Case Study / Scenario-Based MCQs for Sub-Topics of Topic 10: Calculus

Question 1. A company producing smartphones estimates its cost function (in $\textsf{₹}$) per unit as $C(x)$, where $x$ is the number of units produced. As the production level $x$ approaches a large number $L$, the average cost approaches a certain value. The average cost function is given by $A(x) = \frac{C(x)}{x}$. If $C(x) = 500x + 100000$, what is the limit of the average cost as the number of units produced becomes very large?

(A) $\lim\limits_{x \to \infty} \frac{500x + 100000}{x} = 0$

(B) $\lim\limits_{x \to \infty} \frac{500x + 100000}{x} = 100000$

(C) $\lim\limits_{x \to \infty} \frac{500x + 100000}{x} = 500$

(D) The limit does not exist.

Question 2. A tiny object is moving along a straight line. Its position $s(t)$ (in meters) at time $t$ (in seconds) is given by $s(t) = 3t^2 + 2t$. The instantaneous velocity at $t=1$ second is given by the limit $\lim\limits_{h \to 0} \frac{s(1+h) - s(1)}{h}$. Evaluate this limit.

(A) $5 \text{ m/s}$

(B) $8 \text{ m/s}$

(C) $6 \text{ m/s}$

(D) $0 \text{ m/s}$

Question 3. A piece of metal is heated, and its temperature $T$ (in $^\circ$C) at time $t$ (in minutes) approaches a stable temperature of $150^\circ$C as $t \to \infty$. Which of the following limit statements accurately describes this scenario?

(A) $\lim\limits_{t \to 150} T(t) = \infty$

(B) $\lim\limits_{t \to \infty} T(t) = 150$

(C) $\lim\limits_{t \to 0} T(t) = 150$

(D) $\lim\limits_{t \to 150} T(t) = 0$

Question 4. Consider the tax rate function $T(I)$ for an income $I$ (in $\textsf{₹}$) below a certain threshold. If the tax rate is defined as $T(I) = \frac{0.1 \cdot I}{I}$ for $I > 0$, what is the limit of the tax rate as income $I$ approaches $0$ from the right (i.e., for very small positive incomes)?

(A) $0.1$

(B) $0$

(C) $1$

(D) The limit does not exist.

Question 5. A company's production process has a quality control function $Q(p)$ that measures the percentage of good items produced based on the probability $p$ of a machine error. Near a probability of error $p=0$, the function is modeled by $Q(p) = \frac{1 - \cos(\sqrt{p})}{p}$ for $p > 0$. What is the limiting percentage of good items as the error probability approaches zero?

(A) $\lim\limits_{p \to 0^+} \frac{1 - \cos(\sqrt{p})}{p} = 1$

(B) $\lim\limits_{p \to 0^+} \frac{1 - \cos(\sqrt{p})}{p} = 0$

(C) $\lim\limits_{p \to 0^+} \frac{1 - \cos(\sqrt{p})}{p} = \frac{1}{2}$

(D) The limit does not exist.

Question 1. A scientist observes that the concentration of a certain chemical $C(t)$ in a solution at time $t$ approaches $5$ units as $t \to \infty$. Another chemical's concentration $D(t)$ approaches $3$ units as $t \to \infty$. What is the limit of the sum of their concentrations as $t \to \infty$?

(A) $\lim\limits_{t \to \infty} (C(t) + D(t)) = 8$

(B) $\lim\limits_{t \to \infty} (C(t) + D(t)) = 15$

(C) The limit does not exist.

(D) $\lim\limits_{t \to \infty} (C(t) + D(t)) = 5$

Question 2. The population of a certain species in a controlled environment is modeled by $P(t)$. As time $t$ approaches infinity, the population approaches a carrying capacity $K$. A related factor $F(t)$ approaches a value $L$ as $t \to \infty$. If $P(t) = \frac{1000}{2 + e^{-t}}$ and $F(t) = 5 - \frac{1}{t}$, what is the limit of the product $P(t) \cdot F(t)$ as $t \to \infty$? (Assume $\lim\limits_{t \to \infty} P(t) = 500$ and $\lim\limits_{t \to \infty} F(t) = 5$).

(A) $\lim\limits_{t \to \infty} P(t) \cdot F(t) = 500$

(B) $\lim\limits_{t \to \infty} P(t) \cdot F(t) = 2500$

(C) The limit does not exist.

(D) $\lim\limits_{t \to \infty} P(t) \cdot F(t) = \infty$

Question 3. The efficiency of a machine decreases over time. Its efficiency is bounded between $g(t)$ and $h(t)$. For large values of $t$, $g(t) = 0.5 - \frac{1}{t}$ and $h(t) = 0.5 + \frac{\sin t}{t}$. If the efficiency $E(t)$ satisfies $g(t) \leq E(t) \leq h(t)$ for $t > T$ for some large $T$, what is the limit of the efficiency as $t \to \infty$?

(A) $\lim\limits_{t \to \infty} E(t) = 0.5$

(B) $\lim\limits_{t \to \infty} E(t) = 0$

(C) $\lim\limits_{t \to \infty} E(t) = 1$

(D) The limit does not exist.

Question 4. The value of an investment grows according to the formula $A(t) = P(1 + \frac{r}{n})^{nt}$. For continuous compounding, we take the limit as $n \to \infty$. What is the limit of $(1 + \frac{r}{n})^{nt}$ as $n \to \infty$?

(A) $\lim\limits_{n \to \infty} (1 + \frac{r}{n})^{nt} = 1$

(B) $\lim\limits_{n \to \infty} (1 + \frac{r}{n})^{nt} = e^{rt}$

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

(D) $\lim\limits_{n \to \infty} (1 + \frac{r}{n})^{nt} = \infty$

Question 5. In signal processing, the sinc function is defined as $\text{sinc}(x) = \frac{\sin x}{x}$ for $x \neq 0$ and $\text{sinc}(0) = 1$. This definition makes the function continuous at $x=0$. Which standard limit is crucial here?

(A) $\lim\limits_{x \to 0} \frac{1 - \cos x}{x^2}$

(B) $\lim\limits_{x \to 0} (1+x)^{1/x}$

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

(D) $\lim\limits_{x \to \infty} \frac{1}{x}$

Question 1. A switch is designed to turn on a light bulb at time $t=5$ seconds. The voltage supplied to the bulb is modeled by a function $V(t)$. For the light to turn on smoothly without a sudden surge, the voltage function must be continuous at $t=5$. Which of the following conditions must be met for $V(t)$ to be continuous at $t=5$?

(A) $\lim\limits_{t \to 5} V(t)$ must exist.

(B) $V(5)$ must be defined.

