Multiple Correct Answers MCQs for Sub-Topics of Topic 2: Algebra

Question 1. Which of the following represent variables?

(A) $p$

(B) $-7$

(C) $\theta$

(D) $y$

Question 2. In the expression $5x^2 - 2xy + 8$, which of the following are coefficients?

(A) $5$

(B) $-2$

(C) $8$

(D) $x$

Question 3. Identify the constant term(s) in the expression $a^3 - 4a + 12 - b/2 + 5$.

(A) $12$

(B) $-4a$

(C) $5$

(D) $12+5$

Question 4. Which of the following are algebraic expressions?

(A) $3x + 7$

(B) $15$

(C) $p^2 - q^2$

(D) $y/z$

Question 5. Which of the following phrases can be represented by an algebraic expression?

(A) The sum of a number and 5.

(B) Twice a number decreased by 3.

(C) 10 is a prime number.

(D) The product of two different numbers.

Question 6. In the term $7mn^2$, which of the following are factors?

(A) $7$

(B) $m$

(C) $n^2$

(D) $mn$

Question 7. Evaluate the expression $3a - 2b$ for the given values.

(A) $a=1, b=1 \implies 3(1) - 2(1) = 1$

(B) $a=2, b=3 \implies 3(2) - 2(3) = 0$

(C) $a=-1, b=-2 \implies 3(-1) - 2(-2) = -3 + 4 = 1$

(D) $a=0, b=0 \implies 3(0) - 2(0) = 0$

Question 8. In the expression $\frac{x^2}{5} + 3y$, which of the following are true?

(A) The coefficient of $x^2$ is $1/5$.

(B) The coefficient of $y$ is $3$.

(C) There are two terms.

(D) The constant term is $0$.

Question 9. Algebraic expressions are useful for:

(A) Representing unknown quantities.

(B) Formulating general rules or formulas.

(C) Solving equations.

(D) Describing relationships between quantities.

Question 10. Which of the following are like terms?

(A) $2x$ and $-5x$

(B) $3a^2b$ and $7a^2b$

(C) $4xy$ and $4yx$

(D) $6p^2q$ and $6pq^2$

Question 1. Find the sum of the expressions $2x^2 - 3x + 1$ and $x^2 + 5x - 4$.

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

(B) $3x^2 - 2x - 3$

(C) Adding corresponding like terms.

(D) Result has degree 2.

Question 2. Subtract $a - 2b$ from $3a + b$. Which of the following are equivalent ways to represent the result?

(A) $(3a+b) - (a-2b)$

(B) $3a+b - a + 2b$

(C) $2a + 3b$

(D) Subtracting $a$ from $3a$ and $2b$ from $b$.

Question 3. Find the product of $5xy^2$ and $-2x^3y$.

(A) $(5 \times -2) (x \times x^3) (y^2 \times y)$

(B) $-10x^4y^3$

(C) Result is a monomial.

(D) Coefficient is $-10$.

Question 4. Which of the following expressions are equivalent to $3a(a^2 - 4b)$?

(A) $3a \times a^2 - 3a \times 4b$

(B) $3a^3 - 12ab$

(C) Result is a binomial.

(D) $3a^3 - 12b$

Question 5. Expand $(x-5)(x+4)$.

(A) $x^2 + 4x - 5x - 20$

(B) $x^2 - x - 20$

(C) Result is a trinomial.

(D) Constant term is $-20$.

Question 6. Divide $15m^5n^4 - 20m^3n^2$ by $5m^2n$.

(A) $\frac{15m^5n^4}{5m^2n} - \frac{20m^3n^2}{5m^2n}$

(B) $3m^3n^3 - 4mn$

(C) Result is a binomial.

(D) Division of each term by the monomial divisor.

Question 7. Which of the following operations result in a simpler expression by combining like terms?

(A) Adding $3x + 2y$ and $x - y$.

(B) Subtracting $5a^2 - a$ from $7a^2 + 3a$.

(C) Multiplying $(p+1)(p-1)$.

(D) Dividing $6x^2y$ by $2xy$.

Question 8. The product of two binomials can sometimes result in a trinomial. Which of the following multiplications will result in a trinomial?

(A) $(x+2)(x+3)$

(B) $(a-b)(a+b)$

(C) $(y-1)(y-1)$

(D) $(2p+1)(p-4)$

Question 9. Which of the following divisions result in a polynomial?

(A) $(x^3 - 8) \div (x-2)$

(B) $(x^2 + 1) \div x$

(C) $(a^2b^3 + ab^2) \div ab$

(D) $x \div (x+1)$

Question 10. Which of the following statements about algebraic operations are correct?

(A) Addition of expressions is commutative ($A+B = B+A$).

(B) Subtraction of expressions is commutative ($A-B = B-A$).

(C) Multiplication of expressions is associative ($A(BC) = (AB)C$).

(D) Division of expressions is associative ($A \div (B \div C) = (A \div B) \div C$).

Question 1. Which of the following are polynomials?

(A) $x^5 - 2x^3 + \sqrt{2}x + 1$

(B) $y^{1/2} + 3y - 5$

(C) $\frac{3}{z} + 4z^2 - 1$

(D) $10$

Question 2. Consider the polynomial $P(x) = 7x^4 - \frac{1}{2}x^2 + 9$. Which statements are true about this polynomial?

(A) The degree is 4.

(B) It is a trinomial.

(C) The leading coefficient is 7.

(D) The constant term is 9.

Question 3. Classify the following algebraic expressions by type and degree. Which classifications are correct?

(A) $5x^3$ is a monomial of degree 3.

(B) $x^2 - 4$ is a binomial of degree 2.

(C) $2y - 1$ is a binomial of degree 1.

(D) $z^4 + 3z^2 - 7$ is a trinomial of degree 2.

Question 4. For the polynomial $P(x) = x^2 - 4x + 3$, evaluate $P(x)$ at the given values of $x$. Which results are correct?

(A) $P(0) = 3$

(B) $P(1) = 0$

(C) $P(2) = -1$

(D) $P(3) = 0$

Question 5. Which of the following values are zeroes of the polynomial $P(x) = x^2 - x - 6$?

(A) $x=0$

(B) $x=-2$

(C) $x=3$

(D) $x=-3$

Question 6. The graph of a polynomial $y = P(x)$ intersects the x-axis at certain points. The x-coordinates of these intersection points represent:

(A) The value of the polynomial at $x=0$.

(B) The roots of the equation $P(x) = 0$.

(C) The zeroes of the polynomial.

(D) The vertex of the graph (for a quadratic).

Question 7. For a quadratic polynomial $ax^2 + bx + c$, if $\alpha$ and $\beta$ are the zeroes, which statements are true?

(A) $\alpha + \beta = -b/a$

(B) $\alpha \beta = c/a$

(C) $ax^2 + bx + c = a(x-\alpha)(x-\beta)$

(D) The graph is a parabola.

Question 8. A quadratic polynomial has sum of zeroes $-5$ and product of zeroes $6$. Which of the following could be the polynomial?

(A) $x^2 + 5x + 6$

(B) $2x^2 + 10x + 12$

(C) $k(x^2 + 5x + 6)$ for any non-zero constant $k$.

(D) $x^2 - 5x + 6$

Question 9. The graph of a polynomial $y=P(x)$ is shown below. Which statements are likely true based on the graph?

(A) The polynomial is quadratic.

(B) The leading coefficient is positive.

(C) The polynomial has exactly two real zeroes.

(D) The constant term is positive.

Question 10. Which of the following algebraic expressions are NOT polynomials?

(A) $x + 1/x$

(B) $\sqrt{x}$

(C) $x^2 + 3x - 7$

(D) $|x|$

Question 1. According to the Division Algorithm for Polynomials, if $P(x)$ is divided by $D(x)$ (where $D(x) \neq 0$), we can write $P(x) = Q(x) \cdot D(x) + R(x)$. Which statements are true about the remainder $R(x)$?

(A) The degree of $R(x)$ is always less than the degree of $D(x)$.

(B) $R(x)$ can be the zero polynomial.

(C) If the degree of $D(x)$ is 1, the degree of $R(x)$ is 0 or $R(x)=0$.

(D) $R(x)$ is unique.

Question 2. Using the Remainder Theorem, find the remainder when $P(x) = x^3 - 2x^2 + x + 1$ is divided by $(x-1)$. Which statements are correct?

(A) The divisor is of the form $(x-a)$, with $a=1$.

(B) The remainder is $P(1)$.

(C) The remainder is $1^3 - 2(1)^2 + 1 + 1 = 1 - 2 + 1 + 1 = 1$.

(D) The remainder is $P(-1)$.

Question 3. According to the Factor Theorem, if $P(x)$ is a polynomial, which conditions imply that $(x-a)$ is a factor of $P(x)$?

(A) $P(a) = 0$

(B) The remainder when $P(x)$ is divided by $(x-a)$ is 0.

(C) $a$ is a zero of the polynomial $P(x)$.

(D) $P(-a) = 0$

Question 4. Which of the following linear expressions are factors of $P(x) = x^2 - 5x + 6$?

(A) $(x-2)$

(B) $(x-3)$

(C) $(x+2)$

(D) $(x+3)$

Question 5. If $(x+1)$ is a factor of $P(x) = x^3 + kx^2 - x - 2$, which statements are true?

(A) $P(-1) = 0$ by the Factor Theorem.

(B) $(-1)^3 + k(-1)^2 - (-1) - 2 = 0$

(C) $-1 + k + 1 - 2 = 0 \implies k = 2$.

(D) $x=1$ is a zero of $P(x)$.

Question 6. If $P(x)$ is divisible by $(x-a)$, it means:

(A) The remainder is 0 when $P(x)$ is divided by $(x-a)$.

(B) $(x-a)$ is a factor of $P(x)$.

(C) $x=a$ is a zero of $P(x)$.

(D) $a$ is a root of the equation $P(x)=0$.

Question 7. When a polynomial $P(x)$ is divided by a linear polynomial $D(x) = ax+b$, the remainder $R(x)$ will have a degree:

(A) Equal to the degree of the divisor.

(B) Less than the degree of the divisor.

(C) Strictly 0, if the remainder is non-zero.

(D) At most 0.

Question 8. Which theorems are directly used in factorizing polynomials?

(A) Remainder Theorem (identifying potential zeroes/factors).

(B) Factor Theorem (confirming factors/zeroes).

