Single Best Answer MCQs for Sub-Topics of Topic 2: Algebra

Question 1. Which of the following is an example of a variable?

(A) $5$

(B) $\frac{1}{2}$

(C) $x$

(D) $\pi$

Question 2. In the expression $3x + 7$, what is the constant term?

(A) $3$

(B) $x$

(C) $7$

(D) $3x$

Question 3. How many terms are there in the expression $4a^2 - 2ab + b^2 - 5$?

(A) $2$

(B) $3$

(C) $4$

(D) $5$

Question 4. What are the factors of the term $6xy$?

(A) $6, x, y$

(B) $6x, y$

(C) $6y, x$

(D) $6, x, y$, and their products ($6x, 6y, xy, 6xy$)

Question 5. In the term $-5p^2q$, what is the coefficient of $p^2q$?

(A) $5$

(B) $-5$

(C) $p^2$

(D) $q$

Question 6. What is the coefficient of $y$ in the expression $7xy - 3x + 10$?

(A) $7$

(B) $7x$

(C) $-3x$

(D) $y$

Question 7. Which situation best illustrates the use of a variable in algebra?

(A) Counting the number of chairs in a room.

(B) Representing an unknown cost of an item.

(C) Stating that $10$ is greater than $5$.

(D) Calculating the sum of $2$ and $3$.

Question 8. A shopkeeper sells pens for $\textsf{₹}15$ each. If he sells 'n' pens, which expression represents the total revenue?

(A) $15 + n$

(B) $15 \times n$

(C) $15 - n$

(D) $15/n$

Question 9. An algebraic expression is formed by combining variables and constants using which operations?

(A) Addition only

(B) Subtraction only

(C) Multiplication and Division only

(D) Addition, Subtraction, Multiplication, and Division

Question 10. Which of the following is NOT an algebraic expression?

(A) $2x + 3$

(B) $5 - y/4$

(C) $10$

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

Question 11. The phrase "five less than twice a number $m$" can be written as the algebraic expression:

(A) $5 - 2m$

(B) $2m - 5$

(C) $2(m - 5)$

(D) $5m - 2$

Question 12. What is the value of the expression $2x + 5$ when $x = 3$?

(A) $7$

(B) $8$

(C) $11$

(D) $16$

Question 13. Evaluate the expression $a^2 - b$ when $a = 4$ and $b = -2$.

(A) $14$

(B) $18$

(C) $6$

(D) $-6$

Question 14. If the cost of one kilogram of apples is $\textsf{₹}80$, what is the cost of $k$ kilograms of apples? This uses an algebraic expression:

(A) To represent a relationship.

(B) To find a specific value.

(C) To simplify a calculation.

(D) To state a fact.

Question 15. A bus travels at a speed of $v$ km/hour. The distance covered in $t$ hours is given by the expression $vt$. This expression is used for:

(A) Calculating speed.

(B) Calculating time.

(C) Calculating distance.

(D) Calculating average.

Question 16. In the expression $5x^3$, the exponent of $x$ is:

(A) $5$

(B) $x$

(C) $3$

(D) $5x$

Question 17. Which of the following describes 'like terms' in an algebraic expression?

(A) Terms with the same constant.

(B) Terms with the same variable(s) raised to the same powers.

(C) Terms with the same coefficient.

(D) Terms with the same sign.

Question 18. Simplify the expression $5x + 3y - 2x + y$ by combining like terms.

(A) $3x + 4y$

(B) $7x + 4y$

(C) $3x + 2y$

(D) $7x + 2y$

Question 19. If 'age' is represented by the variable $A$, how would you represent "twice the age decreased by $5$ years"?

(A) $2A + 5$

(B) $5 - 2A$

(C) $2(A - 5)$

(D) $2A - 5$

Question 20. A variable allows us to represent:

(A) Only known quantities.

(B) Only constants.

(C) Unknown or changing quantities.

(D) Only fixed numbers.

Question 21. In the term $\frac{p}{3}$, what is the coefficient of $p$?

(A) $3$

(B) $-3$

(C) $\frac{1}{3}$

(D) $p$

Question 22. An expression is formed when algebraic terms are connected by:

(A) Only addition and subtraction signs.

(B) Only multiplication and division signs.

(C) Any of the four fundamental arithmetic signs.

(D) Equality sign.

Question 23. What is the value of the expression $\frac{x+y}{xy}$ when $x = 2$ and $y = 1$?

(A) $\frac{3}{2}$

(B) $3$

(C) $\frac{2}{3}$

(D) $1$

Question 24. Which of the following is a constant?

(A) $m$

(B) $-9$

(C) $z^2$

(D) $ab$

Question 25. The expression for the perimeter of a square with side length $s$ is $4s$. Here, $s$ is a:

(A) Constant

(B) Variable

(C) Term

(D) Coefficient

Question 1. Add the expressions: $(3x + 5y)$ and $(2x - 3y)$.

(A) $5x + 2y$

(B) $x + 2y$

(C) $5x + 8y$

(D) $x + 8y$

Question 2. Subtract $(p - 2q)$ from $(4p + 3q)$.

(A) $3p + 5q$

(B) $3p + q$

(C) $5p + q$

(D) $5p + 5q$

Question 3. Multiply $4a$ by $(2a^2 - 3b)$.

(A) $8a^3 - 12ab$

(B) $8a^3 + 12ab$

(C) $8a^2 - 12ab$

(D) $8a^2 + 12ab$

Question 4. Simplify the expression: $(x+2)(x-3)$.

(A) $x^2 - x - 6$

(B) $x^2 + x - 6$

(C) $x^2 - 5x - 6$

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

Question 5. Divide $18x^4y^3$ by $6x^2y$.

(A) $3x^2y^2$

(B) $3x^6y^4$

(C) $12x^2y^2$

(D) $12x^6y^4$

Question 6. Add: $(7m^2n - 3mn + 5)$ and $(-2m^2n + 5mn - 8)$.

(A) $5m^2n + 2mn - 3$

(B) $9m^2n + 2mn - 3$

(C) $5m^2n - 8mn - 3$

(D) $9m^2n - 8mn - 3$

Question 7. Subtract $(3a^2 - 5a + 2)$ from $(7a^2 + a - 1)$.

(A) $4a^2 + 6a - 3$

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

(C) $10a^2 - 4a + 1$

(D) $10a^2 + 6a - 3$

Question 8. Find the product of $(a - b)$ and $(a^2 + ab + b^2)$.

(A) $a^3 + b^3$

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

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

(D) $a^3 + 2a^2b + 2ab^2 + b^3$

Question 9. Simplify: $2x(3x - 1) - 4(x^2 - 5)$.

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

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

(C) $2x^2 + 6x + 20$

(D) $2x^2 + 6x - 20$

Question 10. Divide $(9p^5q^4 - 12p^3q^2)$ by $3p^2q$.

(A) $3p^3q^3 - 4pq$

(B) $3p^3q^3 - 4p^5q^3$

(C) $3p^{2.5}q^4 - 4p^{1.5}q$

(D) $3p^3q^3 - 4pq$

Question 11. What is the result of adding $5x^2 - 3x + 1$ and $-2x^2 + x - 4$?

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

(B) $3x^2 - 4x - 3$

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

(D) $7x^2 - 4x - 3$

Question 12. What expression should be subtracted from $5a - 3b + 2$ to get $2a + b - 1$?

(A) $3a - 4b + 3$

(B) $3a - 2b + 1$

(C) $7a - 2b + 1$

(D) $3a - 4b + 1$

Question 13. Expand the expression $(m+4)(m-5)$.

(A) $m^2 - m - 20$

(B) $m^2 + m - 20$

(C) $m^2 - 9m - 20$

(D) $m^2 + 9m - 20$

Question 14. Find the quotient when $25x^3y^5$ is divided by $-5xy^2$.

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

(B) $5x^2y^3$

(C) $-5x^4y^7$

(D) $5x^4y^7$

Question 15. The sum of three expressions is $x^2 + 2x + 3$. Two of them are $x^2 - x - 1$ and $-x^2 + 3x - 2$. What is the third expression?

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

(B) $x^2 + 0x + 6$

(C) $x^2 + 2x + 6$

(D) $-x^2 + 0x + 6$

Question 16. What is the product of $(p - q)$ and $(p - q)$?

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

(B) $p^2 + q^2$

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

(D) $p^2 + 2pq + q^2$

Question 17. Simplify: $(ab^2)^3 \times (a^2b)^2$.

(A) $a^7b^8$

(B) $a^5b^5$

(C) $a^6b^6$

(D) $a^8b^7$

Question 18. Divide $(6a^2b + 9ab^2)$ by $3ab$.

(A) $2a + 3b$

(B) $2ab + 3ab$

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

(D) $2a + 3b^2$

Question 19. Find the difference when $10xy - 5xz + 3yz$ is subtracted from $12xy + 2xz - yz$.

(A) $2xy + 7xz - 4yz$

(B) $2xy - 7xz + 4yz$

(C) $-2xy - 7xz + 4yz$

(D) $-2xy + 7xz - 4yz$

Question 20. Multiply $(2x + 3)$ by $(2x - 3)$.

(A) $4x^2 + 9$

(B) $4x^2 - 9$

(C) $4x^2 + 12x - 9$

(D) $4x^2 - 12x - 9$

Question 21. The area of a rectangle is given by the expression $15x^2y^3$. If the length is $5xy$, what is the breadth?

(A) $3xy^2$

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

(C) $10xy^2$

(D) $10x^2y^2$

Question 22. Simplify the expression $-(a - b) + (a + b)$.

(A) $2a$

(B) $2b$

(C) $-2a$

(D) $-2b$

Question 23. What is the result of multiplying $0.5x$ by $20xy$?

(A) $10xy$

(B) $10x^2y$

(C) $100x^2y$

(D) $100xy$

Question 24. The expression $(x^2 + y^2) - (x^2 - y^2)$ simplifies to:

(A) $2x^2$

(B) $2y^2$

(C) $0$

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

Question 25. What is the product of $x$, $y^2$, and $z^3$?

(A) $xyz^6$

(B) $xy^2z^3$

(C) $(xyz)^6$

(D) $x+y^2+z^3$

Question 26. Add $(p+q)$, $(q+r)$, and $(r+p)$.

(A) $p+q+r$

(B) $2p+2q+2r$

(C) $p^2+q^2+r^2$

(D) $p+q+r+3$

Question 27. What must be added to $x^2 + x + 1$ to get $2x^2 + 3x - 5$?

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

(B) $x^2 - 2x + 6$

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

(D) $3x^2 + 4x + 6$

Question 28. Simplify: $(-a^2b) \times (-2ab^2) \times (-3a^3b^3)$.

(A) $6a^6b^6$

(B) $-6a^6b^6$

(C) $6a^5b^5$

(D) $-6a^5b^5$

Question 29. The expression $(3x - 4y) \times (3x + 4y)$ is equal to:

(A) $9x^2 + 16y^2$

(B) $9x^2 - 16y^2$

(C) $9x^2 + 24xy - 16y^2$

(D) $9x^2 - 24xy - 16y^2$

Question 30. Divide $(a^3b^2 - a^2b^3)$ by $a^2b^2$.

(A) $a-b$

(B) $a+b$

(C) $ab$

(D) $a^5b^5$

Question 1. Which of the following is a polynomial?

(A) $x^2 + \frac{1}{x}$

(B) $\sqrt{y} + 3$

(C) $3x^4 - 2x + 1$

(D) $5z^{-2} + z$

Question 2. What is the degree of the polynomial $5x^3 - 2x^5 + 7x - 1$?

(A) $3$

(B) $5$

(C) $1$

(D) $0$

Question 3. A polynomial with exactly three terms is called a:

(A) Monomial

(B) Binomial

(C) Trinomial

(D) Polynomial (general)

Question 4. What is the leading coefficient of the polynomial $-7y^4 + 2y^2 - 9$?

(A) $-7$

(B) $4$

(C) $2$

(D) $-9$

Question 5. Evaluate the polynomial $P(x) = x^2 - 3x + 2$ at $x = -1$.

(A) $0$

(B) $2$

(C) $6$

(D) $-2$

Question 6. A zero of a polynomial $P(x)$ is a value of $x$ such that:

(A) $P(x) > 0$

(B) $P(x) < 0$

(C) $P(x) = 0$

(D) $P(x)$ is undefined

Question 7. Find a zero of the polynomial $P(x) = 2x - 4$.

(A) $x = 0$

(B) $x = 1$

(C) $x = 2$

(D) $x = 4$

Question 8. How many zeroes does a quadratic polynomial generally have?

(A) Exactly one

(B) Exactly two

(C) At most two

(D) At least two

Question 9. The graph of a linear polynomial $P(x) = ax + b$ ($a \neq 0$) is a:

(A) Parabola

(B) Straight line

(C) Circle

(D) Point

Question 10. For a quadratic polynomial $ax^2 + bx + c$ ($a \neq 0$), if the graph intersects the x-axis at two distinct points, how many zeroes does the polynomial have?

(A) One

(B) Two

(C) Zero

(D) Cannot be determined

Question 11. If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $ax^2 + bx + c$, what is the sum of the zeroes ($\alpha + \beta$)?

(A) $b/a$

(B) $-b/a$

(C) $c/a$

(D) $-c/a$

Question 12. If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $ax^2 + bx + c$, what is the product of the zeroes ($\alpha \beta$)?

(A) $b/a$

(B) $-b/a$

(C) $c/a$

(D) $-c/a$

Question 13. Find the sum of the zeroes of the polynomial $2x^2 - 8x + 6$.

(A) $8$

(B) $-8$

(C) $4$

(D) $-4$

Question 14. Find the product of the zeroes of the polynomial $3x^2 + 5x - 2$.

(A) $5/3$

(B) $-5/3$

(C) $-2/3$

(D) $2/3$

Question 15. If the sum of the zeroes of a quadratic polynomial is $3$ and the product is $2$, which of the following could be the polynomial?

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

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

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

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

Question 16. The graph of a quadratic polynomial $y = ax^2 + bx + c$ is called a:

(A) Hyperbola

(B) Parabola

(C) Ellipse

(D) Circle

Question 17. How many zeroes does the polynomial $P(x) = x^2 + 4$ have?

(A) $1$ (real)

(B) $2$ (real)

(C) $0$ (real)

(D) $2$ (complex)

Question 18. If a quadratic polynomial has zeroes $2$ and $-1$, which of the following is the polynomial?

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

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

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

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

Question 19. What is the degree of the polynomial $7$?

(A) $1$

(B) $7$

(C) $0$

(D) Undefined

Question 20. Which of the following is a binomial of degree 2?

(A) $x^2$

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

(C) $x^2 - 5$

(D) $x + 2$

Question 21. If $P(x) = 3x^2 - kx + 5$ has a zero at $x=1$, what is the value of $k$?

(A) $3$

(B) $5$

(C) $8$

(D) $-8$

Question 22. The sum of the zeroes of $3x^2 - 7x - 10$ is:

(A) $7/3$

(B) $-7/3$

(C) $10/3$

(D) $-10/3$

Question 23. The product of the zeroes of $x^2 + 4x + 4$ is:

(A) $4$

(B) $-4$

(C) $2$

(D) $-2$

Question 24. If the graph of a polynomial touches the x-axis at one point, how many real zeroes does it have at that point (counting multiplicity)?

(A) $0$

(B) $1$

(C) $2$

(D) Cannot be determined

Question 25. Which of the following is NOT a polynomial?

(A) $5x^3$

(B) $\frac{1}{2}x^2 - \sqrt{3}x$

(C) $\frac{x^2+1}{x-1}$

(D) $(x+1)(x-1)$

Question 1. According to the Division Algorithm for Polynomials, if $P(x)$ is divided by $D(x)$, then $P(x) = Q(x) \cdot D(x) + R(x)$, where the degree of $R(x)$ is:

(A) Greater than the degree of $D(x)$.

(B) Equal to the degree of $D(x)$.

(C) Less than the degree of $D(x)$.

(D) Equal to the degree of $P(x)$.

Question 2. When a polynomial $P(x)$ is divided by $(x-a)$, the remainder, according to the Remainder Theorem, is:

(A) $P(a)$

(B) $P(-a)$

(C) $a$

(D) $0$

Question 3. If $P(a) = 0$, then according to the Factor Theorem, which of the following is a factor of $P(x)$?

(A) $x$

(B) $a$

(C) $(x-a)$

(D) $(x+a)$

Question 4. Find the remainder when $x^2 - 5x + 6$ is divided by $(x-2)$ using the Remainder Theorem.

(A) $0$

(B) $2$

(C) $-2$

(D) $12$

Question 5. Is $(x-3)$ a factor of $P(x) = x^3 - 4x^2 + 5x - 6$? Use the Factor Theorem.

(A) Yes, because $P(3) \neq 0$

(B) Yes, because $P(3) = 0$

(C) No, because $P(3) \neq 0$

(D) No, because $P(-3) \neq 0$

Question 6. When $x^3 - 1$ is divided by $(x-1)$, the quotient is:

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

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

(C) $x^2 + 1$

(D) $x^2 - 1$

Question 7. Find the remainder when $2x^2 + 3x - 1$ is divided by $(x+1)$.

(A) $0$

(B) $-2$

(C) $2$

(D) $-1$

Question 8. If $(x+2)$ is a factor of $x^2 + kx + 10$, what is the value of $k$?

(A) $5$

(B) $7$

(C) $-7$

(D) $-5$

Question 9. When $x^4 - 16$ is divided by $(x+2)$, the remainder is:

(A) $0$

(B) $16$

(C) $-16$

(D) $32$

Question 10. If $P(x) = x^3 + 3x^2 + 3x + 1$, what is the remainder when $P(x)$ is divided by $(x+1)$?

(A) $0$

(B) $1$

(C) $-1$

(D) $8$

Question 11. For the polynomial $P(x) = x^2 - 5x + 6$, which of the following is NOT a factor?

(A) $(x-2)$

(B) $(x-3)$

(C) $(x+2)$

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

Question 12. If $(2x-1)$ is a factor of $P(x)$, then what is a zero of $P(x)$?

(A) $x = 1$

(B) $x = -1$

(C) $x = 1/2$

(D) $x = -1/2$

Question 13. When $x^2 - 4$ is divided by $(x-2)$, the remainder is $R_1$. When $x^2 - 4$ is divided by $(x+2)$, the remainder is $R_2$. What is $R_1 + R_2$?

(A) $4$

(B) $0$

(C) $8$

(D) $-4$

Question 14. Find the value of 'a' if $(x-a)$ is a factor of $x^2 - a^2 + 2x + 2a - 3$.

(A) $1$

(B) $-1$

(C) $3$

(D) $-3$

Question 15. If $P(x) = x^3 - kx^2 + x + 6$ and $(x+1)$ is a factor of $P(x)$, find the value of $k$.