(C) $\lim\limits_{t \to 5} V(t) = V(5)$.

(D) All of the above.

Question 2. A technician is testing the continuity of a sensor's output voltage $V(T)$ as temperature $T$ changes. The specification requires the output to be continuous at $T = 100^\circ$C. The sensor's output is given by $V(T) = \begin{cases} k T + 5 & , & T < 100 \\ 20 & , & T = 100 \\ 2k T - 5 & , & T > 100 \end{cases}$. What value of $k$ ensures continuity at $T=100^\circ$C?

(A) $k = 0.1$

(B) $k = 0.15$

(C) $k = 0.2$

(D) $k = 0.25$

Question 3. The price $P(t)$ of a certain stock at time $t$ hours during the trading day is observed. A sudden, instantaneous drop in price from $\textsf{₹}100$ to $\textsf{₹}95$ at $t=2$ hours, followed by a continuous movement thereafter, represents what type of discontinuity at $t=2$ in the graph of $P(t)$?

(A) Removable discontinuity

(B) Jump discontinuity

(C) Infinite discontinuity

(D) Oscillating discontinuity

Question 4. A manufacturing process involves two stages. The output quantity from the first stage $g(x)$ (where $x$ is input) is processed in the second stage by a function $f(u)$ (where $u$ is the output from the first stage). If both functions $f$ and $g$ are continuous functions, is the total output function $(f \circ g)(x) = f(g(x))$ necessarily continuous?

(A) Yes, the composition of any two functions is always continuous.

(B) Yes, the composition of continuous functions is continuous (provided the composition is defined).

(C) No, the continuity of $f$ and $g$ does not guarantee the continuity of their composition.

(D) Yes, but only if $f$ and $g$ are linear functions.

Question 5. The cost of electricity (in $\textsf{₹}$) for a consumer is often calculated based on usage slabs. The rate might be $\textsf{₹}5$ per unit for the first 100 units and $\textsf{₹}7$ per unit for usage above 100 units. If $C(u)$ is the total cost for using $u$ units, is $C(u)$ continuous at $u=100$ units? (Assume $C(u) = 5u$ for $0 \leq u \leq 100$ and $C(u) = 5 \cdot 100 + 7(u-100)$ for $u > 100$).

(A) Yes, $C(u)$ is continuous at $u=100$ because the limit from the left equals the limit from the right and equals the function value.

(B) No, $C(u)$ has a jump discontinuity at $u=100$.

(C) Yes, all cost functions are continuous.

(D) No, because the rate changes at $u=100$.

Question 1. A robot arm is moving along a path defined by $y = f(x)$. The smoothness of the robot's movement at a specific point $(a, f(a))$ is related to the differentiability of $f(x)$ at $x=a$. If the path has a sharp corner at $x=a$, is the function differentiable at $x=a$?

(A) Yes, sharp corners do not affect differentiability.

(B) No, a sharp corner indicates that the slope of the tangent is not uniquely defined, so the function is not differentiable.

(C) It depends on whether the function is continuous at $x=a$.

(D) Yes, provided the function is defined at $x=a$.

Question 2. The instantaneous rate of change of the population of bacteria $P(t)$ at time $t$ is given by the derivative $P'(t)$. Which definition from first principles is used to define $P'(t)$?

(A) $\lim\limits_{t \to a} \frac{P(t) - P(a)}{t - a}$

(B) $\lim\limits_{h \to 0} \frac{P(t+h) - P(t)}{h}$

(C) $\lim\limits_{h \to \infty} \frac{P(t+h) - P(t)}{h}$

(D) $P(t)/t$

Question 3. A company's profit function $P(x)$ (in $\textsf{₹}$) for selling $x$ units is given. The company wants to understand the marginal profit, which is the derivative $P'(x)$. If the profit function is continuous for all $x \geq 0$, is it guaranteed to be differentiable for all $x > 0$?

(A) Yes, continuity implies differentiability.

(B) No, continuity is a necessary condition for differentiability, but not sufficient.

(C) Yes, as long as the function is smooth.

(D) It depends on whether $P(x)$ is a polynomial.

Question 4. Consider the function $f(x) = \begin{cases} x^2 & , & x \leq 1 \\ x & , & x > 1 \end{cases}$ representing a process output. Is this function differentiable at $x=1$? ($f'(x) = 2x$ for $x<1$ and $f'(x)=1$ for $x>1$).

(A) Yes, because it is continuous at $x=1$.

(B) No, because the left hand derivative at $x=1$ (limit of $2x$ as $x \to 1^-$ is 2) is not equal to the right hand derivative at $x=1$ (limit of $1$ as $x \to 1^+$ is 1).

(C) Yes, because both pieces are polynomials.

(D) No, because $f(1)$ is defined using the first case.

Question 5. A physical model predicts that the energy of a particle at a point can be described by $E(x) = |x-a|$. At $x=a$, the model shows a sudden change in direction. Based on differentiability, what can be said about the rate of change of energy at $x=a$ according to this model?

(A) The rate of change is 0 at $x=a$.

(B) The rate of change is defined and finite at $x=a$.

(C) The rate of change is not defined (the function is not differentiable) at $x=a$.

(D) The rate of change is infinite at $x=a$.

Question 1. The area of a circular plate is given by $A = \pi r^2$, where $r$ is the radius. What is the rate of change of the area with respect to the radius? (i.e., find $\frac{dA}{dr}$).

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

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

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

(D) $\frac{dA}{dr} = \pi r^3/3$

Question 2. In a simple economic model, the total cost $C(x)$ (in $\textsf{₹}$) of producing $x$ units is given by $C(x) = 1000 + 50x + 0.1x^2$. The marginal cost is given by the derivative $C'(x)$. Find the marginal cost function.

(A) $C'(x) = 50 + 0.1x$

(B) $C'(x) = 1000 + 50 + 0.2x$

(C) $C'(x) = 50 + 0.2x$

(D) $C'(x) = 1000 + 50x + 0.2x$

Question 3. The angle $\theta$ (in radians) of a pendulum from the vertical is modeled by $\theta(t) = A \cos(\omega t + \phi)$, where $t$ is time. The angular velocity is given by $\frac{d\theta}{dt}$. Find the angular velocity function.

(A) $A \omega \sin(\omega t + \phi)$

(B) $-A \omega \sin(\omega t + \phi)$

(C) $-A \sin(\omega t + \phi)$

(D) $A \omega \cos(\omega t + \phi)$

Question 4. The decay of a radioactive substance is modeled by $N(t) = N_0 e^{-\lambda t}$, where $N(t)$ is the amount at time $t$, $N_0$ is the initial amount, and $\lambda$ is the decay constant. The rate of decay is given by $\frac{dN}{dt}$. Find the rate of decay.