(C) Polynomial Division Algorithm (finding quotient and remainder).

(D) Fundamental Theorem of Algebra (existence of roots).

Question 9. If $P(x)$ is divided by $(2x-1)$, the remainder is $P(1/2)$. This is an application of:

(A) Factor Theorem.

(B) Remainder Theorem.

(C) Division Algorithm, setting the divisor to zero.

(D) Evaluating the polynomial at the root of the divisor.

Question 10. Which statements are true about the zero of a linear polynomial $P(x) = ax+b$ ($a \neq 0$)?

(A) The zero is $-b/a$.

(B) It is the root of the equation $ax+b=0$.

(C) By Factor Theorem, $(x - (-b/a)) = (x+b/a)$ is a factor of $ax+b$.

(D) By Remainder Theorem, the remainder when $P(x)$ is divided by $(x - (-b/a))$ is $P(-b/a)=0$.

Question 1. Which of the following are correct expansions of $(a+b)^2$?

(A) $a^2 + 2ab + b^2$

(B) $(a+b)(a+b)$

(C) $a^2 + b^2$

(D) $\text{C}(2, 0)a^2b^0 + \text{C}(2, 1)a^1b^1 + \text{C}(2, 2)a^0b^2$

Question 2. Which of the following represent the factorization of $x^2 - 49$?

(A) $(x-7)(x+7)$

(B) Using the identity $a^2 - b^2 = (a-b)(a+b)$

(C) $(x-7)^2$

(D) $(x+7)^2$

Question 3. Which identities can be used to simplify numerical calculations?

(A) $(a+b)^2 = a^2 + 2ab + b^2$ (e.g., $102^2 = (100+2)^2$)

(B) $(a-b)^2 = a^2 - 2ab + b^2$ (e.g., $99^2 = (100-1)^2$)

(C) $a^2 - b^2 = (a-b)(a+b)$ (e.g., $58^2 - 42^2 = (58-42)(58+42)$)

(D) $(x+a)(x+b) = x^2 + (a+b)x + ab$ (e.g., $105 \times 106 = (100+5)(100+6)$)

Question 4. Expand $(x+y+z)^2$. Which terms are present in the expansion?

(A) $x^2, y^2, z^2$

(B) $2xy, 2yz, 2zx$

(C) $xy, yz, zx$

(D) $x^2+y^2+z^2+2xy+2yz+2zx$

Question 5. Which of the following trinomials are perfect squares that can be factored using identities?

(A) $a^2 + 8a + 16$

(B) $4x^2 - 4x + 1$

(C) $y^2 + 5y + 25$

(D) $9m^2 + 12mn + 4n^2$

Question 6. Which of the following are correct expansions of $(a-b)^3$?

(A) $a^3 - 3a^2b + 3ab^2 - b^3$

(B) $(a-b)(a^2-2ab+b^2)$

(C) $a^3 - b^3$

(D) $(a-b)(a^2+ab+b^2)$

Question 7. Which of the following are correct factorizations of $x^3 + 27$?

(A) $(x+3)(x^2 - 3x + 9)$

(B) Using the identity $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$

(C) $(x+3)^3$

(D) $(x+3)(x^2 + 3x + 9)$

Question 8. Which identity is useful for expanding $(2p-3)(2p+5)$?

(A) $(x+a)(x+b) = x^2 + (a+b)x + ab$

(B) $a^2 - b^2 = (a-b)(a+b)$

(C) $(a+b)^2 = a^2 + 2ab + b^2$

(D) Let $x=2p$, $a=-3$, $b=5$. The identity is $(x+a)(x+b)$.

Question 9. If $a+b+c = 0$, which statements are true?

(A) $a^3+b^3+c^3 = 3abc$

(B) $a^2+b^2+c^2 = -2(ab+bc+ca)$

(C) $(a+b)^2 = c^2$

(D) The identity $a^3+b^3+c^3 - 3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca)$ can be used.

Question 10. Which of the following expressions are equivalent to $a^2 - b^2$?

(A) $(a-b)(a+b)$

(B) $(b-a)(-a-b)$

(C) $(b-a)(b+a)$

(D) $-(b^2 - a^2)$

Question 1. Factorise $12a^2b - 18ab^2$. Which of the following are factors?

(A) $6ab$

(B) $2a - 3b$

(C) $a$

(D) $6a^2b^2$

Question 2. Factorise $x^2 + xy + 8x + 8y$ by grouping.

(A) $x(x+y) + 8(x+y)$

(B) $(x+y)(x+8)$

(C) $(x+8)(x+y)$

(D) $x(x+8) + y(x+8)$

Question 3. Factorise $4m^2 - 25n^2$ using an identity.

(A) $(2m-5n)(2m+5n)$

(B) Using $a^2 - b^2$ identity.

(C) $(2m-5n)^2$

(D) $(4m-25n)(m+n)$

Question 4. Factorise $y^2 + 10y + 25$ using an identity.

(A) $(y+5)^2$

(B) $(y+5)(y+5)$

(C) Using $(a+b)^2$ identity.

(D) $(y-5)^2$

Question 5. Factorise the quadratic trinomial $x^2 - 7x + 10$ by splitting the middle term.

(A) Find two numbers whose sum is $-7$ and product is $10$.

(B) The numbers are $-2$ and $-5$.

(C) $x^2 - 2x - 5x + 10$

(D) $(x-2)(x-5)$

Question 6. Factorise $a^3 - 64$ using an identity.

(A) $(a-4)(a^2 + 4a + 16)$

(B) Using $x^3 - y^3$ identity.

(C) $(a-4)^3$

(D) $(a-4)(a^2 - 4a + 16)$

Question 7. Which of the following are factors of the polynomial $x^3 - 7x + 6$, given that $x=1$ is a zero?

(A) $(x-1)$

(B) Since $x=1$ is a zero, $P(1)=0$.

(C) $(x-2)$ (Since $P(2) = 8-14+6 = 0$)

(D) $(x+3)$ (Since $P(-3) = -27 + 21 + 6 = 0$)

Question 8. Factorise $x^4 - y^4$. Which statements are true?

(A) It is a difference of squares: $(x^2)^2 - (y^2)^2$.

(B) It factors as $(x^2-y^2)(x^2+y^2)$.

(C) It factors completely as $(x-y)(x+y)(x^2+y^2)$.

(D) It factors as $(x-y)^4$

Question 9. Which of the following are correct factorizations of $3x^2 - 10x + 8$?

(A) Splitting the middle term into $-6x$ and $-4x$.

(B) $3x^2 - 6x - 4x + 8 = 3x(x-2) - 4(x-2) = (3x-4)(x-2)$.

(C) $(3x-4)(x-2)$

(D) $(3x-2)(x-4)$

Question 10. Which of the following expressions can be factorised by taking out a common monomial factor?

(A) $5x^2 - 10x$

(B) $a^3b^2 + a^2b^3 - a^4b$

(C) $x^2 + x + 1$

(D) $4p - 8q + 12r$

Question 1. Which of the following are linear equations in one variable?

(A) $2x + 5 = 0$

(B) $3y = 7$

(C) $z^2 - 4 = 0$

(D) $p/2 - 1 = 5$

Question 2. The solution(s) or root(s) of the equation $4x - 8 = 0$ is/are:

(A) A single value of $x$.

(B) $x=2$.

(C) The value of $x$ that makes the equation true.

(D) $x=0$.

Question 3. To solve the equation $5(x-2) = 15$, which steps can be part of the solution process?

(A) Divide both sides by 5: $x-2 = 3$.

(B) Add 2 to both sides: $x = 5$.

(C) Distribute 5: $5x - 10 = 15$. Add 10 to both sides: $5x = 25$. Divide by 5: $x=5$.

(D) Transpose -10 to the right side: $5x = 15 - 10$.

Question 4. A number is increased by 10 and the result is 3 times the original number. Which equation(s) can represent this problem if the number is $n$?

(A) $n + 10 = 3n$

(B) $10 + n = 3 \times n$

(C) $10 = 3n - n$

(D) $10 = 2n$

Question 5. Solve the equation $\frac{x}{2} - \frac{x}{3} = 5$.

(A) Multiply by the LCM of denominators (6): $3x - 2x = 30$.

(B) $x = 30$.

(C) $6(\frac{x}{2} - \frac{x}{3}) = 6(5)$

(D) $x/6 = 5$

Question 6. Which values of $x$ satisfy the equation $2(x-1) + 3 = x + 4$?

(A) $2x - 2 + 3 = x + 4$

(B) $2x + 1 = x + 4$

(C) $x = 3$

(D) $x = 5$

Question 7. A rectangular plot has length $L$ and width $W$. Its perimeter is $2(L+W)$. If the perimeter is 40 m and $L = W+6$, which equations are true?

(A) $2(W+6 + W) = 40$

(B) $2(2W + 6) = 40$

(C) $4W + 12 = 40$

(D) $4W = 28 \implies W = 7$. $L = 13$.

Question 8. The sum of three consecutive even numbers is 66. If the smallest number is $x$, which expressions can represent the sum?

(A) $x + (x+2) + (x+4) = 66$

(B) $3x + 6 = 66$

(C) If $x$ is the middle number, $(x-2) + x + (x+2) = 66 \implies 3x = 66$.

(D) The numbers are 20, 22, 24.

Question 9. The cost of a chair is $\textsf{₹}100$ more than the cost of a table. If the total cost of 2 chairs and 3 tables is $\textsf{₹}1200$, which equations can be used to find the costs? Let cost of a table be $T$ and cost of a chair be $C$.

(A) $C = T + 100$

(B) $2C + 3T = 1200$

(C) $2(T+100) + 3T = 1200$

(D) $5T + 200 = 1200 \implies 5T = 1000 \implies T = 200$. $C = 300$.

Question 10. Solve for $p$: $\frac{p+2}{3} + \frac{p-1}{4} = 1$. Which steps are correct?

(A) Multiply by 12: $4(p+2) + 3(p-1) = 12$.

(B) $4p + 8 + 3p - 3 = 12$.

(C) $7p + 5 = 12 \implies 7p = 7 \implies p = 1$.

(D) Multiply by $3 \times 4 = 12$ to clear denominators.

Question 1. Which of the following are linear equations in two variables?

(A) $3x + 5y - 1 = 0$

(B) $ax + by + c = 0$ where $a, b$ are not both zero.

