x \neq 0 \\ 0 & , & x = 0 \end{cases}$

Question 5. A function is differentiable in an interval $(a, b)$ if which condition holds? (Multiple Correct Answers)

(A) It is continuous throughout $(a, b)$.

(B) The derivative exists at every point $c \in (a, b)$.

(C) The left-hand and right-hand derivatives exist and are equal and finite at every point in $(a, b)$.

(D) The limit $\lim\limits_{h \to 0} \frac{f(x+h) - f(x)}{h}$ exists for every $x \in (a, b)$.

(E) It is differentiable at the endpoints $a$ and $b$.

Question 6. Consider the function $f(x) = x^2$. Which of the following are true about its derivative from the first principle at $x=a$? (Multiple Correct Answers)

(A) $f(a+h) = (a+h)^2 = a^2 + 2ah + h^2$.

(B) $\frac{f(a+h) - f(a)}{h} = \frac{(a^2 + 2ah + h^2) - a^2}{h}$.

(C) $\frac{f(a+h) - f(a)}{h} = 2a + h$.

(D) $f'(a) = \lim\limits_{h \to 0} (2a + h) = 2a$.

(E) The derivative is always positive for $a>0$.

Question 7. If a function $f(x)$ is discontinuous at a point $x=a$, which conclusions can be drawn about its differentiability at $x=a$? (Multiple Correct Answers)

(A) $f(x)$ cannot be differentiable at $x=a$.

(B) $\lim\limits_{h \to 0} \frac{f(a+h) - f(a)}{h}$ cannot exist finitely.

(C) At least one of the conditions for the existence of the limit (and hence the derivative) must fail at $x=a$.

(D) The left-hand and right-hand derivatives will be unequal or at least one will not exist/be infinite.

(E) The function may still have a left-hand or right-hand derivative, but not both finite and equal.

Question 8. Which of the following statements correctly describe the relationship between differentiability and continuity? (Multiple Correct Answers)

(A) Differentiability at a point implies continuity at that point.

(B) Continuity at a point implies differentiability at that point.

(C) A function can be continuous at a point without being differentiable at that point.

(D) A function can be differentiable at a point without being continuous at that point.

(E) If a function is not continuous at a point, it cannot be differentiable at that point.

Question 9. The geometric interpretation of the derivative of $f(x)$ at $x=a$ is the slope of the tangent line to the graph of $y=f(x)$ at $(a, f(a))$. What does this imply? (Multiple Correct Answers)

(A) If $f'(a)$ is positive, the function is increasing at $x=a$.

(B) If $f'(a) = 0$, the tangent line is horizontal.

(C) If $f'(a)$ is undefined, the tangent line might be vertical or there might be a sharp corner/cusp.

(D) The existence of a finite derivative at $a$ means the graph is "smooth" at $(a, f(a))$ and has a non-vertical tangent.

(E) The derivative is always positive if the function is increasing over an interval.

Question 10. Consider the function $f(x) = x |x|$. Which of the following are true? (Multiple Correct Answers)

(A) $f(x) = \begin{cases} x^2 & , & x \geq 0 \\ -x^2 & , & x < 0 \end{cases}$.

(B) $f(x)$ is continuous at $x=0$.

(C) The left-hand derivative at $x=0$ is 0.

(D) The right-hand derivative at $x=0$ is 0.

(E) $f(x)$ is differentiable at $x=0$ and $f'(0) = 0$.

Question 11. Which of the following statements about differentiability are correct? (Multiple Correct Answers)

(A) If $f'(x) > 0$ in an interval, $f(x)$ is strictly increasing in that interval.

(B) If $f'(x) = 0$ at $x=c$, then $f(x)$ has a local extremum at $x=c$.

(C) Differentiation is a linear operator, i.e., $(af+bg)' = af' + bg'$ for constants $a, b$ and differentiable functions $f, g$.

(D) The derivative can be interpreted as the instantaneous rate of change.

(E) The derivative from the first principle is the limit of the average rate of change as the interval length approaches zero.

Question 1. Which of the following are standard differentiation rules? (Multiple Correct Answers)

(A) Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$ (for any real number $n$)

(B) Sum Rule: $\frac{d}{dx}(f(x) + g(x)) = f'(x) + g'(x)$

(C) Product Rule: $\frac{d}{dx}(f(x)g(x)) = f'(x)g(x) + f(x)g'(x)$

(D) Quotient Rule: $\frac{d}{dx}(\frac{f(x)}{g(x)}) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$

(E) Chain Rule: $\frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x)$

Question 2. Find the derivative of $y = 4x^5 - 7x^2 + \sqrt{x} - \frac{1}{x} + 10$. Which terms in the derivative are correct? (Multiple Correct Answers)

(A) The derivative of $4x^5$ is $20x^4$.

(B) The derivative of $-7x^2$ is $-14x$.

(C) The derivative of $\sqrt{x}$ is $\frac{1}{2\sqrt{x}}$.

(D) The derivative of $-\frac{1}{x}$ is $\frac{1}{x^2}$.

(E) The derivative of $10$ is $10x$.

Question 3. Which of the following are correct standard derivatives of trigonometric functions? (Multiple Correct Answers)

(A) $\frac{d}{dx}(\sin x) = \cos x$

(B) $\frac{d}{dx}(\cos x) = -\sin x$

(C) $\frac{d}{dx}(\tan x) = \sec^2 x$

(D) $\frac{d}{dx}(\cot x) = -\text{cosec}^2 x$

(E) $\frac{d}{dx}(\sec x) = \sec x \tan x$

Question 4. Find the derivative of $y = e^x \cos x$. Which steps or final results are correct using the product rule? (Multiple Correct Answers)

(A) Let $u = e^x$ and $v = \cos x$.

(B) $u' = e^x$ and $v' = -\sin x$.

(C) $\frac{dy}{dx} = u'v + uv'$.

(D) $\frac{dy}{dx} = e^x \cos x + e^x (-\sin x)$.

(E) $\frac{dy}{dx} = e^x (\cos x - \sin x)$.

Question 5. Find the derivative of $y = \frac{\sin x}{x}$. Which steps or final results are correct using the quotient rule? (Multiple Correct Answers)

(A) Let $u = \sin x$ and $v = x$.

(B) $u' = \cos x$ and $v' = 1$.

(C) $\frac{dy}{dx} = \frac{u'v - uv'}{v^2}$.

(D) $\frac{dy}{dx} = \frac{\cos x \cdot x - \sin x \cdot 1}{x^2}$.

(E) $\frac{dy}{dx} = \frac{x \cos x - \sin x}{x^2}$.

Question 6. Which of the following are correct standard derivatives of exponential or logarithmic functions? (Multiple Correct Answers)

(A) $\frac{d}{dx}(e^x) = e^x$

(B) $\frac{d}{dx}(a^x) = a^x \log_e a$ (for $a > 0$)

(C) $\frac{d}{dx}(\log_e x) = \frac{1}{x}$ (for $x > 0$)

(D) $\frac{d}{dx}(\log_a x) = \frac{1}{x \log_e a}$ (for $a > 0, a \neq 1, x > 0$)

(E) $\frac{d}{dx}(\log_e |x|) = \frac{1}{|x|}$ (for $x \neq 0$)

Question 7. Find the derivative of $y = \sec x + \text{cosec } x$. Which terms in the derivative are correct? (Multiple Correct Answers)

(A) The derivative of $\sec x$ is $\sec x \tan x$.

(B) The derivative of $\text{cosec } x$ is $\text{cosec } x \cot x$.

(C) The derivative of $\text{cosec } x$ is $-\text{cosec } x \cot x$.

(D) $\frac{dy}{dx} = \sec x \tan x - \text{cosec } x \cot x$.

(E) $\frac{dy}{dx} = \sec x \tan x + \text{cosec } x \cot x$.

Question 8. Which of the following represent the derivative of a polynomial function? (Multiple Correct Answers)

(A) If $P(x) = x^n$, $P'(x) = nx^{n-1}$.

(B) If $P(x) = c$ (constant), $P'(x) = 0$.

(C) If $P(x) = a_n x^n + \dots + a_1 x + a_0$, $P'(x) = n a_n x^{n-1} + \dots + a_1$.

(D) The derivative of a polynomial is always a polynomial of one degree lower (unless the original polynomial is constant).

(E) The derivative of a linear polynomial $ax+b$ is $a$.

Question 9. Find the derivative of $y = \tan x - \cot x$. Which terms in the derivative are correct? (Multiple Correct Answers)

(A) The derivative of $\tan x$ is $\sec^2 x$.

(B) The derivative of $-\cot x$ is $\text{cosec}^2 x$.

(C) The derivative of $-\cot x$ is $-\text{cosec}^2 x$.

(D) $\frac{dy}{dx} = \sec^2 x + \text{cosec}^2 x$.

(E) $\frac{dy}{dx} = \sec^2 x - \text{cosec}^2 x$.

Question 10. Which of the following standard results on derivatives are correct? (Multiple Correct Answers)

(A) $\frac{d}{dx}(c) = 0$

(B) $\frac{d}{dx}(x) = 1$

(C) $\frac{d}{dx}(\sin x) = \cos x$

(D) $\frac{d}{dx}(\log_e x) = 1/x$ (for $x>0$)

(E) $\frac{d}{dx}(e^x) = e^x$

Question 11. Evaluate the derivative of $y = \frac{e^x}{x^2}$. Which steps are correct using the quotient rule? (Multiple Correct Answers)

(A) $u = e^x$, $v = x^2$.

(B) $u' = e^x$, $v' = 2x$.

(C) $\frac{dy}{dx} = \frac{u'v - uv'}{v^2}$.

(D) $\frac{dy}{dx} = \frac{e^x x^2 - e^x 2x}{(x^2)^2}$.

(E) $\frac{dy}{dx} = \frac{e^x (x-2)}{x^3}$.

Question 1. The chain rule is used to find the derivative of composite functions. If $y=f(u)$ and $u=g(x)$, which expressions correctly represent the chain rule? (Multiple Correct Answers)

(A) $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$

(B) $\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$

(C) $\frac{dy}{dx} = f'(u) \cdot u'(x)$

(D) $\frac{dy}{dx} = \frac{df}{dg} \cdot \frac{dg}{dx}$

(E) $\frac{dy}{dx} = \frac{du}{dy} \cdot \frac{dx}{du}$

Question 2. Find the derivative of $y = \sin(x^3)$. Which intermediate steps or final results are correct using the chain rule? (Multiple Correct Answers)

(A) Let $u = x^3$, then $y = \sin u$.

(B) $\frac{dy}{du} = \cos u$.

(C) $\frac{du}{dx} = 3x^2$.

(D) $\frac{dy}{dx} = \cos u \cdot 3x^2$.

(E) $\frac{dy}{dx} = \cos(x^3) \cdot 3x^2$.

Question 3. Find the derivative of $y = (2x^2+5)^4$. Which intermediate steps or final results are correct using the chain rule? (Multiple Correct Answers)

(A) Let $u = 2x^2+5$, then $y = u^4$.

(B) $\frac{dy}{du} = 4u^3$.

(C) $\frac{du}{dx} = 4x$.

(D) $\frac{dy}{dx} = 4u^3 \cdot 4x$.

(E) $\frac{dy}{dx} = 4(2x^2+5)^3 \cdot 4x = 16x(2x^2+5)^3$.

Question 4. Find the derivative of $y = e^{\tan x}$. Which intermediate steps or final results are correct using the chain rule? (Multiple Correct Answers)

(A) Let $u = \tan x$, then $y = e^u$.

(B) $\frac{dy}{du} = e^u$.

(C) $\frac{du}{dx} = \sec^2 x$.

(D) $\frac{dy}{dx} = e^u \cdot \sec^2 x$.

(E) $\frac{dy}{dx} = e^{\tan x} \tan x$.

Question 5. Find the derivative of $y = \log_e(\sin x)$. Which intermediate steps or final results are correct using the chain rule? (Multiple Correct Answers)

(A) Let $u = \sin x$, then $y = \log_e u$.

(B) $\frac{dy}{du} = \frac{1}{u}$.

(C) $\frac{du}{dx} = \cos x$.

(D) $\frac{dy}{dx} = \frac{1}{u} \cdot \cos x$.

(E) $\frac{dy}{dx} = \frac{\cos x}{\sin x} = \cot x$.

Question 6. In Applied Maths, the chain rule is useful for calculating rates of change of composite quantities. If the revenue $R$ is a function of sales $x$, and sales $x$ is a function of time $t$, which expression gives the rate of change of revenue with respect to time? (Multiple Correct Answers)

(A) $\frac{dR}{dt} = \frac{dR}{dx} + \frac{dx}{dt}$

(B) $\frac{dR}{dt} = \frac{dR}{dx} \cdot \frac{dx}{dt}$

(C) Marginal Revenue $\times$ Rate of change of sales.

(D) $\frac{dR}{dt} = R'(x(t)) \cdot x'(t)$

(E) $\frac{dR}{dt} = \frac{dx}{dR} \cdot \frac{dt}{dx}$

Question 7. Find the derivative of $y = \sqrt{\cos x}$. Which intermediate steps or final results are correct using the chain rule? (Multiple Correct Answers)

(A) Let $u = \cos x$, then $y = \sqrt{u} = u^{1/2}$.

(B) $\frac{dy}{du} = \frac{1}{2} u^{-1/2} = \frac{1}{2\sqrt{u}}$.

(C) $\frac{du}{dx} = \sin x$.

(D) $\frac{du}{dx} = -\sin x$.

(E) $\frac{dy}{dx} = \frac{1}{2\sqrt{\cos x}} \cdot (-\sin x) = -\frac{\sin x}{2\sqrt{\cos x}}$.

Question 8. If $y = f(u), u = g(v), v = h(x)$, which expression correctly represents $\frac{dy}{dx}$? (Multiple Correct Answers)

(A) $\frac{dy}{dx} = \frac{dy}{du} + \frac{du}{dv} + \frac{dv}{dx}$

(B) $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dv} \cdot \frac{dv}{dx}$

(C) $\frac{dy}{dx} = f'(u) \cdot g'(v) \cdot h'(x)$

(D) $\frac{dy}{dx} = f'(g(h(x))) \cdot g'(h(x)) \cdot h'(x)$

(E) $\frac{dy}{dx} = \frac{dy}{dv} \cdot \frac{dv}{dx}$ (Incorrect application of Chain rule structure)

Question 9. Find the derivative of $y = \cos^2 x$. Which approaches are correct? (Multiple Correct Answers)

(A) Use the chain rule with $u = \cos x$, $y = u^2$.

(B) Use the identity $\cos^2 x = \frac{1+\cos(2x)}{2}$ and differentiate.

(C) Using the chain rule: $\frac{dy}{dx} = 2 \cos x \cdot (-\sin x) = -2 \sin x \cos x$.

(D) Using the identity: $\frac{d}{dx} (\frac{1}{2} + \frac{1}{2}\cos(2x)) = 0 + \frac{1}{2} (-\sin(2x)) \cdot 2 = -\sin(2x)$.

(E) Note that $-2 \sin x \cos x = -\sin(2x)$, so both approaches yield the same result.

Question 10. Find the derivative of $y = \tan(\sqrt{x^2+1})$. Which steps or results are correct? (Multiple Correct Answers)

(A) This requires multiple applications of the chain rule.

(B) Let $u = \sqrt{x^2+1}$. Then $y = \tan u$.

(C) Let $v = x^2+1$. Then $u = \sqrt{v} = v^{1/2}$.

(D) $\frac{dy}{du} = \sec^2 u$, $\frac{du}{dv} = \frac{1}{2\sqrt{v}}$, $\frac{dv}{dx} = 2x$.

(E) $\frac{dy}{dx} = \sec^2(\sqrt{x^2+1}) \cdot \frac{1}{2\sqrt{x^2+1}} \cdot 2x = \frac{x \sec^2(\sqrt{x^2+1})}{\sqrt{x^2+1}}$.

Question 1. Implicit differentiation is used when an equation relates $x$ and $y$ without explicitly defining $y$ as a function of $x$. When differentiating a term involving $y$, such as $y^n$, with respect to $x$, what rule is applied? (Multiple Correct Answers)

(A) Power Rule: $\frac{d}{dx}(y^n) = ny^{n-1}$.

(B) Chain Rule: $\frac{d}{dx}(y^n) = ny^{n-1} \frac{dy}{dx}$.

(C) Treat $y$ as a function of $x$, say $y=f(x)$.

(D) Differentiate each term in the equation with respect to $x$.

(E) Collect terms with $\frac{dy}{dx}$ and solve for it.

Question 2. Find $\frac{dy}{dx}$ for the equation $x^2 + y^2 = 9$. Which steps or results are correct using implicit differentiation? (Multiple Correct Answers)

(A) Differentiate $x^2$ with respect to $x$: $2x$.

(B) Differentiate $y^2$ with respect to $x$: $2y \frac{dy}{dx}$.

(C) Differentiate 9 with respect to $x$: 0.

(D) The differentiated equation is $2x + 2y \frac{dy}{dx} = 0$.

(E) Solving for $\frac{dy}{dx}$ gives $\frac{dy}{dx} = -x/y$ (for $y \neq 0$).

Question 3. Find $\frac{dy}{dx}$ for the equation $xy = \cos x$. Which steps or results are correct using implicit differentiation? (Multiple Correct Answers)

(A) Use the product rule for $xy$: $1 \cdot y + x \frac{dy}{dx}$.

(B) Differentiate $\cos x$: $\sin x$.

(C) The differentiated equation is $y + x \frac{dy}{dx} = -\sin x$.

(D) Solving for $\frac{dy}{dx}$ gives $\frac{dy}{dx} = \frac{-\sin x - y}{x}$ (for $x \neq 0$).

(E) Solving for $\frac{dy}{dx}$ gives $\frac{dy}{dx} = \frac{\sin x + y}{x}$.

Question 4. If $y = f^{-1}(x)$, and $f$ is differentiable with $f'(y) \neq 0$, then $\frac{dy}{dx} = \frac{1}{f'(y)}$. Which of the following illustrate this rule? (Multiple Correct Answers)

(A) If $y = \sin^{-1} x$, then $x = \sin y$. $\frac{dx}{dy} = \cos y$. So $\frac{dy}{dx} = \frac{1}{\cos y} = \frac{1}{\sqrt{1-\sin^2 y}} = \frac{1}{\sqrt{1-x^2}}$ (for $\cos y > 0$).

(B) If $y = \tan^{-1} x$, then $x = \tan y$. $\frac{dx}{dy} = \sec^2 y$. So $\frac{dy}{dx} = \frac{1}{\sec^2 y} = \frac{1}{1+\tan^2 y} = \frac{1}{1+x^2}$.

(C) If $y = \log_e x$, then $x = e^y$. $\frac{dx}{dy} = e^y$. So $\frac{dy}{dx} = \frac{1}{e^y} = \frac{1}{x}$.

(D) If $y = e^x$, then $x = \log_e y$. $\frac{dx}{dy} = 1/y$. So $\frac{dy}{dx} = y = e^x$.

(E) If $y = x^n$, then $x = y^{1/n}$. $\frac{dx}{dy} = \frac{1}{n} y^{\frac{1}{n}-1}$. So $\frac{dy}{dx} = n y^{1 - \frac{1}{n}} = n y^{\frac{n-1}{n}} = n (x^{1/n})^{\frac{n-1}{n}} = n x^{\frac{n-1}{n^2}}$.

Question 5. Which of the following are correct standard derivatives of inverse trigonometric functions? (Multiple Correct Answers)

(A) $\frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1-x^2}}$ (for $-1 < x < 1$)

(B) $\frac{d}{dx}(\cos^{-1} x) = -\frac{1}{\sqrt{1-x^2}}$ (for $-1 < x < 1$)

(C) $\frac{d}{dx}(\tan^{-1} x) = \frac{1}{1+x^2}$ (for all $x \in \mathbb{R}$)

(D) $\frac{d}{dx}(\cot^{-1} x) = -\frac{1}{1+x^2}$ (for all $x \in \mathbb{R}$)

(E) $\frac{d}{dx}(\sec^{-1} x) = \frac{1}{x\sqrt{x^2-1}}$ (for $|x| > 1$)

Question 6. Find $\frac{dy}{dx}$ for $\sin(x^2 y) = c$. Which steps or results are correct using implicit differentiation? (Multiple Correct Answers)

(A) Differentiate $\sin(x^2 y)$ using the chain rule: $\cos(x^2 y) \cdot \frac{d}{dx}(x^2 y)$.

(B) Differentiate $x^2 y$ using the product rule: $2x y + x^2 \frac{dy}{dx}$.