(A) $6$

(B) $8$

(C) $-6$

(D) $-8$

Question 16. When a polynomial is divided by a linear polynomial $(ax+b)$, the degree of the remainder is always:

(A) $1$

(B) $0$

(C) Less than $1$ (i.e., $0$ or the remainder is $0$)

(D) Greater than or equal to $1$

Question 17. If $x^3 + ax^2 + bx + 6$ is divisible by $(x-2)$, it means that $(x-2)$ is a factor. According to the Factor Theorem, this implies:

(A) $P(2) = 0$

(B) $P(-2) = 0$

(C) $P(6) = 0$

(D) $P(a+b) = 0$

Question 18. When $x^{100} + 2$ is divided by $(x-1)$, the remainder is:

(A) $1$

(B) $2$

(C) $3$

(D) $-1$

Question 19. If $P(x) = x^2 + 5x + r$ is divisible by $(x+2)$, then $r$ is equal to:

(A) $6$

(B) $-6$

(C) $14$

(D) $-14$

Question 20. What is the remainder when $x^3 + 4x^2 + 4x - 1$ is divided by $(x+2)$?

(A) $0$

(B) $1$

(C) $-1$

(D) $2$

Question 21. Which theorem is used to find the remainder without performing long division?

(A) Factor Theorem

(B) Polynomial Division Algorithm

(C) Remainder Theorem

(D) Binomial Theorem

Question 22. If $(x-1)$ and $(x+1)$ are factors of $x^3 + ax^2 + bx - 1$, then:

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

(B) $a=1, b=-1$

(C) $a=-1, b=1$

(D) $a=-1, b=-1$

Question 23. The condition for a polynomial $P(x)$ to be divisible by $(ax+b)$ is:

(A) $P(-b/a) = 0$

(B) $P(b/a) = 0$

(C) $P(a/b) = 0$

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

Question 24. When $x^3 - 6x^2 + 11x - 6$ is divided by $(x-1)$, the remainder is 0. This means $(x-1)$ is a:

(A) Quotient

(B) Remainder

(C) Zero

(D) Factor

Question 25. Which of the following is the remainder when $x^3 - 2x^2 + x - 1$ is divided by $(x-1)$?

(A) $0$

(B) $1$

(C) $-1$

(D) $2$

Question 1. Which of the following is the expansion of $(a+b)^2$?

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

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

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

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

Question 2. The identity $(a-b)^2 = a^2 - 2ab + b^2$ holds true for:

(A) Only specific values of $a$ and $b$

(B) All real values of $a$ and $b$

(C) Only positive values of $a$ and $b$

(D) Only integer values of $a$ and $b$

Question 3. Expand $(x+5)^2$ using an algebraic identity.

(A) $x^2 + 25$

(B) $x^2 + 10x + 25$

(C) $x^2 - 10x + 25$

(D) $x^2 - 25$

Question 4. Simplify $(y-3)^2$ using an algebraic identity.

(A) $y^2 - 9$

(B) $y^2 + 6y + 9$

(C) $y^2 - 6y + 9$

(D) $y^2 + 9$

Question 5. The expression $4p^2 - 9q^2$ can be factorised using which identity?

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

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

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

(D) $(x+a)(x+b)$

Question 6. Factorise $x^2 - 49$ using an identity.

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

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

(C) $(x-7)(x+7)$

(D) $(x-49)(x+1)$

Question 7. Expand $(2x+3y)^2$ using an identity.

(A) $4x^2 + 9y^2$

(B) $4x^2 + 6xy + 9y^2$

(C) $4x^2 + 12xy + 9y^2$

(D) $4x^2 - 12xy + 9y^2$

Question 8. Simplify $(103)^2$ using an algebraic identity.

(A) $100^2 + 3^2$

(B) $(100+3)^2 = 10000 + 600 + 9 = 10609$

(C) $(103)(103)$

(D) $10000 + 9$

Question 9. Evaluate $98^2$ using an algebraic identity.

(A) $(100-2)^2 = 10000 - 400 + 4 = 9604$

(B) $(100-2)^2 = 10000 + 4 = 10004$

(C) $(90+8)^2 = 8100 + 1440 + 64 = 9604$

(D) $98 \times 98$

Question 10. Calculate $55^2 - 45^2$ using an identity.

(A) $(55-45)^2 = 10^2 = 100$

(B) $(55+45)(55-45) = 100 \times 10 = 1000$

(C) $55^2 + 45^2$

(D) $(55-45)(55-45)$

Question 11. Which identity is used for expanding $(x+a)(x+b)$?

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

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

(C) $x^2 + (a-b)x - ab$

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

Question 12. Expand $(x+2)(x+3)$ using the identity $(x+a)(x+b)$.

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

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

(C) $x^2 - x + 6$

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

Question 13. Expand $(x-4)(x+5)$ using an identity.

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

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

(C) $x^2 - 9x - 20$

(D) $x^2 + 9x - 20$

Question 14. The identity $(x+y+z)^2 = x^2+y^2+z^2+2xy+2yz+2zx$ is used to expand trinomial squares. Expand $(a-b+c)^2$.

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

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

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

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

Question 15. Expand $(2x+y-z)^2$.

(A) $4x^2+y^2+z^2+4xy-2yz-4zx$

(B) $4x^2+y^2+z^2+4xy+2yz-4zx$

(C) $4x^2+y^2+z^2-4xy+2yz-4zx$

(D) $4x^2+y^2+z^2-4xy-2yz+4zx$

Question 16. The expression $x^2 + 6x + 9$ is a perfect square. Which identity can be used to factorise it?

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

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

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

(D) $(x+a)(x+b)$

Question 17. Factorise $4y^2 - 12y + 9$ using an identity.

(A) $(2y+3)^2$

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

(C) $(4y-9)^2$

(D) $(2y-3)(2y+3)$

Question 18. Simplify $\frac{x^2 - y^2}{x-y}$.

(A) $x-y$

(B) $x+y$

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

(D) $1$

Question 19. What is the value of $(a+b)^3$?

(A) $a^3 + b^3$

(B) $a^3 + 3a^2b + 3ab^2 + b^3$

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

(D) $a^3 + b^3 + 3ab(a+b)$

Question 20. Expand $(x-2y)^3$.

(A) $x^3 - 8y^3$

(B) $x^3 + 6x^2y + 12xy^2 + 8y^3$

(C) $x^3 - 6x^2y + 12xy^2 - 8y^3$

(D) $x^3 - 6x^2y - 12xy^2 - 8y^3$

Question 21. Which of the following is equal to $a^3 - b^3$?

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

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

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

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

Question 22. Factorise $8x^3 + 27y^3$ using an identity.

(A) $(2x+3y)(4x^2 - 6xy + 9y^2)$

(B) $(2x+3y)(4x^2 + 6xy + 9y^2)$

(C) $(2x-3y)(4x^2 + 6xy + 9y^2)$

(D) $(2x-3y)(4x^2 - 6xy + 9y^2)$

Question 23. If $x+y+z=0$, then $x^3+y^3+z^3$ is equal to:

(A) $xyz$

(B) $3xyz$

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

(D) $0$

Question 24. Without actual calculation, find the value of $(12)^3 + (-7)^3 + (-5)^3$.

(A) $1260$

(B) $0$

(C) $-1260$

(D) $2197$

Question 25. Simplify $(x+1/x)^2$.

(A) $x^2 + 1/x^2$

(B) $x^2 + 2 + 1/x^2$

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

(D) $x^2 + x + 1/x + 1/x^2$

Question 1. Factorise $5x - 15xy$ by taking out the common factor.

(A) $5x(1 - 3y)$

(B) $5(x - 3xy)$

(C) $x(5 - 15y)$

(D) $5xy(1/y - 3)$

Question 2. What is the greatest common factor (GCF) of the terms $12a^2b$ and $18ab^3$?

(A) $6ab$

(B) $6ab^3$

(C) $18a^2b^3$

(D) $36a^2b^3$

Question 3. Factorise $3a(x+y) - b(x+y)$ by grouping.

(A) $(3a+b)(x+y)$

(B) $(3a-b)(x+y)$

(C) $(3a-b)(x-y)$

(D) $(3a+b)(x-y)$

Question 4. Factorise $ab + bc + ax + cx$ by grouping.

(A) $(a+c)(b+x)$

(B) $(a+b)(c+x)$

(C) $(a+x)(b+c)$

(D) $(a+c)(b-x)$

Question 5. Factorise $x^2 - 25$ using an identity.

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

(B) $(x+5)^2$

(C) $(x-5)(x+5)$

(D) $(x-25)(x+1)$

Question 6. Factorise $9a^2 - 30ab + 25b^2$ using an identity.

(A) $(3a+5b)^2$

(B) $(3a-5b)^2$

(C) $(9a-25b)^2$

(D) $(3a-5b)(3a+5b)$

Question 7. Factorise the quadratic trinomial $x^2 + 7x + 12$ by splitting the middle term.

(A) $(x+3)(x+4)$

(B) $(x+2)(x+6)$

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

(D) $(x+1)(x+12)$

Question 8. Factorise $2x^2 + 5x + 3$ by splitting the middle term.

(A) $(2x+1)(x+3)$

(B) $(2x+3)(x+1)$

(C) $(2x-1)(x-3)$

(D) $(2x-3)(x-1)$

Question 9. Factorise $x^3 - 8$ using an identity.

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

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

(C) $(x-2)(x^2-2x+4)$

(D) $(x+2)(x^2+2x+4)$

Question 10. Factorise the cubic polynomial $x^3 - 6x^2 + 11x - 6$, given that $(x-1)$ is a factor.

(A) $(x-1)(x^2 - 5x + 6)$

(B) $(x-1)(x^2 + 5x + 6)$

(C) $(x-1)(x^2 - 5x - 6)$

(D) $(x-1)(x^2 + 5x - 6)$

Question 11. The factors of $x^2 - y^2$ are:

(A) $(x-y), (x-y)$

(B) $(x+y), (x+y)$

(C) $(x-y), (x+y)$

(D) $(x^2), (-y^2)$

Question 12. Factorise $14mn + 21m$ by taking out the common factor.

(A) $7m(2n + 3)$

(B) $7n(2m + 3)$

(C) $7mn(2+3)$

(D) $m(14n + 21)$

Question 13. Factorise $p^2 - 16q^2$ using identities.

(A) $(p-4q)^2$

(B) $(p+4q)^2$

(C) $(p-4q)(p+4q)$

(D) $(p-16q)(p+q)$

Question 14. Factorise $a^2 - 2ab + b^2 - c^2$.

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

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

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

(D) $(a-b-c)(a+b+c)$

Question 15. Factorise $x^2 - 3x - 10$ by splitting the middle term.

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

(B) $(x+5)(x-2)$

(C) $(x-5)(x-2)$

(D) $(x+5)(x+2)$

Question 16. Factorise $6x^2 + 17x + 5$ by splitting the middle term.

(A) $(6x+1)(x+5)$

(B) $(3x+1)(2x+5)$

(C) $(6x+5)(x+1)$

(D) $(2x+1)(3x+5)$

Question 17. Factorise $a^3 + 27b^3$ using identities.

(A) $(a+3b)(a^2 - 3ab + 9b^2)$

(B) $(a+3b)(a^2 + 3ab + 9b^2)$

(C) $(a-3b)(a^2 + 3ab + 9b^2)$

(D) $(a-3b)(a^2 - 3ab + 9b^2)$

Question 18. If $(x-2)$ is a factor of $x^3 - 3x + 2$, the other factors are:

(A) $(x-1), (x+1)$

(B) $(x-1), (x-1)$

(C) $(x+1), (x+1)$

(D) $(x-2), (x-2)$

Question 19. Factorise $xy - yz + xz - z^2$ by grouping.

(A) $(y+z)(x-z)$

(B) $(y-z)(x+z)$

(C) $(y+x)(z-y)$

(D) $(y-x)(z+y)$

Question 20. Factorise $a^4 - b^4$ using identities.

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

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

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

(D) $(a-b)(a+b)(a-b)(a+b)$

Question 21. The factors of $m^2 - 7m + 12$ are:

(A) $(m-3)(m-4)$

(B) $(m+3)(m+4)$

(C) $(m-3)(m+4)$

(D) $(m+3)(m-4)$

Question 22. Factorise $x^3 + x^2 - x - 1$ by grouping.

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

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

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

(D) $(x+1)(x-1)^2$

Question 23. Factorise $121a^2 - 64b^2$ using identities.

(A) $(11a-8b)(11a+8b)$

(B) $(11a-8b)^2$

(C) $(121a-64b)(a+b)$

(D) $(11a+8b)^2$

Question 24. One factor of $a^2+b^2+2ab+2bc+2ca$ is $(a+b+c)$. The other factor is:

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

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

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

(D) $(-a-b-c)$

Question 25. If $(x+1)$ is a factor of $x^2 - kx - 5$, the value of $k$ is:

(A) $-6$

(B) $6$

(C) $-4$

(D) $4$

Question 1. Which of the following is a linear equation in one variable?

(A) $2x + 3y = 5$

(B) $x^2 - 4 = 0$

(C) $3x - 7 = 0$

(D) $\frac{1}{x} + 2 = 3$

Question 2. The solution or root of a linear equation in one variable is the value of the variable that makes the equation:

(A) Greater than zero.

(B) Less than zero.

(C) True.

(D) False.

Question 3. Solve the equation $5x - 10 = 15$.

(A) $x = 1$

(B) $x = 3$

(C) $x = 5$

(D) $x = -1$

Question 4. Using the rule of transposition, if a term is added on one side of an equation, it becomes ________ on the other side.

(A) Added

(B) Subtracted

(C) Multiplied

(D) Divided

Question 5. Solve for $y$: $\frac{y}{2} + 5 = 9$.

(A) $y = 4$

(B) $y = 8$

(C) $y = 10$

(D) $y = 18$

Question 6. The sum of two consecutive integers is 45. If the smaller integer is $x$, which equation represents this?

(A) $x + x + 1 = 45$

(B) $x + x + 2 = 45$

(C) $x + 45 = x + 1$

(D) $2x = 45$

Question 7. A number decreased by 7 is 18. Find the number. Let the number be $n$.

(A) $n - 7 = 18 \implies n = 25$

(B) $n + 7 = 18 \implies n = 11$

(C) $7 - n = 18 \implies n = -11$

(D) $18 - n = 7 \implies n = 11$

Question 8. The perimeter of a rectangle is 30 cm. If the length is 10 cm and the breadth is $b$ cm, which equation can be used to find $b$?

(A) $10 + b = 30$

(B) $2(10) + 2b = 30$

(C) $10b = 30$

(D) $10 + b = 15$

Question 9. A father's age is 5 years more than three times his son's age. If the son's age is $s$ years and the father's age is 44 years, form an equation.

(A) $3s = 44 + 5$

(B) $3s + 5 = 44$

(C) $s + 5 = 3 \times 44$

(D) $s + 3 \times 5 = 44$

Question 10. Solve: $2(x+3) = 10$.

(A) $x = 2$

(B) $x = 3$

(C) $x = 4$

(D) $x = 5$

Question 11. The sum of three consecutive odd numbers is 51. What are the numbers? Let the first odd number be $2n+1$.

(A) 15, 17, 19

(B) 17, 19, 21

(C) 13, 15, 17

(D) 11, 13, 15

Question 12. If you multiply a number by 4 and subtract 5, you get 15. What is the number? Let the number be $N$.

(A) $4N - 5 = 15 \implies 4N = 20 \implies N = 5$

(B) $5 - 4N = 15 \implies -4N = 10 \implies N = -2.5$

(C) $4(N - 5) = 15 \implies 4N - 20 = 15 \implies 4N = 35 \implies N = 8.75$

(D) $4N + 5 = 15 \implies 4N = 10 \implies N = 2.5$

Question 13. Solve for $x$: $3x + 2 = x + 8$.

(A) $x = 3$

(B) $x = 4$

(C) $x = 5$

(D) $x = 6$

Question 14. The cost of a notebook is twice the cost of a pen. If the total cost of one notebook and one pen is $\textsf{₹}45$, what is the cost of the pen? Let the cost of the pen be $p$.

(A) $\textsf{₹}10$

(B) $\textsf{₹}15$

(C) $\textsf{₹}20$

(D) $\textsf{₹}30$

Question 15. A number is such that when you add 10 to it, you get 35. What is the number? Let the number be $n$.

(A) $n + 10 = 35 \implies n = 25$

(B) $n - 10 = 35 \implies n = 45$

(C) $10 - n = 35 \implies n = -25$

(D) $10n = 35 \implies n = 3.5$

Question 16. Solve for $x$: $\frac{x}{3} - \frac{x}{4} = 1$.

(A) $x = 12$

(B) $x = 1$

(C) $x = 7$

(D) $x = -12$

Question 17. If the difference between two numbers is 5 and the larger number is $y$, which equation represents the smaller number in terms of $y$?

(A) $y + 5$

(B) $y - 5$

(C) $5 - y$

(D) $5y$

Question 18. The perimeter of a triangle is 24 cm. If two sides are $x$ cm and $(x+2)$ cm, and the third side is 10 cm, find $x$.

(A) $x + x + 2 + 10 = 24 \implies 2x + 12 = 24 \implies 2x = 12 \implies x = 6$

(B) $x + x + 2 = 24 + 10$

(C) $x(x+2)(10) = 24$

(D) $x + x + 12 = 24$

Question 19. Solve for $m$: $0.5m - 1.2 = 0.3m + 0.8$.

(A) $m = 10$

(B) $m = 1$

(C) $m = 2$

(D) $m = 0.1$

Question 20. A sum of $\textsf{₹}1000$ is divided between Ravi and Mohan. If Ravi gets $\textsf{₹}x$, then Mohan gets $\textsf{₹}(1000-x)$. If Ravi gets $\textsf{₹}200$ more than Mohan, form an equation.

(A) $x = (1000-x) + 200$

(B) $x + (1000-x) = 200$

(C) $(1000-x) = x + 200$

(D) $x + 200 = 1000$

Question 21. Solve for $p$: $\frac{2p+1}{3} = 5$.

(A) $p = 7$

(B) $p = 8$

(C) $p = 6$

(D) $p = 5$

Question 1. Which of the following is a linear equation in two variables?

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

(B) $2x - 3 = 0$

(C) $x y = 10$

(D) $4x + 5y = 20$

Question 2. A linear equation in two variables has how many solutions?

(A) Exactly one solution.

(B) Exactly two solutions.

(C) No solution.

(D) Infinitely many solutions.

Question 3. If $(1, 2)$ is a solution to the equation $ax + by = 5$, which equation can represent the relationship between $a$ and $b$?

(A) $a + 2b = 5$

(B) $2a + b = 5$

(C) $a \times 1 + b \times 2$

(D) $a+b=5$

Question 4. The cost of 2 pens and 3 pencils is $\textsf{₹}30$. Form a linear equation in two variables to represent this, where $x$ is the cost of a pen and $y$ is the cost of a pencil.

(A) $2x + 3y = 30$

(B) $3x + 2y = 30$

(C) $5xy = 30$

(D) $2x = 3y + 30$

Question 5. The standard form of a linear equation in two variables is $Ax + By + C = 0$. What is $A$ in the equation $y = 2x - 5$?

(A) $1$

(B) $-1$

(C) $2$

(D) $-2$

Question 6. For the equation $x + y = 7$, which ordered pair is a solution?