(A) $N_0 e^{-\lambda t}$

(B) $-\lambda N_0 e^{-\lambda t}$

(C) $\lambda N_0 e^{-\lambda t}$

(D) $-\lambda e^{-\lambda t}$

Question 5. The intensity of light $I(x)$ at a depth $x$ in a certain medium is given by $I(x) = I_0 e^{-kx}$, where $I_0$ is the initial intensity and $k$ is a positive constant. What is the rate of change of intensity with respect to depth?

(A) $\frac{dI}{dx} = -k I_0 e^{-kx}$

(B) $\frac{dI}{dx} = I_0 e^{-kx}$

(C) $\frac{dI}{dx} = k I_0 e^{-kx}$

(D) $\frac{dI}{dx} = -k I(x)$

Question 1. The volume $V$ of a spherical balloon is increasing. The radius $r$ (in cm) at time $t$ (in seconds) is $r(t) = 2t+1$. Find the rate of change of volume with respect to time when $t=1$ second. (Volume $V = \frac{4}{3}\pi r^3$).

(A) $36\pi \text{ cm}^3\text{/s}$

(B) $108\pi \text{ cm}^3\text{/s}$

(C) $12\pi \text{ cm}^3\text{/s}$

(D) $144\pi \text{ cm}^3\text{/s}$

Question 2. The pressure $P$ (in Pascals) in a fluid varies with depth $h$ (in meters) according to $P(h) = \rho g h + P_0$. If the depth $h$ is changing with time $t$ (in seconds) as $h(t) = t^2 + 5$, find the rate of change of pressure with respect to time, $\frac{dP}{dt}$. (Assume $\rho$ and $g$ are constants).

(A) $\rho g (t^2+5)$

(B) $2t$

(C) $\rho g (2t)$

(D) $\rho g$

Question 3. The cost $C$ (in $\textsf{₹}$) of producing $x$ units of a product is $C(x) = 1000 + 5\sqrt{x}$. If the number of units produced $x$ depends on the time $t$ (in hours) as $x(t) = 10t^2$, find the rate of change of cost with respect to time when $t=5$ hours.

(A) $\textsf{₹}5/\text{hour}$

(B) $\textsf{₹}10/\text{hour}$

(C) $\textsf{₹}15/\text{hour}$

(D) $\textsf{₹}20/\text{hour}$

Question 4. A patient's body temperature $T$ (in $^\circ$C) after taking medication is given by $T(h) = 37 + \sin(h/2)$, where $h$ is the number of hours after taking the medicine. If the rate of change of hours with respect to some factor $F$ is $\frac{dh}{dF}$, find the rate of change of temperature with respect to $F$, $\frac{dT}{dF}$.

(A) $\cos(h/2) \cdot \frac{dh}{dF}$

(B) $\frac{1}{2} \cos(h/2) \cdot \frac{dh}{dF}$

(C) $\sin(h/2) \cdot \frac{dh}{dF}$

(D) $\frac{1}{2} \sin(h/2) \cdot \frac{dh}{dF}$

Question 5. The concentration $C$ of a drug in the bloodstream is given by $C(t) = 10e^{-0.5t}$, where $t$ is time in hours. If the effectiveness $E$ of the drug depends on the concentration $C$ as $E(C) = \ln(C+1)$, find the rate of change of effectiveness with respect to time, $\frac{dE}{dt}$, at $t=2$ hours.

(A) $-\frac{5e^{-1}}{10e^{-1}+1}$

(B) $\frac{1}{10e^{-1}+1}$

(C) $-5e^{-1}$

(D) $10e^{-1}$

Question 1. A point $(x,y)$ moves along the circle $x^2 + y^2 = 100$. If $\frac{dx}{dt} = 3$ units/s when $x=6$ and $y=8$, find $\frac{dy}{dt}$ at that instant using implicit differentiation with respect to time $t$.

(A) $-2.25$ units/s

(B) $2.25$ units/s

(C) $-1.5$ units/s

(D) $1.5$ units/s

Question 2. The production level $L$ (in units) of a company is related to the capital $K$ (in $\textsf{₹}$) and labour $M$ (in hours) by the equation $L^2 = K M$. If capital is $\textsf{₹}10000$ and labour is 400 hours, and both are increasing such that $\frac{dK}{dt} = 500 \textsf{₹/hour}$ and $\frac{dM}{dt} = 10 \text{ hours/hour}$, find the rate of change of production level $\frac{dL}{dt}$ at that moment.

(A) $25$ units/hour

(B) $50$ units/hour

(C) $75$ units/hour

(D) $100$ units/hour

Question 3. The relationship between the angle of elevation $\theta$ of a camera and the horizontal distance $x$ to a rocket launching vertically is given. The rate of change of $x$ with respect to $\theta$ is related to the derivative of the inverse trigonometric function. If $x = 100 \cot \theta$, what is $\frac{dx}{d\theta}$?

(A) $100 \text{cosec }^2 \theta$

(B) $-100 \text{cosec }^2 \theta$

(C) $100 \sec^2 \theta$

(D) $-100 \cot \theta \text{cosec } \theta$

Question 4. A spherical snowball is melting such that its volume $V$ is decreasing at a constant rate. The radius $r$ is related to the volume by $r = (\frac{3V}{4\pi})^{1/3}$. Find the rate of change of the radius with respect to the volume, $\frac{dr}{dV}$.

(A) $\frac{1}{4\pi r^2}$

(B) $\frac{1}{3} (\frac{3V}{4\pi})^{-2/3} \cdot \frac{3}{4\pi}$

(C) $\frac{1}{4\pi r^2}$ and $\frac{1}{3} (\frac{3V}{4\pi})^{-2/3} \cdot \frac{3}{4\pi}$ are equivalent ways to express $\frac{dr}{dV}$.

(D) $4\pi r^2$

Question 5. The angle of view $\theta$ from a camera observing a vertically rising object is related to the height $y$ of the object above the camera by $\theta = \tan^{-1}(\frac{y}{d})$, where $d$ is the constant horizontal distance. Find the rate of change of the angle of view with respect to height, $\frac{d\theta}{dy}$.

(A) $\frac{d}{y^2+d^2}$

(B) $\frac{y}{y^2+d^2}$

(C) $\frac{1}{\sqrt{y^2+d^2}}$

(D) $\frac{d}{y^2+d^2}$ and $\frac{1/d}{1+(y/d)^2}$ are equivalent.

Question 1. In a biological model, the growth of a cell population is described by $N(t) = (t+1)^{t+1}$, where $t$ is time. To find the rate of growth $\frac{dN}{dt}$, logarithmic differentiation is used. What is $\frac{dN}{dt}$?