(C) Differentiate $c$: 0.

(D) The differentiated equation is $\cos(x^2 y) (2x y + x^2 \frac{dy}{dx}) = 0$.

(E) Solving for $\frac{dy}{dx}$ gives $\frac{dy}{dx} = -\frac{2xy \cos(x^2 y)}{x^2 \cos(x^2 y)} = -\frac{2y}{x}$ (assuming $\cos(x^2 y) \neq 0$ and $x \neq 0$).

Question 7. Which of the following can be evaluated using properties of inverse trigonometric functions before differentiation? (Multiple Correct Answers)

(A) $\sin^{-1}(2x\sqrt{1-x^2})$ (substitute $x=\sin \theta$)

(B) $\tan^{-1}(\frac{2x}{1-x^2})$ (substitute $x=\tan \theta$)

(C) $\cos^{-1}(\frac{1-x^2}{1+x^2})$ (substitute $x=\tan \theta$)

(D) $\sin^{-1}(\frac{1-x^2}{1+x^2})$ (substitute $x=\tan \theta$)

(E) $\sin^{-1} x + \cos^{-1} x$ (use identity $\sin^{-1} x + \cos^{-1} x = \pi/2$ for $|x| \leq 1$)

Question 8. Find $\frac{dy}{dx}$ for $\tan^{-1}(\frac{2x}{1-x^2})$. Which steps or results are correct using substitution $x=\tan \theta$? (Multiple Correct Answers)

(A) Let $x = \tan \theta$, so $y = \tan^{-1}(\frac{2\tan \theta}{1-\tan^2 \theta}) = \tan^{-1}(\tan(2\theta))$.

(B) $y = 2\theta$ (for a certain range of $x$).

(C) Since $x = \tan \theta$, $\theta = \tan^{-1} x$. So $y = 2 \tan^{-1} x$.

(D) $\frac{dy}{dx} = \frac{d}{dx}(2 \tan^{-1} x) = 2 \cdot \frac{1}{1+x^2}$.

(E) This method simplifies the differentiation, avoiding the direct chain rule on $\tan^{-1}(\frac{2x}{1-x^2})$.

Question 9. Find $\frac{dy}{dx}$ for $\tan^{-1}(\frac{x}{1+x^2})$. Which method(s) are suitable? (Multiple Correct Answers)

(A) Direct application of the chain rule.

(B) Substitution with $x = \tan \theta$.

(C) Substitution with $x = \cot \theta$.

(D) Using the property $\tan^{-1} A - \tan^{-1} B = \tan^{-1}(\frac{A-B}{1+AB})$ (with $A=x, B=-x^2$).

(E) Using the property $\tan^{-1} A - \tan^{-1} B = \tan^{-1}(\frac{A-B}{1+AB})$ (with $A=x, B$ is a function of $x$).

Question 10. Find $\frac{dy}{dx}$ for $y = x^y$. Which differentiation method(s) can be used? (Multiple Correct Answers)

(A) Implicit differentiation.

(B) Logarithmic differentiation.

(C) Direct application of the power rule.

(D) Direct application of the exponential rule.

(E) Taking natural logarithm of both sides and then differentiating implicitly.

Question 1. Logarithmic differentiation is particularly useful for differentiating which types of functions? (Multiple Correct Answers)

(A) Functions of the form $y = [f(x)]^{g(x)}$.

(B) Functions that are products or quotients of multiple terms.

(C) Polynomial functions.

(D) Functions with roots involving multiple factors.

(E) Simple trigonometric functions like $\sin x$ or $\cos x$.

Question 2. To differentiate $y = x^{\sin x}$ using logarithmic differentiation, which steps are correct? (Multiple Correct Answers)

(A) Take natural logarithm on both sides: $\log_e y = \sin x \log_e x$.

(B) Differentiate both sides with respect to $x$: $\frac{1}{y} \frac{dy}{dx} = \frac{d}{dx}(\sin x \log_e x)$.

(C) Apply the product rule on the right side: $(-\sin x) \log_e(\cos x) + \cos x \cdot \frac{-\sin x}{\cos x}$.

(D) The right side derivative simplifies to $-\sin x \log_e(\cos x) - \sin x$.

(E) $\frac{dy}{dx} = (\cos x)^{\cos x} (-\sin x) ( 1 + \log_e(\cos x))$.

Question 3. Parametric equations define $x$ and $y$ in terms of a third variable (parameter), e.g., $x = f(t), y = g(t)$. The derivative $\frac{dy}{dx}$ is found using which formula? (Assume $dx/dt \neq 0$). (Multiple Correct Answers)

(A) $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$

(B) $\frac{dy}{dx} = \frac{g'(t)}{f'(t)}$

(C) $\frac{dy}{dx} = \frac{dx/dt}{dy/dt}$

(D) $\frac{dy}{dx} = \frac{dy}{dt} \cdot \frac{dt}{dx}$

(E) $\frac{dy}{dx} = \frac{d}{dt}(\frac{y}{x})$

Question 4. If $x = a \cos \theta$ and $y = b \sin \theta$, where $a, b$ are constants, find $\frac{dy}{dx}$. Which steps or results are correct? (Multiple Correct Answers)

(A) $\frac{dx}{d\theta} = -a \sin \theta$.

(B) $\frac{dy}{d\theta} = b \cos \theta$.

(C) $\frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} = \frac{b \cos \theta}{-a \sin \theta}$.

(D) $\frac{dy}{dx} = -\frac{b}{a} \cot \theta$ (for $\sin \theta \neq 0$).

(E) The derivative depends on the parameter $\theta$.

Question 5. To differentiate $y = \frac{\sqrt{x}(x-1)^2}{(x+2)^3}$ using logarithmic differentiation, which steps are correct? (Multiple Correct Answers)

(A) Take natural logarithm on both sides: $\log_e y = \log_e (\frac{\sqrt{x}(x-1)^2}{(x+2)^3})$.

(B) Use logarithm properties: $\log_e y = \log_e \sqrt{x} + \log_e (x-1)^2 - \log_e (x+2)^3$.

(C) Use logarithm properties: $\log_e y = \frac{1}{2} \log_e x + 2 \log_e |x-1| - 3 \log_e |x+2|$. (Using absolute values is more general for domains).

(D) Differentiate with respect to $x$: $\frac{1}{y} \frac{dy}{dx} = \frac{1}{2x} + \frac{2}{x-1} - \frac{3}{x+2}$.

(E) Solve for $\frac{dy}{dx}$: $\frac{dy}{dx} = y (\frac{1}{2x} + \frac{2}{x-1} - \frac{3}{x+2})$.

Question 6. If $x = t^2$ and $y = t^3$, find $\frac{dy}{dx}$. Which steps or results are correct? (Multiple Correct Answers)

(A) $\frac{dx}{dt} = 2t$.

(B) $\frac{dy}{dt} = 3t^2$.

(C) $\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{3t^2}{2t}$ (for $t \neq 0$).

(D) $\frac{dy}{dx} = \frac{3}{2} t$ (for $t \neq 0$).

(E) Expressing the derivative in terms of $x$: $t = \sqrt{x}$ (for $x \geq 0$), so $\frac{dy}{dx} = \frac{3}{2} \sqrt{x}$ (for $x > 0$).

Question 7. Find the derivative of $y = (\cos x)^{\cos x}$. Which steps are correct using logarithmic differentiation? (Multiple Correct Answers)

(A) $\log_e y = \cos x \log_e(\cos x)$.

(B) Differentiate both sides with respect to $x$ implicitly: $\frac{1}{y} \frac{dy}{dx} = \frac{d}{dx}(\cos x \log_e(\cos x))$.

(C) Apply the product rule on the right side: $(-\sin x) \log_e(\cos x) + \cos x \cdot \frac{-\sin x}{\cos x}$.

(D) The right side derivative simplifies to $-\sin x \log_e(\cos x) - \sin x$.

(E) $\frac{dy}{dx} = (\cos x)^{\cos x} (-\sin x) ( 1 + \log_e(\cos x))$.

Question 8. If $x = \frac{1-t^2}{1+t^2}$ and $y = \frac{2t}{1+t^2}$, which of the following are true? (Multiple Correct Answers)

(A) These are parametric equations for a circle centered at the origin with radius 1 (excluding point $(-1, 0)$). Specifically, for $t \in \mathbb{R}$, it represents the circle $x^2+y^2=1$ except $(-1,0)$.

(B) $\frac{dx}{dt} = \frac{-2t(1+t^2) - (1-t^2)(2t)}{(1+t^2)^2} = \frac{-4t}{(1+t^2)^2}$.

(C) $\frac{dy}{dt} = \frac{2(1+t^2) - 2t(2t)}{(1+t^2)^2} = \frac{2-2t^2}{(1+t^2)^2}$.

(D) $\frac{dy}{dx} = \frac{(2-2t^2)/(1+t^2)^2}{-4t/(1+t^2)^2} = \frac{2(1-t^2)}{-4t} = \frac{t^2-1}{2t}$ (for $t \neq 0$).

(E) The derivative can be expressed in terms of $x$ and $y$ as $-x/y$ (for $y \neq 0$).

Question 9. Find the derivative of $y = (\log_e x)^{x^2}$. Which steps are correct using logarithmic differentiation? (Multiple Correct Answers)

(A) $\log_e y = x^2 \log_e(\log_e x)$.

(B) Differentiate both sides with respect to $x$: $\frac{1}{y} \frac{dy}{dx} = \frac{d}{dx}(x^2 \log_e(\log_e x))$.

(C) Apply the product rule on the right side: $2x \log_e(\log_e x) + x^2 \cdot \frac{1}{\log_e x} \cdot \frac{1}{x}$.

(D) The right side derivative simplifies to $2x \log_e(\log_e x) + \frac{x}{\log_e x}$.

(E) Solve for $\frac{dy}{dx}$: $\frac{dy}{dx} = (\log_e x)^{x^2} (2x \log_e(\log_e x) + \frac{x}{\log_e x})$.

Question 10. If $x = a (\theta - \sin \theta)$ and $y = a (1 - \cos \theta)$, find $\frac{dy}{dx}$. Which steps or results are correct? (Multiple Correct Answers)

(A) $\frac{dx}{d\theta} = a (1 - \cos \theta)$.

(B) $\frac{dy}{d\theta} = a \sin \theta$.

(C) $\frac{dy}{dx} = \frac{a \sin \theta}{a (1 - \cos \theta)} = \frac{\sin \theta}{1 - \cos \theta}$ (for $1-\cos \theta \neq 0$).

(D) Using half-angle formulas: $\sin \theta = 2 \sin(\theta/2) \cos(\theta/2)$ and $1 - \cos \theta = 2 \sin^2(\theta/2)$.

(E) $\frac{dy}{dx} = \frac{2 \sin(\theta/2) \cos(\theta/2)}{2 \sin^2(\theta/2)} = \cot(\theta/2)$ (for $\sin(\theta/2) \neq 0$).

Question 1. The second order derivative of $y=f(x)$ is denoted by $\frac{d^2 y}{dx^2}$ or $y''$. It represents the derivative of the first derivative. Which statements about higher order derivatives are correct? (Multiple Correct Answers)

(A) The third derivative is the derivative of the second derivative.

(B) The $n$-th derivative is denoted by $\frac{d^n y}{dx^n}$ or $y^{(n)}$.

(C) Higher order derivatives are useful in analyzing concavity and inflection points.

(D) The derivative of a polynomial of degree $n$ is a polynomial of degree $n-1$, and the $n$-th derivative is a constant (for $n \geq 0$).

(E) The higher order derivatives of $e^x$ are always $e^x$.

Question 2. Find the second derivative of $y = x^5$. Which steps or results are correct? (Multiple Correct Answers)

(A) The first derivative $\frac{dy}{dx} = 5x^4$.

(B) The second derivative $\frac{d^2 y}{dx^2} = \frac{d}{dx}(5x^4)$.

(C) $\frac{d^2 y}{dx^2} = 20x^3$.

(D) The third derivative $\frac{d^3 y}{dx^3} = 60x^2$.

(E) The fifth derivative $\frac{d^5 y}{dx^5} = 120$.

Question 3. Find the second derivative of $y = \sin(2x)$. Which steps or results are correct? (Multiple Correct Answers)

(A) $\frac{dy}{dx} = \cos(2x) \cdot 2 = 2 \cos(2x)$.

(B) $\frac{d^2 y}{dx^2} = \frac{d}{dx}(2 \cos(2x))$.

(C) $\frac{d^2 y}{dx^2} = 2 (-\sin(2x)) \cdot 2 = -4 \sin(2x)$.

(D) $\frac{d^2 y}{dx^2} = -4y$.

(E) The second derivative is always negative.

Question 4. Find the second derivative of $y = e^{3x} + \log_e x$. Which steps or results are correct? (Multiple Correct Answers)

(A) $\frac{dy}{dx} = 3e^{3x} + \frac{1}{x}$.

(B) $\frac{d^2 y}{dx^2} = \frac{d}{dx}(3e^{3x} + x^{-1})$.

(C) $\frac{d^2 y}{dx^2} = 3(3e^{3x}) + (-1)x^{-2}$.

(D) $\frac{d^2 y}{dx^2} = 9e^{3x} - \frac{1}{x^2}$.

(E) The domain of the second derivative is $(0, \infty)$.

Question 5. If $x = t^2$ and $y = \sin t$, find $\frac{d^2 y}{dx^2}$ at $t=\pi/2$. Which steps or results are correct? (Multiple Correct Answers)

(A) $\frac{dx}{dt} = 2t$, $\frac{dy}{dt} = \cos t$.

(B) $\frac{dy}{dx} = \frac{\cos t}{2t}$ (for $t \neq 0$).

(C) $\frac{d^2 y}{dx^2} = \frac{d}{dx}(\frac{\cos t}{2t}) = \frac{d}{dt}(\frac{\cos t}{2t}) \cdot \frac{dt}{dx}$.

(D) $\frac{d}{dt}(\frac{\cos t}{2t}) = \frac{-\sin t (2t) - \cos t (2)}{(2t)^2} = \frac{-2t \sin t - 2 \cos t}{4t^2} = \frac{-t \sin t - \cos t}{2t^2}$.

(E) $\frac{d^2 y}{dx^2} = \frac{-t \sin t - \cos t}{2t^2} \cdot \frac{1}{2t} = \frac{-t \sin t - \cos t}{4t^3}$. At $t=\pi/2$, $\frac{d^2 y}{dx^2} = \frac{-\pi/2 \cdot 1 - 0}{4(\pi/2)^3} = \frac{-\pi/2}{4\pi^3/8} = \frac{-\pi/2}{\pi^3/2} = -1/\pi^2$.

Question 6. If $y = \tan^{-1} x$, find the second derivative. Which steps or results are correct? (Multiple Correct Answers)

(A) The first derivative $\frac{dy}{dx} = \frac{1}{1+x^2}$.

(B) Rewrite the first derivative as $(1+x^2)^{-1}$.

(C) Differentiate using the power rule and chain rule: $\frac{d^2 y}{dx^2} = -1 (1+x^2)^{-2} \cdot 2x$.

(D) $\frac{d^2 y}{dx^2} = \frac{-2x}{(1+x^2)^2}$.

(E) The second derivative is always negative.

Question 7. If $y = x \log_e x$, find the second derivative. Which steps or results are correct? (Multiple Correct Answers)

(A) The first derivative $\frac{dy}{dx} = 1 \cdot \log_e x + x \cdot \frac{1}{x} = \log_e x + 1$.

(B) The second derivative $\frac{d^2 y}{dx^2} = \frac{d}{dx}(\log_e x + 1)$.

(C) $\frac{d^2 y}{dx^2} = \frac{1}{x} + 0 = \frac{1}{x}$ (for $x>0$).

(D) The domain of the second derivative is $(0, \infty)$.

(E) The third derivative is $-1/x^2$.

Question 8. Consider the function $y = A \cos(\omega t) + B \sin(\omega t)$, where $A, B, \omega$ are constants. Which differential equation does it satisfy? (Multiple Correct Answers)

(A) $\frac{dy}{dt} = -A\omega \sin(\omega t) + B\omega \cos(\omega t)$.

(B) $\frac{d^2 y}{dt^2} = -A\omega^2 \cos(\omega t) - B\omega^2 \sin(\omega t)$.

(C) $\frac{d^2 y}{dt^2} = -\omega^2 (A \cos(\omega t) + B \sin(\omega t))$.

(D) $\frac{d^2 y}{dt^2} = -\omega^2 y$.

(E) $\frac{d^2 y}{dt^2} + \omega^2 y = 0$.

Question 9. If $y = f(x) g(x)$, find the second derivative $y''$. Which terms might appear in $y''$? (Multiple Correct Answers)

(A) $f''(x) g(x)$

(B) $f(x) g''(x)$

(C) $f'(x) g'(x)$

(D) $f(x) g(x)$

(E) $y'' = f''g + 2f'g' + fg''$ (using abbreviated notation).

Question 10. Find the third derivative of $y = e^{ax}$. Which steps or results are correct? (Multiple Correct Answers)

(A) $\frac{dy}{dx} = ae^{ax}$.

(B) $\frac{d^2 y}{dx^2} = a^2 e^{ax}$.

(C) $\frac{d^3 y}{dx^3} = a^3 e^{ax}$.

(D) The $n$-th derivative is $a^n e^{ax}$.

(E) The derivatives are always positive if $a>0$.

Question 1. Which are the conditions for Rolle's Theorem to be applicable to a function $f(x)$ on the closed interval $[a, b]$? (Multiple Correct Answers)

(A) $f(x)$ must be continuous on $[a, b]$.

(B) $f(x)$ must be differentiable on the open interval $(a, b)$.

(C) $f(a)$ must be equal to $f(b)$.

(D) $f(x)$ must be a polynomial function (polynomials satisfy A and B).

(E) There must exist a point $c \in (a, b)$ such that $f'(c) = 0$.

Question 2. Rolle's Theorem guarantees the existence of at least one value $c \in (a, b)$ such that $f'(c) = 0$. What is the geometric interpretation of this result? (Multiple Correct Answers)

(A) There is a point on the curve where the tangent line is horizontal.

(B) The tangent line at $x=c$ is parallel to the x-axis.

(C) If $f(a)=f(b)$, there is a point where the rate of change of the function is zero.

(D) The tangent line at $x=c$ is parallel to the chord joining $(a, f(a))$ and $(b, f(b))$ (since $f(a)=f(b)$, this chord is horizontal).

(E) The function attains a local maximum or minimum in $(a, b)$ (if $f(a)=f(b)$ are not the extrema).

Question 3. Which are the conditions for Lagrange's Mean Value Theorem to be applicable to a function $f(x)$ on the closed interval $[a, b]$? (Multiple Correct Answers)

(A) $f(x)$ must be continuous on $[a, b]$.

(B) $f(x)$ must be differentiable on the open interval $(a, b)$.

(C) $f(a)$ must be equal to $f(b)$.

(D) $f(x)$ must be a polynomial function (polynomials satisfy A and B).

(E) There must exist a point $c \in (a, b)$ such that $f'(c) = \frac{f(b)-f(a)}{b-a}$.

Question 4. Lagrange's Mean Value Theorem guarantees the existence of at least one value $c \in (a, b)$ such that $f'(c) = \frac{f(b)-f(a)}{b-a}$. What is the geometric interpretation? (Multiple Correct Answers)

(A) There is a point on the curve where the tangent line is parallel to the secant line joining $(a, f(a))$ and $(b, f(b))$.

(B) The instantaneous rate of change at $x=c$ equals the average rate of change over $[a, b]$.

(C) If $f'(x)$ is the velocity, the instantaneous velocity at some time $c$ equals the average velocity over the interval $[a, b]$.

(D) The slope of the tangent at $x=c$ is equal to the slope of the chord connecting the endpoints.

(E) There is a point where the tangent line is horizontal.

Question 5. Consider the function $f(x) = x^2$ on the interval $[-2, 2]$. Which of the following statements are true? (Multiple Correct Answers)

(A) $f(x)$ is continuous on $[-2, 2]$.

(B) $f(x)$ is differentiable on $(-2, 2)$.

(C) $f(-2) = f(2)$.

(D) Rolle's Theorem is applicable to $f(x)$ on $[-2, 2]$.

(E) The value of $c$ guaranteed by Rolle's Theorem is $c=0$.

Question 6. Consider the function $f(x) = |x|$ on the interval $[-1, 1]$. Which of the following statements are true? (Multiple Correct Answers)

(A) $f(x)$ is continuous on $[-1, 1]$.