(A) $(0, 0)$

(B) $(1, 7)$

(C) $(3, 4)$

(D) $(7, -1)$

Question 7. The graph of a linear equation in two variables is always a:

(A) Curve

(B) Straight line

(C) Parabola

(D) Point

Question 8. Which point lies on the graph of the equation $2x - y = 4$?

(A) $(0, 0)$

(B) $(2, 0)$

(C) $(0, 2)$

(D) $(1, 1)$

Question 9. The equation of the x-axis is:

(A) $x = 0$

(B) $y = 0$

(C) $x = y$

(D) $x + y = 0$

Question 10. The equation of the y-axis is:

(A) $x = 0$

(B) $y = 0$

(C) $x = y$

(D) $x + y = 0$

Question 11. The graph of the equation $x = 5$ is a line parallel to which axis?

(A) x-axis

(B) y-axis

(C) Both axes

(D) Neither axis

Question 12. The graph of the equation $y = -3$ is a line parallel to which axis?

(A) x-axis

(B) y-axis

(C) Both axes

(D) Neither axis

Question 13. The point where the graph of $x + y = 4$ intersects the x-axis is:

(A) $(0, 4)$

(B) $(4, 0)$

(C) $(0, 0)$

(D) $(2, 2)$

Question 14. The equation of a line passing through the origin is of the form:

(A) $Ax + By = C$ where $C \neq 0$

(B) $Ax + By = 0$

(C) $x = k$

(D) $y = k$

Question 15. How many points are needed to draw the graph of a linear equation in two variables?

(A) Only one point.

(B) Exactly two points are sufficient.

(C) At least three points are recommended for accuracy.

(D) Infinitely many points are needed.

Question 16. Convert the statement "Twice a number $x$ plus three times a number $y$ is 12" into a linear equation in two variables.

(A) $2x + 3y = 12$

(B) $3x + 2y = 12$

(C) $xy + 5 = 12$

(D) $2x(3y) = 12$

Question 17. Which of the following is a solution for $2x + y = 6$?

(A) $(3, 0)$

(B) $(0, 3)$

(C) $(1, 5)$

(D) $(4, -2)$

Question 18. The graph of $y = 2x$ passes through which point?

(A) $(1, 0)$

(B) $(0, 2)$

(C) $(2, 1)$

(D) $(1, 2)$

Question 19. The equation $Ax + By + C = 0$ represents a linear equation in two variables if:

(A) $A=0, B=0$

(B) $A=0$ or $B=0$ (but not both)

(C) $A \neq 0$ and $B \neq 0$

(D) $A \neq 0$ or $B \neq 0$ (or both)

Question 20. Write the equation $3x = 5y - 7$ in the standard form $Ax + By + C = 0$ and find the value of $C$.

(A) $C = -7$

(B) $C = 7$

(C) $C = 3$

(D) $C = -5$

Question 21. A line parallel to the x-axis at a distance of 4 units above it has the equation:

(A) $x = 4$

(B) $x = -4$

(C) $y = 4$

(D) $y = -4$

Question 22. A line parallel to the y-axis passing through the point $(-2, 5)$ has the equation:

(A) $x = -2$

(B) $x = 5$

(C) $y = -2$

(D) $y = 5$

Question 23. The equation $y=0$ represents:

(A) The x-axis

(B) The y-axis

(C) A line parallel to the x-axis

(D) A line parallel to the y-axis

Question 24. If the equation of a line is $3x + 0y = 6$, it simplifies to $3x=6$ or $x=2$. This is a line parallel to the:

(A) x-axis

(B) y-axis

(C) line $y=x$

(D) line $y=-x$

Question 1. A pair of linear equations in two variables represents geometrically:

(A) A single point.

(B) A single line.

(C) A pair of parabolas.

(D) A pair of lines.

Question 2. If a system of linear equations has a unique solution, the lines represented by the equations are:

(A) Intersecting.

(B) Parallel.

(C) Coincident.

(D) Perpendicular.

Question 3. A pair of linear equations is called 'consistent' if it has:

(A) No solution.

(B) Exactly one solution.

(C) Infinitely many solutions.

(D) At least one solution.

Question 4. If the lines represented by a pair of linear equations are parallel, the system is:

(A) Consistent with unique solution.

(B) Consistent with infinitely many solutions.

(C) Inconsistent.

(D) Cannot be determined.

Question 5. For a system $\begin{cases} a_1x + b_1y + c_1 = 0 \\ a_2x + b_2y + c_2 = 0 \end{cases}$, if $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$, the lines are:

(A) Intersecting.

(B) Parallel.

(C) Coincident.

(D) Perpendicular.

Question 6. For a system $\begin{cases} a_1x + b_1y + c_1 = 0 \\ a_2x + b_2y + c_2 = 0 \end{cases}$, if $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$, the system has:

(A) Unique solution.

(B) No solution.

(C) Infinitely many solutions.

(D) Exactly two solutions.

Question 7. The graphical method of solving a pair of linear equations involves finding the point(s) of intersection of the lines. If the lines intersect at a single point, the solution is:

(A) No solution.

(B) Infinitely many solutions.

(C) A unique solution.

(D) Two solutions.

Question 8. Solve the system using substitution method: $\begin{cases} x + y = 5 \\ x - y = 1 \end{cases}$

(A) $x=3, y=2$

(B) $x=2, y=3$

(C) $x=4, y=1$

(D) $x=1, y=4$

Question 9. Solve the system using elimination method: $\begin{cases} 2x + y = 7 \\ x - y = 2 \end{cases}$

(A) $x=3, y=1$

(B) $x=1, y=3$

(C) $x=2, y=3$

(D) $x=3, y=2$

Question 10. For which value of $k$ does the system $\begin{cases} x + 2y = 3 \\ 5x + ky = 15 \end{cases}$ have infinitely many solutions?

(A) $k = 10$

(B) $k = 5$

(C) $k = 3$

(D) $k = -10$

Question 11. For which value of $p$ does the system $\begin{cases} 4x + py = 8 \\ 2x + 2y = 5 \end{cases}$ have no solution?

(A) $p = 4$

(B) $p = -4$

(C) $p = 2$

(D) $p = 5$

Question 12. Which algebraic method involves making the coefficient of one variable the same in both equations?

(A) Substitution method

(B) Elimination method

(C) Cross-multiplication method

(D) Graphical method

Question 13. A system of equations is called 'inconsistent' if it has:

(A) Unique solution.

(B) Infinitely many solutions.

(C) No solution.

(D) At least one solution.

Question 14. If the lines are coincident, the system of equations is:

(A) Consistent with unique solution.

(B) Consistent with infinitely many solutions.

(C) Inconsistent.

(D) Dependent.

Question 15. Solve for $x$ in the system: $\begin{cases} 3x - y = 4 \\ 9x - 3y = 12 \end{cases}$

(A) Unique solution $x=1, y=-1$

(B) Unique solution $x=2, y=2$

(C) No solution

(D) Infinitely many solutions

Question 16. The system $\begin{cases} 2x + 3y = 5 \\ 4x + 6y = 10 \end{cases}$ represents which type of lines?

(A) Intersecting

(B) Parallel

(C) Coincident

(D) Perpendicular

Question 17. Consider the system $\begin{cases} x + y = 5 \\ 2x + 2y = 10 \end{cases}$. How many solutions does it have?

(A) One

(B) Two

(C) None

(D) Infinitely many

Question 18. The point $(2, 1)$ is a solution to which system of equations?

(A) $\begin{cases} x + y = 3 \\ x - y = 1 \end{cases}$

(B) $\begin{cases} x + y = 3 \\ x + y = 1 \end{cases}$

(C) $\begin{cases} x - y = 3 \\ x + y = 1 \end{cases}$

(D) $\begin{cases} x - y = 1 \\ x + y = 4 \end{cases}$

Question 19. Equations like $\frac{1}{x} + \frac{1}{y} = 5$ and $\frac{2}{x} - \frac{3}{y} = -1$ can be reduced to a pair of linear equations by substituting:

(A) $u = x, v = y$

(B) $u = \frac{1}{x}, v = \frac{1}{y}$

(C) $u = xy, v = x/y$

(D) $u = x+y, v = x-y$

Question 20. What is the condition for a system of linear equations to have a unique solution?

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

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

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

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

Question 21. Solve the system $\begin{cases} \frac{1}{x} + \frac{1}{y} = 5 \\ \frac{1}{x} - \frac{1}{y} = 1 \end{cases}$. (Hint: Let $u = 1/x, v = 1/y$).

(A) $x=1/3, y=1/2$

(B) $x=1/2, y=1/3$

(C) $x=3, y=2$

(D) $x=2, y=3$

Question 22. Two lines given by $2x + 3y = 7$ and $(a+b)x + (2a-b)y = 21$ are coincident. Find the values of $a$ and $b$.

(A) $a=1, b=5$

(B) $a=5, b=1$

(C) $a=-1, b=-5$

(D) $a=-5, b=-1$

Question 23. The solution to a system of linear equations found using the graphical method is the coordinate(s) of the point(s) where the lines meet. If the lines are parallel, they don't meet, so there is:

(A) A unique solution.

(B) Infinitely many solutions.

(C) No solution.

(D) Exactly two solutions.

Question 24. The system of equations $x - 2y = 0$ and $3x + 4y = 20$ has a unique solution. Which point represents this solution?

(A) $(4, 2)$

(B) $(2, 4)$

(C) $(0, 0)$

(D) $(10, 5)$

Question 25. If the lines are parallel, the number of common points is:

(A) $1$

(B) $0$

(C) Infinite

(D) $2$

Question 1. Which of the following is a quadratic equation?

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

(B) $x^2 + \frac{1}{x} + 2 = 0$

(C) $2x^2 - 5x + 3 = 0$

(D) $x + 1 = 0$

Question 2. The general form of a quadratic equation is $ax^2 + bx + c = 0$. What is the necessary condition for this to be a quadratic equation?

(A) $a = 0$

(B) $a \neq 0$

(C) $b = 0$

(D) $c = 0$

Question 3. For the quadratic equation $2x^2 - 3x + 1 = 0$, what are the values of $a$, $b$, and $c$?

(A) $a=2, b=3, c=1$

(B) $a=2, b=-3, c=1$

(C) $a=2, b=-3, c=-1$

(D) $a=-2, b=3, c=-1$

Question 4. The roots of a quadratic equation are the values of the variable that:

(A) Make the expression positive.

(B) Make the expression negative.

(C) Satisfy the equation (make it true).

(D) Make the expression zero.

Question 5. The sum of the roots of the quadratic equation $ax^2 + bx + c = 0$ is:

(A) $b/a$

(B) $-b/a$

(C) $c/a$

(D) $-c/a$

Question 6. The product of the roots of the quadratic equation $ax^2 + bx + c = 0$ is:

(A) $b/a$

(B) $-b/a$

(C) $c/a$

(D) $-c/a$

Question 7. Find the sum of the roots of $x^2 - 5x + 6 = 0$.

(A) $5$

(B) $-5$

(C) $6$

(D) $-6$

Question 8. Find the product of the roots of $2x^2 + 7x - 15 = 0$.

(A) $7/2$

(B) $-7/2$

(C) $-15/2$

(D) $15/2$

Question 9. Solve the equation $x^2 - 4x = 0$ by factorisation.

(A) $x=0, x=4$

(B) $x=0, x=-4$

(C) $x=4, x=-4$

(D) $x=0, x=0$

Question 10. Solve $x^2 - 5x + 6 = 0$ by factorisation.

(A) $x=2, x=3$

(B) $x=-2, x=-3$

(C) $x=1, x=6$

(D) $x=-1, x=6$

Question 11. To solve $x^2 + 6x - 7 = 0$ by completing the square, what term should be added to both sides after rearranging to $x^2 + 6x = 7$?

(A) $3^2 = 9$

(B) $6^2 = 36$

(C) $(6/2)^2 = 9$

(D) $(-7/2)^2 = 49/4$

Question 12. Solve $x^2 - 4x + 4 = 0$ by completing the square.

(A) $(x-2)^2 = 0 \implies x=2$ (repeated)

(B) $(x+2)^2 = 0 \implies x=-2$ (repeated)

(C) $(x-4)^2 = 0 \implies x=4$ (repeated)

(D) $(x+4)^2 = 0 \implies x=-4$ (repeated)

Question 13. The quadratic formula to find the roots of $ax^2 + bx + c = 0$ is $x = \frac{-b \pm \sqrt{D}}{2a}$, where $D$ is the discriminant. What is the formula for $D$?

(A) $D = b^2 + 4ac$

(B) $D = b^2 - 4ac$

(C) $D = \sqrt{b^2 - 4ac}$

(D) $D = -b^2 + 4ac$

Question 14. Use the quadratic formula to find the roots of $x^2 + 5x + 6 = 0$.

(A) $x = \frac{-5 \pm \sqrt{25 - 24}}{2} = \frac{-5 \pm 1}{2} \implies x = -2, x = -3$

(B) $x = \frac{5 \pm \sqrt{25 - 24}}{2} = \frac{5 \pm 1}{2} \implies x = 3, x = 2$

(C) $x = \frac{-5 \pm \sqrt{25 + 24}}{2} = \frac{-5 \pm 7}{2} \implies x = 1, x = -6$

(D) $x = \frac{5 \pm \sqrt{25 + 24}}{2} = \frac{5 \pm 7}{2} \implies x = 6, x = -1$

Question 15. If the discriminant $D > 0$, the roots of a quadratic equation are:

(A) Real and distinct.

(B) Real and equal.

(C) Not real (complex).

(D) Cannot be determined.

Question 16. If the discriminant $D = 0$, the roots of a quadratic equation are:

(A) Real and distinct.

(B) Real and equal.

(C) Not real (complex).

(D) Always zero.

Question 17. If the discriminant $D < 0$, the roots of a quadratic equation are:

(A) Real and distinct.

(B) Real and equal.

(C) Not real (complex).

(D) Always positive.

Question 18. Determine the nature of the roots of the equation $x^2 - 4x + 4 = 0$.

(A) Real and distinct.

(B) Real and equal.

(C) Not real.

(D) Real and zero.

Question 19. Determine the nature of the roots of the equation $x^2 + x + 1 = 0$.

(A) Real and distinct.

(B) Real and equal.

(C) Not real.

(D) Rational and distinct.

Question 20. For what value of $k$ does the equation $kx^2 + 4x + 1 = 0$ have real and equal roots?

(A) $k = 4$

(B) $k = -4$

(C) $k = 0$

(D) $k = 16$

Question 21. If the roots of a quadratic equation are $2$ and $-3$, the equation is:

(A) $x^2 - x - 6 = 0$

(B) $x^2 + x - 6 = 0$

(C) $x^2 - 5x - 6 = 0$

(D) $x^2 + 5x - 6 = 0$

Question 22. Solve $x^2 - 9 = 0$ by factorisation.

(A) $x=3, x=-3$

(B) $x=3$ (repeated)

(C) $x=9, x=-9$

(D) $x=9$ (repeated)

Question 23. What constant term should be added to $x^2 - 10x$ to make it a perfect square?

(A) $25$

(B) $100$

(C) $10$

(D) $-25$

Question 24. The equation $x^2 + bx + c = 0$ has roots $\alpha$ and $\beta$. Which statement is correct?

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

(B) $\alpha + \beta = -b$, $\alpha \beta = c$

(C) $\alpha + \beta = b/a$, $\alpha \beta = c/a$

(D) $\alpha + \beta = -b/a$, $\alpha \beta = c/a$

Question 25. If one root of the equation $x^2 - 7x + k = 0$ is 3, what is the value of $k$?

(A) $10$

(B) $12$

(C) $-10$

(D) $-12$

Question 1. The imaginary unit $i$ is defined as:

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

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

(C) $i = -1$

(D) $i = 0$

Question 2. A complex number is generally written in the form $a + bi$, where $a$ and $b$ are:

(A) Imaginary numbers.

(B) Real numbers.

(C) Rational numbers.

(D) Integer numbers.

Question 3. In the complex number $3 - 4i$, the real part is:

(A) $3$

(B) $-4$

(C) $i$

(D) $-4i$

Question 4. In the complex number $3 - 4i$, the imaginary part is:

(A) $3$

(B) $-4$

(C) $i$

(D) $-4i$

Question 5. What is the sum of the complex numbers $(2 + 3i)$ and $(4 - 5i)$?

(A) $6 - 2i$

(B) $6 + 2i$

(C) $-2 - 2i$

(D) $-2 + 8i$

Question 6. Subtract $(1 - i)$ from $(3 + 2i)$.

(A) $2 + 3i$

(B) $2 - 3i$

(C) $-2 - 3i$

(D) $-2 + 3i$

Question 7. Multiply the complex numbers $(2 + i)$ and $(3 - i)$.

(A) $6 - i$

(B) $6 + i$

(C) $7 + i$

(D) $7 - i$

Question 8. What is $i^2$ equal to?

(A) $1$

(B) $-1$

(C) $i$

(D) $-i$

Question 9. What is $i^3$ equal to?

(A) $1$

(B) $-1$

(C) $i$

(D) $-i$

Question 10. What is $i^4$ equal to?

(A) $1$

(B) $-1$

(C) $i$

(D) $-i$

Question 11. Simplify $i^{17}$.

(A) $1$

(B) $-1$

(C) $i$

(D) $-i$

Question 12. Simplify $i^{-9}$.

(A) $i$

(B) $-i$

(C) $1$

(D) $-1$

Question 13. What is the reciprocal of the complex number $i$?

(A) $i$

(B) $-i$

(C) $1/i$

(D) $-1$

Question 14. Simplify $(1+i)^2$.

(A) $1 + i^2$

(B) $2i$

(C) $1+2i-1$

(D) $2$

Question 15. Simplify $\frac{1}{1+i}$.

(A) $\frac{1-i}{2}$

(B) $\frac{1+i}{2}$

(C) $1-i$

(D) $1+i$

Question 16. If $z_1 = 2+3i$ and $z_2 = 1-i$, what is $z_1 + z_2$?

(A) $3+2i$

(B) $3-2i$

(C) $1+4i$

(D) $1-4i$

Question 17. If $z_1 = 2+3i$ and $z_2 = 1-i$, what is $z_1 - z_2$?

(A) $1+4i$

(B) $1-4i$

(C) $3+2i$

(D) $3-2i$

Question 18. What is $i^{4n+1}$ where $n$ is an integer?

(A) $1$

(B) $-1$

(C) $i$

(D) $-i$

Question 19. The identity $(z_1+z_2)^2 = z_1^2 + 2z_1z_2 + z_2^2$ holds true for complex numbers $z_1$ and $z_2$. This property is called:

(A) Commutativity of addition

(B) Associativity of multiplication

(C) Distributivity

(D) The Binomial Theorem (for exponent 2)

Question 20. What is the simplified form of $\sqrt{-4}$?

(A) $2$

(B) $-2$

(C) $2i$

(D) $-2i$

Question 21. If $a+bi = c+di$, where $a, b, c, d$ are real numbers, then:

(A) $a=c$ and $b=-d$

(B) $a=d$ and $b=c$

(C) $a=c$ and $b=d$

(D) $a=-c$ and $b=-d$

Question 22. If $(x+y) + i(x-y) = 5 + 3i$, find the values of $x$ and $y$.