(C) $x^2 + y^2 = 9$

(D) $y = 2x + 4$

Question 2. For the equation $x + 2y = 6$, which of the following are solutions (ordered pairs $(x, y)$)?

(A) $(6, 0)$

(B) $(0, 3)$

(C) $(2, 2)$

(D) $(4, 1)$

Question 3. The cost of a notebook is twice the cost of a pen. If the total cost of one notebook and one pen is $\textsf{₹}45$, which equations can represent this? Let the cost of a notebook be $N$ and the cost of a pen be $P$.

(A) $N = 2P$

(B) $N + P = 45$

(C) If pen cost is $x$, notebook cost is $2x$, then $x + 2x = 45$.

(D) If notebook cost is $y$, pen cost is $y/2$, then $y + y/2 = 45$.

Question 4. Which statements are true about the graph of a linear equation in two variables?

(A) It is always a straight line.

(B) Every point on the line is a solution to the equation.

(C) It can be uniquely determined by plotting any two distinct solutions.

(D) It always passes through the origin.

Question 5. Which of the following points lie on the graph of the equation $x - y = 1$?

(A) $(2, 1)$

(B) $(0, -1)$

(C) $(1, 0)$

(D) $(3, 2)$

Question 6. Which equations represent lines parallel to the coordinate axes?

(A) $x = 5$

(B) $y = -2$

(C) $x + y = 0$

(D) $2x - 3 = 0$

Question 7. Which equations represent the coordinate axes?

(A) $x = 0$ (y-axis)

(B) $y = 0$ (x-axis)

(C) $x = y$

(D) $x^2 + y^2 = 0$ (only the origin)

Question 8. A line is parallel to the x-axis and passes through the point $(3, -4)$. Which statements about this line are true?

(A) Its equation is $y = -4$.

(B) Its y-coordinate is always $-4$.

(C) It is a horizontal line.

(D) It passes through $(0, -4)$.

Question 9. The total number of students in a class is 50. If the number of boys is $b$ and the number of girls is $g$, which equation(s) correctly represent this information?

(A) $b+g=50$

(B) $50 - b = g$

(C) $b/g$ ratio cannot be determined from this.

(D) If there are 10 more boys than girls, then $b=g+10$.

Question 10. The equation $2x - 5 = 0$ can be written as a linear equation in two variables as $2x + 0y - 5 = 0$. Which statements are true about the graph of this equation in the coordinate plane?

(A) It is a line parallel to the y-axis.

(B) It is a line parallel to the x-axis.

(C) It is the line $x = 2.5$.

(D) It passes through the point $(2.5, 5)$.

Question 1. A system of two linear equations in two variables $\begin{cases} a_1x + b_1y + c_1 = 0 \\ a_2x + b_2y + c_2 = 0 \end{cases}$ represents a pair of lines in the Cartesian plane. Which statements are true about the geometrical representation and solutions?

(A) If the lines intersect at a single point, there is a unique solution.

(B) If the lines are parallel and distinct, there is no solution.

(C) If the lines are coincident, there are infinitely many solutions.

(D) The solution(s) correspond to the point(s) where the lines meet.

Question 2. A system of linear equations is called 'consistent' if it has at least one solution. Which types of pairs of lines correspond to consistent systems?

(A) Intersecting lines.

(B) Parallel lines.

(C) Coincident lines.

(D) Perpendicular lines.

Question 3. For the system $\begin{cases} a_1x + b_1y + c_1 = 0 \\ a_2x + b_2y + c_2 = 0 \end{cases}$, which ratio conditions correspond to the lines being parallel and distinct (no solution)?

(A) $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$

(B) The system is inconsistent.

(C) $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$

(D) $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$

Question 4. For which systems are the lines coincident (infinitely many solutions)?

(A) $\begin{cases} x + y = 2 \\ 2x + 2y = 4 \end{cases}$

(B) $\begin{cases} 3x - 2y = 5 \\ 6x - 4y = 10 \end{cases}$

(C) $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ holds.

(D) The system is consistent.

Question 5. Which methods can be used to solve a pair of linear equations in two variables?

(A) Graphical method

(B) Substitution method

(C) Elimination method

(D) Factorisation method

Question 6. Consider the system $\begin{cases} x + y = 3 \\ 2x - y = 0 \end{cases}$. Which statements are true about the solution?

(A) Using substitution: $y = 2x$. Substitute into the first equation: $x + 2x = 3 \implies 3x = 3 \implies x = 1$.

(B) Using elimination: Add the two equations: $(x+y) + (2x-y) = 3+0 \implies 3x = 3 \implies x = 1$.

(C) If $x=1$, then $1+y=3 \implies y=2$. The solution is $(1, 2)$.

(D) The lines are intersecting.

Question 7. Equations like $\frac{5}{x+y} + \frac{2}{x-y} = 4$ and $\frac{15}{x+y} - \frac{5}{x-y} = -1$ can be reduced to a pair of linear equations by making suitable substitutions. Which substitutions would work?

(A) Let $u = x+y$ and $v = x-y$.

(B) Let $u = \frac{1}{x+y}$ and $v = \frac{1}{x-y}$.

(C) The resulting equations would be $5u+2v=4$ and $15u-5v=-1$.

(D) The resulting equations would be $5(x+y) + 2(x-y) = 4$ and $15(x+y) - 5(x-y) = -1$.

Question 8. For which value(s) of $k$ does the system $\begin{cases} kx + 2y = 5 \\ 3x + y = 1 \end{cases}$ have a unique solution?

(A) The condition for a unique solution is $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$.

(B) $\frac{k}{3} \neq \frac{2}{1}$

(C) $k \neq 6$.

(D) $k=6$.

Question 9. For which value(s) of $m$ does the system $\begin{cases} 2x + 3y = 7 \\ 6x + my = 21 \end{cases}$ have infinitely many solutions?

(A) The condition for infinitely many solutions is $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$.

(B) $\frac{2}{6} = \frac{3}{m} = \frac{7}{21}$

(C) $1/3 = 3/m \implies m = 9$.

(D) $1/3 = 7/21$, which is true.

Question 10. Which systems are inconsistent (have no solution)?

(A) $\begin{cases} x + y = 1 \\ x + y = 2 \end{cases}$

(B) $\begin{cases} 2x + 3y = 5 \\ 4x + 6y = 12 \end{cases}$

(C) Systems where the lines are parallel and distinct.

(D) $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$ holds.

Question 1. Which of the following are quadratic equations?

(A) $(x+1)^2 = x^2 - 5$

(B) $3x^2 - 7x + 2 = 0$

(C) $y(y-3) = y^2 + 5$

(D) $z^3 - 4z + 1 = 0$

Question 2. For the quadratic equation $5x^2 - 2x - 3 = 0$, which statements are true about its coefficients in the standard form $ax^2+bx+c=0$?

(A) $a=5$

(B) $b=2$

(C) $c=-3$

(D) $b=-2$

Question 3. If $\alpha$ and $\beta$ are the roots of the quadratic equation $x^2 + bx + c = 0$, which formulas relate the roots and coefficients?

(A) $\alpha + \beta = -b$

(B) $\alpha \beta = c$

(C) $\alpha + \beta = b$

(D) $\alpha \beta = -c$

Question 4. Find the sum and product of the roots of the equation $3x^2 - 9x + 6 = 0$.

(A) Sum of roots $= -(-9)/3 = 9/3 = 3$.

(B) Product of roots $= 6/3 = 2$.

(C) Sum of roots is 9.

(D) Product of roots is 6.

Question 5. Solve the equation $x^2 - 7x + 12 = 0$ by factorisation. Which of the following are correct steps or results?

(A) Find two numbers that multiply to 12 and add to -7.

(B) The numbers are -3 and -4.

(C) $(x-3)(x-4) = 0$.

(D) The roots are $x=3$ and $x=4$.

Question 6. To solve $x^2 - 8x + 15 = 0$ by completing the square, rearrange it to $x^2 - 8x = -15$. What constant term should be added to both sides to complete the square on the LHS?

(A) Square half of the coefficient of $x$.

(B) $(\frac{-8}{2})^2 = (-4)^2 = 16$.

(C) Add 16 to both sides: $x^2 - 8x + 16 = -15 + 16$.

(D) The LHS becomes $(x-4)^2$.

Question 7. For the quadratic equation $ax^2 + bx + c = 0$, the discriminant $D = b^2 - 4ac$. Which statements about the discriminant and the nature of roots are correct?

(A) If $D > 0$, the roots are real and distinct.

(B) If $D = 0$, the roots are real and equal.

(C) If $D < 0$, the roots are not real (complex conjugates, if coefficients are real).

(D) The discriminant determines whether the quadratic expression can be factored into linear factors with real coefficients.

Question 8. Determine the nature of the roots of the equation $4x^2 - 12x + 9 = 0$.

(A) Calculate the discriminant: $D = (-12)^2 - 4(4)(9) = 144 - 144 = 0$.

(B) The roots are real and equal.

(C) The quadratic expression is a perfect square: $(2x-3)^2 = 0$.

(D) The roots are $x = 3/2$ (repeated).

Question 9. For what value(s) of $k$ does the equation $x^2 - kx + 9 = 0$ have real roots?

(A) Real roots means $D \ge 0$.

(B) $D = (-k)^2 - 4(1)(9) = k^2 - 36$.

(C) $k^2 - 36 \ge 0 \implies k^2 \ge 36 \implies k \ge 6$ or $k \le -6$.

(D) $|k| \ge 6$.

Question 10. If the roots of a quadratic equation are $1$ and $-5$, which statements are true?

(A) The sum of the roots is $1 + (-5) = -4$.

(B) The product of the roots is $1 \times (-5) = -5$.

(C) The equation is of the form $x^2 - (\text{Sum})x + (\text{Product}) = 0$, so $x^2 - (-4)x + (-5) = 0$, which is $x^2 + 4x - 5 = 0$.

(D) The factors are $(x-1)$ and $(x+5)$.

Question 1. Which statements are true about the imaginary unit $i$?

(A) $i = \sqrt{-1}$

(B) $i^2 = -1$

(C) $i^4 = 1$

(D) $i^3 = 1$

Question 2. Perform the addition and subtraction of the complex numbers $z_1 = 3 + 2i$ and $z_2 = 1 - 4i$. Which results are correct?

(A) $z_1 + z_2 = (3+1) + (2-4)i = 4 - 2i$.

(B) $z_1 - z_2 = (3-1) + (2-(-4))i = 2 + 6i$.