(B) $f(x)$ is differentiable on $(-1, 1)$.

(C) $f(-1) = f(1)$.

(D) Rolle's Theorem is applicable to $f(x)$ on $[-1, 1]$.

(E) Lagrange's Mean Value Theorem is applicable to $f(x)$ on $[-1, 1]$.

Question 7. Consider the function $f(x) = x^3$ on the interval $[1, 3]$. Which of the following statements are true? (Multiple Correct Answers)

(A) $f(x)$ is continuous on $[1, 3]$.

(B) $f(x)$ is differentiable on $(1, 3)$.

(C) Rolle's Theorem is applicable to $f(x)$ on $[1, 3]$.

(D) Lagrange's Mean Value Theorem is applicable to $f(x)$ on $[1, 3]$.

(E) There exists $c \in (1, 3)$ such that $f'(c) = \frac{3^3 - 1^3}{3 - 1} = \frac{26}{2} = 13$.

Question 8. If a function $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$, and $f'(x) \neq 0$ for all $x \in (a, b)$, what can be concluded? (Multiple Correct Answers)

(A) $f(a) \neq f(b)$.

(B) The function is strictly monotonic on $[a, b]$.

(C) Rolle's Theorem is not applicable.

(D) Lagrange's Mean Value Theorem is not applicable.

(E) There is no horizontal tangent in $(a, b)$.

Question 9. Consider the function $f(x) = \frac{1}{x-2}$ on the interval $[1, 3]$. Which of the following statements are true? (Multiple Correct Answers)

(A) $f(x)$ is continuous on $[1, 3]$.

(B) $f(x)$ is differentiable on $(1, 3)$.

(C) $f(1) = -1$ and $f(3) = 1$.

(D) Rolle's Theorem is applicable.

(E) Lagrange's Mean Value Theorem is applicable.

Question 10. Which of the following functions satisfy the conditions of Rolle's Theorem on the given interval? (Multiple Correct Answers)

(A) $f(x) = \sin x$ on $[0, \pi]$.

(B) $f(x) = \cos x$ on $[0, 2\pi]$.

(C) $f(x) = x^2 - 2x$ on $[0, 2]$.

(D) $f(x) = \tan x$ on $[0, \pi]$.

(E) $f(x) = x(x-1)(x-2)$ on $[0, 2]$.

Question 11. If $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$, and $f'(x) = 0$ for all $x \in (a, b)$, which of the following are true based on the Mean Value Theorem? (Multiple Correct Answers)

(A) For any $x_1, x_2 \in [a, b]$, there exists $c \in (x_1, x_2)$ such that $f'(c) = \frac{f(x_2)-f(x_1)}{x_2-x_1}$.

(B) Since $f'(c)=0$, it implies $\frac{f(x_2)-f(x_1)}{x_2-x_1} = 0$.

(C) This means $f(x_2) - f(x_1) = 0$, so $f(x_1) = f(x_2)$ for any $x_1, x_2 \in [a, b]$.

(D) The function $f(x)$ is a constant function on $[a, b]$.

(E) This result can only be derived from Rolle's Theorem.

Question 12. Which of the following functions satisfy the conditions of Lagrange's Mean Value Theorem on the given interval? (Multiple Correct Answers)

(A) $f(x) = x^2$ on $[1, 5]$.

(B) $f(x) = \log_e x$ on $[1, e]$.

(C) $f(x) = |x|$ on $[-2, 2]$.

(D) $f(x) = e^x$ on $[0, 1]$.

(E) $f(x) = \frac{1}{x}$ on $[1, 4]$.

Question 13. If Rolle's Theorem is applicable to $f(x)$ on $[a, b]$, which of the following are consequences? (Multiple Correct Answers)

(A) $f(x)$ has a local extremum in $(a, b)$.

(B) The tangent at some point $c \in (a, b)$ is parallel to the x-axis.

(C) The maximum and minimum values of $f(x)$ in $[a, b]$ are equal.

(D) $f'(c) = 0$ for at least one $c \in (a, b)$.

(E) The function is a constant on $[a, b]$.

Question 14. In the context of Mean Value Theorems, the interval $(a, b)$ is the open interval excluding the endpoints $a$ and $b$. Why is differentiability typically required on the open interval? (Multiple Correct Answers)

(A) The definition of the derivative involves a limit as $h \to 0$, which requires the function to be defined on both sides of the point (unless it's an endpoint).

(B) Derivatives at endpoints are defined as one-sided limits, which may exist even if the two-sided limit (required for differentiability in an open interval) does not.

(C) The theorem relies on the existence of a tangent line at interior points.

(D) Differentiability at endpoints would impose stronger conditions than necessary for the theorems.

(E) Continuity at endpoints is sufficient for the theorems.

Question 15. If $f(x) = \sin x$ on $[0, \pi]$, find the value(s) of $c$ guaranteed by Rolle's Theorem. Which statements are true? (Multiple Correct Answers)

(A) $f(0)=0, f(\pi)=0$.

(B) $f'(x) = \cos x$.

(C) We need to solve $f'(c) = \cos c = 0$ for $c \in (0, \pi)$.

(D) The value of $c$ is $\pi/2$.

(E) There is exactly one value of $c$ in $(0, \pi)$.

Question 1. If $y = f(x)$, the derivative $\frac{dy}{dx}$ represents the rate of change of $y$ with respect to $x$. If $x$ is a quantity changing with time $t$, and $y$ depends on $x$, the rate of change of $y$ with respect to $t$ is given by which expressions? (Multiple Correct Answers)

(A) $\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}$ (Chain Rule)

(B) The rate of change of $y$ w.r.t. $t$ is $f'(x(t)) \cdot x'(t)$.

(C) This is an application of related rates.

(D) $\frac{dy}{dt} = \frac{dx}{dy} \cdot \frac{dt}{dx}$

(E) If $x$ increases, $y$ always increases.

Question 2. The area $A$ of a circle is given by $A = \pi r^2$, where $r$ is the radius. If the radius is increasing at a rate $\frac{dr}{dt}$, which expressions give the rate of change of the area with respect to time? (Multiple Correct Answers)

(A) $\frac{dA}{dr} = 2\pi r$.

(B) $\frac{dA}{dt} = \frac{dA}{dr} \cdot \frac{dr}{dt}$.

(C) $\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$.

(D) If $\frac{dr}{dt}$ is constant, $\frac{dA}{dt}$ is also constant.

(E) The rate of change of area depends on the current radius $r$.

Question 3. A spherical balloon is being inflated. Its volume $V = \frac{4}{3}\pi r^3$. If the radius is increasing at a rate $\frac{dr}{dt}$, which expressions give the rate of change of the volume with respect to time? (Multiple Correct Answers)

(A) $\frac{dV}{dr} = 4\pi r^2$.

(B) $\frac{dV}{dt} = \frac{dV}{dr} \cdot \frac{dr}{dt}$.

(C) $\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}$.

(D) If the rate of inflation (dV/dt) is constant, the radius increases at a constant rate.

(E) If the rate of inflation (dV/dt) is constant, the radius increases at a decreasing rate.

Question 4. In economics, if $C(x)$ is the total cost of producing $x$ units, the marginal cost is defined as $C'(x)$. If $R(x)$ is the total revenue from selling $x$ units, the marginal revenue is $R'(x)$. Which statements are correct? (Multiple Correct Answers)

(A) Marginal cost is the rate of change of total cost with respect to the number of units produced.

(B) Marginal revenue is the rate of change of total revenue with respect to the number of units sold.

(C) Marginal cost and marginal revenue are calculated using derivatives.

(D) Marginal cost gives the approximate cost of producing one additional unit.

(E) Marginal revenue gives the approximate revenue from selling one additional unit.

Question 5. A ladder 5 m long is leaning against a wall. The base of the ladder is pulled away from the wall along the ground at a rate of 2 cm/sec. Let $x$ be the distance of the base from the wall and $y$ be the height the top of the ladder reaches on the wall. Which statements are true? (Multiple Correct Answers)

(A) $x^2 + y^2 = 5^2 = 25$.

(B) $\frac{dx}{dt} = 2$ cm/sec (given as positive since it's increasing).

(C) Differentiating the equation w.r.t. time gives $2x \frac{dx}{dt} + 2y \frac{dy}{dt} = 0$.

(D) We are looking for $\frac{dy}{dt}$. This will be positive since the height is decreasing.

(E) The rate $\frac{dy}{dt}$ depends on the current values of $x$ and $y$.

Question 6. The total cost function is $C(x) = 0.005x^3 + 0.02x^2 + 30x + 5000$. Which statements about the marginal cost are correct? (Multiple Correct Answers)

(A) The marginal cost function is $C'(x) = 0.015x^2 + 0.04x + 30$.

(B) The marginal cost when $x=100$ is $C'(100) = 0.015(100)^2 + 0.04(100) + 30 = 150 + 4 + 30 = \textsf{₹ }184$.

(C) The marginal cost function represents the additional cost incurred for producing one more unit at production level $x$.

(D) As $x$ increases, the marginal cost increases (check derivative of $C'(x)$).

(E) The fixed cost is $\textsf{₹ }5000$.

Question 7. Sand is pouring from a pipe at the rate of 12 cm$^3$/sec. The falling sand forms a cone on the ground in such a way that the height $h$ of the cone is always one-sixth of the radius $r$ of the base ($h = r/6$ or $r = 6h$). The volume of the cone is $V = \frac{1}{3}\pi r^2 h$. We want to find the rate at which the height of the sand cone is increasing when the height is 4 cm. Which statements are correct? (Multiple Correct Answers)

(A) $\frac{dV}{dt} = 12$ cm$^3$/sec.

(B) Express $V$ in terms of $h$: $V = \frac{1}{3}\pi (6h)^2 h = \frac{1}{3}\pi (36h^2) h = 12\pi h^3$.

(C) Differentiate $V$ with respect to $t$: $\frac{dV}{dt} = \frac{d}{dt}(12\pi h^3) = 12\pi (3h^2) \frac{dh}{dt} = 36\pi h^2 \frac{dh}{dt}$.

(D) Substitute known values: $12 = 36\pi (4^2) \frac{dh}{dt}$.

(E) Solving for $\frac{dh}{dt}$: $12 = 36\pi (16) \frac{dh}{dt} \implies \frac{dh}{dt} = \frac{12}{576\pi} = \frac{1}{48\pi}$ cm/sec.

Question 8. The total revenue function is $R(x) = 3x^2 + 36x + 5$. Which statements about the marginal revenue are correct? (Multiple Correct Answers)

(A) The marginal revenue function is $R'(x) = 6x + 36$.

(B) The marginal revenue when $x=5$ is $R'(5) = 6(5) + 36 = 30 + 36 = \textsf{₹ }66$.

(C) The marginal revenue represents the additional revenue from selling one more unit at sales level $x$.

(D) The marginal revenue is constant as $x$ increases.

(E) The rate of change of marginal revenue is constant.

Question 9. A particle moves along the curve $y = \sqrt{x}$. Find the points on the curve at which the y-coordinate is changing 8 times as fast as the x-coordinate. Which statements are correct? (Multiple Correct Answers)

(A) We are given $\frac{dy}{dt} = 8 \frac{dx}{dt}$.

(B) We need to find $\frac{dy}{dx}$. $\frac{dy}{dx} = \frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}}$.

(C) Using the chain rule, $\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}$.

(D) Substituting the given information, $8 \frac{dx}{dt} = \frac{1}{2\sqrt{x}} \frac{dx}{dt}$ (assuming $\frac{dx}{dt} \neq 0$).

(E) Solving for $x$: $8 = \frac{1}{2\sqrt{x}} \implies 16\sqrt{x} = 1 \implies \sqrt{x} = \frac{1}{16} \implies x = \frac{1}{256}$. The corresponding y-coordinate is $y = \sqrt{1/256} = 1/16$. The point is $(1/256, 1/16)$.

Question 10. The volume of a cube is increasing at a rate of 10 cm$^3$/sec. How fast is the surface area increasing when the length of an edge is 5 cm? Let $s$ be the edge length, $V$ the volume, and $A$ the surface area. Which statements are correct? (Multiple Correct Answers)

(A) $V = s^3$, $A = 6s^2$.

(B) $\frac{dV}{dt} = 3s^2 \frac{ds}{dt}$.

(C) $\frac{dA}{dt} = 12s \frac{ds}{dt}$.

(D) Given $\frac{dV}{dt} = 10$. So $10 = 3s^2 \frac{ds}{dt}$. When $s=5$, $10 = 3(5^2) \frac{ds}{dt} \implies 10 = 75 \frac{ds}{dt} \implies \frac{ds}{dt} = \frac{10}{75} = \frac{2}{15}$ cm/sec.

(E) $\frac{dA}{dt} = 12(5) (\frac{2}{15}) = 60 \cdot \frac{2}{15} = 4 \cdot 2 = 8$ cm$^2$/sec.

Question 11. In the context of Marginal Cost and Marginal Revenue in Applied Maths, which statements are generally true? (Multiple Correct Answers)

(A) Marginal Cost is the derivative of the Total Cost function.

(B) Marginal Revenue is the derivative of the Total Revenue function.

(C) Profit is maximized when Marginal Revenue equals Marginal Cost.

(D) Fixed costs do not affect Marginal Cost.

(E) Marginal quantities can be used to approximate the change in total cost or revenue resulting from a one-unit change in production or sales.

Question 1. The slope of the tangent to the curve $y=f(x)$ at the point $(x_0, y_0)$ is given by $m_T = f'(x_0)$. Which statements about tangents and normals are correct? (Multiple Correct Answers)

(A) The equation of the tangent line is $y - y_0 = m_T (x - x_0)$.

(B) The normal line to the curve at $(x_0, y_0)$ is perpendicular to the tangent line at that point.

(C) The slope of the normal line is $m_N = -\frac{1}{m_T}$, provided $m_T \neq 0$.

(D) The equation of the normal line is $y - y_0 = m_N (x - x_0)$.

(E) If the tangent is horizontal, the normal is vertical.

Question 2. Find the equation of the tangent to the curve $y = x^2 - 2x$ at the point where $x=3$. Which steps or results are correct? (Multiple Correct Answers)

(A) Find the y-coordinate at $x=3$: $y = 3^2 - 2(3) = 9 - 6 = 3$. The point is $(3, 3)$.

(B) Find the derivative: $\frac{dy}{dx} = 2x - 2$.

(C) Find the slope of the tangent at $x=3$: $m_T = 2(3) - 2 = 6 - 2 = 4$.

(D) The equation of the tangent is $y - 3 = 4(x - 3)$.

(E) The equation of the tangent is $y = 4x - 9$.

Question 3. Using differentials, the approximate change in $y=f(x)$ when $x$ changes by a small amount $\Delta x$ is given by $\Delta y \approx dy = f'(x) \Delta x$. Which statements are correct regarding approximations and errors using differentials? (Multiple Correct Answers)

(A) $f(x + \Delta x) \approx f(x) + f'(x) \Delta x$.

(B) The approximate value of $f(x + \Delta x)$ is $f(x) + dy$.

(C) If $x$ is the measured quantity and $\Delta x$ is the error in measurement, $\Delta y$ is the approximate error in the calculated quantity $y=f(x)$.

(D) The relative error in $y$ is approximately $\frac{\Delta y}{y} \approx \frac{dy}{y}$.

(E) The percentage error in $y$ is approximately $\frac{dy}{y} \times 100\%$.

Question 4. Using differentials, approximate the value of $\sqrt{36.6}$. Which steps or results are correct? (Multiple Correct Answers)

(A) Let $f(x) = \sqrt{x}$, $x = 36$, and $\Delta x = 0.6$.

(B) $f(x) = \sqrt{36} = 6$.

(C) $f'(x) = \frac{1}{2\sqrt{x}}$.

(D) $f'(36) = \frac{1}{2\sqrt{36}} = \frac{1}{12}$.

(E) $\sqrt{36.6} \approx f(36) + f'(36) \Delta x = 6 + \frac{1}{12} (0.6) = 6 + \frac{0.6}{12} = 6 + 0.05 = 6.05$.

Question 5. The radius of a sphere is measured as 7 cm with a possible error of 0.03 cm. The volume of the sphere is $V = \frac{4}{3}\pi r^3$. Estimate the maximum error in the volume using differentials. Which statements or results are correct? (Multiple Correct Answers)

(A) $r = 7$ cm, $\Delta r = dr = 0.03$ cm.

(B) $\frac{dV}{dr} = 4\pi r^2$.

(C) $dV = \frac{dV}{dr} dr = 4\pi r^2 dr$.

(D) Substitute values: $dV = 4\pi (7)^2 (0.03) = 4\pi (49) (0.03) = 196\pi (0.03)$.

(E) $dV = 5.88\pi$ cm$^3$.

Question 6. Find the points on the curve $y = x^3 - 11x + 5$ at which the tangent is $y = x - 11$. Which steps or results are correct? (Multiple Correct Answers)

(A) The slope of the given tangent line is $m=1$.

(B) Find the derivative of the curve: $\frac{dy}{dx} = 3x^2 - 11$.

(C) Set the derivative equal to the slope of the tangent: $3x^2 - 11 = 1$.

(D) Solving $3x^2 = 12$ gives $x^2 = 4$, so $x = \pm 2$.

(E) Check if the points $(2, f(2))$ and $(-2, f(-2))$ lie on the line $y=x-11$. For $x=2$, $f(2) = 2^3 - 11(2) + 5 = 8 - 22 + 5 = -9$. Check if $(2, -9)$ is on $y=x-11$: $-9 = 2 - 11$, which is true. For $x=-2$, $f(-2) = (-2)^3 - 11(-2) + 5 = -8 + 22 + 5 = 19$. Check if $(-2, 19)$ is on $y=x-11$: $19 = -2 - 11$, which is false. So the only point is $(2, -9)$.

Question 7. Using differentials, approximate the value of $\cos(61^\circ)$. Use $\pi/180 \approx 0.01745$ radians for $1^\circ$. Which steps or results are correct? (Multiple Correct Answers)

(A) Let $f(x) = \cos x$, $x = 60^\circ = \pi/3$ radians, $\Delta x = 1^\circ = \pi/180$ radians.

(B) $f(x) = \cos(\pi/3) = 1/2$.

(C) $f'(x) = \sin x$.

(D) $f'(\pi/3) = \sin(\pi/3) = \sqrt{3}/2 \approx 0.866$.

(E) $\cos(61^\circ) \approx f(\pi/3) + f'(\pi/3) \Delta x = 0.5 + (-\sqrt{3}/2) (\pi/180) \approx 0.5 - 0.866 \times 0.01745 \approx 0.5 - 0.0151 \approx 0.4849$. (Note: the derivative of $\cos x$ is $-\sin x$, so $f'(x)=-\sin x$)

Question 8. If the percentage error in measuring the diameter of a sphere is $1\%$, what is the percentage error in its volume? Which statements or results are correct? (Multiple Correct Answers)

(A) Let the radius be $r$. The diameter is $D=2r$. If percentage error in $D$ is $1\%$, then $\frac{\Delta D}{D} \times 100\% = 1\%$. So $\frac{2\Delta r}{2r} \times 100\% = \frac{\Delta r}{r} \times 100\% = 1\%$. The percentage error in radius is $1\%$.

(B) Volume $V = \frac{4}{3}\pi r^3$.

(C) $\frac{dV}{dr} = 4\pi r^2$.

(D) The relative error in volume is approximately $\frac{dV}{V} = \frac{4\pi r^2 dr}{\frac{4}{3}\pi r^3} = \frac{3 dr}{r}$.

(E) The percentage error in volume is $\frac{dV}{V} \times 100\% \approx 3 \frac{dr}{r} \times 100\% = 3 \times (1\%) = 3\%$.

Question 9. Find the equation of the normal to the curve $y = \sin x$ at $(0, 0)$. Which steps or results are correct? (Multiple Correct Answers)

(A) The point is $(x_0, y_0) = (0, 0)$.

(B) Find the derivative: $\frac{dy}{dx} = \cos x$.

(C) Find the slope of the tangent at $x=0$: $m_T = \cos(0) = 1$.

(D) The slope of the normal is $m_N = -1/m_T = -1/1 = -1$.

(E) The equation of the normal is $y - 0 = -1 (x - 0)$, which is $y = -x$.

Question 10. Approximate the value of $(25.001)^{1/2}$ using differentials. Which steps or results are correct? (Multiple Correct Answers)

(A) Let $f(x) = x^{1/2}$, $x = 25$, $\Delta x = 0.001$.

(B) $f(x) = \sqrt{25} = 5$.