(A) $x=4, y=1$

(B) $x=1, y=4$

(C) $x=5, y=3$

(D) $x=3, y=5$

Question 23. The sum of a complex number $z$ and its conjugate $\bar{z}$ is always:

(A) A purely real number.

(B) A purely imaginary number.

(C) Zero.

(D) $2i \times (\text{Imaginary part of } z)$.

Question 24. What is the additive inverse of the complex number $2 - 5i$?

(A) $-2 - 5i$

(B) $2 + 5i$

(C) $-2 + 5i$

(D) $-2$

Question 25. What is the multiplicative inverse of the complex number $i$?

(A) $i$

(B) $-i$

(C) $1$

(D) $-1$

Question 1. The Argand plane is used for the geometric representation of complex numbers. In the Argand plane, the x-axis represents the:

(A) Imaginary part.

(B) Real part.

(C) Modulus.

(D) Argument.

Question 2. In the Argand plane, the y-axis represents the:

(A) Imaginary part.

(B) Real part.

(C) Modulus.

(D) Argument.

Question 3. The complex number $2 - 3i$ is represented by which point in the Argand plane?

(A) $(2, 3)$

(B) $(-2, 3)$

(C) $(2, -3)$

(D) $(-2, -3)$

Question 4. The modulus of a complex number $z = a + bi$ is defined as:

(A) $\sqrt{a^2 + b^2}$

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

(C) $\sqrt{a^2 - b^2}$

(D) $a+b$

Question 5. Find the modulus of the complex number $3 + 4i$.

(A) $5$

(B) $7$

(C) $25$

(D) $\sqrt{7}$

Question 6. The conjugate of a complex number $z = a + bi$ is:

(A) $a - bi$

(B) $-a + bi$

(C) $-a - bi$

(D) $b + ai$

Question 7. Find the conjugate of the complex number $5 - 2i$.

(A) $5 + 2i$

(B) $-5 - 2i$

(C) $-5 + 2i$

(D) $2 - 5i$

Question 8. For any complex number $z$, $z \cdot \bar{z}$ is equal to:

(A) $|z|$

(B) $|z|^2$

(C) $z^2$

(D) $\bar{z}^2$

Question 9. The representation of a complex number $z = r(\cos \theta + i \sin \theta)$ is called the:

(A) Cartesian form.

(B) Algebraic form.

(C) Polar form.

(D) Exponential form.

Question 10. In the polar form $r(\cos \theta + i \sin \theta)$, $r$ represents the:

(A) Real part.

(B) Imaginary part.

(C) Modulus.

(D) Argument.

Question 11. In the polar form $r(\cos \theta + i \sin \theta)$, $\theta$ represents the:

(A) Real part.

(B) Imaginary part.

(C) Modulus.

(D) Argument.

Question 12. Find the square root of $i$.

(A) $\pm \frac{1}{\sqrt{2}}(1+i)$

(B) $\pm \frac{1}{\sqrt{2}}(1-i)$

(C) $\pm (1+i)$

(D) $\pm (1-i)$

Question 13. What is the distance of the point representing $a+bi$ from the origin in the Argand plane?

(A) $a+b$

(B) $\sqrt{a^2+b^2}$

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

(D) $|a|+|b|$

Question 14. The conjugate of the conjugate of a complex number $z$ is:

(A) $z$

(B) $-z$

(C) $\bar{z}$

(D) $-\bar{z}$

Question 15. If $z$ is a purely real number, then its imaginary part is $0$. In the Argand plane, purely real numbers lie on the:

(A) Real axis.

(B) Imaginary axis.

(C) Origin.

(D) First quadrant.

Question 16. If $z$ is a purely imaginary number, then its real part is $0$. In the Argand plane, purely imaginary numbers lie on the:

(A) Real axis.

(B) Imaginary axis.

(C) Origin (unless it's $0$).

(D) Fourth quadrant.

Question 17. The modulus of the complex number $z = -1 + i$ is:

(A) $1$

(B) $\sqrt{2}$

(C) $2$

(D) $0$

Question 18. The argument of the complex number $z = 1 + i$ in the range $(-\pi, \pi]$ is:

(A) $\pi/4$

(B) $3\pi/4$

(C) $-\pi/4$

(D) $\pi/2$

Question 19. The polar form of the complex number $z = -1 - i$ is:

(A) $\sqrt{2}(\cos(\pi/4) + i \sin(\pi/4))$

(B) $\sqrt{2}(\cos(-\pi/4) + i \sin(-\pi/4))$

(C) $\sqrt{2}(\cos(-3\pi/4) + i \sin(-3\pi/4))$

(D) $\sqrt{2}(\cos(3\pi/4) + i \sin(3\pi/4))$

Question 20. If $|z| = 5$, the locus of $z$ in the Argand plane is a:

(A) Straight line.

(B) Circle with radius 5 centered at the origin.

(C) Circle with radius $\sqrt{5}$ centered at the origin.

(D) Point.

Question 21. The square root of $-9$ is:

(A) $\pm 3$

(B) $\pm 3i$

(C) $3i$

(D) $-3i$

Question 22. The property $|z_1 z_2| = |z_1| |z_2|$ states that the modulus of the product of two complex numbers is the product of their moduli. This is a property of:

(A) Conjugates.

(B) Moduli.

(C) Arguments.

(D) Real parts.

Question 23. The product of a complex number and its conjugate is always a:

(A) Purely imaginary number.

(B) Negative real number.

(C) Positive real number (or zero).

(D) Complex number with non-zero imaginary part.

Question 24. If $z$ is a complex number such that $z = \bar{z}$, then $z$ is:

(A) Purely imaginary.

(B) Purely real.

(C) Zero.

(D) Not possible for any complex number.

Question 25. The principal argument of a complex number is usually taken in the interval:

(A) $(0, 2\pi]$

(B) $[0, 2\pi)$

(C) $(-\pi, \pi]$

(D) $[-\pi, \pi)$

Question 1. A quadratic equation $ax^2 + bx + c = 0$ with real coefficients ($a, b, c \in \mathbb{R}$) has complex roots if its discriminant ($D$) is:

(A) $D > 0$

(B) $D = 0$

(C) $D < 0$

(D) $D \ge 0$

Question 2. If a quadratic equation with real coefficients has complex roots, these roots always occur as:

(A) Real numbers.

(B) Purely imaginary numbers.

(C) Conjugate pairs.

(D) The same repeated root.

Question 3. Find the roots of the equation $x^2 + 1 = 0$.

(A) $x = 1, x = -1$

(B) $x = i, x = -i$

(C) $x = 1$ (repeated)

(D) $x = -1$ (repeated)

Question 4. Solve the quadratic equation $x^2 + x + 1 = 0$.

(A) $x = \frac{-1 \pm \sqrt{3}i}{2}$

(B) $x = \frac{1 \pm \sqrt{3}i}{2}$

(C) $x = \frac{-1 \pm i}{2}$

(D) $x = \frac{-1 \pm \sqrt{5}}{2}$

Question 5. If one root of a quadratic equation with real coefficients is $2 + 3i$, what is the other root?

(A) $2 - 3i$

(B) $-2 + 3i$

(C) $-2 - 3i$

(D) $3 + 2i$

Question 6. Find the sum and product of the roots of $x^2 - 2x + 5 = 0$.

(A) Sum = 2, Product = 5

(B) Sum = -2, Product = 5

(C) Sum = 2, Product = -5

(D) Sum = -2, Product = -5

Question 7. Solve the equation $x^2 - 6x + 13 = 0$.

(A) $x = 3 \pm 2i$

(B) $x = 3 \pm \sqrt{2}i$

(C) $x = -3 \pm 2i$

(D) $x = -3 \pm \sqrt{2}i$

Question 8. The discriminant of $ax^2 + bx + c = 0$ is $-16$. The roots are:

(A) Real and distinct.

(B) Real and equal.

(C) Complex and conjugate.

(D) Real and zero.

Question 9. Form the quadratic equation whose roots are $1 + i$ and $1 - i$.

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

Product of roots $= (1+i)(1-i) = 1 - i^2 = 1 - (-1) = 2$

Equation is $x^2 - (\text{Sum})x + (\text{Product}) = 0$

(A) $x^2 - 2x + 2 = 0$

(B) $x^2 + 2x + 2 = 0$

(C) $x^2 - 2x - 2 = 0$

(D) $x^2 + 2x - 2 = 0$

Question 10. Solve the equation $4x^2 + 9 = 0$ for complex roots.

(A) $x = \pm \frac{3}{2}$

(B) $x = \pm \frac{2}{3}i$

(C) $x = \pm \frac{3}{2}i$

(D) $x = \pm \frac{2}{3}$

Question 11. If one root of $ax^2 + bx + c = 0$ (with real coefficients) is $p - qi$, where $q \neq 0$, then the other root must be:

(A) $p + qi$

(B) $-p + qi$

(C) $p - qi$ (repeated)

(D) $-p - qi$

Question 12. Find the roots of $x^2 + 2x + 2 = 0$.

(A) $x = -1 \pm i$

(B) $x = 1 \pm i$

(C) $x = -1 \pm 2i$

(D) $x = 1 \pm 2i$

Question 13. If the roots of a quadratic equation are $3i$ and $-3i$, the equation is:

(A) Sum of roots $= 3i + (-3i) = 0$

Product of roots $= (3i)(-3i) = -9i^2 = -9(-1) = 9$

(A) $x^2 + 9 = 0$

(B) $x^2 - 9 = 0$

(C) $x^2 + 6ix - 9 = 0$

(D) $x^2 - 6ix - 9 = 0$

Question 14. The discriminant of $x^2 - 4x + 8 = 0$ is:

(A) $16 - 32 = -16$

(B) $16 + 32 = 48$

(C) $-16$

(D) $16$

Question 15. For the equation $2x^2 - x + 1 = 0$, the roots are:

(A) Real and distinct.

(B) Real and equal.

(C) Complex conjugates.

(D) Real and irrational.

Question 16. Find the roots of $\sqrt{2}x^2 - \sqrt{5}x + \sqrt{2} = 0$.

(A) $\frac{\sqrt{5} \pm \sqrt{5-8}}{2\sqrt{2}} = \frac{\sqrt{5} \pm \sqrt{-3}}{2\sqrt{2}} = \frac{\sqrt{5} \pm \sqrt{3}i}{2\sqrt{2}}$

(A) Real and distinct.

(B) Real and equal.

(C) Complex conjugates.

(D) Purely imaginary.

Question 17. Solve $x^2 - (\sqrt{2} + 1)x + \sqrt{2} = 0$.

(A) Real roots: $x=\sqrt{2}, x=1$

(B) Complex roots.

(C) Real equal roots.

(D) No roots.

Question 18. If $D < 0$ for $ax^2+bx+c=0$, the parabola $y=ax^2+bx+c$:

(A) Intersects the x-axis at two points.

(B) Touches the x-axis at one point.

(C) Does not intersect the x-axis.

(D) Is a straight line.

Question 19. Form the quadratic equation with real coefficients whose roots are $3 \pm 2i$.

(A) Sum of roots $= (3+2i) + (3-2i) = 6$

Product of roots $= (3+2i)(3-2i) = 3^2 - (2i)^2 = 9 - 4i^2 = 9 - 4(-1) = 9 + 4 = 13$

(A) $x^2 - 6x + 13 = 0$

(B) $x^2 + 6x + 13 = 0$

(C) $x^2 - 6x - 13 = 0$

(D) $x^2 + 6x - 13 = 0$

Question 20. The equation $x^2 + ax + b = 0$ has one root $1+i$. If $a$ and $b$ are real, find $a$ and $b$.

(A) Since coefficients are real, the other root is $1-i$.

Sum $= (1+i) + (1-i) = 2 = -a \implies a = -2$

Product $= (1+i)(1-i) = 2 = b \implies b = 2$

(A) $a=-2, b=2$

(B) $a=2, b=-2$

(C) $a=-2, b=-2$

(D) $a=2, b=2$

Question 21. Solve the equation $x^2 + \sqrt{5}x - 5 = 0$.

(A) Real and distinct roots.

(B) Real and equal roots.

(C) Complex roots.

(D) Purely imaginary roots.

Question 22. The nature of the roots of $3x^2 - 2x + 1 = 0$ is:

(A) Real and distinct.

(B) Real and equal.

(C) Complex conjugates.

(D) Real and irrational.

Question 23. Find the roots of $x^2 - (3+i)x + (2+i) = 0$. (Note: Coefficients are not all real).

(A) $x=1, x=2+i$

(B) $x=1+i, x=2$

(C) $x=1, x=2+2i$

(D) $x=i, x=3$

Question 24. The equation $x^2 - px + q = 0$ has roots $\alpha$ and $\beta$. If the roots are complex conjugates, which statement is true about $p$ and $q$ (assuming $p, q$ are real)?

(A) $p^2 - 4q > 0$

(B) $p^2 - 4q = 0$

(C) $p^2 - 4q < 0$

(D) $p$ and $q$ cannot be real.

Question 25. For the equation $x^2 + kx + k = 0$, the roots are complex if:

(A) $k^2 - 4k > 0 \implies k(k-4) > 0 \implies k \in (-\infty, 0) \cup (4, \infty)$

(B) $k^2 - 4k = 0 \implies k=0, k=4$

(C) $k^2 - 4k < 0 \implies k(k-4) < 0 \implies k \in (0, 4)$

(A) $k \in (0, 4)$

(B) $k \in (-\infty, 0) \cup (4, \infty)$

(C) $k = 0$ or $k = 4$

(D) $k > 4$

Question 1. Which of the following is a linear inequality?

(A) $x^2 - 4 < 0$

(B) $2x + 3y \ge 5$

(C) $xy > 1$

(D) $x = 7$

Question 2. The solution set of a linear inequality in one variable can be represented on a:

(A) Coordinate plane.

(B) Number line.

(C) Bar graph.

(D) Pie chart.

Question 3. Solve the linear inequality $2x - 5 > 3$.

(A) $x > 4$

(B) $x < 4$

(C) $x > 8$

(D) $x < 8$

Question 4. When you multiply or divide both sides of an inequality by a negative number, you must:

(A) Keep the inequality sign the same.

(B) Reverse the direction of the inequality sign.

(C) Add the number to both sides.

(D) Subtract the number from both sides.

Question 5. Solve $-3x \le 9$.

(A) $x \le -3$

(B) $x \ge -3$

(C) $x \le 3$

(D) $x \ge 3$

Question 6. The graphical solution of $x \ge 2$ on a number line is:

(A) A ray starting from 2 and extending to the right, with an open circle at 2.

(B) A ray starting from 2 and extending to the right, with a closed circle at 2.

(C) A ray starting from 2 and extending to the left, with an open circle at 2.

(D) A ray starting from 2 and extending to the left, with a closed circle at 2.

Question 7. The graphical solution of $y < 3$ in a two-variable coordinate plane is:

(A) The region above the line $y=3$, including the line.

(B) The region below the line $y=3$, including the line.

(C) The region above the line $y=3$, excluding the line.

(D) The region below the line $y=3$, excluding the line.

Question 8. The graphical solution of $x \ge 0$ in a two-variable coordinate plane is:

(A) The region to the right of the y-axis, excluding the y-axis.

(B) The region to the left of the y-axis, excluding the y-axis.

(C) The region to the right of the y-axis, including the y-axis.

(D) The region to the left of the y-axis, including the y-axis.

Question 9. To graph a linear inequality like $2x + 3y \le 6$, we first graph the boundary line $2x + 3y = 6$. Since the inequality includes '=', the boundary line should be drawn as a:

(A) Dashed line.

(B) Solid line.

(C) Thick line.

(D) Dotted line.

Question 10. The solution region for a system of linear inequalities in two variables is the region that satisfies:

(A) The first inequality only.

(B) The second inequality only.

(C) At least one of the inequalities.

(D) All inequalities in the system simultaneously.

Question 11. The system of inequalities $\begin{cases} x \ge 0 \\ y \ge 0 \end{cases}$ represents which region in the coordinate plane?

(A) First quadrant.

(B) First quadrant, including the axes.

(C) All four quadrants.

(D) The origin only.

Question 12. Which point is in the solution set of $x + y < 5$?

(A) $(3, 3)$

(B) $(2, 2)$

(C) $(5, 0)$

(D) $(3, 2)$

Question 13. Solve for $x$: $5 - 2x \le 11$.

(A) $-2x \le 6 \implies x \ge -3$

(B) $-2x \le 6 \implies x \le -3$

(C) $2x \le 6 \implies x \le 3$

(D) $2x \le -6 \implies x \le -3$

Question 14. If $x \in \mathbb{R}$, the solution to $x+1 > 2$ and $x-1 < 3$ is:

(A) $x > 1$ and $x < 4 \implies 1 < x < 4$

(B) $x > 1$ and $x > 4$

(C) $x < 1$ and $x < 4$

(D) $x < 1$ and $x > 4$

Question 15. A company produces two types of pens, A and B. Pen A requires 2 hours of labour and Pen B requires 3 hours of labour. If the company has a maximum of 30 hours of labour per day, and produces $x$ pens of type A and $y$ pens of type B, which inequality represents the labour constraint?

(A) $2x + 3y \le 30$

(B) $3x + 2y \le 30$

(C) $2x + 3y \ge 30$

(D) $2x + 3y = 30$

Question 16. If $x$ represents the number of items produced, the inequality $x \ge 0$ is an example of a:

(A) Linear equation.

(B) Non-linear inequality.

(C) Non-negativity constraint.

(D) Equality constraint.

Question 17. The solution set for $2 \le x < 5$ is represented on a number line by:

(A) A line segment with open circles at both ends.

(B) A line segment with closed circles at both ends.

(C) A line segment with a closed circle at 2 and an open circle at 5.

(D) A line segment with an open circle at 2 and a closed circle at 5.

Question 18. Which point is NOT in the solution region of $x > 1$ and $y > 2$?

(A) $(2, 3)$

(B) $(1.5, 2.5)$

(C) $(0, 3)$

(D) $(3, 2.1)$

Question 19. Solve $\frac{x}{2} > -1$.

(A) $x > -2$

(B) $x < -2$

(C) $x > 2$

(D) $x < 2$

Question 20. A student needs to score at least 75 marks in a test to get an 'A' grade. If the maximum marks are 100 and the student has scored $m$ marks, the inequality representing this is:

(A) $m > 75$

(B) $m < 75$

(C) $m \le 75$

(D) $m \ge 75$

Question 21. The feasible region in a graphical solution of a system of linear inequalities represents:

(A) The area where at least one inequality is satisfied.

(B) The area where exactly one inequality is satisfied.

(C) The area where all inequalities are satisfied.

(D) The boundary lines only.

Question 22. Which inequality represents "the price $p$ of an item is at most $\textsf{₹}500$"?