(C) $z_2 - z_1 = (1-3) + (-4-2)i = -2 - 6i$.

(D) $z_1 + z_2 = 4 + 6i$.

Question 3. Multiply the complex numbers $(1+i)$ and $(2-i)$. Which results are correct?

(A) $(1+i)(2-i) = 1(2) + 1(-i) + i(2) + i(-i) = 2 - i + 2i - i^2 = 2 + i - (-1) = 2 + i + 1 = 3 + i$.

(B) The product is $3+i$.

(C) Real part of the product is 3.

(D) Imaginary part of the product is $i$.

Question 4. Which of the following are equal to $i^{-1}$?

(A) $\frac{1}{i}$

(B) $\frac{i}{i^2} = \frac{i}{-1} = -i$

(C) $-i$

(D) $i^{3}$

Question 5. Simplify the expression $i^{25}$. Which methods or results are correct?

(A) $i^{25} = i^{4 \times 6 + 1} = (i^4)^6 \times i^1 = 1^6 \times i = i$.

(B) Divide the power by 4 and use the remainder as the new power of $i$.

(C) $25 \div 4$ gives remainder 1, so $i^{25} = i^1 = i$.

(D) $i^{25} = i^{-3}$

Question 6. Simplify $\frac{2}{1+i}$. Which results are correct?

(A) Multiply numerator and denominator by the conjugate of the denominator, which is $1-i$.

(B) $\frac{2}{1+i} \times \frac{1-i}{1-i} = \frac{2(1-i)}{1^2 - i^2} = \frac{2(1-i)}{1 - (-1)} = \frac{2(1-i)}{2} = 1-i$.

(C) The result is $1-i$.

(D) The real part is 1, imaginary part is 1.

Question 7. Which of the following complex numbers are purely real?

(A) $5$

(B) $3i - 3i$

(C) $0$

(D) $2 + 0i$

Question 8. Which of the following complex numbers are purely imaginary?

(A) $2i$

(B) $-5i$

(C) $0$

(D) $0 + 3i$

Question 9. If $(a+b) + (a-b)i = 10 + 4i$, find the values of $a$ and $b$. Which steps or results are correct?

(A) Equate real parts: $a+b = 10$.

(B) Equate imaginary parts: $a-b = 4$.

(C) Solve the system: $(a+b) + (a-b) = 10+4 \implies 2a = 14 \implies a = 7$.

(D) If $a=7$, $7+b=10 \implies b=3$. So $a=7, b=3$.

Question 10. Which statements are true about the algebra of complex numbers?

(A) Addition is commutative: $z_1 + z_2 = z_2 + z_1$.

(B) Multiplication is associative: $z_1(z_2 z_3) = (z_1 z_2)z_3$.

(C) There is an additive identity (0 or $0+0i$).

(D) There is a multiplicative identity (1 or $1+0i$).

Question 1. In the Argand plane, the complex number $a+bi$ is represented by the point $(a, b)$. Which statements are true?

(A) The complex number $5$ is represented by the point $(5, 0)$.

(B) The complex number $3i$ is represented by the point $(0, 3)$.

(C) The complex number $-2-i$ is represented by the point $(-2, -1)$.

(D) The real axis represents purely real numbers.

Question 2. The modulus of a complex number $z=a+bi$ is $|z| = \sqrt{a^2+b^2}$. Which statements are true about the modulus?

(A) $|z|$ is the distance of the point $(a, b)$ from the origin in the Argand plane.

(B) The modulus of $3-4i$ is $\sqrt{3^2+(-4)^2} = \sqrt{9+16} = \sqrt{25} = 5$.

(C) The modulus is always a non-negative real number.

(D) $|z_1 z_2| = |z_1| |z_2|$.

Question 3. The conjugate of a complex number $z=a+bi$ is $\bar{z}=a-bi$. Which statements are true about the conjugate?

(A) The conjugate of $2+5i$ is $2-5i$.

(B) The conjugate of $-3i$ is $3i$.

(C) $z+\bar{z}$ is always purely real.

(D) $z-\bar{z}$ is always purely imaginary.

Question 4. Which statements are true about the complex number $z$ if $z=\bar{z}$?

(A) $z$ must be a real number.

(B) The imaginary part of $z$ is zero.

(C) $z$ lies on the real axis in the Argand plane.

(D) $z$ must be $0$.

Question 5. The polar form of a complex number $z = a+bi$ is $r(\cos \theta + i \sin \theta)$. Which statements are true about the components $r$ and $\theta$?

(A) $r = |z|$.

(B) $\theta$ is the argument of $z$.

(C) $a = r \cos \theta$ and $b = r \sin \theta$.

(D) The principal argument is usually in the interval $[0, 2\pi)$.

Question 6. Find the modulus and principal argument of the complex number $z = 1+i$. Which statements are correct?

(A) $|z| = \sqrt{1^2+1^2} = \sqrt{2}$.

(B) The point $(1, 1)$ is in the first quadrant.

(C) The argument $\theta$ satisfies $\tan \theta = 1/1 = 1$. The principal argument is $\pi/4$.

(D) The polar form is $\sqrt{2}(\cos(\pi/4) + i \sin(\pi/4))$.

Question 7. Find the square roots of $-4$. Which statements are correct?

(A) The square roots are $2i$ and $-2i$.

(B) $\sqrt{-4} = \sqrt{4 \times -1} = \sqrt{4} \times \sqrt{-1} = 2i$. The other root is $-2i$.

(C) The equation $x^2 = -4$ has solutions $x = \pm 2i$.

(D) The principal square root is $2i$.

Question 8. Which of the following represent properties of complex numbers?

(A) $|z| = |\bar{z}|$

(B) $\overline{z_1 + z_2} = \bar{z_1} + \bar{z_2}$

(C) $\overline{z_1 z_2} = \bar{z_1} \bar{z_2}$

(D) $|z_1 + z_2| = |z_1| + |z_2|$ (Triangle inequality)

Question 9. The equation $|z| = k$, where $k$ is a positive real number, represents the locus of a point in the Argand plane. What does this locus represent?

(A) All points at a distance $k$ from the origin.

(B) A circle centered at the origin with radius $k$.

(C) The set of all complex numbers $z$ such that $|z|^2 = k^2$.

(D) A straight line.

Question 10. The argument of a complex number $z$ is the angle $\theta$ it makes with the positive real axis. Which statements are true about the argument?

(A) The argument is unique.

(B) The principal argument of $z=1$ is $0$.

(C) The principal argument of $z=i$ is $\pi/2$.

(D) The principal argument of $z=-1$ is $\pi$.

Question 1. A quadratic equation $ax^2 + bx + c = 0$ with real coefficients has complex roots under which condition(s)?

(A) The discriminant $D = b^2 - 4ac$ is negative.

(B) The equation has no real roots.

(C) The graph of $y=ax^2+bx+c$ does not intersect the x-axis.

(D) $b^2 < 4ac$.

Question 2. If a quadratic equation with real coefficients has one root $p+qi$ (where $q \neq 0$), which statements are true?

(A) The other root must be $p-qi$.

(B) The complex roots occur in conjugate pairs.

(C) The discriminant $D < 0$.

(D) $p+qi$ is a zero of the corresponding quadratic polynomial.

Question 3. Solve the equation $x^2 + 4x + 5 = 0$. Which steps or results are correct?

(A) Discriminant $D = 4^2 - 4(1)(5) = 16 - 20 = -4$.

(B) Roots are $x = \frac{-4 \pm \sqrt{-4}}{2(1)} = \frac{-4 \pm 2i}{2} = -2 \pm i$.

(C) The roots are $-2+i$ and $-2-i$.

(D) The roots are real and distinct.

Question 4. Form a quadratic equation with real coefficients whose roots are $1+2i$ and $1-2i$. Which statements are correct?

(A) Sum of roots $= (1+2i) + (1-2i) = 2$.

(B) Product of roots $= (1+2i)(1-2i) = 1^2 - (2i)^2 = 1 - 4i^2 = 1 - 4(-1) = 5$.

(C) The equation is $x^2 - (\text{Sum})x + (\text{Product}) = 0$, so $x^2 - 2x + 5 = 0$.

(D) The coefficients are real.

Question 5. If one root of the quadratic equation $2x^2 - 6x + k = 0$ is $3+i$, and the coefficients are real, which statements are true about the equation?

(A) The other root is $3-i$.

(B) The sum of the roots is 6.

(C) The product of the roots is $(3+i)(3-i) = 10$.

(D) The equation is of the form $x^2 - 6x + 10 = 0$ (or a multiple thereof).

Question 6. Solve the equation $x^2 + 9 = 0$. Which statements are correct?

(A) $x^2 = -9$.

(B) $x = \pm \sqrt{-9} = \pm 3i$.

(C) The roots are $3i$ and $-3i$.

(D) The discriminant is $0^2 - 4(1)(9) = -36 < 0$.

Question 7. Form a quadratic equation with real coefficients whose roots are $\pm 5i$. Which statements are correct?

(A) The roots are $5i$ and $-5i$.

(B) Sum of roots $= 5i + (-5i) = 0$.

(C) Product of roots $= (5i)(-5i) = -25i^2 = -25(-1) = 25$.

(D) The equation is $x^2 - 0x + 25 = 0$, i.e., $x^2 + 25 = 0$.

Question 8. Which statements are true about the roots of the equation $x^2 - 2x + 3 = 0$?

(A) Discriminant $D = (-2)^2 - 4(1)(3) = 4 - 12 = -8 < 0$.

(B) The roots are complex and conjugate.

(C) Roots are $x = \frac{-(-2) \pm \sqrt{-8}}{2(1)} = \frac{2 \pm \sqrt{8}i}{2} = \frac{2 \pm 2\sqrt{2}i}{2} = 1 \pm \sqrt{2}i$.

(D) The roots are real and distinct.

Question 9. If the roots of a quadratic equation with real coefficients are $p+qi$ and $r+si$, where $q \neq 0$, which statements must be true?

(A) $p=r$

(B) $q=-s$

(C) The roots are $p+qi$ and $p-qi$.

(D) The discriminant is negative.

Question 10. Which of the following quadratic equations with real coefficients have complex roots?