(C) $f'(x) = \frac{1}{2} x^{-1/2} = \frac{1}{2\sqrt{x}}$.

(D) $f'(25) = \frac{1}{2\sqrt{25}} = \frac{1}{10} = 0.1$.

(E) $(25.001)^{1/2} \approx f(25) + f'(25) \Delta x = 5 + 0.1 \times 0.001 = 5 + 0.0001 = 5.0001$.

Question 1. A function $f(x)$ is strictly increasing on an interval $I$ if for any $x_1, x_2 \in I$ with $x_1 < x_2$, $f(x_1) < f(x_2)$. Which of the following are conditions for a function to be increasing/decreasing based on the first derivative? (Multiple Correct Answers)

(A) If $f'(x) > 0$ for all $x$ in an interval, then $f(x)$ is strictly increasing on that interval.

(B) If $f'(x) < 0$ for all $x$ in an interval, then $f(x)$ is strictly decreasing on that interval.

(C) If $f'(x) \geq 0$ for all $x$ in an interval, then $f(x)$ is increasing on that interval.

(D) If $f'(x) \leq 0$ for all $x$ in an interval, then $f(x)$ is decreasing on that interval.

(E) If $f'(x) = 0$ for all $x$ in an interval, then $f(x)$ is constant on that interval.

Question 2. Find the intervals where $f(x) = x^3 - 3x^2 + 2$ is strictly increasing or strictly decreasing. Which statements are correct? (Multiple Correct Answers)

(A) Find the derivative: $f'(x) = 3x^2 - 6x = 3x(x-2)$.

(B) Critical points are where $f'(x) = 0$, i.e., $x=0$ and $x=2$.

(C) $f'(x) > 0$ when $x \in (-\infty, 0) \cup (2, \infty)$, so $f(x)$ is strictly increasing on $(-\infty, 0]$ and $[2, \infty)$.

(D) $f'(x) < 0$ when $x \in (0, 2)$, so $f(x)$ is strictly decreasing on $[0, 2]$.

(E) The function is monotonic over its entire domain $\mathbb{R}$.

Question 3. Find the intervals where $f(x) = 2x^3 - 9x^2 + 12x + 15$ is strictly increasing or strictly decreasing. Which statements are correct? (Multiple Correct Answers)

(A) $f'(x) = 6x^2 - 18x + 12 = 6(x^2 - 3x + 2) = 6(x-1)(x-2)$.

(B) Critical points are $x=1$ and $x=2$.

(C) $f'(x) > 0$ when $x \in (-\infty, 1) \cup (2, \infty)$, so $f(x)$ is strictly increasing on $(-\infty, 1]$ and $[2, \infty)$.

(D) $f'(x) < 0$ when $x \in (1, 2)$, so $f(x)$ is strictly decreasing on $[1, 2]$.

(E) The function is monotonic over its entire domain $\mathbb{R}$.

Question 4. Which of the following functions are strictly increasing on their entire domain? (Multiple Correct Answers)

(A) $f(x) = e^x$

(B) $f(x) = \log_e x$ (on its domain)

(C) $f(x) = x^3$

(D) $f(x) = x^2$

(E) $f(x) = x + \sin x$

Question 5. Find the intervals where $f(x) = \sin x - \cos x$ is strictly increasing in $[0, 2\pi]$. Which statements are correct? (Multiple Correct Answers)

(A) $f'(x) = \cos x + \sin x$.

(B) $f'(x) = 0$ when $\cos x + \sin x = 0$, i.e., $\tan x = -1$. In $[0, 2\pi]$, $x = 3\pi/4, 7\pi/4$.

(C) $f'(x) > 0$ when $\cos x + \sin x > 0$. This occurs in $(0, 3\pi/4) \cup (7\pi/4, 2\pi)$ in $[0, 2\pi]$.

(D) $f'(x) < 0$ when $\cos x + \sin x < 0$. This occurs in $(3\pi/4, 7\pi/4)$ in $[0, 2\pi]$.

(E) $f(x)$ is strictly increasing on $[0, 3\pi/4]$ and $[7\pi/4, 2\pi]$ and strictly decreasing on $[3\pi/4, 7\pi/4]$.

Question 6. A function is monotonic on an interval if it is either entirely increasing or entirely decreasing on that interval. Which statements about monotonic functions are true? (Multiple Correct Answers)

(A) A strictly monotonic function is always one-to-one (injective).

(B) A continuous strictly monotonic function has a continuous inverse function.

(C) If $f'(x) > 0$ on an interval, the function is monotonic on that interval.

(D) If a function is monotonic on an interval, it must be differentiable on that interval.

(E) Examples of monotonic functions include $x^3$, $e^x$, $\log_e x$, $\tan^{-1} x$ on their respective domains.

Question 7. Find the intervals where $f(x) = \frac{x}{x^2 + 1}$ is strictly increasing or strictly decreasing. Which statements are correct? (Multiple Correct Answers)

(A) $f'(x) = \frac{1(x^2+1) - x(2x)}{(x^2+1)^2} = \frac{x^2+1-2x^2}{(x^2+1)^2} = \frac{1-x^2}{(x^2+1)^2}$.

(B) $f'(x) = 0$ when $1-x^2 = 0$, so $x = \pm 1$.

(C) $f'(x) > 0$ when $1-x^2 > 0$, i.e., $-1 < x < 1$. So $f(x)$ is strictly increasing on $[-1, 1]$.

(D) $f'(x) < 0$ when $1-x^2 < 0$, i.e., $x < -1$ or $x > 1$. So $f(x)$ is strictly decreasing on $(-\infty, -1]$ and $[1, \infty)$.

(E) The function is strictly increasing on $(-1, 1)$.

Question 8. Which of the following statements are true about the function $f(x) = x + \frac{1}{x}$? (Multiple Correct Answers)

(A) $f'(x) = 1 - \frac{1}{x^2}$.

(B) $f'(x) = 0$ when $x^2=1$, so $x=\pm 1$.

(C) For $x > 0$, $f'(x) > 0$ when $x^2 > 1$, i.e., $x > 1$. So $f(x)$ is strictly increasing on $(1, \infty)$.

(D) For $x > 0$, $f'(x) < 0$ when $x^2 < 1$, i.e., $0 < x < 1$. So $f(x)$ is strictly decreasing on $(0, 1)$.

(E) For $x < 0$, $f'(x) > 0$ when $x^2 < 1$, i.e., $-1 < x < 0$. So $f(x)$ is strictly increasing on $(-1, 0)$.

Question 9. Which of the following functions are strictly decreasing on their entire domain? (Multiple Correct Answers)

(A) $f(x) = -x^3$

(B) $f(x) = e^{-x}$

(C) $f(x) = \cot^{-1} x$

(D) $f(x) = 1/x$ (on $(-\infty, 0)$ and $(0, \infty)$ separately)

(E) $f(x) = \frac{1}{x^2}$ (on $(0, \infty)$)

Question 10. For the function $f(x) = \cos x$, which intervals are correct for monotonicity? (Multiple Correct Answers)

(A) Strictly decreasing on $[0, \pi]$.

(B) Strictly increasing on $[\pi, 2\pi]$.

(C) Strictly decreasing on $[2n\pi, (2n+1)\pi]$ for integer $n$.

(D) Strictly increasing on $[(2n+1)\pi, (2n+2)\pi]$ for integer $n$.

(E) Monotonic over its entire domain $\mathbb{R}$.

Question 11. If a function $f(x)$ is differentiable on an interval $(a, b)$ and $f'(x) > 0$ for all $x$ in $(a, b)$ except for a finite number of points where $f'(x)=0$, which statements are true? (Multiple Correct Answers)

(A) The function is strictly increasing on $(a, b)$.

(B) The function is increasing on $(a, b)$.

(C) The tangent lines are horizontal only at isolated points.

(D) The function may have local extrema at the points where $f'(x)=0$.

(E) Example: $f(x) = x^3$ on $(-1, 1)$. $f'(x)=3x^2 > 0$ for $x \neq 0$, $f'(0)=0$. The function is strictly increasing on $(-1, 1)$.

Question 1. A function $f(x)$ has a local maximum at $x=c$ if there exists an interval $(c-\delta, c+\delta)$ such that for all $x$ in this interval (except possibly $c$), $f(x) \leq f(c)$. Which statements about local extrema are correct? (Multiple Correct Answers)

(A) Local extrema occur at critical points.

(B) A critical point is where $f'(c)=0$ or $f'(c)$ is undefined.

(C) If $f'(c)=0$, $x=c$ is necessarily a local extremum.

(D) Local maximum is the largest value of the function on its entire domain.

(E) Local minimum is the smallest value of the function on its entire domain.

Question 2. According to the First Derivative Test, which conditions indicate a local extremum at a critical point $x=c$ where $f(c)$ is defined? (Multiple Correct Answers)

(A) If $f'(x)$ changes sign from positive to negative as $x$ increases through $c$, there is a local maximum at $c$.

(B) If $f'(x)$ changes sign from negative to positive as $x$ increases through $c$, there is a local minimum at $c$.

(C) If $f'(x)$ does not change sign as $x$ increases through $c$, there is no local extremum at $c$ (it might be an inflection point).

(D) The First Derivative Test can be applied even if $f'(c)$ is undefined, as long as $c$ is a critical point where $f(c)$ is defined.

(E) The First Derivative Test requires $f(x)$ to be differentiable at $c$.

Question 3. According to the Second Derivative Test, which conditions indicate a local extremum at a critical point $x=c$ where $f'(c)=0$? (Multiple Correct Answers)

(A) If $f''(c) > 0$, there is a local minimum at $c$.

(B) If $f''(c) < 0$, there is a local maximum at $c$.

(C) If $f''(c) = 0$, the test is inconclusive.

(D) If the test is inconclusive, there is no local extremum at $c$.

(E) The Second Derivative Test requires $f(x)$ to be twice differentiable at $c$.

Question 4. Find the local extrema of $f(x) = x^3 - 6x^2 + 5$. Which statements are correct? (Multiple Correct Answers)

(A) $f'(x) = 3x^2 - 12x = 3x(x-4)$.

(B) Critical points are $x=0$ and $x=4$.

(C) $f''(x) = 6x - 12$.

(D) $f''(0) = -12 < 0$, so local maximum at $x=0$. The local maximum value is $f(0) = 5$.

(E) $f''(4) = 6(4) - 12 = 24 - 12 = 12 > 0$, so local minimum at $x=4$. The local minimum value is $f(4) = 4^3 - 6(4^2) + 5 = 64 - 96 + 5 = -27$.

Question 5. Find the absolute maximum and minimum values of $f(x) = x^2 - 4$ on the interval $[0, 3]$. Which statements are correct? (Multiple Correct Answers)

(A) Critical points: $f'(x) = 2x$. $f'(x) = 0 \implies x=0$. $x=0$ is a critical point.

(B) Evaluate $f(x)$ at critical points in $(0, 3)$ and at endpoints. Critical point in $(0, 3)$ is none. Endpoints are $x=0, x=3$.

(C) Evaluate at endpoints and critical point: $f(0) = 0^2 - 4 = -4$. $f(3) = 3^2 - 4 = 9 - 4 = 5$.

(D) The absolute maximum value is 5.

(E) The absolute minimum value is -4.

Question 6. In optimization problems (Applied Maths), which steps are typically involved in finding the maximum or minimum value of a quantity? (Multiple Correct Answers)

(A) Define the quantity to be maximized or minimized as a function of one variable.

(B) Determine the domain of the function.

(C) Find the critical points of the function within the domain.

(D) Use the First or Second Derivative Test, or evaluate the function at critical points and endpoints of the domain, to find the extrema.

(E) Interpret the result in the context of the original problem.

Question 7. Find two positive numbers whose sum is 20 and whose product is maximum. Which statements are correct? (Multiple Correct Answers)

(A) Let the numbers be $x$ and $y$. Given $x+y=20$, so $y = 20-x$. The numbers must be positive, so $x>0$ and $y>0 \implies 20-x > 0 \implies x < 20$. Domain is $(0, 20)$.

(B) The product is $P = xy = x(20-x) = 20x - x^2$.

(C) Find the derivative of the product: $P'(x) = 20 - 2x$.

(D) Set $P'(x) = 0$ to find critical points: $20 - 2x = 0 \implies x = 10$. $x=10$ is in the domain $(0, 20)$.

(E) Use the Second Derivative Test: $P''(x) = -2 < 0$. So $x=10$ corresponds to a maximum. When $x=10$, $y = 20-10 = 10$. The numbers are 10 and 10.

Question 8. Find the dimensions of the rectangle with perimeter 100 cm that has the maximum area. Which statements are correct? (Multiple Correct Answers)

(A) Let the sides be $l$ and $w$. Perimeter $2(l+w) = 100$, so $l+w=50$, $w = 50-l$. Since dimensions are positive, $l>0$ and $w>0 \implies 50-l>0 \implies l<50$. Domain is $(0, 50)$.

(B) Area $A = lw = l(50-l) = 50l - l^2$.

(C) Find the derivative of the area: $A'(l) = 50 - 2l$.

(D) Set $A'(l) = 0$: $50 - 2l = 0 \implies l=25$. $l=25$ is in the domain $(0, 50)$.

(E) Use the Second Derivative Test: $A''(l) = -2 < 0$. So $l=25$ corresponds to a maximum. When $l=25$, $w = 50-25 = 25$. The dimensions are 25 cm by 25 cm (a square).

Question 9. Find the minimum value of $f(x) = 2x^2 - 12x + 18$. Which statements are correct? (Multiple Correct Answers)

(A) $f'(x) = 4x - 12$.

(B) Critical point: $4x - 12 = 0 \implies x=3$.

(C) $f''(x) = 4$.

(D) Since $f''(3) = 4 > 0$, $x=3$ corresponds to a local minimum.

(E) Since the function is a parabola opening upwards, the local minimum is also the absolute minimum. The minimum value is $f(3) = 2(3^2) - 12(3) + 18 = 18 - 36 + 18 = 0$.

Question 10. The profit function of a company is given by $P(x) = 41 + 24x - 18x^2$, where $x$ is the number of units sold. Find the number of units to be sold to maximize the profit. Which statements are correct? (Multiple Correct Answers)

(A) The number of units $x$ to maximize profit is found by setting the derivative of the profit function $P'(x)$ equal to zero.

(B) $P'(x) = 24 - 36x$. Setting $P'(x) = 0$ gives $x = 24/36 = 2/3$.

(C) The Second Derivative $P''(x) = -36$ is negative, indicating a maximum at $x=2/3$.

(D) The maximum profit occurs when $x=2/3$ units are sold (for the continuous function).

(E) If $x$ must be an integer, we should check the integers closest to $2/3$ (i.e., $x=0$ and $x=1$).

Question 11. Find the absolute maximum and minimum values of $f(x) = x^3 - 3x$ on the interval $[0, 2]$. Which statements are correct? (Multiple Correct Answers)

(A) $f'(x) = 3x^2 - 3 = 3(x^2-1) = 3(x-1)(x+1)$.

(B) Critical points are where $f'(x)=0$ or is undefined. $x=\pm 1$. Only $x=1$ is in the open interval $(0, 2)$.

(C) Evaluate $f(x)$ at the critical point $x=1$ and the endpoints $x=0, x=2$.

(D) $f(0) = 0^3 - 3(0) = 0$. $f(1) = 1^3 - 3(1) = 1 - 3 = -2$. $f(2) = 2^3 - 3(2) = 8 - 6 = 2$.

(E) The absolute maximum value is 2, and the absolute minimum value is -2.

Question 12. Find two positive numbers $x$ and $y$ such that their sum is 60 and $xy^3$ is maximum. Which statements are correct? (Multiple Correct Answers)

(A) Given $x+y=60$, so $x=60-y$. Since $x, y > 0$, $y>0$ and $60-y>0 \implies y<60$. Domain for $y$ is $(0, 60)$.

(B) We want to maximize $P = xy^3 = (60-y)y^3 = 60y^3 - y^4$.

(C) Find the derivative of $P$ with respect to $y$: $P'(y) = 180y^2 - 4y^3 = 4y^2(45-y)$.

(D) Critical points in $(0, 60)$ are where $P'(y) = 0$, so $4y^2(45-y)=0$. Since $y \neq 0$, $45-y=0 \implies y=45$.

(E) Use the First Derivative Test: For $0 < y < 45$, $P'(y) > 0$. For $45 < y < 60$, $P'(y) < 0$. Since $P'(y)$ changes from positive to negative at $y=45$, this is a local maximum. Since it's the only critical point in the open interval, it's the absolute maximum. When $y=45$, $x=60-45=15$. The numbers are 15 and 45.

Question 1. The indefinite integral of a function $f(x)$, denoted by $\int f(x) dx$, represents the process of finding its antiderivative(s). Which statements are correct? (Multiple Correct Answers)

(A) If $F'(x) = f(x)$, then $F(x)$ is an antiderivative of $f(x)$.

(B) The indefinite integral $\int f(x) dx = F(x) + C$, where $C$ is the constant of integration.

(C) The constant of integration $C$ accounts for the fact that the derivative of a constant is zero, so there are infinitely many antiderivatives for a given function, differing by a constant.

(D) Finding the indefinite integral is the reverse process of differentiation.

(E) The indefinite integral represents a single function.

Question 2. Which of the following are correct standard formulas for indefinite integrals? (Multiple Correct Answers)

(A) $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$)

(B) $\int \frac{1}{x} dx = \log_e |x| + C$ (for $x \neq 0$)

(C) $\int \sin x dx = \cos x + C$

(D) $\int \cos x dx = \sin x + C$

(E) $\int e^x dx = e^x + C$

Question 3. Evaluate $\int (3x^2 - 4x + 5) dx$. Which steps or results are correct using the properties of indefinite integrals? (Multiple Correct Answers)

(A) $\int (3x^2 - 4x + 5) dx = \int 3x^2 dx - \int 4x dx + \int 5 dx$.

(B) $\int 3x^2 dx = 3 \int x^2 dx = 3 \frac{x^3}{3} + C_1 = x^3 + C_1$.

(C) $\int 4x dx = 4 \int x dx = 4 \frac{x^2}{2} + C_2 = 2x^2 + C_2$.

(D) $\int 5 dx = 5x + C_3$.

(E) The result is $x^3 - 2x^2 + 5x + C$, where $C = C_1 - C_2 + C_3$ is the combined constant of integration.

Question 4. In Applied Maths, if $MC(x)$ is the marginal cost function, then the total cost function $C(x)$ can be found by integration. If $MC(x) = 6x + 2$, and the fixed cost is $\textsf{₹ }500$, which statements are correct? (Multiple Correct Answers)

(A) $C(x) = \int MC(x) dx = \int (6x + 2) dx$.

(B) $\int (6x + 2) dx = 3x^2 + 2x + C$.

(C) The fixed cost is the cost when $x=0$, so $C(0) = 500$.

(D) Using $C(0) = 500$: $3(0)^2 + 2(0) + C = 500$, so $C = 500$.

(E) The total cost function is $C(x) = 3x^2 + 2x + 500$.

Question 5. Which of the following are correct standard formulas for indefinite integrals? (Multiple Correct Answers)

(A) $\int \sec x \tan x dx = \sec x + C$

(B) $\int \text{cosec } x \cot x dx = \text{cosec } x + C$

(C) $\int \sec^2 x dx = \tan x + C$

(D) $\int \text{cosec}^2 x dx = -\cot x + C$

(E) $\int a^x dx = a^x \log_e a + C$ (for $a>0, a \neq 1$)

Question 6. Evaluate $\int (\sin x + \sec x \tan x) dx$. Which statements or results are correct? (Multiple Correct Answers)

(A) $\int (\sin x + \sec x \tan x) dx = \int \sin x dx + \int \sec x \tan x dx$.

(B) $\int \sin x dx = -\cos x + C_1$.

(C) $\int \sec x \tan x dx = \sec x + C_2$.

(D) The result is $-\cos x + \sec x + C$.

(E) The derivative of $-\cos x + \sec x + C$ is $\sin x + \sec x \tan x$.

Question 7. If $\int f(x) dx = G(x) + C$, then which properties are true? (Multiple Correct Answers)

(A) $G'(x) = f(x)$.

(B) $\int c f(x) dx = c G(x) + C$ for a constant $c$.

(C) $\int (f(x) + g(x)) dx = G(x) + \int g(x) dx + C$.

(D) $\frac{d}{dx} (\int f(x) dx) = f(x)$.

(E) $\int G'(x) dx = G(x) + C$.

Question 8. The graph of $y = F(x) + C$ for different values of $C$ represents a family of curves. Which statements are correct? (Multiple Correct Answers)

(A) These curves are vertical translations of each other.