(A) $p < 500$

(B) $p > 500$

(C) $p \le 500$

(D) $p \ge 500$

Question 23. If a number $x$ is between 3 and 7 (inclusive of 3, exclusive of 7), the inequality is:

(A) $3 < x < 7$

(B) $3 \le x \le 7$

(C) $3 \le x < 7$

(D) $3 < x \le 7$

Question 24. The solution set of the inequality $x > 5$ in integer form is:

(A) $\{5, 6, 7, \dots\}$

(B) $\{6, 7, 8, \dots\}$

(C) $\{\dots, 3, 4, 5\}$

(D) $\{\dots, 4, 5\}$

Question 25. Which point is on the boundary line but NOT in the solution set of $2x + y < 4$?

(A) $(0, 0)$

(B) $(2, 0)$

(C) $(0, 4)$

(D) $(1, 1)$

Question 1. A sequence is a set of numbers arranged in a definite order according to some rule. A series is the:

(A) Product of the terms of a sequence.

(B) Sum of the terms of a sequence.

(C) Difference between consecutive terms.

(D) Ratio between consecutive terms.

Question 2. In an Arithmetic Progression (AP), the difference between any term and its preceding term is constant. This constant is called the:

(A) Common ratio.

(B) Common difference.

(C) First term.

(D) Last term.

Question 3. The $n$-th term of an AP with first term $a$ and common difference $d$ is given by $a_n = a + (n-1)d$. Find the 10th term of the AP: $2, 5, 8, \dots$

(A) $a=2, d=3$. $a_{10} = 2 + (10-1)3 = 2 + 9 \times 3 = 2 + 27 = 29$

(A) $29$

(B) $32$

(C) $26$

(D) $30$

Question 4. The sum of the first $n$ terms of an AP is given by $S_n = \frac{n}{2}[2a + (n-1)d]$. Find the sum of the first 5 terms of the AP: $3, 7, 11, \dots$

(A) $a=3, d=4$. $S_5 = \frac{5}{2}[2(3) + (5-1)4] = \frac{5}{2}[6 + 4 \times 4] = \frac{5}{2}[6 + 16] = \frac{5}{2}[22] = 55$

(A) $50$

(B) $55$

(C) $60$

(D) $45$

Question 5. If $a, b, c$ are in AP, then $b$ is called the Arithmetic Mean (AM) of $a$ and $c$. The formula for AM is:

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

(B) $b = \sqrt{ac}$

(C) $b = \frac{2ac}{a+c}$

(D) $b = a+c$

Question 6. Insert one AM between 8 and 18.

(A) $\frac{8+18}{2} = \frac{26}{2} = 13$

(A) $10$

(B) $13$

(C) $15$

(D) $16$

Question 7. In a Geometric Progression (GP), the ratio of any term to its preceding term is constant. This constant is called the:

(A) Common difference.

(B) Common ratio.

(C) First term.

(D) Last term.

Question 8. The $n$-th term of a GP with first term $a$ and common ratio $r$ is given by $a_n = ar^{n-1}$. Find the 5th term of the GP: $2, 6, 18, \dots$

(A) $a=2, r=3$. $a_5 = 2 \times 3^{5-1} = 2 \times 3^4 = 2 \times 81 = 162$

(A) $54$

(B) $81$

(C) $162$

(D) $243$

Question 9. If $a, b, c$ are positive numbers in GP, then $b$ is called the Geometric Mean (GM) of $a$ and $c$. The formula for GM is:

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

(B) $b = \sqrt{ac}$

(C) $b = \frac{2ac}{a+c}$

(D) $b = a+c$

Question 10. Insert one positive GM between 4 and 9.

(A) $\sqrt{4 \times 9} = \sqrt{36} = 6$

(A) $5$

(B) $6$

(C) $6.5$

(D) $7$

Question 11. For any two positive numbers $a$ and $b$, the relationship between their AM and GM is always:

(A) $AM \le GM$

(B) $AM \ge GM$

(C) $AM = GM$

(D) $AM < GM$

Question 12. The sum of the first $n$ natural numbers is given by the series $1 + 2 + 3 + \dots + n$. The formula for this sum is:

(A) $n(n+1)$

(B) $\frac{n(n-1)}{2}$

(C) $\frac{n(n+1)}{2}$

(D) $n^2$

Question 13. The sum of the first $n$ odd numbers is $1 + 3 + 5 + \dots + (2n-1)$. The formula for this sum is:

(A) $n(n+1)$

(B) $n^2$

(C) $n^2 + 1$

(D) $\frac{n(n+1)}{2}$

Question 14. If the first term of an AP is 5 and the common difference is -2, the first three terms are:

(A) 5, 3, 1

(B) 5, 7, 9

(C) 5, -3, -1

(D) 5, 10, 15

Question 15. If the first term of a GP is 3 and the common ratio is 2, the first three terms are:

(A) 3, 5, 7

(B) 3, 6, 9

(C) 3, 6, 12

(D) 3, 1.5, 0.75

Question 16. In an AP, if $a_k$ is the $k$-th term, then $a_k = \frac{a_{k-1} + a_{k+1}}{2}$ for $k > 1$. This property relates to the concept of:

(A) Common ratio.

(B) Common difference.

(C) Arithmetic Mean.

(D) Geometric Mean.

Question 17. In a GP, if $a_k$ is the $k$-th term (and all terms are positive), then $a_k = \sqrt{a_{k-1} \cdot a_{k+1}}$ for $k > 1$. This property relates to the concept of:

(A) Common ratio.

(B) Common difference.

(C) Arithmetic Mean.

(D) Geometric Mean.

Question 18. If the $n$-th term of a sequence is given by $a_n = 2n+1$, is it an AP or a GP? If AP, find the common difference.

(A) AP, d=2

(B) AP, d=1

(C) GP, r=2

(D) GP, r=3

Question 19. If the $n$-th term of a sequence is given by $a_n = 3^{n-1}$, is it an AP or a GP? If GP, find the common ratio.

(A) AP, d=3

(B) GP, r=3

(C) GP, r=1/3

(D) Neither AP nor GP

Question 20. The sum of the first $n$ terms of a GP is $S_n = a\frac{(r^n - 1)}{r-1}$ (for $r \neq 1$). Find the sum of the first 4 terms of the GP: $1, 2, 4, 8, \dots$

(A) $a=1, r=2$. $S_4 = 1 \times \frac{(2^4 - 1)}{2-1} = \frac{16-1}{1} = 15$

(A) $10$

(B) $15$

(C) $16$

(D) $30$

Question 21. A sum of money is invested at a simple interest rate. The amounts at the end of each year form a sequence. This sequence is an:

(A) AP

(B) GP

(C) Neither AP nor GP

(D) Harmonic Progression

Question 22. A sum of money is invested at a compound interest rate. The amounts at the end of each year form a sequence. This sequence is a:

(A) AP

(B) GP

(C) Neither AP nor GP

(D) Harmonic Progression

Question 23. What is the sum of the first $n$ even numbers ($2 + 4 + 6 + \dots + 2n$)?

(A) $n^2$

(B) $n(n+1)$

(C) $\frac{n(n+1)}{2}$

(D) $2n^2$

Question 24. If the terms of a sequence are $1, -1, 1, -1, \dots$, is it an AP, GP, or neither?

(A) AP

(B) GP

(C) Neither AP nor GP

(D) Both AP and GP

Question 25. The sum of the first $n$ terms of a GP is $S_n = \frac{a(1-r^n)}{1-r}$ (for $r \neq 1$). This formula is preferred when:

(A) $|r| > 1$

(B) $|r| < 1$

(C) $r > 1$

(D) $r < 1$

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

(A) Real numbers.

(B) Complex numbers.

(C) Positive integers.

(D) All integers.

Question 2. A mathematical statement is denoted by $P(n)$, where $n$ is a positive integer. The first step in the Principle of Mathematical Induction is to prove that:

(A) $P(k)$ is true for some arbitrary positive integer $k$.

(B) $P(k+1)$ is true whenever $P(k)$ is true.

(C) $P(1)$ is true.

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

Question 3. The second step in the Principle of Mathematical Induction is the Inductive Hypothesis, which involves assuming that:

(A) $P(1)$ is true.

(B) $P(k)$ is true for some positive integer $k$.

(C) $P(k+1)$ is true.

(D) $P(n)$ is true for all positive integers $n \ge k$.

Question 4. The third step in the Principle of Mathematical Induction is the Inductive Step, which involves proving that:

(A) $P(1)$ is true.

(B) $P(k)$ is false.

(C) $P(k+1)$ is true whenever $P(k)$ is assumed true.

(D) $P(n)$ is true for all $n$.

Question 5. If the base case $P(1)$ is true and the inductive step (if $P(k)$ is true, then $P(k+1)$ is true) is proven, what can be concluded by the Principle of Mathematical Induction?

(A) $P(n)$ is true for $n=1$ only.

(B) $P(n)$ is true for $n=k$ only.

(C) $P(n)$ is true for all positive integers $n$.

(D) No conclusion can be drawn.

Question 6. Consider the statement $P(n): 1 + 2 + \dots + n = \frac{n(n+1)}{2}$. What is $P(1)$?

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

(B) $1 + 2 = \frac{2(2+1)}{2}$

(C) $1 = \frac{n(n+1)}{2}$

(D) $1 + 2 + \dots + k = \frac{k(k+1)}{2}$

Question 7. If we are proving $P(n): n^2 > n$ for all integers $n \ge 2$, the base case would be proving $P(2)$ is true. What is $P(2)$?

(A) $1^2 > 1$

(B) $2^2 > 2$

(C) $k^2 > k$

(D) $(k+1)^2 > k+1$

Question 8. In the inductive step for proving $P(n)$, we assume $P(k)$ is true for some $k \ge n_0$ (where $n_0$ is the base case starting value). We then need to prove:

(A) $P(k-1)$ is true.

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

(C) $P(k+1)$ is true.

(D) $P(n)$ is true.

Question 9. Consider the statement $P(n): 1 + 3 + 5 + \dots + (2n-1) = n^2$. Assuming $P(k)$ is true, i.e., $1 + 3 + \dots + (2k-1) = k^2$, what is the Left Hand Side (LHS) of $P(k+1)$?

(A) $1 + 3 + \dots + (2(k+1)-1)$

(B) $1 + 3 + \dots + (2k-1) + (2k+1)$

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

(D) $(k+1)^2$

Question 10. Consider the statement $P(n): n(n+1)$ is even. Which of the following is the correct base case ($n=1$) check?

(A) $1(1+1) = 2$, which is even. True.

(B) $2(2+1) = 6$, which is even. True.

(C) $k(k+1)$ is even. Assume true.

(D) $(k+1)(k+2)$ is even. To be proven.

Question 11. Which of the following types of statements can be proven using Mathematical Induction?

(A) Statements about real numbers like $\sqrt{2}$ is irrational.

(B) Statements about divisibility involving positive integers, like $2^n > n$ for all positive integers $n$.

(C) Statements about the sum of a finite series where the number of terms depends on $n$.

(D) Statements about inequalities involving positive integers.

Select the most appropriate general categories.

(A) A and B only

(B) B and C only

(C) C and D only

(D) B, C, and D

Question 12. To prove $P(n): 2^n > n$ for all positive integers $n \ge 1$. The base case $P(1)$ is $2^1 > 1$, which is $2 > 1$ (True). In the inductive step, we assume $2^k > k$ for some $k \ge 1$. We need to prove $2^{k+1} > k+1$. Starting with $2^{k+1} = 2 \cdot 2^k$, using the inductive hypothesis, we have $2 \cdot 2^k > 2k$. What is the next step?

(A) Since $2k = k+k$ and for $k \ge 1$, $k \ge 1$, so $k+k \ge k+1$. Thus $2^{k+1} > 2k \ge k+1$, so $2^{k+1} > k+1$. True.

(B) We need to show $2k \ge k+1$. This is true if $k \ge 1$. Thus $2^{k+1} > k+1$. True.

(C) $2k = k + k$. We need to show $k \ge 1$. Since $k \ge 1$, $2k \ge k+1$. True.

(D) All of the above represent correct reasoning for the inductive step.

Question 13. Let $P(n)$ be the statement $n^2 + n + 41$ is prime. If we check $P(1) = 1+1+41=43$ (prime), $P(2) = 4+2+41=47$ (prime), does this prove $P(n)$ is true for all positive integers $n$?

(A) Yes, checking a few cases is sufficient for induction.

(B) No, induction requires proving the inductive step, not just checking examples.

(C) Yes, if it works for the first two cases, it will work for all.

(D) No, we need to check if $P(41)$ is prime to be sure.

Question 14. The statement $P(n): n! \ge 2^n$ is true for $n \ge 4$. What is the base case for this statement?

(A) $n=1$, $1! \ge 2^1 \implies 1 \ge 2$ (False)

(B) $n=2$, $2! \ge 2^2 \implies 2 \ge 4$ (False)

(C) $n=3$, $3! \ge 2^3 \implies 6 \ge 8$ (False)

(D) $n=4$, $4! \ge 2^4 \implies 24 \ge 16$ (True)

Question 15. Which of the following is NOT a standard step in the Principle of Mathematical Induction?

(A) Proving the base case $P(n_0)$ is true.

(B) Assuming $P(k)$ is true for some $k \ge n_0$.

(C) Proving $P(k+1)$ is true using the assumption $P(k)$.

(D) Proving $P(n)$ is true by checking a sufficient number of values of $n$.

Question 16. Consider the statement $P(n): n^3 + 2n$ is divisible by 3 for all positive integers $n$. What do we need to prove in the inductive step, assuming $k^3 + 2k$ is divisible by 3?

(A) $k^3 + 2k + 1$ is divisible by 3.

(B) $(k+1)^3 + 2(k+1)$ is divisible by 3.

(C) $n^3 + 2n$ is divisible by 3 for $n=k+1$.

(D) Both B and C are correct interpretations of the goal for the inductive step.

Question 17. To prove the sum formula $1 + 2 + \dots + n = \frac{n(n+1)}{2}$ by induction, if we assume $1 + 2 + \dots + k = \frac{k(k+1)}{2}$, we need to show that $1 + 2 + \dots + k + (k+1) = \frac{(k+1)((k+1)+1)}{2}$. The LHS can be written as $\frac{k(k+1)}{2} + (k+1)$. Simplify this expression.

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

(B) $\frac{k(k+1)}{2} + k+1$

(C) $\frac{k^2+k+2k+2}{2} = \frac{k^2+3k+2}{2}$

(D) All options represent equivalent forms, but (A) directly shows the desired RHS form.

Question 18. Consider the statement $P(n): n! > 2^n$ for $n \ge 4$. Assuming $k! > 2^k$ for some $k \ge 4$, we multiply both sides by $(k+1)$: $(k+1)k! > (k+1)2^k$, which is $(k+1)! > (k+1)2^k$. To prove $(k+1)! > 2^{k+1}$, we need to show that $(k+1)2^k \ge 2^{k+1}$. This is equivalent to showing $(k+1)2^k \ge 2 \cdot 2^k$, which simplifies to $k+1 \ge 2$. This is true for $k \ge 1$. Since our assumption is for $k \ge 4$, the inequality holds. This demonstrates:

(A) The base case is true.

(B) The inductive hypothesis is valid.

(C) The inductive step is successful.

(D) The statement is true for $n \ge 4$ by PMI.

Question 19. Which of the following is a valid application of the Principle of Mathematical Induction?

(A) Proving that the sum of the angles in any triangle is $180^\circ$.

(B) Proving that the sum of the first $n$ terms of an arithmetic sequence is given by a specific formula, for all positive integers $n$.

(C) Proving that all even numbers are divisible by 2.

(D) Proving that the next prime number after a given prime exists.

Question 20. The Principle of Mathematical Induction is based on the property of positive integers that if a set of positive integers contains 1, and also contains $k+1$ whenever it contains $k$, then it contains all positive integers. This property is called:

(A) The Well-Ordering Principle.

(B) The Archimedean Property.

(C) The Principle of Mathematical Induction (Peano Axiom related).

(D) The Fundamental Theorem of Arithmetic.

Question 21. When applying induction to an inequality, say $P(n): f(n) > g(n)$, assuming $f(k) > g(k)$, we often need to show $f(k+1) > g(k+1)$. The step involves relating $f(k+1)$ and $g(k+1)$ back to $f(k)$ and $g(k)$. Which transformation is often useful?

(A) Expressing $f(k+1)$ in terms of $f(k)$ and showing $f(k+1) - f(k) > g(k+1) - g(k)$.

(B) Expressing $f(k+1)$ in terms of $f(k)$ and showing $f(k+1) > g(k) + (\text{some value})$.

(C) Manipulating $f(k+1) > g(k+1)$ directly without using the inductive hypothesis.

(D) Both A and B are often useful strategies depending on the specific functions $f$ and $g$.

Question 22. Consider the statement $P(n): 7^n - 3^n$ is divisible by 4 for all positive integers $n$. Base case $P(1): 7^1 - 3^1 = 4$, which is divisible by 4. Assume $7^k - 3^k = 4m$ for some integer $m$. Now consider $P(k+1): 7^{k+1} - 3^{k+1} = 7 \cdot 7^k - 3 \cdot 3^k$. Which is a useful algebraic manipulation in the inductive step?

(A) $7 \cdot 7^k - 3 \cdot 3^k = 7(7^k - 3^k) + 7 \cdot 3^k - 3 \cdot 3^k = 7(4m) + 4 \cdot 3^k = 4(7m + 3^k)$

(B) $7 \cdot 7^k - 3 \cdot 3^k = 3(7^k - 3^k) + 4 \cdot 7^k = 3(4m) + 4 \cdot 7^k = 4(3m + 7^k)$

(C) $7^{k+1} - 3^{k+1} = (7^k - 3^k)(7+3) = 10(7^k - 3^k)$

(D) Both A and B are useful manipulations showing divisibility by 4.

Question 23. If you are proving $P(n)$ for $n \ge n_0$, the base case must be proven for $n=n_0$. If you prove $P(1)$ but the statement is only true for $n \ge 5$, what does the induction process tell you if the inductive step holds for $k \ge 1$?

(A) The statement is true for all $n \ge 1$.

(B) The induction fails.

(C) The statement is only true for $n=1$ and $n \ge 5$.

(D) The base case must be chosen carefully; if the inductive step holds for $k \ge n_0$, you only need to prove $P(n_0)$.

Question 24. Mathematical induction can be used to prove that the sum of the interior angles of a convex $n$-sided polygon is $(n-2) \times 180^\circ$ for $n \ge 3$. What is the base case?

(A) $n=1$, sum of angles of a 1-sided polygon.

(B) $n=2$, sum of angles of a 2-sided polygon.

(C) $n=3$, sum of angles of a triangle is $(3-2) \times 180^\circ = 180^\circ$.

(D) $n=4$, sum of angles of a quadrilateral is $(4-2) \times 180^\circ = 360^\circ$.

Question 25. Suppose you are proving $P(n): 1 + 2 + \dots + n = \frac{n(n+1)}{2}$ by induction, and you have successfully shown $P(1)$ is true. If you assume $P(k)$ is true for some $k \ge 1$ and correctly deduce $P(k+1)$ is true, this means:

(A) If the formula is true for sum up to $k$, it is also true for sum up to $k+1$.