(A) $x^2 - x + 1 = 0$ ($D = 1-4 = -3 < 0$)

(B) $2x^2 + 5x + 4 = 0$ ($D = 25 - 32 = -7 < 0$)

(C) $x^2 + 4 = 0$ ($D = 0 - 16 = -16 < 0$)

(D) $x^2 - 3x + 2 = 0$ ($D = 9 - 8 = 1 > 0$)

Question 1. Which of the following are linear inequalities?

(A) $3x - 5 \ge 0$

(B) $2x + y < 4$

(C) $x^2 + 2x - 3 > 0$

(D) $y > 5$

Question 2. Solve the inequality $-2x + 6 \le 10$. Which steps and results are correct?

(A) Subtract 6 from both sides: $-2x \le 4$.

(B) Divide both sides by -2 and reverse the inequality sign: $x \ge -2$.

(C) The solution set on a number line is a ray starting from -2 and going to the right, including -2.

(D) The solution set is $(- \infty, -2]$.

Question 3. Consider the inequality $3 < x \le 7$. Which representations of the solution set are correct?

(A) On a number line, a segment from 3 to 7 with an open circle at 3 and a closed circle at 7.

(B) Interval notation: $(3, 7]$.

(C) $\{x \in \mathbb{R} \mid 3 < x \le 7\}$

(D) All real numbers $x$ such that $x > 3$ and $x \le 7$.

Question 4. To graph the linear inequality $x + y > 4$ in the coordinate plane, which steps are correct?

(A) Graph the boundary line $x+y=4$.

(B) Draw the boundary line as a solid line.

(C) Choose a test point not on the line, e.g., $(0, 0)$. Substitute into the inequality: $0+0 > 4 \implies 0 > 4$ (False).

(D) Since $(0, 0)$ results in a false statement, shade the region that does NOT contain $(0, 0)$.

Question 5. Which points are in the solution set of the inequality $2x + 3y \ge 6$?

(A) $(0, 0)$

(B) $(3, 0)$

(C) $(0, 2)$

(D) $(2, 1)$

Question 6. The graphical solution region for a system of linear inequalities is called the feasible region. Which statements about the feasible region are true?

(A) It is the intersection of the solution sets of all inequalities in the system.

(B) Any point within the feasible region is a solution to the system.

(C) For inequalities with $\le$ or $\ge$, the boundary lines are included in the feasible region.

(D) For inequalities with $<$ or $>$, the boundary lines are included in the feasible region.

Question 7. Solve the system of inequalities: $x+1 > 0$ and $x-2 < 0$. Assume $x \in \mathbb{R}$. Which statements are correct?

(A) From $x+1 > 0$, we get $x > -1$.

(B) From $x-2 < 0$, we get $x < 2$.

(C) The solution is the intersection of $x > -1$ and $x < 2$, which is $-1 < x < 2$.

(D) The solution set is $(-1, 2)$.

Question 8. A store sells apples ($x$) and oranges ($y$). An apple costs $\textsf{₹}10$ and an orange costs $\textsf{₹}15$. A customer wants to spend no more than $\textsf{₹}100$ and buy at least 3 apples. Which inequalities represent these conditions?

(A) $10x + 15y \le 100$

(B) $x \ge 3$

(C) $x \ge 0, y \ge 0$ (assuming cannot buy negative fruit)

(D) $10x + 15y > 100$

Question 9. Which statements are true about multiplying or dividing an inequality by a number?

(A) Multiplying by a positive number keeps the inequality sign the same.

(B) Dividing by a positive number keeps the inequality sign the same.

(C) Multiplying by a negative number reverses the inequality sign.

(D) Dividing by a negative number reverses the inequality sign.

Question 10. The system of inequalities $\begin{cases} x \le 5 \\ y \ge 2 \\ x \ge 0 \\ y \ge 0 \end{cases}$ defines a feasible region. Which statements are true about this region?

(A) It is bounded.

(B) It lies in the first quadrant (including boundaries).

(C) The point $(3, 3)$ is in the region.

(D) The point $(6, 2)$ is in the region.

Question 1. Which of the following are arithmetic progressions (AP)?

(A) 2, 5, 8, 11, ...

(B) 10, 8, 6, 4, ...

(C) 3, 6, 12, 24, ...

(D) 1, 1, 1, 1, ...

Question 2. For an AP with first term $a$ and common difference $d$, which statements are true?

(A) The $n$-th term is $a_n = a + (n-1)d$.

(B) The common difference is $a_n - a_{n-1}$ for $n>1$.

(C) If $a_k$ is any term, $a_k = \frac{a_{k-1} + a_{k+1}}{2}$.

(D) The sum of the first $n$ terms is $S_n = \frac{n}{2}(a + a_n)$.

Question 3. Which of the following are geometric progressions (GP)?

(A) 1, 3, 9, 27, ...

(B) 16, 8, 4, 2, ...

(C) $-2, 4, -8, 16, ...$

(D) 5, 10, 15, 20, ...

Question 4. For a GP with first term $a$ and common ratio $r$, which statements are true?

(A) The $n$-th term is $a_n = ar^{n-1}$.

(B) The common ratio is $a_n / a_{n-1}$ for $n>1$ (provided $a_{n-1} \neq 0$).

(C) If $a_k$ is any non-zero term, $a_k^2 = a_{k-1} \cdot a_{k+1}$.

(D) The sum of the first $n$ terms is $S_n = \frac{a(1-r^n)}{1-r}$ for $r \neq 1$.

Question 5. For two positive numbers $a$ and $b$, which statements about their Arithmetic Mean (AM) and Geometric Mean (GM) are true?

(A) $AM = \frac{a+b}{2}$

(B) $GM = \sqrt{ab}$

(C) $AM \ge GM$

(D) $AM = GM$ only if $a=b$.

Question 6. Calculate the 8th term of the AP: 5, 9, 13, ... Which statements are correct?

(A) First term $a=5$, common difference $d=4$.

(B) $a_8 = a + (8-1)d = 5 + 7(4) = 5 + 28 = 33$.

(C) The 8th term is 33.

(D) The sequence is an AP.

Question 7. Calculate the 6th term of the GP: 3, 6, 12, ... Which statements are correct?

(A) First term $a=3$, common ratio $r=2$.

(B) $a_6 = ar^{6-1} = 3 \times 2^5 = 3 \times 32 = 96$.

(C) The 6th term is 96.

(D) The sequence is a GP.

Question 8. Which formulae represent the sum of specific series?

(A) Sum of first $n$ natural numbers: $\sum\limits_{k=1}^{n} k = \frac{n(n+1)}{2}$.

(B) Sum of first $n$ odd numbers: $1 + 3 + \dots + (2n-1) = n^2$.

(C) Sum of first $n$ even numbers: $2 + 4 + \dots + 2n = n(n+1)$.

(D) Sum of first $n$ natural numbers squared: $\sum\limits_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$.

Question 9. If $a, b, c$ are in AP, and also in GP, which statements must be true?

(A) They must all be equal ($a=b=c$).

(B) If $a, b, c$ are in AP, $b-a = c-b \implies 2b = a+c$.

(C) If $a, b, c$ are in GP, $b/a = c/b \implies b^2 = ac$.

(D) If $2b=a+c$ and $b^2=ac$, then $(a+c)/2 = \pm \sqrt{ac}$. This implies $a=c$, and thus $a=b=c$.

Question 10. A sequence of numbers is $5, 5, 5, 5, \dots$. Which statements are true about this sequence?

(A) It is an AP with common difference 0.

(B) It is a GP with common ratio 1.

(C) It is both an AP and a GP.

(D) The $n$-th term is $a_n = 5$ for all $n \ge 1$.

Question 1. The Principle of Mathematical Induction is primarily used to prove statements about:

(A) Sets of objects.

(B) Properties of positive integers.

(C) Divisibility rules for integers.

(D) Formulas for sums or products of sequences.

Question 2. The steps in the Principle of Mathematical Induction (standard form starting from $n=1$) are:

(A) Prove the base case, $P(1)$, is true.

(B) Assume the inductive hypothesis, that $P(k)$ is true for some positive integer $k$.

(C) Prove the inductive step, that if $P(k)$ is true, then $P(k+1)$ is true.

(D) Conclude that $P(n)$ is true for all positive integers $n$.

Question 3. Suppose you want to prove the statement $P(n): n^2 > 2n + 1$ for all integers $n \ge 4$. Which statements are true about the base case?

(A) The base case is $P(1)$.

(B) The base case is $P(4)$.

(C) $P(4): 4^2 > 2(4) + 1 \implies 16 > 8 + 1 \implies 16 > 9$, which is true.

(D) You must prove $P(1), P(2), P(3)$ as well.

Question 4. In the inductive step of proving $P(n)$, we assume $P(k)$ is true for some integer $k \ge n_0$ (where $n_0$ is the base case value) and then prove $P(k+1)$. This process demonstrates:

(A) That the truth of the statement for one integer implies its truth for the next integer.

(B) The inductive hypothesis.

(C) The chain reaction effect of the induction principle.

(D) That $P(k)$ is true for all $k \ge n_0$.

Question 5. Which of the following statements about the Principle of Mathematical Induction are correct?

(A) If $P(1)$ is true and $P(k) \implies P(k+1)$ for all $k \ge 1$, then $P(n)$ is true for all $n \in \mathbb{N}$.

(B) If $P(n_0)$ is true and $P(k) \implies P(k+1)$ for all $k \ge n_0$, then $P(n)$ is true for all integers $n \ge n_0$.

(C) The base case is always $n=1$.

(D) The inductive step requires assuming $P(k+1)$ and proving $P(k)$.

Question 6. Consider the statement $P(n): 1 + 2 + \dots + n = \frac{n(n+1)}{2}$. Assume $P(k)$ is true for some $k \ge 1$, i.e., $1 + 2 + \dots + k = \frac{k(k+1)}{2}$. To prove $P(k+1)$, which steps are valid starting from the LHS of $P(k+1)$?

(A) LHS of $P(k+1) = (1 + 2 + \dots + k) + (k+1)$.

(B) Substitute the inductive hypothesis: $\frac{k(k+1)}{2} + (k+1)$.

(C) Simplify the expression: $(k+1)(\frac{k}{2} + 1) = (k+1)(\frac{k+2}{2}) = \frac{(k+1)(k+2)}{2}$.

(D) This final expression is the RHS of $P(k+1)$.

Question 7. The statement $P(n): 2^n > n$ for all positive integers $n$. Which statements are true about the proof by induction?

(A) Base case $P(1): 2^1 > 1 \implies 2 > 1$, which is true.