(B) At any given $x$, the slopes of these curves are the same, equal to $f(x)$.

(C) This family of curves represents the indefinite integral of $f(x)$.

(D) The constant $C$ determines the specific curve in the family.

(E) The curves can intersect each other.

Question 9. Evaluate $\int \frac{\sin x}{\cos^2 x} dx$. Which methods or results are correct? (Multiple Correct Answers)

(A) Rewrite the integrand as $\frac{1}{\cos x} \cdot \frac{\sin x}{\cos x} = \sec x \tan x$.

(B) The integral of $\sec x \tan x$ is $\sec x + C$.

(C) Use substitution $u = \cos x$, $du = -\sin x dx$. The integral becomes $\int \frac{-du}{u^2} = -\int u^{-2} du$.

(D) $-\int u^{-2} du = - \frac{u^{-1}}{-1} + C = \frac{1}{u} + C$.

(E) Substitute back $u = \cos x$: $\frac{1}{\cos x} + C = \sec x + C$. Both methods give the same result.

Question 10. If the marginal revenue function is $MR(x) = 10 - 4x$, and the total revenue from selling 0 units is $\textsf{₹ }0$, find the total revenue function $R(x)$. Which statements are correct? (Multiple Correct Answers)

(A) $R(x) = \int MR(x) dx = \int (10 - 4x) dx$.

(B) $\int (10 - 4x) dx = 10x - 2x^2 + C$.

(C) $R(0) = 0$ is the initial condition to find $C$.

(D) Using $R(0) = 0$: $10(0) - 2(0)^2 + C = 0$, so $C=0$.

(E) The total revenue function is $R(x) = 10x - 2x^2$.

Question 11. Which of the following are general formulas obtained by integrating standard derivatives? (Multiple Correct Answers)

(A) $\int \cos x dx = \sin x + C$

(B) $\int \sec^2 x dx = \tan x + C$

(C) $\int e^x dx = e^x + C$

(D) $\int \frac{1}{x} dx = \log_e x + C$

(E) $\int \frac{1}{\sqrt{1-x^2}} dx = \sin^{-1} x + C$

Question 1. The method of integration by substitution is based on the chain rule for differentiation. It is useful when the integrand is of the form $f(g(x)) \cdot g'(x)$. Which of the following integrals can be solved using a simple substitution? (Multiple Correct Answers)

(A) $\int 2x \sin(x^2) dx$ (Substitute $u=x^2$)

(B) $\int e^{\tan x} \sec^2 x dx$ (Substitute $u=\tan x$)

(C) $\int \frac{\log_e x}{x} dx$ (Substitute $u=\log_e x$)

(D) $\int x e^x dx$

(E) $\int \sin^3 x \cos x dx$ (Substitute $u=\sin x$)

Question 2. Evaluate $\int x \cos(x^2) dx$ using substitution $u=x^2$. Which steps or results are correct? (Multiple Correct Answers)

(A) Let $u = x^2$, then $du = 2x dx$.

(B) The integral becomes $\int \cos(x^2) \cdot (x dx)$.

(C) Rewrite the integral in terms of $u$: $\int \cos u \cdot \frac{1}{2} du = \frac{1}{2} \int \cos u du$.

(D) $\frac{1}{2} \int \cos u du = \frac{1}{2} \sin u + C$.

(E) Substitute back $u=x^2$: $\frac{1}{2} \sin(x^2) + C$.

Question 3. The integration by parts formula is $\int u dv = uv - \int v du$. How do you choose $u$ and $dv$? (Multiple Correct Answers)

(A) Choose $dv$ to be a part of the integrand that can be easily integrated to find $v$.

(B) Choose $u$ to be a part of the integrand that becomes simpler when differentiated.

(C) A common mnemonic for choosing $u$ is LIATE (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential).

(D) Choose $u$ as the function that comes earlier in the LIATE order.

(E) Choose $dv$ as the remaining part of the integrand, including $dx$.

Question 4. Evaluate $\int x e^x dx$ using integration by parts. Which choices for $u$ and $dv$ and resulting steps are correct? (Multiple Correct Answers)

(A) Choose $u = x$ and $dv = e^x dx$.

(B) Then $du = dx$ and $v = e^x$.

(C) $\int u dv = uv - \int v du = x e^x - \int e^x dx$.

(D) $\int e^x dx = e^x + C_1$.

(E) The result is $x e^x - e^x + C$.

Question 5. Evaluate $\int \log_e x dx$ using integration by parts. Which choices for $u$ and $dv$ and resulting steps are correct? (Multiple Correct Answers)

(A) Choose $u = \log_e x$ and $dv = dx$.

(B) Then $du = \frac{1}{x} dx$ and $v = x$.

(C) $\int u dv = uv - \int v du = (\log_e x)(x) - \int x \cdot \frac{1}{x} dx$.

(D) $\int x \cdot \frac{1}{x} dx = \int 1 dx = x + C_1$.

(E) The result is $x \log_e x - x + C$.

Question 6. The integral $\int e^x (f(x) + f'(x)) dx = e^x f(x) + C$ is a standard result often solved using integration by parts. Which steps demonstrate this for $f(x) = \tan x$? (Multiple Correct Answers)

(A) The integral is $\int e^x (\tan x + \sec^2 x) dx$. Here $f(x) = \tan x$ and $f'(x) = \sec^2 x$.

(B) Using the formula, the result is $e^x \tan x + C$.

(C) To prove the formula: Use integration by parts on $\int e^x f(x) dx$. Let $u=f(x), dv=e^x dx$. Then $du=f'(x) dx, v=e^x$.

(D) $\int e^x f(x) dx = f(x) e^x - \int e^x f'(x) dx$.

(E) So $\int e^x (f(x) + f'(x)) dx = \int e^x f(x) dx + \int e^x f'(x) dx = (e^x f(x) - \int e^x f'(x) dx) + \int e^x f'(x) dx = e^x f(x) + C$.

Question 7. Evaluate $\int \sin^5 x \cos x dx$ using substitution. Which steps or results are correct? (Multiple Correct Answers)

(A) Let $u = \sin x$.

(B) Then $du = \cos x dx$.

(C) The integral becomes $\int u^5 du$.

(D) $\int u^5 du = \frac{u^6}{6} + C$.

(E) Substitute back $u=\sin x$: $\frac{\sin^6 x}{6} + C$.

Question 8. Evaluate $\int x^2 e^x dx$. This integral requires integration by parts multiple times. Which statements are correct? (Multiple Correct Answers)

(A) Using LIATE, choose $u=x^2$ and $dv=e^x dx$. Then $du=2x dx$ and $v=e^x$.

(B) $\int x^2 e^x dx = x^2 e^x - \int e^x (2x) dx = x^2 e^x - 2 \int x e^x dx$.

(C) The integral $\int x e^x dx$ must be solved using integration by parts again.

(D) From Q170, $\int x e^x dx = xe^x - e^x + C_1$.

(E) Substituting this back: $\int x^2 e^x dx = x^2 e^x - 2 (xe^x - e^x) + C = x^2 e^x - 2xe^x + 2e^x + C = e^x (x^2 - 2x + 2) + C$.

Question 9. Evaluate $\int \tan x dx$. Which steps or results are correct? (Multiple Correct Answers)

(A) Rewrite $\tan x$ as $\frac{\sin x}{\cos x}$.

(B) Use substitution $u = \cos x$, $du = -\sin x dx$.

(C) The integral becomes $\int \frac{-du}{u} = -\int \frac{1}{u} du$.

(D) $-\int \frac{1}{u} du = -\log_e |u| + C$.

(E) Substitute back $u = \cos x$: $-\log_e |\cos x| + C = \log_e |(\cos x)^{-1}| + C = \log_e |\sec x| + C$.

Question 10. Evaluate $\int \sin^{-1} x dx$. Which steps or results are correct using integration by parts? (Multiple Correct Answers)

(A) Choose $u = \sin^{-1} x$ and $dv = dx$.

(B) Then $du = \frac{1}{\sqrt{1-x^2}} dx$ and $v = x$.

(C) $\int u dv = uv - \int v du = x \sin^{-1} x - \int x \frac{1}{\sqrt{1-x^2}} dx$.

(D) The integral $\int \frac{x}{\sqrt{1-x^2}} dx$ can be solved by substitution, e.g., $w = 1-x^2$, $dw = -2x dx$.

(E) $\int \frac{x}{\sqrt{1-x^2}} dx = \int \frac{-1/2 dw}{\sqrt{w}} = -\frac{1}{2} \int w^{-1/2} dw = -\frac{1}{2} \frac{w^{1/2}}{1/2} + C_1 = -\sqrt{w} + C_1 = -\sqrt{1-x^2} + C_1$. So $\int \sin^{-1} x dx = x \sin^{-1} x - (-\sqrt{1-x^2}) + C = x \sin^{-1} x + \sqrt{1-x^2} + C$.

Question 1. The method of partial fractions is used to integrate rational functions $\frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are polynomials. What conditions should the rational function satisfy for this method to be directly applicable? (Multiple Correct Answers)

(A) The degree of $P(x)$ must be less than the degree of $Q(x)$ (proper rational function).

(B) The denominator $Q(x)$ must be factorizable into linear or quadratic factors.

(C) If the degree of $P(x)$ is greater than or equal to the degree of $Q(x)$, long division must be performed first.

(D) The factors of $Q(x)$ must not be repeated.

(E) The method decomposes the rational function into simpler fractions that are easier to integrate.

Question 2. To integrate $\int \frac{x}{(x-1)(x-2)} dx$ using partial fractions, which steps are correct? (Multiple Correct Answers)

(A) Write $\frac{x}{(x-1)(x-2)} = \frac{A}{x-1} + \frac{B}{x-2}$.

(B) Multiply by $(x-1)(x-2)$: $x = A(x-2) + B(x-1)$.

(C) Set $x=1$: $1 = A(1-2) + B(1-1) \implies 1 = -A \implies A = -1$.

(D) Set $x=2$: $2 = A(2-2) + B(2-1) \implies 2 = B \implies B = 2$.

(E) The integral is $\int (\frac{-1}{x-1} + \frac{2}{x-2}) dx = -\log_e |x-1| + 2\log_e |x-2| + C = \log_e |\frac{(x-2)^2}{x-1}| + C$.

Question 3. Which of the following are standard integrals of the form $\int \frac{dx}{\sqrt{a^2 \pm x^2}}$ or $\int \frac{dx}{\sqrt{x^2 \pm a^2}}$? (Assume $a>0$). (Multiple Correct Answers)

(A) $\int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}(\frac{x}{a}) + C$

(B) $\int \frac{dx}{\sqrt{x^2 - a^2}} = \log_e |x + \sqrt{x^2 - a^2}| + C$

(C) $\int \frac{dx}{\sqrt{x^2 + a^2}} = \log_e |x + \sqrt{x^2 + a^2}| + C$

(D) $\int \frac{dx}{\sqrt{a^2 + x^2}} = \sinh^{-1}(\frac{x}{a}) + C$

(E) $\int \frac{dx}{\sqrt{x^2 - a^2}} = \cosh^{-1}(\frac{x}{a}) + C$

Question 4. Which of the following are standard integrals of the form $\int \frac{dx}{a^2 \pm x^2}$ or $\int \frac{dx}{x^2 - a^2}$? (Assume $a>0$). (Multiple Correct Answers)

(A) $\int \frac{dx}{a^2 + x^2} = \frac{1}{a} \tan^{-1}(\frac{x}{a}) + C$

(B) $\int \frac{dx}{x^2 + a^2} = \frac{1}{a} \cot^{-1}(\frac{x}{a}) + C$

(C) $\int \frac{dx}{a^2 - x^2} = \frac{1}{2a} \log_e |\frac{a+x}{a-x}| + C$

(D) $\int \frac{dx}{x^2 - a^2} = \frac{1}{2a} \log_e |\frac{x-a}{x+a}| + C$

(E) $\int \frac{dx}{x^2 - a^2} = \frac{1}{2a} \log_e |\frac{a+x}{a-x}| + C$

Question 5. Which of the following are standard integrals involving $\sqrt{a^2 \pm x^2}$ or $\sqrt{x^2 \pm a^2}$ in the numerator? (Assume $a>0$). (Multiple Correct Answers)

(A) $\int \sqrt{a^2 - x^2} dx = \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1}(\frac{x}{a}) + C$

(B) $\int \sqrt{x^2 - a^2} dx = \frac{x}{2}\sqrt{x^2 - a^2} - \frac{a^2}{2} \log_e |x + \sqrt{x^2 - a^2}| + C$

(C) $\int \sqrt{x^2 + a^2} dx = \frac{x}{2}\sqrt{x^2 + a^2} + \frac{a^2}{2} \log_e |x + \sqrt{x^2 + a^2}| + C$

(D) $\int \sqrt{a^2 + x^2} dx = \frac{x}{2}\sqrt{a^2 + x^2} + \frac{a^2}{2} \sinh^{-1}(\frac{x}{a}) + C$

(E) $\int \sqrt{a^2 - x^2} dx = \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2} \cos^{-1}(\frac{x}{a}) + C$

Question 6. To integrate rational functions of $\sin x$ and $\cos x$, the substitution $t = \tan(x/2)$ is often used. Which resulting identities for $\sin x$, $\cos x$, and $dx$ are correct in terms of $t$? (Multiple Correct Answers)

(A) $\sin x = \frac{2t}{1-t^2}$

(B) $\sin x = \frac{2t}{1+t^2}$

(C) $\cos x = \frac{1-t^2}{1+t^2}$

(D) $\cos x = \frac{1+t^2}{1-t^2}$

(E) $dx = \frac{2dt}{1+t^2}$

Question 7. Evaluate $\int \frac{dx}{x^2 + 2x + 2}$. Which steps or results are correct? (Multiple Correct Answers)

(A) Complete the square in the denominator: $x^2 + 2x + 2 = (x+1)^2 + 1$.

(B) The integral becomes $\int \frac{dx}{(x+1)^2 + 1^2}$.

(C) Use substitution $u = x+1$, $du = dx$. The integral becomes $\int \frac{du}{u^2 + 1^2}$.

(D) Use the standard formula $\int \frac{du}{u^2 + a^2} = \frac{1}{a} \tan^{-1}(\frac{u}{a}) + C$, with $a=1$.

(E) The result is $\tan^{-1}(x+1) + C$.

Question 8. Evaluate $\int \frac{dx}{\sqrt{9 - x^2}}$. Which steps or results are correct? (Multiple Correct Answers)

(A) This is of the form $\int \frac{dx}{\sqrt{a^2 - x^2}}$ with $a=3$.

(B) The standard formula is $\sin^{-1}(\frac{x}{a}) + C$.

(C) The result is $\sin^{-1}(\frac{x}{3}) + C$.

(D) The domain for the integrand to be real is $|x| \leq 3$.

(E) The integral is defined for $|x| < 3$.

Question 9. Evaluate $\int \frac{x+1}{\sqrt{x^2+2x+5}} dx$. Which steps or results are correct? (Multiple Correct Answers)

(A) Notice that the numerator $x+1$ is proportional to the derivative of the expression inside the square root $x^2+2x+5$.

(B) Let $u = x^2+2x+5$. Then $du = (2x+2) dx = 2(x+1) dx$.

(C) So $(x+1) dx = \frac{1}{2} du$.

(D) The integral becomes $\int \frac{1}{\sqrt{u}} \frac{1}{2} du = \frac{1}{2} \int u^{-1/2} du$.

(E) $\frac{1}{2} \int u^{-1/2} du = \frac{1}{2} \frac{u^{1/2}}{1/2} + C = u^{1/2} + C = \sqrt{x^2+2x+5} + C$.

Question 10. Which of the following rational functions require long division before applying partial fraction decomposition? (Multiple Correct Answers)

(A) $\frac{x^3+1}{x^2-4}$

(B) $\frac{x^2}{x^2+1}$

(C) $\frac{x-5}{x^2+2x+1}$

(D) $\frac{2x^4+x^2-1}{x^3+2x}$

(E) $\frac{1}{(x+1)(x-2)}$

Question 1. The definite integral $\int_a^b f(x) dx$ is defined as the limit of a Riemann sum, provided the limit exists and is finite. Which statements about the definite integral are correct? (Multiple Correct Answers)

(A) The definite integral represents the signed area between the curve $y=f(x)$ and the x-axis from $x=a$ to $x=b$.

(B) If $f(x) \geq 0$ on $[a, b]$, the definite integral equals the area under the curve.

(C) The value of the definite integral is a single real number.

(D) The definite integral is independent of the variable of integration (dummy variable).

(E) The Fundamental Theorem of Calculus provides a method for evaluating definite integrals using antiderivatives.

Question 2. The First Fundamental Theorem of Calculus relates differentiation and integration. It states that if $f$ is continuous on $[a, b]$ and $F(x) = \int_a^x f(t) dt$, then $F'(x) = f(x)$ for all $x \in (a, b)$. Which statements are correct? (Multiple Correct Answers)

(A) The derivative of an integral with a variable upper limit is the integrand evaluated at that limit.

(B) This theorem shows that integration is the reverse process of differentiation (up to a constant).

(C) If the upper limit is a function of $x$, say $u(x)$, then $\frac{d}{dx} \int_a^{u(x)} f(t) dt = f(u(x)) \cdot u'(x)$.

(D) If the lower limit is a function of $x$, say $v(x)$, then $\frac{d}{dx} \int_{v(x)}^b f(t) dt = -f(v(x)) \cdot v'(x)$.

(E) If both limits are functions of $x$, $\frac{d}{dx} \int_{v(x)}^{u(x)} f(t) dt = f(u(x)) u'(x) - f(v(x)) v'(x)$.

Question 3. The Second Fundamental Theorem of Calculus (Evaluation Theorem) provides a way to compute definite integrals. It states that if $f$ is continuous on $[a, b]$ and $F$ is any antiderivative of $f$ on $[a, b]$, then $\int_a^b f(x) dx = F(b) - F(a)$. Which statements are correct? (Multiple Correct Answers)

(A) To evaluate a definite integral, find an antiderivative of the integrand.

(B) Subtract the value of the antiderivative at the lower limit from its value at the upper limit.

(C) The constant of integration $C$ cancels out when evaluating $F(b) - F(a)$.

(D) This theorem applies even if $f$ has jump discontinuities on $[a, b]$.

(E) This theorem connects the concept of area under a curve (definite integral) with the concept of antiderivatives (indefinite integral).

Question 4. Evaluate $\int_1^3 x^2 dx$. Which steps or results are correct using the Fundamental Theorem? (Multiple Correct Answers)

(A) An antiderivative of $f(x) = x^2$ is $F(x) = \frac{x^3}{3}$.

(B) We need to compute $F(3) - F(1)$.

(C) $F(3) = \frac{3^3}{3} = \frac{27}{3} = 9$.

(D) $F(1) = \frac{1^3}{3} = \frac{1}{3}$.

(E) The value of the integral is $9 - \frac{1}{3} = \frac{27-1}{3} = \frac{26}{3}$.

Question 5. Evaluate $\int_0^{\pi/2} \cos x dx$. Which steps or results are correct using the Fundamental Theorem? (Multiple Correct Answers)

(A) An antiderivative of $f(x) = \cos x$ is $F(x) = \sin x$.

(B) We need to compute $F(\pi/2) - F(0)$.

(C) $F(\pi/2) = \sin(\pi/2) = 1$.

(D) $F(0) = \sin(0) = 0$.

(E) The value of the integral is $1 - 0 = 1$.

Question 6. What does $\int_a^b f(x) dx$ represent geometrically? (Assume $f(x)$ is continuous on $[a, b]$). (Multiple Correct Answers)

(A) The length of the curve from $x=a$ to $x=b$.

(B) If $f(x) \geq 0$, it's the area under the curve $y=f(x)$ above the x-axis, bounded by $x=a$ and $x=b$.

(C) If $f(x) \leq 0$, it's the negative of the area between the curve and the x-axis, bounded by $x=a$ and $x=b$.

(D) It's always the total area bounded by the curve and the x-axis.

(E) It represents the net signed area (area above x-axis minus area below x-axis).

Question 7. Evaluate $\int_0^1 e^x dx$. Which steps or results are correct using the Fundamental Theorem? (Multiple Correct Answers)

(A) An antiderivative of $f(x) = e^x$ is $F(x) = e^x$.

(B) We need to compute $F(1) - F(0)$.

(C) $F(1) = e^1 = e$.

(D) $F(0) = e^0 = 1$.

(E) The value of the integral is $e - 1$.