(B) The formula is true for all positive integers $n$.

(C) The inductive step implies that the property "propagates" from $k$ to $k+1$.

(D) All of the above.

Question 26. Consider the statement $P(n): \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \dots + \frac{1}{n(n+1)} = \frac{n}{n+1}$. What is the base case $P(1)$?

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

(B) $\frac{1}{2} = \frac{1}{2}$

(C) $P(1): \frac{1}{2} = \frac{1}{1+1}$

(D) Options A and B are equivalent, but C is the complete statement check.

Question 27. If you are proving an inequality $P(n): f(n) > g(n)$ for $n \ge n_0$ and the inductive step from $k$ to $k+1$ requires the condition $k \ge m$ where $m > n_0$, what does this imply about the proof?

(A) The induction is valid only for $n \ge m$. You need to check base cases $P(n_0), P(n_0+1), \dots, P(m-1)$ separately.

(B) The induction fails completely.

(C) The base case must be $P(m)$.

(D) The inductive hypothesis should be assumed for all $k \ge n_0$.

Question 28. Strong induction is a variation of mathematical induction where, in the inductive step, you assume $P(j)$ is true for all $j$ such that $n_0 \le j \le k$, rather than just $P(k)$. When might strong induction be necessary or more convenient than standard induction?

(A) When proving formulas for sums or products.

(B) When the truth of $P(k+1)$ depends not only on $P(k)$ but on earlier values like $P(k-1), P(k-2)$, etc.

(C) When proving inequalities.

(D) Strong induction is always required; standard induction is insufficient.

Question 29. Consider the statement $P(n): 1 + 2 + \dots + n = \frac{n(n+1)}{2}$. In the inductive step, we showed $\frac{k(k+1)}{2} + (k+1) = \frac{(k+1)(k+2)}{2}$. This step relies on which algebraic operation?

(A) Factorisation

(B) Common denominator

(C) Distributive property

(D) Associative property

Question 30. A common mistake in mathematical induction is assuming $P(k+1)$ is true in the inductive step. The correct approach is to:

(A) Use $P(k)$ to derive $P(k+1)$.

(B) Prove $P(k+1)$ independently of $P(k)$.

(C) Show that $P(k) \implies P(k+1)$.

(D) Both A and C describe the correct approach.

Question 1. If there are $m$ ways to do one thing and $n$ ways to do another thing, then there are $m \times n$ ways to do both things. This is known as the:

(A) Addition Principle of Counting.

(B) Multiplication Principle of Counting.

(C) Permutation Principle.

(D) Combination Principle.

Question 2. How many different outfits can you create if you have 3 shirts, 2 pairs of trousers, and 4 pairs of shoes?

(A) $3+2+4=9$

(B) $3 \times 2 \times 4 = 24$

(C) $3^2 \times 4 = 36$

(D) $4^3 \times 2 = 128$

Question 3. Factorial notation $n!$ represents the product of the first $n$ positive integers. What is the value of $5!$?

(A) $5 \times 4 \times 3 \times 2 \times 1 = 120$

(B) $5+4+3+2+1=15$

(C) $5$

(D) $100$

Question 4. By definition, $0!$ is equal to:

(A) $0$

(B) $1$

(C) Undefined

(D) $\infty$

Question 5. A permutation is an arrangement of objects in a specific order. The number of permutations of $n$ distinct objects taken $r$ at a time is given by $\text{P}(n, r)$ or $^nP_r = \frac{n!}{(n-r)!}$. What is the number of ways to arrange 3 distinct books on a shelf?

(A) $3! = 6$

(B) $\text{P}(3, 1) = 3$

(C) $\text{C}(3, 3) = 1$

(D) $3 \times 2 = 6$

Question 6. Calculate $\text{P}(5, 2)$.

(A) $\frac{5!}{(5-2)!} = \frac{120}{3!} = \frac{120}{6} = 20$

(B) $5 \times 2 = 10$

(C) $\frac{5!}{2!} = \frac{120}{2} = 60$

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

Question 7. The number of permutations of $n$ objects where $p_1$ objects are of one kind, $p_2$ are of a second kind, ..., $p_k$ are of a $k$-th kind is $\frac{n!}{p_1! p_2! \dots p_k!}$. How many distinct permutations are there of the letters in the word 'INDIA'?

(A) $5! = 120$

(B) $\frac{5!}{2!} = \frac{120}{2} = 60$ (I appears twice)

(C) $\frac{5!}{3!} = \frac{120}{6} = 20$

(D) $5 \times 4 \times 3 \times 2 \times 1 / 2 = 60$

Question 8. A combination is a selection of objects where the order does not matter. The number of combinations of $n$ distinct objects taken $r$ at a time is given by $\text{C}(n, r)$ or $^nC_r = \frac{n!}{r!(n-r)!}$. In how many ways can you choose a committee of 2 people from a group of 5 people?

(A) $\text{P}(5, 2) = 20$

(B) $\text{C}(5, 2) = \frac{5!}{2!(5-2)!} = \frac{120}{2 \times 6} = \frac{120}{12} = 10$

(C) $5 \times 2 = 10$

(D) $5+2=7$

Question 9. Calculate $\text{C}(7, 3)$.

(A) $\frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35$

(B) $\frac{7!}{3!} = \frac{5040}{6} = 840$

(C) $\frac{7!}{4!} = \frac{5040}{24} = 210$

(D) $\text{P}(7, 3) = 210$

Question 10. The relationship between permutations and combinations is $\text{P}(n, r) = \text{C}(n, r) \times r!$. What is the value of $\text{C}(n, n)$?

(A) $0$

(B) $1$

(C) $n$

(D) $n!$

Question 11. What is the value of $\text{C}(n, 0)$?

(A) $0$

(B) $1$

(C) $n$

(D) Undefined

Question 12. The property $\text{C}(n, r) = \text{C}(n, n-r)$ means that selecting $r$ objects from $n$ is the same as selecting $n-r$ objects to leave behind. What is the value of $\text{C}(10, 8)$?

(A) $\text{C}(10, 8) = \text{C}(10, 10-8) = \text{C}(10, 2) = \frac{10 \times 9}{2 \times 1} = 45$

(B) $\text{C}(10, 8) = \frac{10!}{8!} = 90$

(C) $\text{P}(10, 8)$

(D) $10 \times 8 = 80$

Question 13. A coin is tossed 3 times. How many different sequences of heads and tails are possible?

(A) $3$

(B) $2 \times 3 = 6$

(C) $2^3 = 8$

(D) $3^2 = 9$

Question 14. How many 4-digit numbers can be formed using the digits 1, 2, 3, 4, 5 if repetition of digits is allowed?

(A) $5! = 120$

(B) $\text{P}(5, 4) = 120$

(C) $5^4 = 625$

(D) $4^5 = 1024$

Question 15. How many 4-digit numbers can be formed using the digits 1, 2, 3, 4, 5 if repetition of digits is NOT allowed?

(A) $5^4 = 625$

(B) $\text{P}(5, 4) = \frac{5!}{(5-4)!} = \frac{120}{1} = 120$

(C) $\text{C}(5, 4) = 5$

(D) $4! = 24$

Question 16. A group consists of 7 men and 5 women. In how many ways can a committee of 3 men and 2 women be formed?

(A) Choose 2 girls from 4: $\text{C}(4, 2)$ ways. Choose 3 boys from 6: $\text{C}(6, 3)$ ways. Total = $\text{C}(4, 2) \times \text{C}(6, 3) = 6 \times 20 = 120$

(B) $\text{C}(12, 5)$

(C) $\text{P}(7, 3) \times \text{P}(5, 2)$

(D) $\text{C}(7, 3) \times \text{C}(5, 2)$

Question 17. $\text{C}(7, 3) = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35$. $\text{C}(5, 2) = \frac{5 \times 4}{2 \times 1} = 10$. Total ways = $35 \times 10 = 350$.

(A) $350$

(B) $792$

(C) $2520$

(D) $45$

Question 18. In how many ways can the letters of the word 'ASSASSINATION' be arranged?

The letters are A($\times$3), S($\times$4), I($\times$2), N($\times$2), T($\times$1), O($\times$1). Total letters = 13.

Number of arrangements = $\frac{13!}{3! 4! 2! 2! 1! 1!}$

(A) $\frac{13!}{3! 4! 2! 2!}$

(B) $\frac{13!}{3! 4! 2!}$

(C) $13!$

(D) $\frac{13!}{4!}$

Question 19. Calculate the value of $\frac{8!}{6! \times 2!}$.

(A) $\frac{8 \times 7 \times 6!}{6! \times 2} = \frac{8 \times 7}{2} = 28$

(B) $\frac{8 \times 7}{2} = 28$

(C) $\frac{8 \times 7 \times 6 \times 5 \times 4 \times 3}{2} = 20160$

(D) $\frac{8!}{12} = \frac{40320}{12} = 3360$

Question 20. In how many ways can 5 boys and 3 girls be arranged in a row so that no two girls are together?

(A) Arrange the 5 boys in $5!$ ways.

This creates 6 spaces for the girls (including ends): \_ B \_ B \_ B \_ B \_ B \_

Choose 3 spaces out of 6 for the girls in $\text{C}(6, 3)$ ways.

Arrange the 3 girls in the chosen spaces in $3!$ ways.

Total ways = $5! \times \text{C}(6, 3) \times 3! = 120 \times 20 \times 6 = 14400$

(A) $5! \times 3!$

(B) $\text{P}(8, 8)$

(C) $5! \times \text{P}(6, 3)$

(D) $5! \times \text{C}(6, 3)$

Question 21. $\text{P}(n, n)$ is the number of ways to arrange $n$ distinct objects, which is $n!$. What is $\text{C}(n, 1)$?

(A) $0$

(B) $1$

(C) $n$

(D) $n!$

Question 22. How many chords can be drawn through 21 points on a circle?

A chord is formed by selecting 2 points from 21. The order does not matter.

(A) $\text{P}(21, 2) = 21 \times 20 = 420$

(B) $\text{C}(21, 2) = \frac{21 \times 20}{2 \times 1} = 210$

(C) $21^2 = 441$

(D) $21 \times 2 = 42$

Question 23. A committee of 5 is to be formed from 6 boys and 4 girls. In how many ways can this be done if the committee must include exactly 2 girls?

(A) Choose 2 girls from 4: $\text{C}(4, 2)$ ways. Choose 3 boys from 6: $\text{C}(6, 3)$ ways. Total = $\text{C}(4, 2) \times \text{C}(6, 3) = 6 \times 20 = 120$

(B) $\text{C}(10, 5)$

(C) $\text{C}(4, 2) + \text{C}(6, 3)$

(D) $\text{P}(4, 2) \times \text{P}(6, 3)$

Question 24. Calculate $\text{C}(n, r)$ if $n = 8$ and $r = 5$.

(A) $\text{C}(8, 5) = \text{C}(8, 3) = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56$

(B) $\text{P}(8, 5) = \frac{8!}{3!} = 8 \times 7 \times 6 \times 5 \times 4 = 6720$

(C) $\frac{8!}{5!} = 8 \times 7 \times 6 = 336$

(D) $8 \times 5 = 40$

Question 25. In a competition, there are 10 participants. Trophies are awarded for the first, second, and third place. In how many ways can the trophies be distributed?

This is an arrangement where order matters (1st, 2nd, 3rd are distinct positions).

(A) $\text{C}(10, 3)$

(B) $\text{P}(10, 3) = 10 \times 9 \times 8 = 720$

(C) $10^3$

(D) $10!$

Question 1. The Binomial Theorem for a positive integral index $n$ gives the expansion of:

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

(B) $(a-b)^n$

(C) $(a \times b)^n$

(D) Both (A) and (B)

Question 2. According to the Binomial Theorem, the expansion of $(x+y)^n$ is given by $\sum_{k=0}^{n} \text{C}(n, k) x^{n-k} y^k$. How many terms are there in the expansion of $(x+y)^n$?

(A) $n$

(B) $n+1$

(C) $2n$

(D) $2^{n+1}$

Question 3. What is the coefficient of the term $x^2y^3$ in the expansion of $(x+y)^5$?

The term is of the form $\text{C}(n, k) x^{n-k} y^k$. Here $n=5$, $n-k=2 \implies k=3$. The coefficient is $\text{C}(5, 3)$.

(A) $\text{C}(5, 2) = 10$

(B) $\text{C}(5, 3) = 10$

(C) $5$

(D) $2 \times 3 = 6$

Question 4. The general term (or $(r+1)$-th term) in the expansion of $(a+b)^n$ is given by $T_{r+1} = \text{C}(n, r) a^{n-r} b^r$. What is the 4th term in the expansion of $(x+2)^5$?

For the 4th term, $r+1=4 \implies r=3$. Here $n=5$, $a=x$, $b=2$.

$T_4 = \text{C}(5, 3) x^{5-3} 2^3 = 10 x^2 \times 8 = 80x^2$

(A) $\text{C}(5, 4) x^{5-4} 2^4$

(B) $\text{C}(5, 3) x^{5-3} 2^3$

(C) $10 x^2 \cdot 8 = 80x^2$

(D) $10 x^3 \cdot 4 = 40x^3$

Question 5. In the expansion of $(a-b)^n$, the terms have alternating signs. The general term $T_{r+1}$ is given by $\text{C}(n, r) a^{n-r} (-b)^r = (-1)^r \text{C}(n, r) a^{n-r} b^r$. What is the coefficient of $p^2q^4$ in the expansion of $(p-q)^6$?

The term is $T_{r+1} = \text{C}(6, r) p^{6-r} (-q)^r$. We need $p^{6-r} q^r = p^2q^4$, so $r=4$.

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

(A) $\text{C}(6, 4)$

(B) $-\text{C}(6, 4)$

(C) $15$

(D) $-15$

Question 6. The middle term(s) in the expansion of $(x+y)^n$ depends on whether $n$ is even or odd. If $n$ is an even integer, say $n=2m$, there is one middle term. Which term is it?

(A) $(m)$-th term

(B) $(m+1)$-th term

(C) $(2m)$-th term

(D) $(2m+1)$-th term

Question 7. If $n$ is an odd integer, say $n=2m+1$, there are two middle terms. Which terms are they?

(A) $m$-th and $(m+1)$-th terms

(B) $(m+1)$-th and $(m+2)$-th terms

(C) $m$-th and $(m+2)$-th terms

(D) $(2m)$-th and $(2m+1)$-th terms

Question 8. Find the middle term(s) in the expansion of $(a+b)^6$.

Here $n=6$ (even). $m = n/2 = 3$. The middle term is the $(m+1)$-th = 4th term.

$T_4 = \text{C}(6, 3) a^{6-3} b^3 = 20 a^3 b^3$

(A) 3rd term

(B) 4th term

(C) 3rd and 4th terms

(D) 4th and 5th terms

Question 9. Find the coefficient of the middle term in the expansion of $(x+y)^8$.

Here $n=8$ (even). Middle term is $(8/2 + 1)$th = 5th term. $T_5 = \text{C}(8, 4) x^4 y^4$.

Coefficient = $\text{C}(8, 4) = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70$

(A) $\text{C}(8, 3) = 56$

(B) $\text{C}(8, 4) = 70$

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

(D) $\text{C}(8, 8) = 1$

Question 10. Find the middle terms in the expansion of $(x-y)^7$.

Here $n=7$ (odd). $n=2m+1 \implies 7 = 2m+1 \implies 2m=6 \implies m=3$.

Middle terms are $(m+1)$th and $(m+2)$th, i.e., 4th and 5th terms.

$T_4 = \text{C}(7, 3) x^4 (-y)^3 = -35 x^4 y^3$

$T_5 = \text{C}(7, 4) x^3 (-y)^4 = 35 x^3 y^4$

(A) 3rd and 4th terms

(B) 4th and 5th terms

(C) $35 x^4 y^3$ and $-35 x^3 y^4$

(D) $-35 x^4 y^3$ and $35 x^3 y^4$

Question 11. The sum of the binomial coefficients in the expansion of $(1+x)^n$ is $\sum_{k=0}^{n} \text{C}(n, k)$. This sum is equal to:

(A) $n$

(B) $n!$

(C) $2^n$

(D) $2n$

Question 12. What is the sum of the coefficients in the expansion of $(2x - 3y)^5$?

Set $x=1, y=1$. Sum of coefficients is $(2(1) - 3(1))^5 = (2-3)^5 = (-1)^5 = -1$

(A) $1$

(B) $-1$

(C) $(2-3)^5 = -1$

(D) $(2+3)^5 = 3125$

Question 13. Find the term independent of $x$ in the expansion of $(x + \frac{1}{x})^{10}$.

General term $T_{r+1} = \text{C}(10, r) x^{10-r} (\frac{1}{x})^r = \text{C}(10, r) x^{10-r} x^{-r} = \text{C}(10, r) x^{10-2r}$.

For the term independent of $x$, the power of $x$ is 0, so $10-2r = 0 \implies 2r = 10 \implies r = 5$.

The term is $T_{5+1} = T_6 = \text{C}(10, 5) x^0 = \text{C}(10, 5) = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = 252$

(A) $\text{C}(10, 5)$

(B) $252$

(C) $\text{C}(10, 0)$

(D) $\text{C}(10, 10)$

Question 14. Which of the following is NOT a property of binomial coefficients $\text{C}(n, r)$?

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

(B) $\text{C}(n, r) + \text{C}(n, r-1) = \text{C}(n+1, r)$ (Pascal's Identity)

(C) $\sum_{r=0}^{n} \text{C}(n, r) = 2^n$

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

Question 15. Find the number of terms in the expansion of $(1+2x+x^2)^5$.

Note that $1+2x+x^2 = (1+x)^2$. So the expression is $((1+x)^2)^5 = (1+x)^{10}$.

The expansion of $(1+x)^{10}$ has $10+1=11$ terms.

(A) $5$

(B) $6$

(C) $10$

(D) $11$

Question 16. The coefficient of $x^r$ in the expansion of $(1+x)^n$ is:

(A) $nCr$

(B) $\text{C}(n, r)$

(C) $\text{P}(n, r)$

(D) $r!$

Question 17. Find the coefficient of $x^3$ in the expansion of $(3x - 1)^4$.

General term $T_{r+1} = \text{C}(4, r) (3x)^{4-r} (-1)^r = \text{C}(4, r) 3^{4-r} x^{4-r} (-1)^r$.

We need $4-r = 3$, so $r=1$.

Coefficient = $\text{C}(4, 1) 3^{4-1} (-1)^1 = 4 \times 3^3 \times (-1) = 4 \times 27 \times (-1) = -108$

(A) $\text{C}(4, 3) 3^3 (-1)^1$

(B) $\text{C}(4, 1) 3^3 (-1)^3$

(C) $-108$

(D) $108$

Question 18. Find the middle term in the expansion of $(2x + \frac{1}{x})^6$.

Here $n=6$ (even). Middle term is $(6/2 + 1)$th = 4th term. $r=3$.