(A) $(t+1)^{t+1} \ln(t+1)$

(B) $(t+1)^{t+1} (1 + \ln(t+1))$

(C) $(t+1)^{t+1}$

(D) $(t+1) \ln(t+1)$

Question 2. The position of a particle in a 2D plane at time $t$ is given by parametric equations $x(t) = 5 \cos t$ and $y(t) = 5 \sin t$. Find the slope of the tangent to the path of the particle at $t = \pi/4$.

(A) $-1$

(B) $1$

(C) $0$

(D) Undefined

Question 3. A complex formula for predicting market trends involves the expression $f(x) = \frac{(x^2+1)^{3/2} \sqrt{x-1}}{(x+4)^5}$. To find $f'(x)$, logarithmic differentiation is used. Which expression is obtained after taking the logarithm and differentiating implicitly?

(A) $\frac{f'(x)}{f(x)} = \frac{3}{2} \ln(x^2+1) + \frac{1}{2} \ln(x-1) - 5 \ln(x+4)$

(B) $\frac{f'(x)}{f(x)} = \frac{3}{2} \frac{2x}{x^2+1} + \frac{1}{2} \frac{1}{x-1} - 5 \frac{1}{x+4}$

(C) $\ln f(x) = \frac{3}{2} \ln(x^2+1) + \frac{1}{2} \ln(x-1) - 5 \ln(x+4)$

(D) $\ln f(x) = \frac{3}{2} (x^2+1) + \frac{1}{2} (x-1) - 5 (x+4)$

Question 4. The trajectory of a projectile is given by $x(t) = 20t$ and $y(t) = 20t - 5t^2$, where $x$ and $y$ are in meters and $t$ is in seconds. Find the horizontal rate of change of vertical position (i.e., $\frac{dy}{dx}$).

(A) $20 - 10t$

(B) $\frac{20 - 10t}{20}$

(C) $20t - 5t^2$

(D) $20$

Question 5. A variable resistance $R$ in an electronic circuit depends on voltage $V$ as $R = V^{\cos V}$. To find the sensitivity $\frac{dR}{dV}$, logarithmic differentiation is suitable. What is the derivative $\frac{dR}{dV}$?

(A) $V^{\cos V} (-\sin V \ln V + \cos V \frac{1}{V})$

(B) $\cos V \cdot V^{\cos V - 1}$

(C) $-\sin V \cdot V^{\cos V} \ln V$

(D) $V^{\cos V} \cos V \ln V$

Question 1. A car's position $s(t)$ (in km) at time $t$ (in hours) is given by $s(t) = t^3 - 6t^2 + 9t$. The velocity is $v(t) = s'(t)$ and the acceleration is $a(t) = v'(t) = s''(t)$. Find the acceleration of the car at $t=2$ hours.

(A) $-12 \text{ km/h}^2$

(B) $0 \text{ km/h}^2$

(C) $6 \text{ km/h}^2$

(D) $12 \text{ km/h}^2$

Question 2. The temperature $T$ (in $^\circ$C) of a chemical reaction changes with time $t$ (in minutes) according to $T(t) = t e^{-t}$. Find the rate of change of the rate of change of temperature with respect to time (i.e., the second derivative $\frac{d^2 T}{dt^2}$) at $t=1$ minute.

(A) $e^{-1}$

(B) $-e^{-1}$

(C) $0$

(D) $2e^{-1}$

Question 3. In mechanics, the jerk is the rate of change of acceleration with respect to time, which is the third derivative of position $s(t)$. If $s(t) = \sin(\omega t)$, where $\omega$ is a constant, find the jerk.

(A) $-\omega^2 \sin(\omega t)$

(B) $-\omega^3 \cos(\omega t)$

(C) $\omega^3 \cos(\omega t)$

(D) $\omega^2 \cos(\omega t)$

Question 4. The deflection $y$ of a beam at a distance $x$ from one end is governed by an equation involving higher order derivatives. If $y = A \sin(\frac{\pi x}{L})$, where $A$ and $L$ are constants, find the second derivative $\frac{d^2 y}{dx^2}$.

(A) $\frac{\pi}{L} A \cos(\frac{\pi x}{L})$

(B) $-\frac{\pi^2}{L^2} A \sin(\frac{\pi x}{L})$

(C) $-\frac{\pi}{L} A \sin(\frac{\pi x}{L})$

(D) $\frac{\pi^2}{L^2} A \cos(\frac{\pi x}{L})$

Question 5. In a circuit, the charge $q(t)$ on a capacitor is related to the voltage $V(t)$ and current $I(t)$. The rate of change of current is $\frac{dI}{dt}$, which is related to the second derivative of charge $\frac{d^2 q}{dt^2}$. If $q(t) = C V_0 (1 - e^{-t/RC})$, find $\frac{d^2 q}{dt^2}$. (Assume $C, V_0, R$ are constants).

(A) $-\frac{V_0}{R} e^{-t/RC}$

(B) $\frac{V_0}{RC^2} e^{-t/RC}$

(C) $-\frac{V_0}{R^2 C} e^{-t/RC}$

(D) $\frac{V_0}{R^2 C} e^{-t/RC}$

Question 1. A car travels from City A to City B, a distance of 200 km, in 4 hours. According to the Mean Value Theorem, what can be said about the car's instantaneous velocity during this trip?

(A) The car's velocity was constant throughout the trip.

(B) The car's maximum velocity was exactly 50 km/h.

(C) At some point during the trip, the car's instantaneous velocity was exactly 50 km/h.

(D) The average velocity was 50 km/h, but nothing can be said about the instantaneous velocity.

Question 2. A company's accumulated profit $P(t)$ (in $\textsf{₹}$) at the end of $t$ years is given by a differentiable function. The profit was $\textsf{₹}10$ Lakhs at $t=2$ years and $\textsf{₹}16$ Lakhs at $t=5$ years. According to the Mean Value Theorem, there was a point in time between $t=2$ and $t=5$ years when the instantaneous rate of profit increase (marginal profit rate) was equal to the average rate of profit increase over that period. What was that average rate?

(A) $\textsf{₹}2$ Lakhs/year

(B) $\textsf{₹}3$ Lakhs/year

(C) $\textsf{₹}6$ Lakhs/year

(D) $\textsf{₹}10$ Lakhs/year

Question 3. A projectile is launched vertically upwards. Its height $h(t)$ at time $t$ is given by $h(t) = 40t - 5t^2$. The projectile hits the ground at $t=8$. Which theorem guarantees that at some point during its flight, the instantaneous velocity was zero? (Note: $h(0)=0, h(8)=0$).