Question 8. Consider the limit of a sum definition for $\int_a^b f(x) dx = \lim\limits_{n \to \infty} \sum\limits_{i=1}^n f(x_i^*) \Delta x$, where $\Delta x = \frac{b-a}{n}$ and $x_i^*$ is a point in the $i$-th subinterval. Which statements are correct? (Multiple Correct Answers)

(A) As $n \to \infty$, $\Delta x \to 0$.

(B) The sum $\sum\limits_{i=1}^n f(x_i^*) \Delta x$ represents the sum of areas of $n$ rectangles.

(C) The choice of $x_i^*$ (left endpoint, right endpoint, midpoint, etc.) affects the value of the limit.

(D) For a continuous function, the limit exists and is unique regardless of the choice of $x_i^*$.

(E) This definition relates the integral to the geometric concept of area.

Question 9. Which of the following are properties of definite integrals? (Assume $f, g$ are continuous on the relevant intervals, and $k$ is a constant). (Multiple Correct Answers)

(A) $\int_a^b f(x) dx = -\int_b^a f(x) dx$

(B) $\int_a^a f(x) dx = 0$

(C) $\int_a^b k f(x) dx = k \int_a^b f(x) dx$

(D) $\int_a^b (f(x) + g(x)) dx = \int_a^b f(x) dx + \int_a^b g(x) dx$

(E) $\int_a^b f(x) dx + \int_b^c f(x) dx = \int_a^c f(x) dx$

Question 10. If $f(x)$ is an odd function, i.e., $f(-x) = -f(x)$, which property of definite integrals is true? (Multiple Correct Answers)

(A) $\int_{-a}^a f(x) dx = 2 \int_0^a f(x) dx$

(B) $\int_{-a}^a f(x) dx = 0$

(C) Example: $\int_{-\pi/2}^{\pi/2} \sin x dx = 0$.

(D) Example: $\int_{-1}^1 x^3 dx = 0$.

(E) If $f(x)$ is an even function, $f(-x)=f(x)$, then $\int_{-a}^a f(x) dx = 0$.

Question 1. Evaluate $\int_0^{\pi/2} \sin^2 x dx$. Which methods or properties can be used? (Multiple Correct Answers)

(A) Use the identity $\sin^2 x = \frac{1 - \cos(2x)}{2}$.

(B) Integrate using substitution.

(C) Evaluate $\int \frac{1 - \cos(2x)}{2} dx = \frac{1}{2}x - \frac{1}{4}\sin(2x) + C$.

(D) Use the property $\int_0^a f(x) dx = \int_0^a f(a-x) dx$. Let $I = \int_0^{\pi/2} \sin^2 x dx$. $I = \int_0^{\pi/2} \sin^2 (\pi/2 - x) dx = \int_0^{\pi/2} \cos^2 x dx$. Then $2I = \int_0^{\pi/2} (\sin^2 x + \cos^2 x) dx = \int_0^{\pi/2} 1 dx = [x]_0^{\pi/2} = \pi/2$. So $I = \pi/4$.

(E) The value is $\pi/4$.

Question 2. Evaluate $\int_0^1 x e^{x^2} dx$ using substitution. Which steps or results are correct? (Multiple Correct Answers)

(A) Let $u = x^2$. Then $du = 2x dx$.

(B) The integral becomes $\int e^u \frac{1}{2} du$.

(C) When $x=0$, $u=0^2=0$. When $x=1$, $u=1^2=1$. The limits change from $0$ to $1$.

(D) The definite integral is $\int_0^1 e^u \frac{1}{2} du = \frac{1}{2} [e^u]_0^1 = \frac{1}{2} (e^1 - e^0)$.

(E) The value is $\frac{1}{2}(e - 1)$.

Question 3. Which of the following properties are useful for evaluating definite integrals? (Assume $f$ is continuous where needed). (Multiple Correct Answers)

(A) $\int_0^{2a} f(x) dx = 2 \int_0^a f(x) dx$ if $f(2a-x) = f(x)$.

(B) $\int_0^{2a} f(x) dx = 0$ if $f(2a-x) = -f(x)$.

(C) $\int_{-a}^a f(x) dx = 2 \int_0^a f(x) dx$ if $f(x)$ is an even function.

(D) $\int_{-a}^a f(x) dx = 0$ if $f(x)$ is an odd function.

(E) $\int_a^b f(x) dx = \int_a^b f(a+b-x) dx$.

Question 4. Evaluate $\int_0^{\pi} \sin^3 x dx$. Which methods or properties are useful? (Multiple Correct Answers)

(A) Use the identity $\sin^3 x = \frac{3\sin x - \sin(3x)}{4}$.

(B) Use substitution $u = \cos x$. Then $\sin^3 x dx = (1-\cos^2 x) \sin x dx = -(1-u^2) du = (u^2-1) du$. When $x=0, u=1$. When $x=\pi, u=-1$. The integral is $\int_1^{-1} (u^2-1) du = -\int_{-1}^1 (u^2-1) du$.

(C) Using the identity method: $\int_0^{\pi} \frac{3\sin x - \sin(3x)}{4} dx = \frac{1}{4} [-3\cos x + \frac{\cos(3x)}{3}]_0^{\pi}$.

(D) Evaluate the result from (C): $\frac{1}{4} [(-3\cos\pi + \frac{\cos(3\pi)}{3}) - (-3\cos 0 + \frac{\cos 0}{3})] = \frac{1}{4} [(-3(-1) + \frac{-1}{3}) - (-3(1) + \frac{1}{3})] = \frac{1}{4} [(3 - 1/3) - (-3 + 1/3)] = \frac{1}{4} [6 - 2/3] = \frac{1}{4} [16/3] = 4/3$.

(E) Evaluate the result from (B): $-\int_{-1}^1 (u^2-1) du = -[\frac{u^3}{3}-u]_{-1}^1 = -[(\frac{1}{3}-1) - (\frac{-1}{3}-(-1))] = -[-2/3 - (-1/3+1)] = -[-2/3 - 2/3] = -[-4/3] = 4/3$.

Question 5. Evaluate $\int_0^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx$ using properties. Which steps or results are correct? (Multiple Correct Answers)

(A) Let $I = \int_0^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx$.

(B) Use the property $\int_0^a f(x) dx = \int_0^a f(a-x) dx$. So $I = \int_0^{\pi/2} \frac{\sqrt{\sin(\pi/2 - x)}}{\sqrt{\sin(\pi/2 - x)} + \sqrt{\cos(\pi/2 - x)}} dx = \int_0^{\pi/2} \frac{\sqrt{\cos x}}{\sqrt{\cos x} + \sqrt{\sin x}} dx$.

(C) Add the two expressions for $I$: $2I = \int_0^{\pi/2} \left( \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} + \frac{\sqrt{\cos x}}{\sqrt{\cos x} + \sqrt{\sin x}} \right) dx$.

(D) $2I = \int_0^{\pi/2} \frac{\sqrt{\sin x} + \sqrt{\cos x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx = \int_0^{\pi/2} 1 dx = [x]_0^{\pi/2} = \pi/2 - 0 = \pi/2$.

(E) $I = \pi/4$.

Question 6. Which of the following integrals evaluate to 0? (Multiple Correct Answers)

(A) $\int_{-1}^1 x^3 dx$ (Odd function over symmetric interval)

(B) $\int_{-\pi/2}^{\pi/2} \sin x dx$ (Odd function over symmetric interval)

(C) $\int_{-1}^1 e^x dx$

(D) $\int_{-\pi}^{\pi} \tan x dx$ (Odd function over symmetric interval, but discontinuous at $\pm \pi/2$)

(E) $\int_{-2}^2 (x^5 - x) dx$ (Odd function over symmetric interval)

Question 7. Evaluate $\int_0^1 \frac{\tan^{-1} x}{1+x^2} dx$. Which steps or results are correct using substitution? (Multiple Correct Answers)

(A) Let $u = \tan^{-1} x$. Then $du = \frac{1}{1+x^2} dx$.

(B) When $x=0$, $u = \tan^{-1}(0) = 0$. When $x=1$, $u = \tan^{-1}(1) = \pi/4$. The limits change from $0$ to $\pi/4$.

(C) The integral becomes $\int_0^{\pi/4} u du$.

(D) $\int_0^{\pi/4} u du = [\frac{u^2}{2}]_0^{\pi/4} = \frac{(\pi/4)^2}{2} - \frac{0^2}{2}$.

(E) The value is $\frac{\pi^2/16}{2} = \frac{\pi^2}{32}$.

Question 8. Which definite integrals can be evaluated using the property $\int_0^a f(x) dx = \int_0^a f(a-x) dx$? (Multiple Correct Answers)

(A) $\int_0^{\pi/2} \frac{\sin x}{\sin x + \cos x} dx$

(B) $\int_0^1 \frac{\log_e(1+x)}{1+x^2} dx$

(C) $\int_0^{\pi/2} \log_e(\tan x) dx$

(D) $\int_0^{\pi} x \sin x dx$

(E) $\int_0^{\pi/2} \sin^n x dx$

Question 9. Evaluate $\int_0^\pi x \sin x dx$. Which method(s) or steps are useful? (Multiple Correct Answers)

(A) Use integration by parts: Let $u=x, dv = \sin x dx$. Then $du=dx, v=-\cos x$.

(B) $\int x \sin x dx = -x \cos x - \int (-\cos x) dx = -x \cos x + \int \cos x dx = -x \cos x + \sin x + C$.

(C) Evaluate the definite integral: $[-x \cos x + \sin x]_0^\pi = (-\pi \cos\pi + \sin\pi) - (-0 \cos 0 + \sin 0) = (-\pi(-1) + 0) - (0 + 0) = \pi$.

(D) Use the property $\int_0^a x f(x) dx = \int_0^a (a-x) f(a-x) dx$. Let $I = \int_0^\pi x \sin x dx$. $I = \int_0^\pi (\pi-x) \sin(\pi-x) dx = \int_0^\pi (\pi-x) \sin x dx = \int_0^\pi \pi \sin x dx - \int_0^\pi x \sin x dx = \pi \int_0^\pi \sin x dx - I$.

(E) $2I = \pi \int_0^\pi \sin x dx = \pi [-\cos x]_0^\pi = \pi (-\cos\pi - (-\cos 0)) = \pi (-(-1) - (-1)) = \pi(1+1) = 2\pi$. So $I = \pi$.

Question 10. Evaluate $\int_0^{\pi/2} \log_e(\tan x) dx$. Which statements are correct? (Multiple Correct Answers)

(A) Let $I = \int_0^{\pi/2} \log_e(\tan x) dx$.

(B) Use the property $\int_0^a f(x) dx = \int_0^a f(a-x) dx$. $I = \int_0^{\pi/2} \log_e(\tan(\pi/2 - x)) dx = \int_0^{\pi/2} \log_e(\cot x) dx$.

(C) Add the two integrals: $2I = \int_0^{\pi/2} (\log_e(\tan x) + \log_e(\cot x)) dx$.

(D) Using logarithm properties: $\log_e(\tan x) + \log_e(\cot x) = \log_e(\tan x \cdot \cot x) = \log_e(1) = 0$.

(E) $2I = \int_0^{\pi/2} 0 dx = 0$. So $I = 0$.

Question 1. If $f(x)$ is continuous on $[a, b]$ and $f(x) \geq 0$ for all $x \in [a, b]$, the area of the region bounded by $y = f(x)$, the x-axis, and the lines $x=a$ and $x=b$ is given by $\int_a^b f(x) dx$. Which statements are correct? (Multiple Correct Answers)

(A) The area under the curve $y=x^2$ from $x=0$ to $x=2$ is $\int_0^2 x^2 dx = [\frac{x^3}{3}]_0^2 = \frac{8}{3}$ square units.

(B) The area under the curve $y=\sin x$ from $x=0$ to $x=\pi$ is $\int_0^\pi \sin x dx = [-\cos x]_0^\pi = (-\cos\pi) - (-\cos 0) = -(-1) - (-1) = 1+1=2$ square units.

(C) The area bounded by the curve $y=x^3$, the x-axis, $x=0$ and $x=1$ is $\int_0^1 x^3 dx = [\frac{x^4}{4}]_0^1 = 1/4$ square unit.

(D) If $f(x)$ is sometimes positive and sometimes negative on $[a, b]$, the total area is $\int_a^b |f(x)| dx$.

(E) The definite integral $\int_a^b f(x) dx$ always gives the total area regardless of the sign of $f(x)$.

Question 2. The area of the region bounded by two curves $y=f(x)$ and $y=g(x)$ between $x=a$ and $x=b$, where $f(x) \geq g(x)$ on $[a, b]$, is given by $\int_a^b (f(x) - g(x)) dx$. Which statements are correct? (Multiple Correct Answers)

(A) The area between $y=x$ and $y=x^2$ from $x=0$ to $x=1$. In $[0, 1]$, $x \geq x^2$. The area is $\int_0^1 (x - x^2) dx$.

(B) $\int_0^1 (x - x^2) dx = [\frac{x^2}{2} - \frac{x^3}{3}]_0^1 = (\frac{1}{2} - \frac{1}{3}) - (0 - 0) = \frac{3-2}{6} = \frac{1}{6}$ square unit.

(C) The area between $y=x^2$ and $y=x$ is $1/6$ square unit.

(D) The area between $y=x^2$ and $y=4$ is bounded by $x=\pm 2$. The area is $\int_{-2}^2 (4 - x^2) dx$.

(E) $\int_{-2}^2 (4 - x^2) dx = [4x - \frac{x^3}{3}]_{-2}^2 = (4(2) - \frac{2^3}{3}) - (4(-2) - \frac{(-2)^3}{3}) = (8 - 8/3) - (-8 + 8/3) = 16 - 16/3 = (48-16)/3 = 32/3$ square units.

Question 3. Find the area of the region bounded by the parabola $y^2 = 4x$ and the line $x=3$. Which statements are correct? (Multiple Correct Answers)

(A) The parabola $y^2 = 4x$ is symmetric about the x-axis.

(B) The points of intersection are when $y^2 = 4(3) = 12$, so $y = \pm \sqrt{12} = \pm 2\sqrt{3}$. The points are $(3, 2\sqrt{3})$ and $(3, -2\sqrt{3})$.

(C) The area can be calculated as $\int_0^3 2\sqrt{4x} dx = \int_0^3 4\sqrt{x} dx$.

(D) $\int_0^3 4\sqrt{x} dx = 4 \int_0^3 x^{1/2} dx = 4 [\frac{x^{3/2}}{3/2}]_0^3 = 4 [\frac{2}{3} x^{3/2}]_0^3 = \frac{8}{3} [3^{3/2} - 0^{3/2}] = \frac{8}{3} (3\sqrt{3}) = 8\sqrt{3}$ square units.

(E) The area can also be calculated by integrating with respect to $y$: $x = y^2/4$. The integral is $\int_{-2\sqrt{3}}^{2\sqrt{3}} (3 - y^2/4) dy$.

Question 4. Find the area of the circle $x^2 + y^2 = a^2$ using integration. Which statements or steps are correct? (Multiple Correct Answers)

(A) The upper half of the circle is given by $y = \sqrt{a^2 - x^2}$.

(B) The x-values range from $-a$ to $a$.

(C) The area of the circle is $2 \int_{-a}^a \sqrt{a^2 - x^2} dx$.

(D) Use the standard integral $\int \sqrt{a^2 - x^2} dx = \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1}(\frac{x}{a}) + C$.

(E) Evaluate the definite integral: $2 [\frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1}(\frac{x}{a})]_{-a}^a = 2 [(\frac{a}{2}\sqrt{a^2 - a^2} + \frac{a^2}{2} \sin^{-1}(\frac{a}{a})) - (\frac{-a}{2}\sqrt{a^2 - (-a)^2} + \frac{a^2}{2} \sin^{-1}(\frac{-a}{a}))] = 2 [(0 + \frac{a^2}{2} \sin^{-1}(1)) - (0 + \frac{a^2}{2} \sin^{-1}(-1))] = 2 [\frac{a^2}{2} (\pi/2) - \frac{a^2}{2} (-\pi/2)] = 2 [\frac{\pi a^2}{4} + \frac{\pi a^2}{4}] = 2 [\frac{\pi a^2}{2}] = \pi a^2$.

Question 5. Find the area bounded by the curve $y = \cos x$, the x-axis, from $x=0$ to $x=2\pi$. Which statements are correct? (Multiple Correct Answers)

(A) The area is given by $\int_0^{2\pi} \cos x dx$.

(B) The integral $\int_0^{2\pi} \cos x dx = [\sin x]_0^{2\pi} = \sin(2\pi) - \sin(0) = 0 - 0 = 0$. This represents the net signed area.

(C) To find the total area, we need to consider the absolute value: $\int_0^{2\pi} |\cos x| dx$.

(D) $|\cos x| = \cos x$ for $x \in [0, \pi/2]$ and $x \in [3\pi/2, 2\pi]$. $|\cos x| = -\cos x$ for $x \in [\pi/2, 3\pi/2]$.

(E) The total area is $\int_0^{\pi/2} \cos x dx + \int_{\pi/2}^{3\pi/2} (-\cos x) dx + \int_{3\pi/2}^{2\pi} \cos x dx = [\sin x]_0^{\pi/2} - [\sin x]_{\pi/2}^{3\pi/2} + [\sin x]_{3\pi/2}^{2\pi} = (1-0) - (-1-1) + (0-(-1)) = 1 - (-2) + 1 = 1+2+1 = 4$ square units.

Question 6. Find the area of the region bounded by the parabola $y = x^2$ and the line $y=2x$. Which steps or results are correct? (Multiple Correct Answers)

(A) Find the points of intersection by setting $x^2 = 2x$. $x^2 - 2x = 0 \implies x(x-2) = 0$. Points of intersection at $x=0$ and $x=2$. The corresponding y-values are $y=0$ and $y=4$. Points are $(0,0)$ and $(2,4)$.

(B) The area is given by $\int_a^b (\text{upper curve} - \text{lower curve}) dx$. In $[0, 2]$, the line $y=2x$ is above the parabola $y=x^2$ (e.g., check $x=1$: $2(1)=2$, $1^2=1$).

(C) The area is $\int_0^2 (2x - x^2) dx$.

(D) $\int_0^2 (2x - x^2) dx = [x^2 - \frac{x^3}{3}]_0^2 = (2^2 - \frac{2^3}{3}) - (0^2 - \frac{0^3}{3}) = 4 - 8/3 = (12-8)/3 = 4/3$ square units.

(E) The area is $4/3$ square units.

Question 7. Find the area bounded by the lines $y=x$, $y=2x$ and $x=1$. Which steps or results are correct? (Multiple Correct Answers)

(A) The region is in the first quadrant, bounded by three lines.

(B) The upper curve is $y=2x$, and the lower curve is $y=x$ from $x=0$ to $x=1$. (They intersect at (0,0)).

(C) The area is given by $\int_0^1 (2x - x) dx = \int_0^1 x dx$.

(D) $\int_0^1 x dx = [\frac{x^2}{2}]_0^1 = \frac{1^2}{2} - \frac{0^2}{2} = 1/2$ square unit.

(E) The area is $1/2$ square unit.

Question 8. Find the area bounded by the curves $y = x^2$ and $y = \sqrt{x}$. Which steps or results are correct? (Multiple Correct Answers)

(A) Find points of intersection: $x^2 = \sqrt{x}$. Square both sides: $x^4 = x$. $x^4 - x = 0 \implies x(x^3 - 1) = 0 \implies x(x-1)(x^2+x+1)=0$. Real solutions are $x=0$ and $x=1$. Points are $(0,0)$ and $(1,1)$.

(B) In the interval $[0, 1]$, $\sqrt{x} \geq x^2$ (e.g., check $x=1/4$: $\sqrt{1/4}=1/2$, $(1/4)^2=1/16$, and $1/2 \geq 1/16$). The upper curve is $y=\sqrt{x}$, lower is $y=x^2$.

(C) The area is $\int_0^1 (\sqrt{x} - x^2) dx$.

(D) $\int_0^1 (x^{1/2} - x^2) dx = [\frac{x^{3/2}}{3/2} - \frac{x^3}{3}]_0^1 = [\frac{2}{3} x^{3/2} - \frac{x^3}{3}]_0^1 = (\frac{2}{3} - \frac{1}{3}) - (0 - 0) = \frac{1}{3}$ square unit.

(E) The area is $1/3$ square unit.

Question 9. In Applied Maths, integration is used for accumulation. Which statements are correct applications of integration? (Multiple Correct Answers)

(A) If $v(t)$ is the velocity of an object, $\int_{t_1}^{t_2} v(t) dt$ gives the displacement from time $t_1$ to $t_2$.