$T_4 = \text{C}(6, 3) (2x)^{6-3} (\frac{1}{x})^3 = \text{C}(6, 3) (2x)^3 (\frac{1}{x})^3 = 20 \times 8x^3 \times \frac{1}{x^3} = 20 \times 8 = 160$

(A) 4th term, $160$

(B) 3rd term, $80x^2$

(C) 4th term, $160x^6$

(D) 3rd and 4th terms

Question 19. The value of $\text{C}(n, r)$ is maximum when $r$ is:

(A) $0$ or $n$

(B) $1$ or $n-1$

(C) Close to $n/2$

(D) Cannot be determined

Question 20. The binomial expansion of $(1+x)^n$ can be derived using the concept of selecting terms from each of the $n$ factors. The term $\text{C}(n, r) x^r$ arises from choosing $x$ from $r$ factors and $1$ from the remaining $(n-r)$ factors. This relates binomial expansion to:

(A) Permutations.

(B) Combinations.

(C) Factorials.

(D) Principle of Mathematical Induction.

Question 1. A matrix is a rectangular arrangement of numbers or functions. The numbers or functions are called the:

(A) Rows.

(B) Columns.

(C) Elements or entries.

(D) Order.

Question 2. The order of a matrix with $m$ rows and $n$ columns is given by:

(A) $m+n$

(B) $m \times n$

(C) $n \times m$

(D) $mn$

Question 3. Consider the matrix $A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$. What is the order of matrix A?

(A) $3 \times 2$

(B) $2 \times 3$

(C) $6$

(D) $2+3=5$

Question 4. In the matrix $A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$, what is the element $a_{21}$ (element in the 2nd row and 1st column)?

(A) $2$

(B) $4$

(C) $1$

(D) $5$

Question 5. A matrix with only one row is called a:

(A) Column matrix.

(B) Row matrix.

(C) Square matrix.

(D) Identity matrix.

Question 6. A square matrix is a matrix where the number of rows is equal to the number of columns. Which of the following is a square matrix?

(A) $\begin{pmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{pmatrix}$

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

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

(D) $\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$

Question 7. Two matrices A and B are equal if and only if they have the same order and their corresponding elements are equal. If $\begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} 2 & 3 \\ 4 & 5 \end{pmatrix}$, what is the value of $c$?

(A) $2$

(B) $3$

(C) $4$

(D) $5$

Question 8. For matrix addition or subtraction, the matrices must have the same:

(A) Number of rows.

(B) Number of columns.

(C) Order.

(D) Elements.

Question 9. If $A = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 0 \\ -1 & 2 \end{pmatrix}$, what is $A+B$?

(A) $\begin{pmatrix} 3 & 1 \\ 2 & 6 \end{pmatrix}$

(B) $\begin{pmatrix} 3 & 1 \\ 4 & 6 \end{pmatrix}$

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

(D) $\begin{pmatrix} 1 & 1 \\ 2 & 2 \end{pmatrix}$

Question 10. If $A = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}$ and $k=3$, what is $kA$ (scalar multiplication)?

(A) $\begin{pmatrix} 6 & 3 \\ 9 & 12 \end{pmatrix}$

(B) $\begin{pmatrix} 5 & 4 \\ 6 & 7 \end{pmatrix}$

(C) $\begin{pmatrix} 6 & 1 \\ 3 & 12 \end{pmatrix}$

(D) $\begin{pmatrix} 2 & 3 \\ 3 & 12 \end{pmatrix}$

Question 11. For matrix multiplication $AB$, the number of columns in matrix A must be equal to the number of rows in matrix B. If A is $m \times n$ and B is $p \times q$, for AB to be defined, we must have:

(A) $m=p$

(B) $m=q$

(C) $n=p$

(D) $n=q$

Question 12. If matrix A is of order $2 \times 3$ and matrix B is of order $3 \times 4$, what is the order of the product matrix AB?

(A) $3 \times 3$

(B) $2 \times 4$

(C) $3 \times 2$

(D) $4 \times 2$

Question 13. If $A = \begin{pmatrix} 1 & 2 \end{pmatrix}$ and $B = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$, what is the product AB?

A is $1 \times 2$, B is $2 \times 1$. AB is $1 \times 1$.

AB = $(1 \times 3 + 2 \times 4) = (3 + 8) = (11)$

(A) $\begin{pmatrix} 3 & 6 \\ 4 & 8 \end{pmatrix}$

(B) $\begin{pmatrix} 11 \end{pmatrix}$

(C) $\begin{pmatrix} 7 \end{pmatrix}$

(D) $\begin{pmatrix} 3 \\ 8 \end{pmatrix}$

Question 14. If $A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & 3 \\ 4 & 5 \end{pmatrix}$, what is AB?

A is the $2 \times 2$ Identity matrix. Multiplying by the Identity matrix results in the original matrix.

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

(B) $\begin{pmatrix} 2 & 3 \\ 4 & 5 \end{pmatrix}$

(C) $\begin{pmatrix} 2 & 0 \\ 0 & 5 \end{pmatrix}$

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

Question 15. A matrix where all elements are zero is called a:

(A) Identity matrix.

(B) Diagonal matrix.

(C) Zero matrix (or Null matrix).

(D) Square matrix.

Question 16. An identity matrix (denoted by $I_n$ or $I$) is a square matrix with ones on the main diagonal and zeros elsewhere. What is the $3 \times 3$ identity matrix?

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

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

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

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

Question 17. Matrix multiplication is generally not commutative, meaning that for two matrices A and B, $AB \neq BA$. When is $AB = BA$ always true?

(A) For any two square matrices of the same order.

(B) If one of the matrices is a zero matrix.

(C) If one of the matrices is an identity matrix (and multiplication is defined).

(D) Matrix multiplication is always commutative.

Question 18. If $A = \begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, calculate $AB$.

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

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

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

(C) $\begin{pmatrix} 0 & -1 \\ 3 & 0 \end{pmatrix}$

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

Question 19. A matrix where the number of columns is equal to the number of rows is a square matrix. The elements $a_{ii}$ (where the row and column indices are the same) form the:

(A) Diagonal elements.

(B) Off-diagonal elements.

(C) Trace of the matrix.

(D) Principal diagonal.

Question 20. A diagonal matrix is a square matrix where all non-diagonal elements are zero. Which of the following is a diagonal matrix?

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

(B) $\begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix}$

(C) $\begin{pmatrix} 1 & 0 \\ 2 & 3 \end{pmatrix}$

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

Question 21. If $A$ and $B$ are matrices of the same order, then $(A+B)' = A' + B'$, where $A'$ denotes the transpose of matrix $A$. This is a property of matrix addition and transpose. Matrix addition is commutative, i.e., $A+B = B+A$. Matrix multiplication is associative, i.e., $A(BC) = (AB)C$. Matrix multiplication distributes over addition, i.e., $A(B+C) = AB + AC$. Which property is NOT always true for matrix multiplication?

(A) Commutativity

(B) Associativity

(C) Distributivity

(D) Existence of multiplicative inverse for any non-zero matrix

Question 22. The product of a zero matrix of any order and any other matrix (for which the product is defined) is always:

(A) The identity matrix.

(B) The zero matrix.

(C) The original other matrix.

(D) A non-zero matrix.

Question 23. If $A = \begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix}$, what is $A^2$?

$A^2 = \begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix} \begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix} = \begin{pmatrix} 1(1)+(-1)(-1) & 1(-1)+(-1)(1) \\ (-1)(1)+1(-1) & (-1)(-1)+1(1) \end{pmatrix} = \begin{pmatrix} 1+1 & -1-1 \\ -1-1 & 1+1 \end{pmatrix} = \begin{pmatrix} 2 & -2 \\ -2 & 2 \end{pmatrix}$

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

(B) $\begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix}$

(C) $\begin{pmatrix} 2 & -2 \\ -2 & 2 \end{pmatrix}$

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

Question 24. The trace of a square matrix is the sum of the elements on its main diagonal. What is the trace of the matrix $\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}$?

(A) $1+5+9 = 15$

(B) $3+6+9 = 18$

(C) $1+4+7 = 12$

(D) $1+2+3+4+5+6+7+8+9 = 45$

Question 25. If $A = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 0 \\ 0 & 4 \end{pmatrix}$, what is AB?

Product of diagonal matrices is a diagonal matrix with diagonal entries as the product of corresponding entries.

AB = $\begin{pmatrix} 2 \times 1 & 0 \\ 0 & 3 \times 4 \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ 0 & 12 \end{pmatrix}$

(A) $\begin{pmatrix} 3 & 0 \\ 0 & 7 \end{pmatrix}$

(B) $\begin{pmatrix} 2 & 0 \\ 0 & 12 \end{pmatrix}$

(C) $\begin{pmatrix} 2 & 0 \\ 0 & 12 \end{pmatrix}$

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

Question 1. The transpose of a matrix $A$, denoted by $A'$, $A^T$, or $A^t$, is obtained by interchanging the rows and columns of A. If $A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$, what is $A'$?

(A) $\begin{pmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{pmatrix}$

(B) $\begin{pmatrix} 4 & 5 & 6 \\ 1 & 2 & 3 \end{pmatrix}$

(C) $\begin{pmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{pmatrix}$

(D) $\begin{pmatrix} 6 & 5 & 4 \\ 3 & 2 & 1 \end{pmatrix}$

Question 2. A square matrix $A$ is called a symmetric matrix if $A' = A$. Which of the following is a symmetric matrix?

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

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

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

(D) $\begin{pmatrix} 1 & 1 \\ 2 & 2 \end{pmatrix}$

Question 3. A square matrix $A$ is called a skew-symmetric matrix if $A' = -A$. For a skew-symmetric matrix, the diagonal elements are always:

(A) $1$

(B) $-1$

(C) $0$

(D) Any real number

Question 4. If $A$ is a square matrix, then $A + A'$ is always a:

(A) Symmetric matrix.

(B) Skew-symmetric matrix.

(C) Identity matrix.

(D) Zero matrix.

Question 5. If $A$ is a square matrix, then $A - A'$ is always a:

(A) Symmetric matrix.

(B) Skew-symmetric matrix.

(C) Identity matrix.

(D) Zero matrix.

Question 6. Any square matrix $A$ can be expressed as the sum of a symmetric matrix and a skew-symmetric matrix. This is given by $A = \underbrace{\frac{1}{2}(A+A')}_{\text{Symmetric}} + \underbrace{\frac{1}{2}(A-A')}_{\text{Skew-symmetric}}$. What is the skew-symmetric part of the matrix $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$?

$A' = \begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix}$

$A - A' = \begin{pmatrix} 1-1 & 2-3 \\ 3-2 & 4-4 \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$

Skew-symmetric part $= \frac{1}{2}(A-A') = \begin{pmatrix} 0 & -1/2 \\ 1/2 & 0 \end{pmatrix}$

(A) $\begin{pmatrix} 1 & 2.5 \\ 2.5 & 4 \end{pmatrix}$

(B) $\begin{pmatrix} 0 & -0.5 \\ 0.5 & 0 \end{pmatrix}$

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

(D) $\begin{pmatrix} 0 & 0.5 \\ -0.5 & 0 \end{pmatrix}$

Question 7. Elementary operations (or transformations) on a matrix include interchanging any two rows/columns, multiplying a row/column by a non-zero scalar, and adding a multiple of one row/column to another. Which operation is NOT an elementary operation?

(A) $R_i \leftrightarrow R_j$ (Interchange row i and row j)

(B) $k R_i \to R_i$, where $k \neq 0$ (Multiply row i by a non-zero scalar k)

(C) $R_i + R_j \to R_i$ (Add row j to row i)

(D) $R_i + k R_j \to R_i$, where k is any scalar (Add a multiple of row j to row i)

Question 8. An invertible matrix (or non-singular matrix) is a square matrix $A$ for which there exists a matrix $B$ such that $AB = BA = I$, where $I$ is the identity matrix of the same order. The matrix $B$ is called the inverse of A, denoted by $A^{-1}$. For a matrix to be invertible, it must be:

(A) A row matrix.

(B) A column matrix.

(C) A square matrix.

(D) A zero matrix.

Question 9. If $A = \begin{pmatrix} 2 & 3 \\ 1 & 2 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & -3 \\ -1 & 2 \end{pmatrix}$, is B the inverse of A? Check if $AB = I$.

$AB = \begin{pmatrix} 2 & 3 \\ 1 & 2 \end{pmatrix} \begin{pmatrix} 2 & -3 \\ -1 & 2 \end{pmatrix} = \begin{pmatrix} 2(2)+3(-1) & 2(-3)+3(2) \\ 1(2)+2(-1) & 1(-3)+2(2) \end{pmatrix} = \begin{pmatrix} 4-3 & -6+6 \\ 2-2 & -3+4 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = I$.

Also $BA = \begin{pmatrix} 2 & -3 \\ -1 & 2 \end{pmatrix} \begin{pmatrix} 2 & 3 \\ 1 & 2 \end{pmatrix} = \begin{pmatrix} 2(2)+(-3)(1) & 2(3)+(-3)(2) \\ (-1)(2)+2(1) & (-1)(3)+2(2) \end{pmatrix} = \begin{pmatrix} 4-3 & 6-6 \\ -2+2 & -3+4 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = I$.

(A) Yes, because $AB=I$.

(B) Yes, because $BA=I$.

(C) Yes, because $AB=BA=I$.

(D) No, because $AB \neq BA$.

Question 10. If A is a square matrix, under what condition is $A'A$ a symmetric matrix?

$(A'A)' = A''(A') = A A'$. For $A'A$ to be symmetric, we need $(A'A)' = A'A$, i.e., $AA' = A'A$.

However, the question is flawed; $A'A$ is *always* symmetric, regardless of whether $AA'=A'A$. Let $B = A'A$. Then $B' = (A'A)' = A''(A') = AA'$. If $AA' = A'A$, then $B'=B$. But consider $A = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$. $A' = \begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix}$. $A'A = \begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 2 \\ 2 & 5 \end{pmatrix}$, which is symmetric. $AA' = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix} = \begin{pmatrix} 5 & 2 \\ 2 & 1 \end{pmatrix}$, which is also symmetric, but $A'A \neq AA'$.

So $A'A$ is always symmetric.

(A) If A is symmetric.

(B) If A is skew-symmetric.

(C) If $AA' = A'A$ (A is normal).

(D) $A'A$ is always a symmetric matrix.

Question 11. If A is a square matrix of order $n$, then $(A')' = A$. This means taking the transpose twice returns the original matrix. Which property is true for $(AB)'$?

(A) $A'B'$

(B) $B'A'$

(C) $(AB)^{-1}$

(D) $AB$

Question 12. If A is an invertible matrix, then $(A^{-1})' = (A')^{-1}$. This property states that the transpose of the inverse is the inverse of the transpose. Which property is true for $(AB)^{-1}$ when A and B are invertible matrices of the same order?

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

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

(C) $AB$

(D) $(BA)^{-1}$

Question 13. An elementary row operation on a matrix results in a new matrix that is row equivalent to the original matrix. Applying elementary operations is a common method for finding the inverse of a matrix. Which elementary row operation on $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ transforms it to $\begin{pmatrix} 1 & 0 \\ 3 & 1 \end{pmatrix}$?

(A) $R_2 \to R_2 + 3R_1$

(B) $R_1 \to R_1 + 3R_2$

(C) $R_2 \to 3R_1$

(D) $R_1 \leftrightarrow R_2$

Question 14. If a matrix A is expressed as $A = P + Q$, where P is symmetric and Q is skew-symmetric, then $P = \frac{1}{2}(A+A')$ and $Q = \frac{1}{2}(A-A')$. What is $P+Q$?

(A) $A'$

(B) $A$

(C) $2A$

(D) $A - A'$

Question 15. For a square matrix A, if $A^2 = I$, then A is called an involutory matrix. If $A^2 = A$, then A is called an idempotent matrix. Which of the following is an identity matrix?

(A) Symmetric matrix

(B) Diagonal matrix

(C) Both A and B

(D) Neither A nor B

Question 16. If A and B are symmetric matrices of the same order, then $AB - BA$ is always a:

(A) Symmetric matrix.

(B) Skew-symmetric matrix.

(C) Identity matrix.

(D) Zero matrix.

Question 17. If A is a square matrix such that $A^T A = I$, then A is called an orthogonal matrix. Which is a property of orthogonal matrices?

(A) $A = A^T$

(B) $A = -A^T$

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

(D) $A^2 = I$

Question 18. If a matrix A is invertible, its inverse $A^{-1}$ is unique. This is a fundamental property of invertible matrices. If $A$ is invertible, which statement is true about $A'$(transpose of A)?

(A) $A'$ is always invertible.

(B) $A'$ is never invertible.

(C) $A'$ is invertible only if A is symmetric.

(D) The invertibility of $A'$ depends on the elements of A.

Question 19. If A and B are invertible matrices of the same order, which statement is true about $(A+B)^{-1}$?

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

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

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

(D) There is no general formula relating $(A+B)^{-1}$ to $A^{-1}$ and $B^{-1}$.

Question 20. Applying elementary row operations to a matrix $A$ to transform it into the identity matrix $I$, while simultaneously applying the same operations to $I$, yields the inverse matrix $A^{-1}$. This method is called:

(A) Cramer's Rule.

(B) Matrix inversion by adjoint method.

(C) Gauss-Jordan method (using elementary row operations).

(D) Matrix multiplication.

Question 21. For any square matrix A, $A + A'$ is symmetric and $A - A'$ is skew-symmetric. What is $(A + A')'$?

(A) $A - A'$

(B) $A + A'$

(C) $-A - A'$

(D) $A' - A$

Question 22. If A is a matrix such that $A=A'$, then A is symmetric. If $A=-A'$, then A is skew-symmetric. Consider a matrix $A$ such that $A = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$. Is it symmetric, skew-symmetric, or neither?

$A' = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} = - \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} = -A$. So A is skew-symmetric.

(A) Symmetric

(B) Skew-symmetric

(C) Neither

(D) Both symmetric and skew-symmetric

Question 23. If A and B are symmetric matrices of the same order, is $AB$ necessarily symmetric?

$(AB)' = B'A'$. If A and B are symmetric, $B'=B$ and $A'=A$. So $(AB)' = BA$. For $AB$ to be symmetric, we need $(AB)' = AB$, i.e., $BA = AB$. This is not always true for matrix multiplication.

(A) Yes, always.

(B) No, not necessarily.

(C) Yes, only if A and B commute ($AB=BA$).

(D) Yes, only if A and B are diagonal matrices.

Question 24. The inverse of a matrix exists only if its determinant is non-zero. Such a matrix is called non-singular. An invertible matrix is always non-singular. What is the determinant of a singular matrix?

(A) $1$

(B) $0$

(C) Undefined

(D) Non-zero

Question 25. If A is an invertible matrix, which equation is correct?

(A) $AA^{-1} = O$ (Zero matrix)

(B) $AA^{-1} = A$

(C) $AA^{-1} = I$ (Identity matrix)

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

Question 1. A determinant is a scalar value associated with a square matrix. The determinant of a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is denoted by $|A|$ or $\det(A)$ and calculated as:

(A) $ad + bc$

(B) $ad - bc$

(C) $ac - bd$

(D) $ab - cd$

Question 2. Calculate the determinant of the matrix $\begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix}$.