(A) Mean Value Theorem

(B) Intermediate Value Theorem

(C) Extreme Value Theorem

(D) Rolle's Theorem

Question 4. A cyclist starts a race and finishes at the same point after completing a loop. The distance covered is a function of time. If the cyclist's speed was always finite and the path was smooth (differentiable), what does Rolle's Theorem imply about the cyclist's instantaneous velocity at some point during the loop?

(A) The velocity was constant.

(B) The velocity vector was zero (i.e., the cyclist was momentarily stopped).

(C) The speed was at its maximum.

(D) The average velocity was zero.

Question 5. Consider a continuous and differentiable function $f(x)$ representing the concentration of a substance over the interval $[0, 5]$. If $f(0)=f(5)$, what is the implication according to Rolle's Theorem?

(A) The maximum concentration occurred at either $x=0$ or $x=5$.

(B) The average concentration over the interval is zero.

(C) At some point $c$ between 0 and 5, the rate of change of concentration $f'(c)$ was zero.

(D) The concentration was constant over the interval.

Question 1. A spherical balloon is being inflated. The volume $V$ is increasing at a constant rate of $100$ cm$^3$/s. At the instant the radius $r$ is $5$ cm, what is the rate at which the radius is increasing?

(A) $\frac{1}{\pi} \text{ cm/s}$

(B) $\frac{1}{25\pi} \text{ cm/s}$

(C) $\frac{1}{4\pi} \text{ cm/s}$

(D) $\frac{1}{5\pi} \text{ cm/s}$

Question 2. Sand is being poured onto a conical pile at a rate of $10$ m$^3$/min. The height of the pile is always equal to the radius of the base. How fast is the height increasing when the height is $8$ m? (Volume of a cone $V = \frac{1}{3}\pi r^2 h$).

(A) $\frac{5}{32\pi} \text{ m/min}$

(B) $\frac{5}{64\pi} \text{ m/min}$

(C) $\frac{15}{64\pi} \text{ m/min}$

(D) $\frac{10}{9\pi} \text{ m/min}$

Question 3. The total cost $C(x)$ (in $\textsf{₹}$) of producing $x$ units of a product is given by $C(x) = 0.005x^3 - 0.02x^2 + 30x + 5000$. Find the marginal cost when 100 units are produced.

(A) $\textsf{₹}130.6$

(B) $\textsf{₹}132$

(C) $\textsf{₹}150$

(D) $\textsf{₹}131.6$

Question 4. The total revenue $R(x)$ (in $\textsf{₹}$) received from the sale of $x$ units of a product is given by $R(x) = 13x + 0.02x^2$. Find the marginal revenue when $x=500$ units.

(A) $\textsf{₹}33$

(B) $\textsf{₹}23$

(C) $\textsf{₹}13$

(D) $\textsf{₹}43$

Question 5. A ladder 5 meters long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at a rate of $2$ m/s. How fast is the top of the ladder sliding down the wall when the bottom is 4 meters from the wall?

(A) $-8/3$ m/s

(B) $-4/3$ m/s

(C) $-16/3$ m/s

(D) $-2/3$ m/s

Question 1. Find the equation of the tangent line to the curve $y = x^3 - x$ at the point where $x=2$.

(A) $y = 11x - 18$

(B) $y = 11x - 22$

(C) $y = 11x - 20$

(D) $y = 11x - 16$

Question 2. Find the equation of the normal line to the curve $y = \sin x$ at the point $(0, 0)$.

(A) $y = x$

(B) $y = -x$

(C) $x = 0$

(D) $y = 0$

Question 3. Use differentials to approximate the value of $\sqrt{49.5}$.

(A) $7.035$

(B) $7.007$

(C) $7.071$

(D) $7.005$

Question 4. The side of a square is measured as $5$ cm with a possible error of $0.02$ cm. Estimate the maximum error in the calculated area of the square using differentials.

(A) $0.1 \text{ cm}^2$

(B) $0.2 \text{ cm}^2

(C) $0.04 \text{ cm}^2

(D) $0.004 \text{ cm}^2

Question 5. The radius of a sphere is measured to be $10$ cm with a possible percentage error of $1\%$. Estimate the maximum percentage error in the calculation of the surface area of the sphere.

(A) $1\%$

(B) $2\%$

(C) $3\%$

(D) $4\%$

Question 1. A company's profit $P(x)$ (in $\textsf{₹}$) as a function of production $x$ (in units) is given by $P(x) = 100x - 0.5x^2$ for $x \geq 0$. Find the interval where the profit is increasing.

(A) $(0, 50)$

(B) $(50, \infty)$

(C) $(-\infty, 50)$

(D) $(0, 100)$

Question 2. The temperature $T(t)$ of a cooling object at time $t$ is given by $T(t) = T_m + (T_0 - T_m)e^{-kt}$, where $T_m, T_0, k$ are constants and $k > 0$. Is the temperature increasing or decreasing over time?

(A) Increasing, because $\frac{dT}{dt} > 0$.

(B) Decreasing, because $\frac{dT}{dt} < 0$.

(C) Constant, because the rate of change is zero.

(D) It depends on the value of $T_0 - T_m$.

Question 3. Find the interval where the function $f(x) = x^2 - 6x + 5$ is strictly decreasing.

(A) $(-\infty, 3)$

(B) $(3, \infty)$

(C) $(-\infty, -3)$

(D) $(-3, \infty)$

Question 4. The concentration $C(t)$ of a chemical in a reaction is given by $C(t) = t e^{-t}$ for $t \geq 0$. Determine if the concentration is initially increasing or decreasing.

(A) Initially increasing.

(B) Initially decreasing.

(C) Initially constant.

(D) Cannot be determined.

Question 5. A manufacturer's efficiency index $E(x)$ depends on the expenditure $x$ (in $\textsf{₹}$ Lakhs) on quality control, given by $E(x) = 100x - x^2$ for $0 \leq x \leq 100$. Is the efficiency index increasing or decreasing when the expenditure is $\textsf{₹}40$ Lakhs?

(A) Increasing

(B) Decreasing

(C) Neither increasing nor decreasing

(D) Cannot be determined from the given function.

Question 1. A farmer wants to fence a rectangular field adjacent to a river. He has 400 meters of fencing and does not need fencing along the river. What are the dimensions of the field that maximize the area?

(A) 100 m $\times$ 200 m

(B) 100 m $\times$ 100 m

(C) 200 m $\times$ 200 m

(D) 50 m $\times$ 300 m

Question 2. The total revenue $R(x)$ (in $\textsf{₹}$) from the sale of $x$ units is given by $R(x) = 500x - x^2$. Find the number of units $x$ that maximizes the total revenue.