(B) Assume $P(k): 2^k > k$ for some $k \ge 1$.

(C) In the inductive step, we need to show $2^{k+1} > k+1$. $2^{k+1} = 2 \cdot 2^k > 2k$. If $2k \ge k+1$, then $2^{k+1} > k+1$.

(D) $2k \ge k+1$ simplifies to $k \ge 1$, which holds since we assumed $k \ge 1$. So $P(k+1)$ is true.

Question 8. Consider the statement $P(n): n^2 - n + 41$ is a prime number for all positive integers $n$. You check $P(1)=41, P(2)=43, P(3)=47, \dots, P(40)=1601$. All are prime. Then you check $P(41)=41^2 - 41 + 41 = 41^2$, which is not prime. What does this example demonstrate?

(A) Checking many cases is not sufficient to prove a statement for all positive integers.

(B) Mathematical induction is a valid proof technique.

(C) The inductive step must hold for *all* $k \ge n_0$.

(D) This statement $P(n)$ is false for $n=41$.

Question 9. Strong induction is a variant of PMI. In strong induction, to prove $P(k+1)$, the inductive hypothesis is that $P(j)$ is true for all integers $j$ such that $n_0 \le j \le k$. Which statements are true about strong induction?

(A) It is a less powerful proof technique than standard induction.

(B) It is useful when the truth of $P(k+1)$ depends on multiple previous cases, not just $P(k)$.

(C) If you can prove a statement by standard induction, you can also prove it by strong induction.

(D) The base case is still required in strong induction.

Question 10. Which of the following types of statements can be proven using the Principle of Mathematical Induction?

(A) Divisibility properties (e.g., $a^n - b^n$ is divisible by $a-b$).

(B) Summation formulas (e.g., sum of geometric series).

(C) Inequalities involving integers (e.g., Bernoulli's inequality $(1+x)^n \ge 1+nx$ for $x \ge -1$).

(D) Properties of recursive sequences.

Question 1. You want to travel from City A to City B and then to City C. There are 3 routes from A to B and 4 routes from B to C. Which statements are true about the number of ways to travel from A to B to C?

(A) This is an application of the Fundamental Principle of Counting (Multiplication Principle).

(B) The total number of ways is $3+4=7$.

(C) The total number of ways is $3 \times 4 = 12$.

(D) For each route from A to B, there are 4 options to go from B to C.

Question 2. Which statements about factorial notation are correct?

(A) $n! = n \times (n-1) \times \dots \times 2 \times 1$ for a positive integer $n$.

(B) $5! = 120$.

(C) $0! = 1$.

(D) $n! = n \times (n-1)!$ for $n \ge 1$.

Question 3. Distinguish between permutations and combinations. Which statements are true?

(A) Permutations are about arrangement where order matters.

(B) Combinations are about selection where order does not matter.

(C) $\text{P}(n, r)$ is the number of ways to choose $r$ objects from $n$ and arrange them.

(D) $\text{C}(n, r)$ is the number of ways to choose $r$ objects from $n$ without regard to order.

Question 4. Calculate $\text{P}(6, 3)$. Which results or methods are correct?

(A) Number of ways to arrange 3 objects out of 6 distinct objects.

(B) $6 \times 5 \times 4 = 120$.

(C) $\frac{6!}{(6-3)!} = \frac{720}{6} = 120$.

(D) $\frac{6!}{3!} = \frac{720}{6} = 120$.

Question 5. Calculate $\text{C}(5, 2)$. Which results or methods are correct?

(A) Number of ways to choose 2 objects out of 5 distinct objects.

(B) $\frac{5 \times 4}{2 \times 1} = 10$.

(C) $\frac{5!}{2!(5-2)!} = \frac{120}{2 \times 6} = 10$.

(D) $\text{P}(5, 2) / 2! = 20 / 2 = 10$.

Question 6. Which properties of combinations are correct?

(A) $\text{C}(n, r) = \text{C}(n, n-r)$

(B) $\text{C}(n, 0) = 1$

(C) $\text{C}(n, n) = 1$

(D) $\text{C}(n, 1) = 1$

Question 7. How many distinct permutations are there of the letters in the word 'MATHEMATICS'?

Letters: M($\times$2), A($\times$2), T($\times$2), H($\times$1), E($\times$1), I($\times$1), C($\times$1), S($\times$1). Total letters = 11.

Number of arrangements = $\frac{11!}{2! 2! 2!}$

(A) $\frac{11!}{2! 2! 2!}$

(B) There are 3 letters that repeat twice.

(C) $\frac{11!}{8}$

(D) $11!$

Question 8. How many 3-digit numbers can be formed using the digits 1, 2, 3, 4, 5, 6 if repetition of digits IS allowed?

(A) For each of the 3 positions, there are 6 choices.

(B) $6 \times 6 \times 6 = 216$.

(C) $6^3 = 216$.

(D) $\text{P}(6, 3) = 120$.

Question 9. How many 3-digit numbers can be formed using the digits 1, 2, 3, 4, 5, 6 if repetition of digits is NOT allowed?

(A) This is a permutation problem: arranging 3 digits out of 6.

(B) $\text{P}(6, 3) = 6 \times 5 \times 4 = 120$.

(C) $\frac{6!}{(6-3)!} = 120$.

(D) $6^3 = 216$.

Question 10. From a group of 8 people, in how many ways can a committee of 3 people be selected?

(A) This is a combination problem: selecting 3 people without regard to order.

(B) $\text{C}(8, 3) = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56$.

(C) $\text{P}(8, 3) = 8 \times 7 \times 6 = 336$.

(D) $\text{C}(8, 3) = \text{C}(8, 5) = 56$.

Question 1. The Binomial Theorem for a positive integral index $n$ provides the expansion of $(a+b)^n$. Which of the following are correct representations of this expansion?

(A) $\sum_{k=0}^{n} \text{C}(n, k) a^{n-k} b^k$

(B) $\text{C}(n, 0)a^n + \text{C}(n, 1)a^{n-1}b + \text{C}(n, 2)a^{n-2}b^2 + \dots + \text{C}(n, n)b^n$

(C) $(a+b)(a+b)\dots(a+b)$ (n times)

(D) $a^n + b^n$

Question 2. What is the number of terms in the expansion of $(2x - 3y)^7$?

(A) The index is $n=7$.

(B) The number of terms is $n+1$.

(C) The number of terms is 8.

(D) The terms alternate in sign.

Question 3. The general term (or $(r+1)$-th term) in the expansion of $(a+b)^n$ is $T_{r+1} = \text{C}(n, r) a^{n-r} b^r$. What are the components for finding the 5th term in the expansion of $(p+q)^{10}$?

(A) $n=10$

(B) $r+1 = 5 \implies r=4$.

(C) $a=p$, $b=q$.

(D) The 5th term is $\text{C}(10, 5) p^5 q^5$.

Question 4. Find the coefficient of $x^4$ in the expansion of $(1+x)^6$. Which statements are correct?

(A) Using $T_{r+1} = \text{C}(n, r) x^r$, where $n=6$. We need $x^4$, so $r=4$.

(B) The coefficient is $\text{C}(6, 4)$.

(C) $\text{C}(6, 4) = \frac{6 \times 5}{2 \times 1} = 15$.

(D) The term is $15x^4$.

Question 5. Find the term independent of $x$ in the expansion of $(x^2 + \frac{1}{x})^9$. Which steps or results are correct?

(A) General term $T_{r+1} = \text{C}(9, r) (x^2)^{9-r} (\frac{1}{x})^r = \text{C}(9, r) x^{18-2r} x^{-r} = \text{C}(9, r) x^{18-3r}$.

(B) For the term independent of $x$, $18-3r = 0 \implies 3r = 18 \implies r = 6$.

(C) The term independent of $x$ is the 6th term ($r+1=6$).

(D) The term is $T_7 = \text{C}(9, 6) = \text{C}(9, 3) = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84$.

Question 6. Find the middle term(s) in the expansion of $(a+b)^{11}$. Which statements are correct?

(A) The index $n=11$ is odd.

(B) There are two middle terms.

(C) The middle terms are the $\frac{11+1}{2} = 6$th term and the $(6+1) = 7$th term.

(D) The middle terms are $T_5$ and $T_6$.

Question 7. Find the middle term in the expansion of $(x-y)^8$. Which statements are correct?

(A) The index $n=8$ is even.

(B) There is one middle term, which is the $\frac{8}{2} + 1 = 5$th term.

(C) The middle term is $T_5$.

(D) The coefficient of the middle term is $\text{C}(8, 4) = 70$.

Question 8. What is the sum of the coefficients in the expansion of $(x+y)^n$?

(A) Set $x=1$ and $y=1$ in the expansion.

(B) The sum is $(1+1)^n = 2^n$.

(C) The sum is $n!$.

(D) The sum is $\sum_{k=0}^{n} \text{C}(n, k)$.

Question 9. What is the sum of the coefficients in the expansion of $(3a - 2b)^4$?

(A) Set $a=1, b=1$.

(B) The sum is $(3(1) - 2(1))^4 = (3-2)^4 = 1^4 = 1$.

(C) The sum is 1.

(D) The sum is $(3+2)^4 = 5^4 = 625$.

Question 10. Which of the following statements about the coefficients in $(a+b)^n$ are correct?

(A) The coefficients are symmetric, i.e., $\text{C}(n, r) = \text{C}(n, n-r)$.

(B) The coefficients are the numbers in the $n$-th row of Pascal's triangle.

(C) The sum of the coefficients is $2^n$.

(D) The coefficients are all integers.

Question 1. Consider the matrix $A = \begin{pmatrix} 2 & 3 \\ 4 & 5 \\ 6 & 7 \end{pmatrix}$. Which statements are true?

(A) The order of the matrix is $3 \times 2$.

(B) The element $a_{21}$ is 4.

(C) It has 3 rows and 2 columns.

(D) It is a square matrix.

Question 2. Which of the following are examples of square matrices?

(A) $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$

(B) A $3 \times 3$ matrix.

(C) A matrix where the number of rows equals the number of columns.

(D) $\begin{pmatrix} 5 \end{pmatrix}$

Question 3. Which of the following are examples of diagonal matrices?

(A) $\begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}$

(B) $\begin{pmatrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{pmatrix}$ (Scalar matrix)

(C) $\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ (Identity matrix)

(D) $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$

Question 4. For matrices A and B to be equal ($A=B$), which conditions must be met?