(B) If $a(t)$ is the acceleration, $\int_{t_1}^{t_2} a(t) dt$ gives the change in velocity from $t_1$ to $t_2$.

(C) If $r(t)$ is the rate of flow of a liquid, $\int_{t_1}^{t_2} r(t) dt$ gives the total amount of liquid flowed from $t_1$ to $t_2$.

(D) If $MR(x)$ is the marginal revenue, $\int_{x_1}^{x_2} MR(x) dx$ gives the change in total revenue from producing $x_1$ to $x_2$ units.

(E) The total cost given the marginal cost $C'(x)$ and fixed cost $C_0$ is $C(x) = \int C'(x) dx + C_0$.

Question 10. The area bounded by the curve $y=f(x)$, the y-axis, and the lines $y=c, y=d$ is given by $\int_c^d x dy = \int_c^d f^{-1}(y) dy$ (if $x=f^{-1}(y)$). Which statements are correct? (Multiple Correct Answers)

(A) For the parabola $y^2 = 4x$, $x = y^2/4$. The area bounded by $y^2 = 4x$, the y-axis, and the lines $y=0, y=2$ is $\int_0^2 \frac{y^2}{4} dy$.

(B) $\int_0^2 \frac{y^2}{4} dy = \frac{1}{4} [\frac{y^3}{3}]_0^2 = \frac{1}{4} (\frac{8}{3} - 0) = \frac{2}{3}$ square unit.

(C) This is equivalent to the area under $y = \sqrt{4x} = 2\sqrt{x}$ from $x=0$ to $x=1$. $\int_0^1 2\sqrt{x} dx = 2 [\frac{x^{3/2}}{3/2}]_0^1 = 2 \cdot \frac{2}{3} [x^{3/2}]_0^1 = \frac{4}{3} (1-0) = 4/3$. Wait, the region is different. The integral w.r.t y is correct for the bounded region.

(D) The area of the region bounded by $x = y^2$, the y-axis, and $y=1$ is $\int_0^1 y^2 dy = [y^3/3]_0^1 = 1/3$ square unit.

(E) When finding the area between two curves, if integrating w.r.t. $x$, we use $\int (y_{upper} - y_{lower}) dx$. If integrating w.r.t. $y$, we use $\int (x_{right} - x_{left}) dy$.

Question 1. A differential equation is an equation involving an independent variable, a dependent variable, and derivatives of the dependent variable with respect to the independent variable. Which of the following are differential equations? (Multiple Correct Answers)

(A) $\frac{dy}{dx} = x^2 \sin x$

(B) $\frac{d^2 y}{dx^2} + 5 \frac{dy}{dx} + 6y = 0$

(C) $x dy + y dx = 0$

(D) $(\frac{dy}{dx})^2 = y$

(E) $y = x^2 + 3x + 2$ (This is an algebraic equation, not a differential equation).

Question 2. The order of a differential equation is the order of the highest derivative appearing in it. The degree is the highest power of the highest order derivative, provided the equation is a polynomial in derivatives. What are the order and degree of the equation $(\frac{d^3 y}{dx^3})^2 + x (\frac{dy}{dx})^4 + y = 0$? (Multiple Correct Answers)

(A) The highest derivative is $\frac{d^3 y}{dx^3}$, so the order is 3.

(B) The highest power of the highest order derivative is 2, so the degree is 2.

(C) The order is 3.

(D) The degree is 4.

(E) The degree is defined because the equation is a polynomial in derivatives $\frac{dy}{dx}$ and $\frac{d^3 y}{dx^3}$.

Question 3. What are the order and degree of the differential equation $\frac{d^2 y}{dx^2} = \sqrt{1 + (\frac{dy}{dx})^2}$? (Multiple Correct Answers)

(A) The highest derivative is $\frac{d^2 y}{dx^2}$, so the order is 2.

(B) Square both sides to make it a polynomial in derivatives: $(\frac{d^2 y}{dx^2})^2 = 1 + (\frac{dy}{dx})^2$.

(C) The highest power of the highest order derivative ($\frac{d^2 y}{dx^2}$) is 2.

(D) The degree is 1.

(E) The order is 2 and the degree is 2.

Question 4. A solution of a differential equation is a function that satisfies the equation. Which statements about solutions are correct? (Multiple Correct Answers)

(A) A general solution contains arbitrary constants equal to the order of the equation.

(B) A particular solution is obtained by substituting specific values for the arbitrary constants in the general solution, often determined by initial or boundary conditions.

(C) A singular solution is a solution that cannot be obtained from the general solution.

(D) Verifying a solution involves substituting the function and its derivatives into the differential equation and checking if the equation holds true.

(E) For a first-order DE, the general solution contains one arbitrary constant.

Question 5. Form the differential equation of the family of curves $y = mx + c$, where $m$ and $c$ are arbitrary constants. Which steps or results are correct? (Multiple Correct Answers)

(A) There are two arbitrary constants, so the order of the resulting DE will be 2.

(B) Differentiate once with respect to $x$: $\frac{dy}{dx} = m$.

(C) Differentiate again with respect to $x$: $\frac{d^2 y}{dx^2} = 0$.

(D) The differential equation is $\frac{d^2 y}{dx^2} = 0$.

(E) This DE is of order 2 and degree 1.

Question 6. Form the differential equation of the family of circles passing through the origin and having their centres on the x-axis. The equation of such a family is $(x-a)^2 + y^2 = a^2$, where $a$ is the arbitrary constant. Which steps or results are correct? (Multiple Correct Answers)

(A) There is one arbitrary constant ($a$), so the order of the DE will be 1.

(B) Expand the equation: $x^2 - 2ax + a^2 + y^2 = a^2 \implies x^2 - 2ax + y^2 = 0$.

(C) Differentiate with respect to $x$: $2x - 2a + 2y \frac{dy}{dx} = 0 \implies x - a + y \frac{dy}{dx} = 0$.

(D) From the differentiated equation, express $a$: $a = x + y \frac{dy}{dx}$.

(E) Substitute $a$ back into the original equation $x^2 - 2ax + y^2 = 0$: $x^2 - 2(x + y \frac{dy}{dx})x + y^2 = 0 \implies x^2 - 2x^2 - 2xy \frac{dy}{dx} + y^2 = 0 \implies y^2 - x^2 = 2xy \frac{dy}{dx}$.

Question 7. Which of the following differential equations are of the first order? (Multiple Correct Answers)

(A) $\frac{dy}{dx} = x+y$

(B) $(\frac{dy}{dx})^2 + y = x$

(C) $\frac{d^2 y}{dx^2} = y$ (This is second order)

(D) $y' = \frac{x^2-y^2}{x y}$

(E) $x dy + y dx = 0$ (Rewrite as $\frac{dy}{dx} = -y/x$, which is first order)

Question 8. In Applied Maths, differential equations are used to model systems where quantities change over time or space. Which types of phenomena can be modeled using differential equations? (Multiple Correct Answers)

(A) Population growth/decay.

(B) Radioactive decay.

(C) Newton's Law of Cooling.

(D) Motion of objects (e.g., velocity and acceleration).

(E) Chemical reactions.

Question 9. Form the differential equation of the family of parabolas $y = a x^2$. Which steps or results are correct? (Multiple Correct Answers)

(A) There is one arbitrary constant ($a$), so the order of the resulting DE will be 1.

(B) Differentiate with respect to $x$: $\frac{dy}{dx} = 2ax$.

(C) From the differentiated equation, express $a$: $a = \frac{1}{2x} \frac{dy}{dx}$ (for $x \neq 0$).

(D) Substitute $a$ back into the original equation: $y = (\frac{1}{2x} \frac{dy}{dx}) x^2 = \frac{x}{2} \frac{dy}{dx}$ (for $x \neq 0$).

(E) The differential equation is $2y = x \frac{dy}{dx}$ (for $x \neq 0$).

Question 10. Which of the following are solutions to the differential equation $\frac{dy}{dx} = 2x$? (Multiple Correct Answers)

(A) $y = x^2$

(B) $y = x^2 + 5$

(C) $y = x^2 - 10$

(D) $y = x^2 + C$ (general solution)

(E) $y = x^2 + f(C)$ for any constant $C$ (equivalent to D).

Question 11. What is the order and degree of the differential equation $y''' + (y'')^2 + y' = 0$? (Multiple Correct Answers)

(A) The highest derivative is $y'''$, so the order is 3.

(B) The highest power of the highest order derivative ($y'''$) is 1.

(C) The degree is 2.

(D) The equation is a polynomial in derivatives.

(E) The order is 3 and the degree is 1.

Question 12. Form the differential equation of the family of exponential functions $y = A e^{3x}$, where $A$ is an arbitrary constant. Which steps or results are correct? (Multiple Correct Answers)

(A) There is one arbitrary constant, so the order of the DE is 1.

(B) Differentiate: $\frac{dy}{dx} = 3 A e^{3x}$.

(C) Substitute $A e^{3x} = y$ into the differentiated equation: $\frac{dy}{dx} = 3y$.

(D) The differential equation is $\frac{dy}{dx} - 3y = 0$.

(E) This is a first-order linear differential equation.

Question 13. Which of the following statements about the solution of a differential equation are correct? (Multiple Correct Answers)

(A) A general solution represents a family of curves.

(B) A particular solution corresponds to a specific curve from the family.

(C) Initial conditions provide information about the state of the system at a specific point, which helps in finding a particular solution.

(D) The number of arbitrary constants in the general solution of a DE is equal to its order.

(E) A singular solution is always included in the general solution.

Question 14. What is the order and degree of the differential equation $\sqrt{(\frac{dy}{dx})^3} = \frac{d^2 y}{dx^2}$? (Multiple Correct Answers)

(A) The highest derivative is $\frac{d^2 y}{dx^2}$, so the order is 2.

(B) Square both sides to remove the radical: $(\frac{d^2 y}{dx^2})^2 = (\frac{dy}{dx})^3$.

(C) The highest power of the highest order derivative ($\frac{d^2 y}{dx^2}$) is 2.

(D) The degree is 3.

(E) The order is 2 and the degree is 2.

Question 15. Form the differential equation of the family of straight lines $y = mx$, where $m$ is an arbitrary constant. Which steps or results are correct? (Multiple Correct Answers)

(A) There is one arbitrary constant ($m$), so the order of the DE will be 1.

(B) Differentiate with respect to $x$: $\frac{dy}{dx} = m$.

(C) Substitute $m = \frac{dy}{dx}$ into the original equation: $y = x \frac{dy}{dx}$.

(D) The differential equation is $y = x \frac{dy}{dx}$.

(E) This is a first-order differential equation.

Question 16. Which of the following are examples of first-order differential equations? (Multiple Correct Answers)

(A) $\frac{dy}{dx} = \cos x$

(B) $y' = 5y$

(C) $\frac{dV}{dt} = -kV$

(D) $(y')^2 + y = 0$

(E) $\frac{d^2 y}{dx^2} = x$

Question 17. Form the differential equation of the family of circles $x^2 + y^2 = r^2$, where $r$ is the arbitrary constant. Which steps or results are correct? (Multiple Correct Answers)

(A) There is one arbitrary constant ($r$), so the order of the DE will be 1.

(B) Differentiate with respect to $x$: $2x + 2y \frac{dy}{dx} = 2r \frac{dr}{dx}$. (Incorrect differentiation)

(C) Differentiate with respect to $x$: $2x + 2y \frac{dy}{dx} = 0$.

(D) The differential equation is $x + y \frac{dy}{dx} = 0$.

(E) This is a first-order differential equation.

Question 18. In Applied Maths, the formulation of a differential equation often arises from describing how a quantity changes with respect to another. Which statements are correct? (Multiple Correct Answers)

(A) The rate of change of population is proportional to the current population: $\frac{dP}{dt} = kP$.

(B) The acceleration of an object is proportional to the applied force: $m \frac{dv}{dt} = F$.

(C) The rate of decay of a radioactive substance is proportional to its amount: $\frac{dA}{dt} = -kA$.

(D) The marginal cost is the rate of change of total cost: $MC = \frac{dC}{dx}$.

(E) The slope of the tangent to a curve is the derivative: $\frac{dy}{dx}$.

Question 19. Which of the following represent the general solution of a differential equation? (Multiple Correct Answers)

(A) A single function that satisfies the DE.

(B) A family of functions that satisfy the DE.

(C) A solution that contains arbitrary constants.

(D) A solution obtained by substituting specific values into the arbitrary constants.

(E) The set of all possible solutions to the differential equation.

Question 1. A first-order differential equation is separable if it can be written in the form $g(y) dy = f(x) dx$. Which of the following equations are separable? (Multiple Correct Answers)

(A) $\frac{dy}{dx} = \frac{x^2}{y}$

(B) $\frac{dy}{dx} = e^{x+y}$

(C) $\frac{dy}{dx} = x+y$

(D) $\frac{dy}{dx} = y \cot x$

(E) $y dx - x dy = 0$

Question 2. Solve the separable differential equation $\frac{dy}{dx} = y$. Which steps or results are correct? (Multiple Correct Answers)

(A) Rewrite as $\frac{dy}{y} = dx$ (for $y \neq 0$).

(B) Integrate both sides: $\int \frac{dy}{y} = \int dx$.

(C) $\log_e |y| = x + C'$.

(D) Exponentiate both sides: $|y| = e^{x+C'} = e^x e^{C'}$. Let $e^{C'} = C_1 > 0$. So $|y| = C_1 e^x$.

(E) The general solution is $y = C e^x$, where $C$ is any non-zero constant. (Note: $y=0$ is also a solution, covered by letting $C=0$).

Question 3. A first-order differential equation is homogeneous if it can be written in the form $\frac{dy}{dx} = f(\frac{y}{x})$. Which of the following equations are homogeneous? (Multiple Correct Answers)

(A) $\frac{dy}{dx} = \frac{y^2 - x^2}{xy}$

(B) $\frac{dy}{dx} = \frac{y}{x} + \tan(\frac{y}{x})$

(C) $\frac{dy}{dx} = x+y$

(D) $(x^2 + y^2) dx - 2xy dy = 0$ (Rewrite as $\frac{dy}{dx} = \frac{x^2+y^2}{2xy}$)

(E) $\frac{dy}{dx} = \frac{x+y+1}{x-y-1}$

Question 4. To solve a homogeneous differential equation $\frac{dy}{dx} = f(\frac{y}{x})$, the substitution $y = vx$ is used. Which statements are correct about this substitution? (Multiple Correct Answers)

(A) Differentiating $y=vx$ with respect to $x$ using the product rule gives $\frac{dy}{dx} = v + x \frac{dv}{dx}$.

(B) The original equation transforms into $v + x \frac{dv}{dx} = f(v)$.

(C) The transformed equation $x \frac{dv}{dx} = f(v) - v$ is always a variable separable equation in terms of $x$ and $v$.

(D) Separate variables: $\frac{dv}{f(v)-v} = \frac{dx}{x}$ (for $f(v)-v \neq 0, x \neq 0$).

(E) Integrate both sides to find the solution in terms of $v$ and $x$, then substitute back $v=y/x$.

Question 5. Solve the homogeneous differential equation $\frac{dy}{dx} = \frac{y}{x} + 1$. Which steps or results are correct? (Multiple Correct Answers)

(A) Using the substitution $y=vx$, we get $v + x \frac{dv}{dx} = v + 1$.

(B) $x \frac{dv}{dx} = 1$.

(C) Separate variables: $dv = \frac{dx}{x}$.

(D) Integrate both sides: $\int dv = \int \frac{dx}{x} \implies v = \log_e |x| + C$.

(E) Substitute back $v=y/x$: $\frac{y}{x} = \log_e |x| + C \implies y = x (\log_e |x| + C)$.

Question 6. Some equations are reducible to homogeneous form. For $\frac{dy}{dx} = \frac{ax+by+c}{Ax+By+C}$, when $aB \neq Ab$, the substitution $x=X+h, y=Y+k$ transforms the equation into a homogeneous one if $(h, k)$ is the solution to $ax+by+c=0$ and $Ax+By+C=0$. Which statements are correct? (Multiple Correct Answers)

(A) The transformation $\frac{dy}{dx} = \frac{dY}{dX}$.

(B) The new equation is $\frac{dY}{dX} = \frac{aX+bY}{AX+BY}$.

(C) The transformed equation is homogeneous in $X$ and $Y$.

(D) After solving the homogeneous equation for $Y$ in terms of $X$, substitute back $X=x-h$ and $Y=y-k$.

(E) This method is useful when the lines $ax+by+c=0$ and $Ax+By+C=0$ are parallel.

Question 7. Which of the following are methods for solving first-order differential equations? (Multiple Correct Answers)

(A) Variable Separable Method.

(B) Homogeneous Equations Method.

(C) Linear Differential Equations Method.

(D) Integration by Parts.

(E) Factorization.

Question 8. Solve the differential equation $e^x \tan y dx + (1-e^x) \sec^2 y dy = 0$. Which steps or results are correct? (Multiple Correct Answers)

(A) This is a variable separable equation. Separate variables: $\frac{\sec^2 y}{\tan y} dy = \frac{-e^x}{1-e^x} dx$ (for $\tan y \neq 0, 1-e^x \neq 0$).

(B) Integrate both sides: $\int \frac{\sec^2 y}{\tan y} dy = \int \frac{-e^x}{1-e^x} dx$.

(C) On the left side, let $u = \tan y$, $du = \sec^2 y dy$. $\int \frac{du}{u} = \log_e |u| = \log_e |\tan y|$.

(D) On the right side, let $v = 1-e^x$, $dv = -e^x dx$. $\int \frac{dv}{v} = \log_e |v| = \log_e |1-e^x|$.

(E) The general solution is $\log_e |\tan y| = \log_e |1-e^x| + C'$. Exponentiate: $|\tan y| = e^{\log_e |1-e^x| + C'} = |1-e^x| e^{C'}$. Let $e^{C'} = C_1 > 0$. $|\tan y| = C_1 |1-e^x|$. The general solution is $\tan y = C (1-e^x)$ for any constant $C \neq 0$. (Note: $y=n\pi$ and $e^x=1$ also need consideration for full rigor).

Question 9. Solve the differential equation $x \frac{dy}{dx} = y \log_e (\frac{y}{x})$. Which steps or results are correct? (Multiple Correct Answers)

(A) This is a homogeneous equation, as $\frac{dy}{dx} = \frac{y}{x} \log_e(\frac{y}{x}) = f(\frac{y}{x})$.

(B) Use substitution $y=vx$, $\frac{dy}{dx} = v + x \frac{dv}{dx}$. The equation becomes $v + x \frac{dv}{dx} = v \log_e v$.

(C) $x \frac{dv}{dx} = v \log_e v - v = v (\log_e v - 1)$.

(D) Separate variables: $\frac{dv}{v(\log_e v - 1)} = \frac{dx}{x}$ (for $v \neq 0, \log_e v - 1 \neq 0, x \neq 0$).

(E) Integrate: $\int \frac{dv}{v(\log_e v - 1)} = \int \frac{dx}{x}$. On the left, let $w = \log_e v - 1$, $dw = \frac{1}{v} dv$. $\int \frac{dw}{w} = \log_e |w| = \log_e |\log_e v - 1|$. On the right, $\int \frac{dx}{x} = \log_e |x| + C'$. So $\log_e |\log_e v - 1| = \log_e |x| + C'$. Exponentiate: $|\log_e v - 1| = e^{\log_e |x| + C'} = |x| e^{C'}$. Let $e^{C'} = C_1 > 0$. $|\log_e v - 1| = C_1 |x|$. $\log_e v - 1 = C x$ for non-zero C. Substitute back $v=y/x$: $\log_e (y/x) - 1 = Cx$. $\log_e (y/x) = Cx+1$. $y/x = e^{Cx+1}$. $y = x e^{Cx+1}$.

Question 10. Which of the following differential equations can be solved by the method of variable separation? (Multiple Correct Answers)

(A) $\frac{dy}{dx} = \sin(x+y)$

(B) $\frac{dy}{dx} = xy + x + y + 1 = x(y+1) + (y+1) = (x+1)(y+1)$. This is separable.

(C) $\frac{dy}{dx} = \frac{x-y}{x+y}$ (This is homogeneous, not separable).

(D) $\frac{dy}{dx} = (x^2+1)(y^2+1)$. This is separable.

(E) $\frac{dy}{dx} = \sqrt{\frac{1-\sin y}{1+\cos x}}$. This is separable: $\frac{dy}{\sqrt{1-\sin y}} = \sqrt{1+\cos x} dx$.