(A) 100 units

(B) 250 units

(C) 500 units

(D) 1000 units

Question 3. Find the local maximum value of the function $f(x) = x^3 - 6x^2 + 5$ using derivative tests.

(A) 5 (at $x=0$)

(B) -27 (at $x=4$)

(C) 0

(D) There is no local maximum.

Question 4. A company wants to minimize the average cost per unit. If the total cost function is $C(x) = 100 + 50x + x^2$, find the production level $x$ that minimizes the average cost $A(x) = C(x)/x$.

(A) 5 units

(B) 10 units

(C) 100 units

(D) 20 units

Question 5. Find the absolute maximum and minimum values of the function $f(x) = x^2 - 4x + 3$ on the interval $[0, 3]$.

(A) Max=3, Min=-1

(B) Max=3, Min=3

(C) Max=0, Min=-1

(D) Max=3, Min=0

Question 1. The marginal cost of producing $x$ units of a product is given by $MC(x) = 3x^2 - 2x + 5$. If the fixed cost is $\textsf{₹}100$, find the total cost function $C(x)$.

(A) $x^3 - x^2 + 5x + 100$

(B) $6x - 2 + 100$

(C) $x^3 - x^2 + 5x + C$

(D) $x^3 - x^2 + 5x$

Question 2. The velocity of a particle moving along a straight line is given by $v(t) = 2t + 3$. If the initial position of the particle at $t=0$ is $s(0) = 5$, find the position function $s(t)$.

(A) $t^2 + 3t$

(B) $t^2 + 3t + 5$

(C) $2 + 3t + 5$

(D) $t^2 + 3t + C$

Question 3. A company's marginal revenue function is $MR(x) = 100 - 0.4x$. Find the total revenue function $R(x)$, assuming that revenue is zero when no units are sold ($R(0)=0$).

(A) $100x - 0.4x^2$

(B) $100x - 0.2x^2$

(C) $100 - 0.2x^2 + C$

(D) $100x - 0.2x^2 + C$

Question 4. The acceleration of an object is given by $a(t) = 6t - 4$. If the initial velocity is $v(0) = 5$ and the initial position is $s(0) = 10$, find the velocity function $v(t)$.

(A) $3t^2 - 4t$

(B) $3t^2 - 4t + 5$

(C) $6 - 4t + 5$

(D) $3t^2 - 4t + C$

Question 5. The rate of growth of a plant's height (in cm/day) is given by $\frac{dh}{dt} = \sqrt{t}$. If the initial height at $t=0$ is $10$ cm, find the height after 4 days.

(A) $18.67$ cm

(B) $16$ cm

(C) $18$ cm

(D) $14.67$ cm

Question 1. The rate of flow of water into a tank is given by $R(t) = t \sqrt{t^2+1}$ liters/minute. If the tank is initially empty, find the total amount of water in the tank after $\sqrt{3}$ minutes.

(A) $8/3$ liters

(B) $16/3$ liters

(C) $4/3$ liters

(D) $2/3$ liters

Question 2. The force $F(x)$ acting on an object varies with its position $x$ as $F(x) = x e^x$. The work done in moving the object from $x=1$ to $x=2$ is given by $\int_{1}^{2} F(x) dx$. Evaluate this integral using integration by parts.

(A) $e^2$

(B) $2e^2 - e$

(C) $e^2 - e$

(D) $2e^2$

Question 3. The concentration of a pollutant in a lake is changing at a rate given by $\frac{dC}{dt} = t \cos(\pi t)$. The total change in concentration from $t=0$ to $t=1$ hour is given by $\int_{0}^{1} t \cos(\pi t) dt$. Evaluate this integral using integration by parts.

(A) $1/\pi$

(B) $-2/\pi^2$

(C) $0$

(D) $-1/\pi$

Question 4. The rate of chemical reaction is sometimes proportional to the concentration of the reactant raised to a power. If the rate is proportional to $(C+1)^2$ and the rate constant leads to the rate $\frac{dC}{dt} = -k(C+1)^2$, integration by substitution can be used. If the rate is $\frac{dC}{dt} = -C^2$ and $C(0)=1$, find $C(t)$.

(A) $C(t) = \frac{1}{t+1}$

(B) $C(t) = 1-t$

(C) $C(t) = e^{-t}$

(D) $C(t) = \frac{1}{t}$

Question 5. The current $I(t)$ in a circuit changes according to $\frac{dI}{dt} = e^{2t} \sin t$. Find the function $I(t)$ by integrating $\int e^{2t} \sin t dt$ using integration by parts (twice).

(A) $\frac{1}{5} e^{2t} (2 \sin t - \cos t) + C$

(B) $\frac{1}{5} e^{2t} (\sin t + 2 \cos t) + C$

(C) $\frac{1}{5} e^{2t} (2 \sin t + \cos t) + C$

(D) $\frac{1}{5} e^{2t} (\sin t - 2 \cos t) + C$

Question 1. A reaction rate is given by $R(C) = \frac{1}{(C-1)(C+2)}$, where $C$ is the concentration. The total change in concentration over a time interval can be found by integrating this rate. Evaluate $\int \frac{1}{(C-1)(C+2)} dC$ using partial fractions.

(A) $\frac{1}{3} \ln|\frac{C-1}{C+2}| + K$

(B) $3 \ln|\frac{C-1}{C+2}| + K$

(C) $\frac{1}{3} \ln|(C-1)(C+2)| + K$

(D) $\frac{1}{3} (\ln|C-1| - \ln|C+2|)$

Question 2. The velocity of a particle is given by $v(t) = \frac{t+1}{t^2-4}$. Find the displacement of the particle from $t=3$ to $t=4$ by evaluating $\int_{3}^{4} \frac{t+1}{t^2-4} dt$ using partial fractions.

(A) $\frac{3}{4} \ln(\frac{4}{3}) + \frac{1}{4} \ln(\frac{6}{5})$

(B) $\frac{1}{4} \ln(\frac{4}{3}) + \frac{3}{4} \ln(\frac{6}{5})$

(C) $\frac{1}{4} \ln(\frac{6}{5}) + \frac{3}{4} \ln(\frac{4}{3})$

(D) $\ln|\frac{(t-2)^{3/4}}{(t+2)^{1/4}}|$

Question 3. Find the area under the curve $y = \frac{1}{\sqrt{9-x^2}}$ from $x=0$ to $x=3$.

(A) $\pi/2$

(B) $\pi/6$

(C) $\pi/3$

(D) $\pi$

Question 4. Evaluate the integral $\int \frac{dx}{x^2+16}$, which might arise in finding the time period in certain physical systems.