(A) They must have the same order.

(B) Their corresponding elements must be equal.

(C) They must be square matrices.

(D) They must have the same sum of elements.

Question 5. If $A = \begin{pmatrix} 1 & -1 \\ 2 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} -1 & 1 \\ 0 & -2 \end{pmatrix}$, find $A+B$. Which statements are correct?

(A) The order of both matrices is $2 \times 2$.

(B) $A+B = \begin{pmatrix} 1+(-1) & -1+1 \\ 2+0 & 3+(-2) \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 2 & 1 \end{pmatrix}$.

(C) The sum is $\begin{pmatrix} 0 & 0 \\ 2 & 1 \end{pmatrix}$.

(D) The sum is $\begin{pmatrix} 2 & -2 \\ 2 & 5 \end{pmatrix}$.

Question 6. If $A = \begin{pmatrix} 2 & 0 \\ 1 & -1 \end{pmatrix}$ and $k=-2$, find $kA$. Which statements are correct?

(A) Multiply each element of A by -2.

(B) $kA = \begin{pmatrix} -4 & 0 \\ -2 & 2 \end{pmatrix}$.

(C) The resulting matrix has the same order as A.

(D) The resulting matrix is the additive inverse if $k=-1$.

Question 7. For matrix multiplication $AB$ to be defined, the number of columns in A must equal the number of rows in B. If A is $m \times n$ and B is $p \times q$, for which product(s) is the multiplication defined?

(A) AB if $n=p$

(B) BA if $q=m$

(C) $A^2$ if $m=n$

(D) $B^2$ if $p=q$

Question 8. If matrix A is of order $3 \times 2$ and matrix B is of order $2 \times 4$, which statements are true?

(A) The product AB is defined.

(B) The order of AB is $3 \times 4$.

(C) The product BA is defined.

(D) The order of BA is $2 \times 3$.

Question 9. If $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, calculate $AB$. Which statements are correct?

(A) $AB = \begin{pmatrix} 1(0)+2(1) & 1(1)+2(0) \\ 3(0)+4(1) & 3(1)+4(0) \end{pmatrix} = \begin{pmatrix} 2 & 1 \\ 4 & 3 \end{pmatrix}$.

(B) The result is $\begin{pmatrix} 2 & 1 \\ 4 & 3 \end{pmatrix}$.

(C) $BA = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = \begin{pmatrix} 0(1)+1(3) & 0(2)+1(4) \\ 1(1)+0(3) & 1(2)+0(4) \end{pmatrix} = \begin{pmatrix} 3 & 4 \\ 1 & 2 \end{pmatrix}$.

(D) $AB = BA$.

Question 10. Which of the following matrices are zero matrices?

(A) A matrix of any order where all elements are 0.

(B) $\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$

(C) $\begin{pmatrix} 0 & 0 & 0 \end{pmatrix}$

(D) $\begin{pmatrix} 0 \\ 0 \end{pmatrix}$

Question 1. Find the transpose of the matrix $A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$. Which statements are correct?

(A) The transpose is $A' = \begin{pmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{pmatrix}$.

(B) The rows of A become the columns of $A'$.

(C) If A is $m \times n$, then $A'$ is $n \times m$.

(D) $(A')' = A$.

Question 2. Which of the following are properties of the transpose of matrices A and B (assuming operations are defined)?

(A) $(A+B)' = A' + B'$

(B) $(kA)' = kA'$ for scalar $k$.

(C) $(AB)' = A'B'$

(D) $(AB)' = B'A'$

Question 3. A square matrix A is symmetric if $A' = A$ and skew-symmetric if $A' = -A$. Which statements are true?

(A) If A is symmetric, then $a_{ij} = a_{ji}$ for all $i, j$.

(B) If A is skew-symmetric, then $a_{ij} = -a_{ji}$ for all $i, j$.

(C) The diagonal elements of a skew-symmetric matrix are always zero.

(D) The identity matrix is symmetric.

Question 4. If A is a square matrix, which matrices are always symmetric or skew-symmetric?

(A) $A + A'$ is always symmetric.

(B) $A - A'$ is always skew-symmetric.

(C) $\frac{1}{2}(A+A')$ is the symmetric part of A.

(D) $\frac{1}{2}(A-A')$ is the skew-symmetric part of A.

Question 5. Any square matrix A can be uniquely expressed as the sum of a symmetric matrix and a skew-symmetric matrix, $A = P+Q$. Which statements are true about P and Q?

(A) $P = \frac{1}{2}(A+A')$

(B) $Q = \frac{1}{2}(A-A')$

(C) P is symmetric.

(D) Q is symmetric.

Question 6. Which of the following are elementary row operations on a matrix?

(A) Interchanging two rows ($R_i \leftrightarrow R_j$).

(B) Multiplying a row by any scalar ($k R_i \to R_i$).

(C) Adding a multiple of one row to another row ($R_i + k R_j \to R_i$).

(D) Multiplying a row by a non-zero scalar ($k R_i \to R_i$, $k \neq 0$).

Question 7. A square matrix A is invertible if there exists a matrix B of the same order such that $AB = BA = I$. Which statements are true about an invertible matrix A?

(A) Its inverse $A^{-1}$ is unique.

(B) $A$ must be a non-singular matrix (det(A) $\neq 0$).

(C) $(A^{-1})^{-1} = A$.

(D) If A is invertible, then $A'$ is also invertible.

Question 8. If A and B are invertible matrices of the same order, which statements are true?

(A) $AB$ is invertible.

(B) $(AB)^{-1} = B^{-1}A^{-1}$.

(C) $A+B$ is always invertible.

(D) $(A+B)^{-1} = A^{-1}+B^{-1}$.

Question 9. Consider the matrix $A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$. Which statements are true?

(A) It is a diagonal matrix.

(B) It is a symmetric matrix.

(C) It is an invertible matrix.

(D) Its determinant is 0.

Question 10. Elementary row operations are applied to a matrix. These operations:

(A) Preserve the rank of the matrix.

(B) Can be used to find the inverse of an invertible matrix.

(C) Change the determinant of the matrix in a predictable way.

(D) Change the order of the matrix.

Question 1. Calculate the determinant of the matrix $\begin{pmatrix} 5 & -2 \\ 3 & 1 \end{pmatrix}$. Which statements are correct?

(A) Determinant $= (5)(1) - (-2)(3)$.

(B) Determinant $= 5 - (-6) = 5 + 6 = 11$.

(C) The value is 11.

(D) The matrix is singular.

Question 2. Which properties of determinants are correct for a square matrix A?

(A) If two rows (or columns) are interchanged, the determinant changes sign.

(B) If any row (or column) is multiplied by a scalar $k$, the determinant is multiplied by $k$.

(C) If a multiple of one row (or column) is added to another row (or column), the determinant does not change.

(D) If a row (or column) is a zero vector, the determinant is 1.

Question 3. Which statements are true about the determinant of a matrix?

(A) The determinant of the identity matrix $I_n$ is always 1.

(B) The determinant of a zero matrix is always 0.

(C) $\det(A') = \det(A)$.

(D) $\det(A^2) = (\det(A))^2$.

Question 4. If A and B are square matrices of the same order, which properties of determinants are correct?

(A) $\det(AB) = \det(A) \det(B)$.

(B) $\det(A+B) = \det(A) + \det(B)$.

(C) $\det(kA) = k \det(A)$, where $k$ is a scalar.

(D) $\det(kA) = k^n \det(A)$, where A is of order $n$ and $k$ is a scalar.

Question 5. The area of a triangle with vertices $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ is given by $\frac{1}{2} \left| \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \right|$. Which statements are true?

(A) The area is always non-negative.

(B) If the three points are collinear, the area is 0.

(C) The determinant value can be positive or negative depending on the order of vertices.

(D) The formula involves calculating the determinant of a $3 \times 3$ matrix.

Question 6. Consider the matrix $A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}$. Find the minor of the element $a_{23}$ (which is 6). Which statements are correct?

(A) Delete the 2nd row and 3rd column.

(B) The resulting submatrix is $\begin{pmatrix} 1 & 2 \\ 7 & 8 \end{pmatrix}$.

(C) The minor is the determinant of the submatrix, $1 \times 8 - 2 \times 7 = 8 - 14 = -6$.

(D) The cofactor of $a_{23}$ is $(-1)^{2+3} \times (-6) = -1 \times -6 = 6$.

Question 7. The adjoint of a square matrix A is the transpose of the cofactor matrix. Which statements are true about the adjoint?

(A) For a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $\text{adj}(A) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$.

(B) $A \cdot \text{adj}(A) = |A| I$.

(C) $\text{adj}(A) \cdot A = |A| I$.

(D) If $|A|=0$, then $\text{adj}(A)$ is the zero matrix.

Question 8. Calculate the adjoint of the matrix $A = \begin{pmatrix} 1 & -1 \\ 2 & 3 \end{pmatrix}$. Which statements are correct?

(A) Cofactor matrix $C = \begin{pmatrix} C_{11} & C_{12} \\ C_{21} & C_{22} \end{pmatrix}$. $C_{11}=3$, $C_{12}=-2$, $C_{21}=-(-1)=1$, $C_{22}=1$. $C = \begin{pmatrix} 3 & -2 \\ 1 & 1 \end{pmatrix}$.

(B) $\text{adj}(A) = C' = \begin{pmatrix} 3 & 1 \\ -2 & 1 \end{pmatrix}$.

(C) The adjoint is $\begin{pmatrix} 3 & 1 \\ -2 & 1 \end{pmatrix}$.

(D) $|A| = 1(3) - (-1)(2) = 3+2 = 5$.

Question 9. If A is a non-singular matrix of order $n$, which properties are true?

(A) $|\text{adj}(A)| = |A|^{n-1}$.

(B) $|A^{-1}| = 1/|A|$.

(C) $\text{adj}(A)$ is also non-singular.

(D) $|A| \neq 0$.

Question 10. For what value(s) of $k$ is the matrix $A = \begin{pmatrix} 1 & k \\ 2 & 4 \end{pmatrix}$ singular? Which statements are correct?

(A) A is singular if $|A|=0$.

(B) $|A| = 1(4) - k(2) = 4 - 2k$.

(C) $4 - 2k = 0 \implies 2k = 4 \implies k = 2$.

(D) If $k=2$, the columns (and rows) are linearly dependent.