Question 11. Which of the following equations are homogeneous? (Multiple Correct Answers)

(A) $\frac{dy}{dx} = \frac{x^2+y^2}{x^2-y^2}$

(B) $\frac{dy}{dx} = \frac{x-y}{x+y}$

(C) $(x^3 + xy^2) dx - (x^2 y + y^3) dy = 0$. Rewrite as $\frac{dy}{dx} = \frac{x^3+xy^2}{x^2 y + y^3} = \frac{x(x^2+y^2)}{y(x^2+y^2)}$. For $x^2+y^2 \neq 0$, $\frac{dy}{dx} = x/y = 1/(y/x)$, which is homogeneous.

(D) $\frac{dy}{dx} = \frac{x^2+y}{x}$. (Not homogeneous).

(E) $\frac{dy}{dx} = \sin(y/x)$. (Homogeneous).

Question 12. To solve a first-order differential equation of the form $\frac{dy}{dx} = f(ax+by+c)$, which substitution is often useful? (Multiple Correct Answers)

(A) Let $v = ax+by+c$.

(B) Differentiate $v$ with respect to $x$: $\frac{dv}{dx} = a + b \frac{dy}{dx}$.

(C) So $\frac{dy}{dx} = \frac{1}{b} (\frac{dv}{dx} - a)$.

(D) The differential equation becomes $\frac{1}{b} (\frac{dv}{dx} - a) = f(v)$.

(E) This transforms the original equation into a variable separable equation in terms of $v$ and $x$: $\frac{dv}{dx} = bf(v) + a$.

Question 13. Solve the differential equation $\frac{dy}{dx} = (x+y)^2$. Which steps or results are correct? (Multiple Correct Answers)

(A) This is of the form $\frac{dy}{dx} = f(x+y)$. Let $v = x+y$.

(B) $\frac{dv}{dx} = 1 + \frac{dy}{dx}$. So $\frac{dy}{dx} = \frac{dv}{dx} - 1$.

(C) The equation becomes $\frac{dv}{dx} - 1 = v^2$. $\frac{dv}{dx} = v^2 + 1$.

(D) Separate variables: $\frac{dv}{v^2+1} = dx$.

(E) Integrate both sides: $\int \frac{dv}{v^2+1} = \int dx \implies \tan^{-1} v = x + C$. Substitute back $v=x+y$: $\tan^{-1} (x+y) = x + C$.

Question 14. Which of the following equations are reducible to homogeneous form using the substitution $x=X+h, y=Y+k$? (Multiple Correct Answers)

(A) $\frac{dy}{dx} = \frac{x+y-1}{x-y+2}$ (Lines $x+y-1=0$ and $x-y+2=0$ intersect, $aB-Ab = 1(-1)-1(1) = -2 \neq 0$).

(B) $\frac{dy}{dx} = \frac{2x+3y}{x-y}$ (Already homogeneous).

(C) $\frac{dy}{dx} = \frac{x+2y+3}{2x+4y+5}$ (Lines $x+2y+3=0$ and $2x+4y+5=0$ are parallel, $aB-Ab = 1(4)-2(2)=0$).

(D) $\frac{dy}{dx} = \frac{x-y}{x+y+1}$ (Lines $x-y=0$ and $x+y+1=0$ intersect, $aB-Ab=1(1)-(-1)(1)=2 \neq 0$).

(E) $\frac{dy}{dx} = \sin(x+y)$ (Reducible using $v=x+y$, not this method).

Question 15. Which statements about first-order differential equations are correct? (Multiple Correct Answers)

(A) They involve only the first derivative of the dependent variable.

(B) The general solution involves one arbitrary constant.

(C) They can be classified as separable, homogeneous, linear, etc.

(D) Solving them often involves integration.

(E) Initial conditions are used to find a particular solution.

Question 16. Solve the differential equation $x dy + y dx = 0$. Which steps or results are correct? (Multiple Correct Answers)

(A) This is a variable separable equation: $\frac{dy}{y} = -\frac{dx}{x}$ (for $x, y \neq 0$).

(B) Integrate both sides: $\int \frac{dy}{y} = -\int \frac{dx}{x}$.

(C) $\log_e |y| = -\log_e |x| + C' = \log_e |x|^{-1} + C'$.

(D) $\log_e |y| - \log_e |x|^{-1} = C' \implies \log_e |y x| = C'$.

(E) Exponentiate: $|yx| = e^{C'}$. Let $e^{C'} = C_1 > 0$. The general solution is $yx = C$ for any constant $C \neq 0$. (Note: $x=0$ or $y=0$ are also solutions, covered by $C=0$).

Question 17. Solve the differential equation $\frac{dy}{dx} = \frac{x^2+y^2}{xy}$. Which method and steps are correct? (Multiple Correct Answers)

(A) This is a homogeneous equation. Use $y=vx, \frac{dy}{dx} = v + x \frac{dv}{dx}$.

(B) $v + x \frac{dv}{dx} = \frac{x^2 + (vx)^2}{x(vx)} = \frac{x^2(1+v^2)}{vx^2} = \frac{1+v^2}{v}$ (for $x, v \neq 0$).

(C) $x \frac{dv}{dx} = \frac{1+v^2}{v} - v = \frac{1+v^2-v^2}{v} = \frac{1}{v}$.

(D) Separate variables: $v dv = \frac{dx}{x}$ (for $v \neq 0, x \neq 0$).

(E) Integrate: $\int v dv = \int \frac{dx}{x} \implies \frac{v^2}{2} = \log_e |x| + C'$. Substitute back $v=y/x$: $\frac{(y/x)^2}{2} = \log_e |x| + C' \implies \frac{y^2}{2x^2} = \log_e |x| + C' \implies y^2 = 2x^2 (\log_e |x| + C')$.

Question 1. A first-order linear differential equation is of the form $\frac{dy}{dx} + P(x)y = Q(x)$ or $\frac{dx}{dy} + P(y)x = Q(y)$. Which of the following equations are linear? (Multiple Correct Answers)

(A) $\frac{dy}{dx} + y = x^2$

(B) $\frac{dy}{dx} + y^2 = \sin x$ (Non-linear due to $y^2$).

(C) $x \frac{dy}{dx} + y = \log_e x$. This can be written as $\frac{dy}{dx} + \frac{1}{x} y = \frac{\log_e x}{x}$, which is linear (for $x \neq 0$).

(D) $\frac{dy}{dx} = \frac{x-y}{x+y}$ (Homogeneous, not linear).

(E) $\frac{dx}{dy} + x \cos y = y$. This is linear in $x$.

Question 2. The integrating factor (IF) for the linear differential equation $\frac{dy}{dx} + P(x)y = Q(x)$ is $e^{\int P(x) dx}$. Which statements are correct about the integrating factor? (Multiple Correct Answers)

(A) Multiplying the linear DE by the IF makes the left side the derivative of a product.

(B) The left side of the equation becomes $\frac{d}{dx} (y \cdot \text{IF})$.

(C) The general solution is $y \cdot (\text{IF}) = \int Q(x) \cdot (\text{IF}) dx + C$.

(D) The choice of the constant of integration for $\int P(x) dx$ does not affect the final general solution.

(E) The IF is always a positive function (since $e$ raised to any real power is positive).

Question 3. Find the integrating factor for the equation $\frac{dy}{dx} - y = e^x$. Which steps or results are correct? (Multiple Correct Answers)

(A) This is a linear DE with $P(x) = -1$ and $Q(x) = e^x$.

(B) $\int P(x) dx = \int -1 dx = -x$.

(C) The integrating factor is $e^{\int P(x) dx} = e^{-x}$.

(D) Multiply the equation by IF: $e^{-x} (\frac{dy}{dx} - y) = e^x e^{-x} = 1$.

(E) The left side is $\frac{d}{dx}(y e^{-x})$.

Question 4. Solve the differential equation $\frac{dy}{dx} - y = e^x$. Which steps or results are correct? (Multiple Correct Answers)

(A) From Q3, IF $= e^{-x}$, and the equation becomes $\frac{d}{dx}(y e^{-x}) = 1$.

(B) Integrate both sides: $\int \frac{d}{dx}(y e^{-x}) dx = \int 1 dx$.

(C) $y e^{-x} = x + C$.

(D) Solve for $y$: $y = (x+C) e^x$.

(E) The general solution is $y = xe^x + Ce^x$.

Question 5. Find the integrating factor for $x \frac{dy}{dx} + 2y = x^2$. Which steps or results are correct? (Multiple Correct Answers)

(A) Rewrite in standard form: $\frac{dy}{dx} + \frac{2}{x} y = x$ (for $x \neq 0$). Here $P(x) = 2/x$ and $Q(x) = x$.

(B) $\int P(x) dx = \int \frac{2}{x} dx = 2 \log_e |x| = \log_e |x|^2 = \log_e x^2$.

(C) The integrating factor is $e^{\int P(x) dx} = e^{\log_e x^2} = x^2$.

(D) Multiply the standard form by IF: $x^2 (\frac{dy}{dx} + \frac{2}{x} y) = x^2 \cdot x$.

(E) $\frac{d}{dx}(y x^2) = x^3$.

Question 6. Solve the differential equation $x \frac{dy}{dx} + 2y = x^2$. Which steps or results are correct? (Multiple Correct Answers)

(A) From Q5, IF $= x^2$, and the equation becomes $\frac{d}{dx}(y x^2) = x^3$.

(B) Integrate both sides: $\int \frac{d}{dx}(y x^2) dx = \int x^3 dx$.

(C) $y x^2 = \frac{x^4}{4} + C$.

(D) Solve for $y$: $y = \frac{x^2}{4} + \frac{C}{x^2}$.

(E) The general solution is $y = \frac{x^4}{4x^2} + \frac{C}{x^2}$.

Question 7. Find the integrating factor for $\frac{dy}{dx} + y \tan x = \sec x$. Which steps or results are correct? (Multiple Correct Answers)

(A) This is a linear DE with $P(x) = \tan x$ and $Q(x) = \sec x$.

(B) $\int P(x) dx = \int \tan x dx = \log_e |\sec x|$.

(C) The integrating factor is $e^{\int P(x) dx} = e^{\log_e |\sec x|} = |\sec x|$.

(D) Multiply the equation by IF: $|\sec x| (\frac{dy}{dx} + y \tan x) = |\sec x| \sec x$.

(E) The left side is $\frac{d}{dx}(y |\sec x|)$.

Question 8. Solve the differential equation $\frac{dy}{dx} + y \tan x = \sec x$. Which steps or results are correct? (Assume $\sec x > 0$). (Multiple Correct Answers)

(A) From Q7, IF $= \sec x$, and the equation becomes $\frac{d}{dx}(y \sec x) = \sec^2 x$.

(B) Integrate both sides: $\int \frac{d}{dx}(y \sec x) dx = \int \sec^2 x dx$.

(C) $y \sec x = \tan x + C$.

(D) Solve for $y$: $y = \frac{\tan x + C}{\sec x}$.

(E) The general solution is $y = \sin x + C \cos x$. (Since $\frac{\tan x}{\sec x} = \frac{\sin x/\cos x}{1/\cos x} = \sin x$ and $\frac{C}{\sec x} = C \cos x$).

Question 9. Which of the following equations are linear first-order differential equations? (Multiple Correct Answers)

(A) $y' + xy = x^2$

(B) $\frac{dy}{dx} + \frac{y}{x} = \cos x$

(C) $\frac{dy}{dx} = y$

(D) $y \frac{dy}{dx} = x$ (Non-linear)

(E) $\frac{dx}{dy} + x \cos y = y$. This is linear in $x$.

Question 10. Which statements about the integrating factor method are correct? (Multiple Correct Answers)

(A) It is used to solve linear first-order differential equations.

(B) It transforms the left side of the standard form into the derivative of a product.

(C) The integrating factor depends on $x$ (for $\frac{dy}{dx} + P(x)y = Q(x)$).

(D) The integrating factor depends on $y$ (for $\frac{dx}{dy} + P(y)x = Q(y)$).

(E) The method involves multiplying the entire equation by the integrating factor and then integrating both sides.

Question 1. Mathematical modeling using differential equations involves translating real-world problems into mathematical terms. Which steps are typically involved? (Multiple Correct Answers)

(A) Identify the dependent and independent variables.

(B) Express the relationships between the quantities and their rates of change as a differential equation.

(C) Solve the differential equation to find the function describing the phenomenon.

(D) Use initial or boundary conditions to find a particular solution.

(E) Interpret the solution in the context of the original problem.

Question 2. The rate of change of a quantity $Q$ is proportional to the quantity itself. This can be modeled by the differential equation $\frac{dQ}{dt} = kQ$. Which real-world phenomena can be modeled by this equation? (Multiple Correct Answers)

(A) Population growth (under ideal conditions).

(B) Radioactive decay (rate is proportional to the amount present, $k<0$).

(C) Compound interest compounded continuously.

(D) Newton's Law of Cooling (rate of change of temperature difference is proportional to the difference).

(E) The velocity of an object falling with air resistance proportional to velocity.

Question 3. Solve the differential equation $\frac{dQ}{dt} = kQ$ with the initial condition $Q(0) = Q_0$. Which steps or results are correct? (Multiple Correct Answers)

(A) Separate variables: $\frac{dQ}{Q} = k dt$ (for $Q \neq 0$).

(B) Integrate both sides: $\int \frac{dQ}{Q} = \int k dt$.

(C) $\log_e |Q| = kt + C'$.

(D) Exponentiate: $|Q| = e^{kt+C'} = e^{kt} e^{C'}$. Let $e^{C'} = C_1 > 0$. $|Q| = C_1 e^{kt}$. The general solution is $Q(t) = C e^{kt}$.

(E) Apply the initial condition: $Q(0) = C e^{k(0)} = C e^0 = C$. Since $Q(0) = Q_0$, $C = Q_0$. The particular solution is $Q(t) = Q_0 e^{kt}$.

Question 4. In radioactive decay, the rate of decay is proportional to the amount present. If the half-life is $T_{1/2}$, this means the amount reduces by half in time $T_{1/2}$. Which statements are correct? (Multiple Correct Answers)

(A) The model is $\frac{dN}{dt} = -kN$, where $N(t)$ is the amount at time $t$ and $k>0$ is the decay constant.

(B) The solution is $N(t) = N_0 e^{-kt}$, where $N_0$ is the initial amount.

(C) At $t = T_{1/2}$, $N(T_{1/2}) = N_0/2$.

(D) $N_0/2 = N_0 e^{-kT_{1/2}} \implies 1/2 = e^{-kT_{1/2}} \implies \log_e(1/2) = -kT_{1/2}$.

(E) $-\log_e 2 = -kT_{1/2} \implies k = \frac{\log_e 2}{T_{1/2}}$. The decay constant $k$ is related to the half-life $T_{1/2}$.

Question 5. Newton's Law of Cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings. If $T(t)$ is the temperature of the object at time $t$ and $T_a$ is the constant ambient temperature, the differential equation is $\frac{dT}{dt} = -k(T - T_a)$ for some constant $k>0$. Which statements are correct? (Multiple Correct Answers)

(A) Let $y(t) = T(t) - T_a$. Then $\frac{dy}{dt} = \frac{dT}{dt}$.

(B) The differential equation becomes $\frac{dy}{dt} = -ky$, which is a simple proportional decay model.

(C) The solution for $y(t)$ is $y(t) = y_0 e^{-kt}$, where $y_0 = T(0) - T_a$.

(D) The solution for $T(t)$ is $T(t) - T_a = (T(0) - T_a) e^{-kt}$.

(E) The temperature of the object approaches the ambient temperature as $t \to \infty$.

Question 6. In a logistic growth model, the rate of population growth is limited by carrying capacity. The model is $\frac{dP}{dt} = k P (M-P)$, where $P(t)$ is the population, $k$ is the growth rate constant, and $M$ is the carrying capacity. Which statements are correct? (Multiple Correct Answers)

(A) This is a first-order differential equation.

(B) This is a non-linear differential equation.

(C) If $P$ is small compared to $M$, the growth is approximately exponential ($\frac{dP}{dt} \approx kMP$).

(D) If $P$ is close to $M$, the growth rate is close to zero.

(E) The equilibrium solutions (where $dP/dt = 0$) are $P=0$ and $P=M$.

Question 7. The velocity $v(t)$ of an object satisfies the differential equation $m \frac{dv}{dt} = F - kv$, where $F$ is a constant applied force and $k$ is a resistance constant ($m, k > 0$). Which statements are correct? (Multiple Correct Answers)

(A) This is a first-order linear differential equation in $v(t)$.

(B) The equation can be written as $\frac{dv}{dt} + \frac{k}{m} v = \frac{F}{m}$.

(C) The integrating factor is $e^{\int (k/m) dt} = e^{kt/m}$.

(D) The general solution is $v(t) e^{kt/m} = \int \frac{F}{m} e^{kt/m} dt + C$.

(E) $\int \frac{F}{m} e^{kt/m} dt = \frac{F}{m} \frac{e^{kt/m}}{k/m} + C' = \frac{F}{k} e^{kt/m} + C'$. The general solution is $v(t) = \frac{F}{k} + C e^{-kt/m}$.

Question 8. In financial applications, if the rate of change of an investment $A(t)$ is proportional to the current amount with an interest rate $r$, plus a constant deposit rate $D$, the model is $\frac{dA}{dt} = rA + D$. Which statements are correct? (Multiple Correct Answers)

(A) This is a first-order linear differential equation.

(B) The equation can be written as $\frac{dA}{dt} - rA = D$. Here $P(t)=-r, Q(t)=D$.

(C) The integrating factor is $e^{\int -r dt} = e^{-rt}$.

(D) The general solution is $A(t) e^{-rt} = \int D e^{-rt} dt + C = D \frac{e^{-rt}}{-r} + C = -\frac{D}{r} e^{-rt} + C$.

(E) The general solution is $A(t) = -\frac{D}{r} + C e^{rt}$.

Question 9. In chemical kinetics, a first-order reaction, where the rate of disappearance of a reactant A is proportional to the concentration of A, $[A](t)$, is modeled by $\frac{d[A]}{dt} = -k[A]$. Which statements are correct? (Multiple Correct Answers)

(A) This is the same mathematical model as radioactive decay.

(B) If $[A](0) = [A]_0$, the solution is $[A](t) = [A]_0 e^{-kt}$.

(C) The time taken for the concentration to reduce to half is the half-life, $t_{1/2}$.

(D) $t_{1/2} = \frac{\log_e 2}{k}$.

(E) The units of the rate constant $k$ are typically units of inverse time (e.g., s$^{-1}$, min$^{-1}$).

Question 10. Which of the following statements are true regarding the formulation and solution of differential equations in Applied Maths? (Multiple Correct Answers)

(A) The differential equation captures the instantaneous rate of change.

(B) The solution to the differential equation predicts the value of the quantity over time or space.

(C) Initial conditions are necessary to find a unique particular solution from the general solution.

(D) Different real-world problems can sometimes be described by the same type of differential equation.

(E) Solving a differential equation is the core step in predictive modeling using rates of change.

Question 11. A sum of $\textsf{₹ }50,000$ is invested at an annual interest rate of 6% compounded continuously. The differential equation for the amount $A(t)$ after $t$ years is $\frac{dA}{dt} = 0.06 A$. Which statements are correct? (Multiple Correct Answers)

(A) The initial condition is $A(0) = 50000$.

(B) This is a first-order linear separable differential equation.

(C) The solution is $A(t) = 50000 e^{0.06t}$.

(D) The amount grows exponentially.

(E) The rate constant is $k=0.06$.

Question 12. The rate of sales $S$ of a new product is proportional to the number of potential customers $M$ who have not yet purchased the product. If $N(t)$ is the number of people who have purchased the product at time $t$, and $M$ is the total market size, the number of potential customers is $M-N(t)$. The model is $\frac{dN}{dt} = k(M - N)$. Which statements are correct? (Multiple Correct Answers)

(A) This is a first-order linear differential equation in $N(t)$.

(B) The equation can be written as $\frac{dN}{dt} + kN = kM$. Here $P(t)=k, Q(t)=kM$.

(C) The integrating factor is $e^{\int k dt} = e^{kt}$.

(D) The general solution is $N(t) e^{kt} = \int kM e^{kt} dt + C = kM \frac{e^{kt}}{k} + C = Me^{kt} + C$.

(E) The general solution is $N(t) = M + C e^{-kt}$. If $N(0)=N_0$, $N_0 = M + C \implies C = N_0 - M$. So $N(t) = M + (N_0 - M)e^{-kt}$.