(A) $\frac{1}{16} \tan^{-1}(\frac{x}{16}) + C$

(B) $\frac{1}{4} \tan^{-1}(\frac{x}{4}) + C$

(C) $\frac{1}{8} \tan^{-1}(\frac{x}{4}) + C$

(D) $\frac{1}{4} \tan^{-1}(4x) + C$

Question 5. Integrate $\int \frac{dx}{\sqrt{x^2+25}}$, which is related to the distance formula in certain coordinate systems.

(A) $\ln|x + \sqrt{x^2+25}| + C$

(B) $\sin^{-1}(\frac{x}{5}) + C$

(C) $\ln|\sqrt{x^2+25}| + C$

(D) $\frac{1}{5} \tan^{-1}(\frac{x}{5}) + C$

Question 1. The velocity of a particle is given by $v(t) = t^2$ m/s. The displacement of the particle from $t=1$ to $t=3$ seconds can be found using a definite integral. Which of the following represents the displacement?

(A) $\int_1^3 t^2 dt = [t^3/3]_1^3 = (3^3/3) - (1^3/3) = 9 - 1/3 = 26/3$ meters.

(B) $\int t^2 dt = t^3/3 + C$

(C) $\lim\limits_{n \to \infty} \sum_{i=1}^n (t_i^*)^2 \Delta t_i$

(D) The area under the velocity-time graph.

Question 2. A factory's production rate (in units per hour) is given by $P(t) = 10t + 20$. The total production from $t=0$ to $t=5$ hours is given by the definite integral $\int_{0}^{5} (10t + 20) dt$. Evaluate this total production.

(A) 125 units

(B) 225 units

(C) 325 units

(D) 425 units

Question 3. The rate of change of the amount of water in a reservoir is given by $R(t)$ liters/day. The total change in the amount of water from day $a$ to day $b$ is given by $\int_{a}^{b} R(t) dt$. If $R(t) = 100 \sin(\frac{\pi t}{10})$ and we want the total change from $t=0$ to $t=10$, which theorem is used for evaluation?

(A) Mean Value Theorem

(B) Integration by Parts

(C) Fundamental Theorem of Integral Calculus

(D) Squeeze Theorem

Question 4. The concentration of a drug in an organ at time $x$ is given by $C(x)$. The total accumulation of the drug up to time $T$ is $\int_{0}^{T} C(x) dx$. If $F(T) = \int_{0}^{T} C(x) dx$, the rate at which the total accumulation is changing with respect to $T$ is $F'(T)$. According to the Fundamental Theorem of Calculus, what is $F'(T)$?

(A) $C'(T)$

(B) $C(T)$

(C) $\int_0^T C'(x) dx$

(D) $0$

Question 5. The area under the curve $y = e^x$ from $x=0$ to $x=1$ can be calculated as a definite integral. Evaluate this area.

(A) $e$ square units

(B) $e-1$ square units

(C) $1-e$ square units

(D) $1$ square unit

Question 1. Evaluate the definite integral $\int_{0}^{\pi/4} \tan x dx$.

(A) $\ln \sqrt{2}$

(B) $\ln 2$

(C) $1$

(D) $0$

Question 2. Evaluate the definite integral $\int_{0}^{1} \frac{x}{x^2+1} dx$ using substitution.

(A) $\frac{1}{2} \ln 2$

(B) $\ln 2$

(C) $\frac{1}{2} \ln 1 = 0$

(D) $\frac{\pi}{4}$

Question 3. Evaluate the definite integral $\int_{-\pi/2}^{\pi/2} \sin^3 x dx$.

(A) $\pi/2$

(B) $1$

(C) $0$

(D) $2$

Question 4. Evaluate $\int_{0}^{2\pi} |\sin x| dx$.

(A) $0

(B) $2

(C) $4$

(D) $1$

Question 5. Evaluate the integral $\int_{0}^{\pi/2} \frac{\sin x}{\sin x + \cos x} dx$ using the property $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx$.

(A) $\pi/4$

(B) $\pi/2$

(C) $0$

(D) $1$

Question 1. Find the area of the region bounded by the parabola $y = x^2$, the x-axis, and the line $x=3$.

(A) 3 square units

(B) 6 square units

(C) 9 square units

(D) $27/3$ square units

Question 2. Find the area of the region bounded by the curve $y = \sin x$, the x-axis, from $x=0$ to $x=\pi$.

(A) 0 square units

(B) 1 square unit

(C) 2 square units

(D) $\pi$ square units

Question 3. Find the area of the region bounded by the line $y=x$ and the parabola $y=x^2$ in the first quadrant.

(A) $1/6$ square units

(B) $1/3$ square units

(C) $1/2$ square units

(D) $1$ square unit

Question 4. Find the area enclosed by the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.

(A) $\pi a^2$

(B) $\pi b^2

(C) $\pi ab$

(D) $ab$

Question 5. Find the area of the region bounded by the curves $y^2 = 4x$ and $x = 1$.

(A) $4/3$ square units

(B) $8/3$ square units

(C) $2/3$ square units

(D) $16/3$ square units

Question 1. A scientist is modeling the rate of change of temperature $T$ of a liquid with respect to time $t$. The rate is proportional to the difference between the liquid's temperature and the ambient temperature $T_a$. Which of the following is a differential equation representing this situation?

(A) $\frac{dT}{dt} = k(T - T_a)$

(B) $\frac{dT}{dt} = k T$

(C) $T = k(T - T_a)$

(D) $\frac{dT}{T - T_a} = k dt$

Question 2. A model for the spread of a disease states that the rate of change of the number of infected people $I$ with respect to time $t$ is proportional to the product of the number of infected people and the number of susceptible people ($S$). If the total population is $N$, then $S = N-I$. Which of the following is a differential equation for the number of infected people?

(A) $\frac{dI}{dt} = k I$

(B) $\frac{dI}{dt} = k (N-I)$

(C) $\frac{dI}{dt} = k I (N-I)$

(D) $\frac{dI}{dt} = k (I + S)$

Question 3. The position $x(t)$ of a mass attached to a spring is described by an equation where the acceleration is proportional to the negative of the displacement. This leads to a second-order differential equation. What is the order of the differential equation that represents simple harmonic motion, given by $m\frac{d^2x}{dt^2} = -kx$?

(A) 1

(B) 2

(C) 3

(D) 0

Question 4. Formulate the differential equation representing the family of circles passing through the origin and having their centers on the x-axis. The equation of such a family is $(x-a)^2 + y^2 = a^2$.

(A) $y^2 = x^2 + 2xy \frac{dy}{dx}$

(B) $y' = \frac{2xy}{x^2-y^2}$

(C) $2(x-a) + 2y y' = 0$

(D) $y^2 = -x^2 - 2xy \frac{dy}{dx}$

Question 5. Consider the differential equation $(\frac{dy}{dx})^2 = y$. What is its order and degree?