Question 1. The inverse of a square matrix A exists if and only if:

(A) $|A| \neq 0$.

(B) A is non-singular.

(C) $\text{adj}(A)$ is non-zero.

(D) The matrix is invertible.

Question 2. The formula for the inverse of an invertible matrix A is $A^{-1} = \frac{1}{|A|} \text{adj}(A)$. Which statements are true?

(A) This formula applies to any square matrix.

(B) It requires $|A| \neq 0$.

(C) $\text{adj}(A)$ is the transpose of the cofactor matrix.

(D) $A^{-1} A = I$.

Question 3. Consider the matrix $A = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$. Which statements about its inverse are correct?

(A) $|A| = 2(1) - 1(1) = 1$. So the inverse exists.

(B) $\text{adj}(A) = \begin{pmatrix} 1 & -1 \\ -1 & 2 \end{pmatrix}$.

(C) $A^{-1} = \frac{1}{|A|} \text{adj}(A) = \frac{1}{1} \begin{pmatrix} 1 & -1 \\ -1 & 2 \end{pmatrix} = \begin{pmatrix} 1 & -1 \\ -1 & 2 \end{pmatrix}$.

(D) $A^{-1} = \begin{pmatrix} 1 & -1 \\ -1 & 2 \end{pmatrix}$.

Question 4. A system of linear equations $a_{11}x + a_{12}y = b_1$, $a_{21}x + a_{22}y = b_2$ can be written in matrix form as $AX = B$. Which statements are correct?

(A) $A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}$ is the coefficient matrix.

(B) $X = \begin{pmatrix} x \\ y \end{pmatrix}$ is the variable matrix.

(C) $B = \begin{pmatrix} b_1 \\ b_2 \end{pmatrix}$ is the constant matrix.

(D) If $|A| \neq 0$, the system has a unique solution $X = A^{-1}B$.

Question 5. Consider the system $\begin{cases} x + y = 3 \\ 2x - y = 0 \end{cases}$. In matrix form $AX=B$, $A = \begin{pmatrix} 1 & 1 \\ 2 & -1 \end{pmatrix}$, $X = \begin{pmatrix} x \\ y \end{pmatrix}$, $B = \begin{pmatrix} 3 \\ 0 \end{pmatrix}$. $|A| = 1(-1) - 1(2) = -1 - 2 = -3$. Since $|A| \neq 0$, a unique solution exists. Which statements are correct about solving this system using matrix inverse?

(A) The inverse of A exists.

(B) The solution is given by $X = A^{-1}B$.

(C) $A^{-1} = \frac{1}{-3} \text{adj}(A) = \frac{1}{-3} \begin{pmatrix} -1 & -1 \\ -2 & 1 \end{pmatrix} = \begin{pmatrix} 1/3 & 1/3 \\ 2/3 & -1/3 \end{pmatrix}$.

(D) $X = \begin{pmatrix} 1/3 & 1/3 \\ 2/3 & -1/3 \end{pmatrix} \begin{pmatrix} 3 \\ 0 \end{pmatrix} = \begin{pmatrix} 1/3(3)+1/3(0) \\ 2/3(3)+(-1/3)(0) \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$. So $x=1, y=2$.

Question 6. Cramer's Rule provides a formula for the solution of a system $AX=B$ when $|A| \neq 0$. For a $2 \times 2$ system, $x = \frac{|A_x|}{|A|}$ and $y = \frac{|A_y|}{|A|}$. Which statements are true?

(A) $A_x$ is formed by replacing the first column of A with B.

(B) $A_y$ is formed by replacing the second column of A with B.

(C) This rule is applicable only when the determinant of the coefficient matrix is non-zero.

(D) This rule can be used for systems with any number of variables (provided $|A| \neq 0$).

Question 7. Consider the system $\begin{cases} 3x - 2y = 4 \\ 6x - 4y = 8 \end{cases}$. Which statements are true?

(A) $A = \begin{pmatrix} 3 & -2 \\ 6 & -4 \end{pmatrix}$, $B = \begin{pmatrix} 4 \\ 8 \end{pmatrix}$.

(B) $|A| = 3(-4) - (-2)(6) = -12 + 12 = 0$. The matrix is singular.

(C) $(\text{adj}(A))B = \begin{pmatrix} -4 & 2 \\ -6 & 3 \end{pmatrix} \begin{pmatrix} 4 \\ 8 \end{pmatrix} = \begin{pmatrix} -16+16 \\ -24+24 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$.

(D) Since $|A|=0$ and $(\text{adj}(A))B = O$, the system has infinitely many solutions.

Question 8. Which statements are true about the system $AX=B$ where $|A| = 0$?

(A) If $(\text{adj}(A))B \neq O$, the system is inconsistent (no solution).

(B) If $(\text{adj}(A))B = O$, the system is consistent (infinitely many solutions).

(C) If $|A|=0$, the system has no unique solution.

(D) If $|A|=0$, the system always has a solution.

Question 9. Consider the homogeneous system $AX=O$. Which statements are true?

(A) It is always consistent (has the trivial solution $X=O$).

(B) It has a unique solution (the trivial solution) if and only if $|A| \neq 0$.

(C) It has infinitely many solutions (including non-trivial solutions) if and only if $|A| = 0$.

(D) The solution $X=O$ corresponds to the origin in the case of 2 or 3 variables.

Question 10. Which statements are true about the properties of matrix inverse for invertible matrices A and B of the same order?

(A) $(A^{-1})^{-1} = A$

(B) $(A')^{-1} = (A^{-1})'$

(C) $(AB)^{-1} = B^{-1}A^{-1}$

(D) $|A^{-1}| = 1/|A|$

Question 1. Which of the following word problems can be solved using a linear equation in one variable?

(A) The sum of two consecutive integers is 51. Find the integers.

(B) The age of a father is three times his son's age. The sum of their ages is 48. Find their ages.

(C) The product of two consecutive positive integers is 210. Find the integers.

(D) A number increased by 7 is equal to twice the number decreased by 3. Find the number.

Question 2. Which of the following word problems can be modeled using a system of linear equations in two variables?

(A) The sum of two numbers is 15 and their difference is 5. Find the numbers.

(B) The cost of 2 pens and 3 notebooks is $\textsf{₹}80$. The cost of 3 pens and 4 notebooks is $\textsf{₹}110$. Find the cost of each pen and notebook.

(C) The sum of the digits of a two-digit number is 11. If the digits are reversed, the new number is 9 more than the original number. Find the number.

(D) A boat travels upstream and downstream. Find the speed of the boat and the speed of the stream.

Question 3. Which of the following word problems can be modeled using a quadratic equation?

(A) The product of two consecutive positive integers is 156. Find the integers.

(B) The area of a rectangular field is 528 m$^2$. The length is one more than twice the width. Find the dimensions.

(C) The sum of the squares of two consecutive odd numbers is 290. Find the numbers.

(D) A train travels a certain distance at a uniform speed. If the speed were more/less, the time taken would be different. (Problems leading to $\frac{D}{v} - \frac{D}{v \pm \Delta v} = \Delta t$ forms).

Question 4. A part of the monthly hostel charges is fixed and the remaining depends on the number of days one has taken food in the mess. When a student A takes food for 20 days, she has to pay $\textsf{₹}1000$ as hostel charges, whereas a student B, who takes food for 26 days, pays $\textsf{₹}1180$. Which equations can represent this? Let fixed charge be $x$ and cost per day be $y$.

(A) $x + 20y = 1000$

(B) $x + 26y = 1180$

(C) This is a system of linear equations.

(D) Subtracting the equations gives $6y = 180 \implies y=30$.

Question 5. The sum of the areas of two squares is 400 cm$^2$. The difference of their areas is 200 cm$^2$. Which statements are true about finding the sides of the squares? Let the sides be $x$ and $y$.

(A) $x^2 + y^2 = 400$

(B) $x^2 - y^2 = 200$ (assuming $x>y$)

(C) Adding the equations: $2x^2 = 600 \implies x^2 = 300 \implies x = \sqrt{300} = 10\sqrt{3}$.

(D) Subtracting the equations: $2y^2 = 200 \implies y^2 = 100 \implies y = 10$.

Question 6. A two-digit number is such that the product of its digits is 14. If 45 is added to the number, the digits are reversed. Which statements are true about setting up the equations? Let the number be $10t + u$.

(A) $tu = 14$

(B) $(10t + u) + 45 = 10u + t$

(C) The first equation is quadratic.

(D) The second equation is linear.

Question 7. Two pipes A and B can fill a tank in $x$ and $y$ hours respectively. Together, they fill the tank in $T$ hours. Which equations can represent this relationship?

(A) Rate of A is $1/x$ tank/hour, rate of B is $1/y$ tank/hour.

(B) Combined rate is $1/x + 1/y$.

(C) $T \times (\frac{1}{x} + \frac{1}{y}) = 1$.

(D) $\frac{1}{x} + \frac{1}{y} = \frac{1}{T}$.

Question 8. A sum of $\textsf{₹}A$ is divided between two persons P and Q. If P gets $\textsf{₹}x$, and Q gets $\textsf{₹}y$, and P gets $\textsf{₹}100$ more than Q, which equations are true?

(A) $x+y=A$

(B) $x = y + 100$

(C) $y = x - 100$

(D) $x - y = 100$

Question 9. The numerator of a fraction is 3 less than its denominator. If 1 is added to the denominator, the fraction becomes 1/2. Which statements are true about setting up the equations? Let the numerator be $n$ and the denominator be $d$.

(A) $n = d - 3$

(B) The fraction is $n/d$.

(C) $\frac{n}{d+1} = \frac{1}{2}$

(D) Substituting $n=d-3$ into the second equation gives $\frac{d-3}{d+1} = \frac{1}{2}$.

Question 10. In a race of 100 metres, A beats B by 10 metres and B beats C by 10 metres. If A, B, and C run at constant speeds $v_A, v_B, v_C$, which statements are true?

(A) When A runs 100m, B runs 90m. $\frac{100}{v_A} = \frac{90}{v_B} \implies v_A/v_B = 10/9$.

(B) When B runs 100m, C runs 90m. $\frac{100}{v_B} = \frac{90}{v_C} \implies v_B/v_C = 10/9$.

(C) $v_A/v_C = (v_A/v_B) \times (v_B/v_C) = (10/9) \times (10/9) = 100/81$.

(D) When A runs 100m, C runs 80m.