(A) Order 1, Degree 1

(B) Order 1, Degree 2

(C) Order 2, Degree 1

(D) Order 2, Degree 2

Question 1. The rate of population growth of a species is proportional to its current population $P$. This is modeled by $\frac{dP}{dt} = kP$. Solve this differential equation using the variable separable method.

(A) $P = kt + C$

(B) $\ln P = kt + C$

(C) $P = C e^{kt}$

(D) $t = kP + C$

Question 2. Solve the differential equation $\frac{dy}{dx} = \frac{x^2 + y^2}{xy}$, which arises in some physics problems involving scaling.

(A) $y^2 = x^2 (\ln x^2 + C)$

(B) $y = Cx^2$

(C) $y^2 = Cx$

(D) $y^2 = x^2 (Cx^2)$

Question 3. The decay of a radioactive substance is described by $\frac{dA}{dt} = -0.1 A$, where $A$ is the amount in grams and $t$ is in years. If the initial amount is 100 grams, find the amount after 5 years.

(A) $100 e^{-0.5}$ grams

(B) $100 e^{-5}$ grams

(C) $100 - 0.5$ grams

(D) $100 e^{0.5}$ grams

Question 4. Solve the differential equation $\frac{dy}{dx} = \frac{x+y+1}{x+y}$, which can be solved by substituting $v = x+y$.

(A) $\ln(x+y) + (x+y) = x + C$

(B) $x+y = C e^x$

(C) $(x+y)^2 = 2x + C$

(D) $y = Cx - x$

Question 5. The velocity of a particle is inversely proportional to its position $x$, given by $\frac{dx}{dt} = \frac{k}{x}$. Find the position $x(t)$ if the initial position is $x(0) = x_0$.

(A) $x = kt + x_0$

(B) $x^2 = 2kt + x_0^2$

(C) $x = C e^{kt}$

(D) $x = \sqrt{kt + x_0^2}$

Question 1. The current $I(t)$ in an electrical circuit with resistance $R$ and inductance $L$ is given by $L\frac{dI}{dt} + RI = V(t)$, where $V(t)$ is the applied voltage. This is a first-order linear differential equation. If $R, L, V$ are constants, find the integrating factor for this equation.

(A) $e^{Rt/L}$

(B) $e^{Lt/R}$

(C) $e^{V t}$

(D) $e^{t/RL}$

Question 2. Solve the linear differential equation $\frac{dy}{dx} + y \cot x = \text{cosec } x$.

(A) $y = \sin x (x + C)$

(B) $y = \text{cosec } x (x + C)$

(C) $y \sin x = x + C$

(D) $y \text{cosec } x = x + C$

Question 3. The concentration $C(t)$ of a pollutant in a tank changes according to $\frac{dC}{dt} + \frac{1}{10} C = f(t)$, where $f(t)$ represents the rate at which the pollutant is added. Find the integrating factor for this equation.

(A) $e^t$

(B) $e^{t/10}$

(C) $e^{-t/10}$

(D) $e^{10t}$

Question 4. Solve the linear differential equation $x \frac{dy}{dx} + y = x^3$ for $x > 0$.

(A) $xy = x^4/4 + C$

(B) $y = x^3/4 + C/x$

(C) $y = x^2/4 + C/x$

(D) $y = x^3/4 + C$

Question 5. Solve the initial value problem $\frac{dy}{dx} - y = e^x$ with $y(0)=1$.

(A) $y = xe^x + e^x$

(B) $y = xe^x + 1$

(C) $y = e^x + C$

(D) $y = xe^x + 1 + e^x$

Question 1. A population of fish in a lake grows at a rate proportional to the current population, but is harvested at a constant rate. If $P(t)$ is the population at time $t$, $k$ is the growth constant, and $H$ is the harvesting rate, which differential equation models this situation?

(A) $\frac{dP}{dt} = kP + H$

(B) $\frac{dP}{dt} = kP - H$

(C) $\frac{dP}{dt} = k - HP$

(D) $\frac{dP}{dt} = H - kP$

Question 2. The rate at which a substance dissolves in a liquid is proportional to the product of the amount of undissolved substance and the difference between the saturation concentration and the current concentration. If $S$ is the saturation concentration, $A(t)$ is the amount dissolved at time $t$, $A_0$ is the initial amount of substance, and $k$ is the proportionality constant, formulate a differential equation for $A(t)$. (Note: Undissolved amount is $A_0 - A(t)$, current concentration depends on $A(t)$ and volume $V$, but let's assume concentration is $A(t)/V$). A simplified model might use amount directly for simplicity.

(A) $\frac{dA}{dt} = k A(t)$

(B) $\frac{dA}{dt} = k (A_0 - A(t))$

(C) $\frac{dA}{dt} = k (A_0 - A(t)) (S - A(t)/V)$

(D) $\frac{dA}{dt} = k (S - A(t))$

Question 3. A certain commodity's price $P(t)$ changes over time $t$. The rate of change of price is proportional to the difference between the demand $D$ and the supply $S$. If $D = a - bP$ and $S = c + dP$ (where $a,b,c,d$ are positive constants), which differential equation models the price change?

(A) $\frac{dP}{dt} = k (D + S)$

(B) $\frac{dP}{dt} = k (S - D)$

(C) $\frac{dP}{dt} = k (D - S)$

(D) $\frac{dP}{dt} = kP$

Question 4. In a simple model of a learning process, the rate at which a person learns a skill is proportional to the amount of the skill yet to be learned. If $S_{max}$ is the maximum amount of the skill and $S(t)$ is the amount learned at time $t$, formulate a differential equation for $S(t)$.

(A) $\frac{dS}{dt} = k S(t)$

(B) $\frac{dS}{dt} = k (S_{max} - S(t))$

(C) $\frac{dS}{dt} = k S_{max}$

(D) $\frac{dS}{dt} = k (S(t) - S_{max})$

Question 5. A tank initially contains 100 liters of pure water. A solution containing 1 kg of salt per liter enters the tank at a rate of 5 liters/minute. The well-mixed solution leaves the tank at the same rate. If $A(t)$ is the amount of salt in the tank at time $t$, which differential equation models the amount of salt in the tank?

(A) $\frac{dA}{dt} = 5 \cdot 1 - 5 \frac{A(t)}{100}$

(B) $\frac{dA}{dt} = 5 - \frac{A(t)}{20}$

(C) $\frac{dA}{dt} = 5 - 5A(t)$

(D) Both (A) and (B) are correct forms of the same equation.