(A) $3 \times 4 - 2 \times 1 = 12 - 2 = 10$

(B) $3 \times 1 - 2 \times 4 = 3 - 8 = -5$

(C) $12 + 2 = 14$

(D) $10$

Question 3. If any two rows (or columns) of a determinant are identical, the value of the determinant is:

(A) $1$

(B) $-1$

(C) $0$

(D) Depends on the elements

Question 4. If any row (or column) of a determinant consists entirely of zeros, the value of the determinant is:

(A) $1$

(B) $-1$

(C) $0$

(D) Depends on the size of the matrix

Question 5. The determinant of the identity matrix $I_n$ is always:

(A) $0$

(B) $1$

(C) $n$

(D) Depends on $n$

Question 6. If $A$ and $B$ are square matrices of the same order, then $\det(AB) = \det(A) \det(B)$. This property relates the determinant of a product to the product of determinants. What is $\det(A')$ (determinant of the transpose)?

(A) $-\det(A)$

(B) $1/\det(A)$

(C) $\det(A)$

(D) $0$

Question 7. The area of a triangle with vertices $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ can be calculated using determinants as $\frac{1}{2} |\det(M)|$, where $M = \begin{pmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{pmatrix}$. If the area of the triangle is 0, what does it mean about the three points?

(A) They form a right-angled triangle.

(B) They are collinear (lie on the same straight line).

(C) They form an equilateral triangle.

(D) They form an isosceles triangle.

Question 8. The adjoint of a square matrix $A$, denoted by $\text{adj}(A)$, is the transpose of the cofactor matrix of A. If $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the adjoint of A is:

(A) $\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$

(B) $\begin{pmatrix} d & b \\ c & a \end{pmatrix}$

(C) $\begin{pmatrix} a & c \\ b & d \end{pmatrix}$

(D) $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$

Question 9. Calculate the adjoint of the matrix $A = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}$.

Adjoint is $\begin{pmatrix} 4 & -3 \\ -1 & 2 \end{pmatrix}$

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

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

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

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

Question 10. For any square matrix A, $A \cdot (\text{adj}(A)) = (\text{adj}(A)) \cdot A$ is equal to:

(A) $A$

(B) $I$

(C) $|A| \cdot I$

(D) $O$

Question 11. If $|A| \neq 0$, then $A^{-1} = \frac{1}{|A|} \text{adj}(A)$. This formula gives the inverse using the adjoint. For what value of $x$ is the matrix $A = \begin{pmatrix} 2 & x \\ 4 & 5 \end{pmatrix}$ singular?

A is singular if $|A| = 0$. $|A| = 2 \times 5 - x \times 4 = 10 - 4x$.

$10 - 4x = 0 \implies 4x = 10 \implies x = 10/4 = 5/2 = 2.5$

(A) $2$

(B) $5$

(C) $8$

(D) $2.5$

Question 12. If the determinant of a matrix is zero, the matrix is:

(A) Invertible.

(B) Non-singular.

(C) Singular.

(D) Identity matrix.

Question 13. If $A$ is a non-singular matrix of order $n$, then $|\text{adj}(A)| = |A|^{n-1}$. This property relates the determinant of the adjoint to the determinant of the matrix. What is $|\text{adj}(A)|$ if $A$ is a $3 \times 3$ matrix with $|A| = 5$?

(A) $5^{3-1} = 5^2 = 25$

(B) $5^3 = 125$

(C) $5$

(D) $1/5$

Question 14. The area of a triangle with vertices $(1, 2), (3, 4), (5, 6)$ is given by $\frac{1}{2} |\det(M)|$, where $M = \begin{pmatrix} 1 & 2 & 1 \\ 3 & 4 & 1 \\ 5 & 6 & 1 \end{pmatrix}$.

$\det(M) = 1(4-6) - 2(3-5) + 1(18-20) = 1(-2) - 2(-2) + 1(-2) = -2 + 4 - 2 = 0$.

(A) The area is 0.

(B) The area is positive, but requires calculation.

(C) The points are collinear.

(D) Both A and C are correct.

Question 15. If $A$ is a non-singular matrix, then $\text{adj}(A') = (\text{adj}(A))'$. The adjoint of the transpose is the transpose of the adjoint. What is the adjoint of the identity matrix $I_n$?

(A) $O$

(B) $I_n$

(C) $-I_n$

(D) Cannot be determined

Question 16. For a $3 \times 3$ matrix, the cofactor of an element $a_{ij}$ is $(-1)^{i+j}$ times the determinant of the submatrix obtained by deleting the $i$-th row and $j$-th column. In the matrix $A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}$, what is the minor of the element $a_{11}$ (which is 1)?

Minor of $a_{11}$ is the determinant of the submatrix $\begin{pmatrix} 5 & 6 \\ 8 & 9 \end{pmatrix}$, which is $5 \times 9 - 6 \times 8 = 45 - 48 = -3$.

(A) $\begin{pmatrix} 5 & 6 \\ 8 & 9 \end{pmatrix}$

(B) $-3$

(C) $3$

(D) $-1 \times 3$

Question 17. Using the previous matrix, what is the cofactor of the element $a_{11}$?

Cofactor of $a_{11}$ is $(-1)^{1+1} \times (\text{Minor of } a_{11}) = 1 \times (-3) = -3$.

(A) $3$

(B) $-3$

(C) $\begin{pmatrix} 5 & 6 \\ 8 & 9 \end{pmatrix}$

(D) $45$

Question 18. The sum of the products of elements of any row or column with their corresponding cofactors is equal to the value of the determinant. For the matrix $A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}$, $|A| = a_{11} C_{11} + a_{12} C_{12}$, where $C_{ij}$ is the cofactor of $a_{ij}$. What is $C_{12}$?

$C_{12} = (-1)^{1+2} \times (\text{Minor of } a_{12}) = -1 \times (\det \begin{pmatrix} a_{21} \end{pmatrix}) = -a_{21}$.

(A) $a_{22}$

(B) $-a_{22}$

(C) $a_{21}$

(D) $-a_{21}$

Question 19. The sum of the products of elements of any row with the cofactors of the corresponding elements of *another* row is always:

(A) $0$

(B) $1$

(C) $|A|$

(D) $|A| I$

Question 20. If A is a $3 \times 3$ matrix such that $|A| = 4$, what is $|2A|$?

For a scalar $k$ and matrix A of order $n$, $|kA| = k^n |A|$. Here $k=2$, $n=3$, $|A|=4$.

$|2A| = 2^3 |A| = 8 \times 4 = 32$.

(A) $2|A| = 8$

(B) $4|A| = 16$

(C) $8|A| = 32$

(D) $2 \times 4 = 8$

Question 1. The inverse of a square matrix A exists if and only if:

(A) A is a symmetric matrix.

(B) A is a singular matrix.

(C) A is a non-singular matrix.

(D) A is an identity matrix.

Question 2. The formula for the inverse of a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ (if $|A| \neq 0$) is:

(A) $A^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$

(B) $A^{-1} = \frac{1}{ad-bc} \begin{pmatrix} a & b \\ c & d \end{pmatrix}$

(C) $A^{-1} = \frac{1}{ad-bc} \begin{pmatrix} -a & -b \\ -c & -d \end{pmatrix}$

(D) $A^{-1} = \frac{1}{ad+bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$

Question 3. Find the inverse of the matrix $A = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}$.

$|A| = 2 \times 4 - 1 \times 3 = 8 - 3 = 5$. Since $|A| \neq 0$, inverse exists.

$A^{-1} = \frac{1}{5} \begin{pmatrix} 4 & -1 \\ -3 & 2 \end{pmatrix} = \begin{pmatrix} 4/5 & -1/5 \\ -3/5 & 2/5 \end{pmatrix}$

(A) $\begin{pmatrix} 4/5 & -1/5 \\ -3/5 & 2/5 \end{pmatrix}$

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

(C) $\begin{pmatrix} 2/5 & 1/5 \\ 3/5 & 4/5 \end{pmatrix}$

(D) $\begin{pmatrix} 1/5 & 0 \\ 0 & 1/5 \end{pmatrix}$

Question 4. A system of linear equations can be written in matrix form as $AX = B$, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix. If A is a non-singular matrix, the unique solution to the system is given by:

(A) $X = AB^{-1}$

(B) $X = BA^{-1}$

(C) $X = A^{-1}B$

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

Question 5. Consider the system of equations $\begin{cases} 2x + 3y = 5 \\ x + 2y = 3 \end{cases}$. In matrix form $AX=B$, what is the matrix A?

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

(B) $\begin{pmatrix} x \\ y \end{pmatrix}$

(C) $\begin{pmatrix} 5 \\ 3 \end{pmatrix}$

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

Question 6. Using the previous system, solve for X using the matrix inverse method, given that the inverse of $A = \begin{pmatrix} 2 & 3 \\ 1 & 2 \end{pmatrix}$ is $A^{-1} = \begin{pmatrix} 2 & -3 \\ -1 & 2 \end{pmatrix}$.

$X = A^{-1}B = \begin{pmatrix} 2 & -3 \\ -1 & 2 \end{pmatrix} \begin{pmatrix} 5 \\ 3 \end{pmatrix} = \begin{pmatrix} 2(5)+(-3)(3) \\ (-1)(5)+2(3) \end{pmatrix} = \begin{pmatrix} 10-9 \\ -5+6 \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$.

So $x=1, y=1$.

(A) $\begin{pmatrix} 1 \\ 1 \end{pmatrix}$

(B) $\begin{pmatrix} 5 \\ 3 \end{pmatrix}$

(C) $\begin{pmatrix} 2 & 3 \\ 1 & 2 \end{pmatrix} \begin{pmatrix} 5 \\ 3 \end{pmatrix}$

(D) $\begin{pmatrix} 1 \\ -1 \end{pmatrix}$

Question 7. Cramer's Rule is another method to solve a system of linear equations using determinants. For a system with coefficient matrix A, if $|A| \neq 0$, the solution for $x$ is given by $x = \frac{|A_x|}{|A|}$, where $A_x$ is the matrix obtained by replacing the first column of A with the constant matrix B. For the system $\begin{cases} 2x + 3y = 5 \\ x + 2y = 3 \end{cases}$, what is the matrix $A_x$?

Coefficient matrix $A = \begin{pmatrix} 2 & 3 \\ 1 & 2 \end{pmatrix}$. Constant matrix $B = \begin{pmatrix} 5 \\ 3 \end{pmatrix}$.

$A_x$ replaces column 1 of A with B: $\begin{pmatrix} 5 & 3 \\ 3 & 2 \end{pmatrix}$.

(A) $\begin{pmatrix} 2 & 5 \\ 1 & 3 \end{pmatrix}$

(B) $\begin{pmatrix} 5 & 3 \\ 3 & 2 \end{pmatrix}$

(C) $\begin{pmatrix} 2 & 3 \\ 5 & 3 \end{pmatrix}$

(D) $\begin{pmatrix} 5 & 3 \\ 1 & 2 \end{pmatrix}$

Question 8. Using Cramer's Rule for the system $\begin{cases} 2x + 3y = 5 \\ x + 2y = 3 \end{cases}$, we found $A = \begin{pmatrix} 2 & 3 \\ 1 & 2 \end{pmatrix}$, $A_x = \begin{pmatrix} 5 & 3 \\ 3 & 2 \end{pmatrix}$. $|A| = 5$, $|A_x| = 5(2) - 3(3) = 10 - 9 = 1$. Calculate the value of $x$.

(A) $x = |A_x| / |A| = 1 / 5$

(B) $x = |A| / |A_x| = 5 / 1 = 5$

(C) $x = |A_x| \times |A| = 1 \times 5 = 5$

(D) $x = 1/5$

Question 9. For a system of linear equations $AX=B$, if $|A| = 0$ and $(\text{adj}(A))B \neq O$, the system is:

(A) Consistent with unique solution.

(B) Consistent with infinitely many solutions.

(C) Inconsistent (no solution).

(D) Cannot be determined.

Question 10. For a system of linear equations $AX=B$, if $|A| = 0$ and $(\text{adj}(A))B = O$, the system is:

(A) Consistent with unique solution.

(B) Consistent with infinitely many solutions.

(C) Inconsistent (no solution).

(D) Cannot be determined (could be consistent with infinite solutions or inconsistent).

Question 11. Which method is applicable only for systems of linear equations where the number of equations is equal to the number of variables and the coefficient matrix is non-singular?

(A) Substitution method

(B) Elimination method

(C) Matrix Inverse Method

(D) Graphical method

Question 12. If $A = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$, does $A^{-1}$ exist?

$|A| = 1 \times 1 - 1 \times 1 = 0$. Since $|A|=0$, the matrix is singular.

(A) Yes, and it is $\begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix}$.

(B) Yes, and it is $\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$.

(C) No, because the determinant is zero.

(D) Cannot be determined.

Question 13. Consider the homogeneous system of equations $AX = O$. This system is always consistent (it always has the trivial solution $X=O$). When does it have a non-trivial solution (a solution other than $x=0, y=0, \dots$)?

(A) If $|A| \neq 0$

(B) If $|A| = 0$

(C) It never has a non-trivial solution.

(D) It always has a non-trivial solution.

Question 14. What is $(\text{adj}(I_n))$, where $I_n$ is the identity matrix of order $n$?

(A) $O_n$

(B) $I_n$

(C) $n I_n$

(D) Cannot be determined

Question 15. If A is an invertible matrix, what is $|A^{-1}|$?

We know $AA^{-1} = I$. Taking determinants on both sides, $|AA^{-1}| = |I|$, so $|A||A^{-1}| = 1$. Thus $|A^{-1}| = 1/|A|$.

(A) $|A|$

(B) $-|A|$

(C) $1/|A|$

(D) $0$

Question 16. For the system $\begin{cases} x - y = 1 \\ 2x - 2y = 2 \end{cases}$. What is the determinant of the coefficient matrix A?

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

(A) $0$

(B) $1$

(C) $2$

(D) $-2$

Question 17. Based on the determinant of the coefficient matrix being 0 for the system $\begin{cases} x - y = 1 \\ 2x - 2y = 2 \end{cases}$, does it have a unique solution, no solution, or infinitely many solutions?

Since $|A|=0$, it is either no solution or infinite solutions. The second equation is just 2 times the first equation ($2(x-y) = 2(1)$). So the equations are dependent (represent the same line). This indicates infinitely many solutions.

Using the $(\text{adj}(A))B$ check: $\text{adj}(A) = \begin{pmatrix} -2 & 1 \\ -2 & 1 \end{pmatrix}$. $B = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$.

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

Since $|A|=0$ and $(\text{adj}(A))B = O$, the system has infinitely many solutions.

(A) Unique solution

(B) No solution

(C) Infinitely many solutions

(D) Cannot be determined using the determinant alone

Question 18. For a system of non-homogeneous linear equations ($B \neq O$), if $|A| \neq 0$, the system is:

(A) Consistent and has a unique solution.

(B) Consistent and has infinitely many solutions.

(C) Inconsistent (no solution).

(D) Always consistent.

Question 19. The matrix method ($X = A^{-1}B$) and Cramer's Rule are equivalent for solving a system $AX=B$ when $|A| \neq 0$. What is the main limitation of these methods for practical larger systems?

(A) They cannot handle more than 3 variables.

(B) Calculating determinants and inverses for large matrices is computationally expensive.

(C) They only work for homogeneous systems.

(D) They only find one solution, even if there are infinitely many.

Question 20. If A is an invertible matrix, then $(\text{adj}(A^{-1}))$ is equal to:

We know $A^{-1} = \frac{1}{|A|} \text{adj}(A)$.

Let $B = A^{-1}$. Then $|B| = |A^{-1}| = 1/|A|$. $A = B^{-1}$.

Using $B^{-1} = \frac{1}{|B|} \text{adj}(B)$, we get $A = \frac{1}{|A^{-1}|} \text{adj}(A^{-1}) = \frac{1}{1/|A|} \text{adj}(A^{-1}) = |A| \text{adj}(A^{-1})$.

So $\text{adj}(A^{-1}) = \frac{1}{|A|} A$.

(A) $|A| A$

(B) $\frac{1}{|A|} A$

(C) $A$

(D) $A^{-1}$

Question 1. The sum of two numbers is 25 and their difference is 13. What are the two numbers?

(A) 18 and 7

(B) 20 and 5

(C) 19 and 6

(D) 17 and 8

Question 2. The perimeter of a rectangular garden is 50 metres. If the length is 5 metres more than the width, find the dimensions of the garden.

(A) 15 m by 10 m

(B) 20 m by 5 m

(C) 18 m by 7 m

(D) 16 m by 9 m

Question 3. A sum of $\textsf{₹}500$ is in the form of denominations of $\textsf{₹}5$ and $\textsf{₹}10$. If the total number of notes is 90, find the number of $\textsf{₹}5$ notes.

(A) 10

(B) 50

(C) 80

(D) 40

Question 4. The product of two consecutive positive integers is 306. Find the integers.

(A) 16 and 17

(B) 17 and 18

(C) 18 and 19

(D) 15 and 16

Question 5. A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, it would have taken 3 hours more to cover the same distance. Find the original speed of the train.

(A) 40 km/h

(B) 48 km/h

(C) 50 km/h

(D) 60 km/h

Question 6. A boat goes 30 km upstream and 44 km downstream in 10 hours. Also, it goes 40 km upstream and 55 km downstream in 13 hours. Find the speed of the stream.

(A) 2 km/h

(B) 3 km/h

(C) 4 km/h

(D) 5 km/h

Question 7. The sum of the digits of a two-digit number is 9. If 27 is added to the number, the digits are reversed. Find the number.

(A) 27

(B) 36

(C) 45

(D) 54

Question 8. A man is 3 times as old as his son. In 10 years, he will be twice as old as his son. What are their current ages?

(A) Son = 8, Man = 24

(B) Son = 9, Man = 27

(C) Son = 10, Man = 30

(D) Son = 12, Man = 36

Question 9. The hypotenuse of a right-angled triangle is 13 cm. If the difference between the other two sides is 7 cm, find the lengths of the other two sides.

(A) 5 cm and 12 cm

(B) 6 cm and 13 cm

(C) 7 cm and 14 cm

(D) 8 cm and 15 cm

Question 10. A piece of cloth costs $\textsf{₹}200$. If the piece was 5 m longer and each metre of cloth cost $\textsf{₹}2$ less, the cost would have remained the same. Find the original length of the piece and the original rate per metre.

(A) Length = 20 m, Rate = $\textsf{₹}10$/m

(B) Length = 25 m, Rate = $\textsf{₹}8$/m

(C) Length = 15 m, Rate = $\textsf{₹}13.33$/m

(D) Length = 10 m, Rate = $\textsf{₹}20$/m

Question 11. If the price of petrol increases by $\textsf{₹}5$ per litre, a person has to buy 2 litres less petrol for $\textsf{₹}1000$. What was the original price per litre?

(A) $\textsf{₹}40$

(B) $\textsf{₹}45$

(C) $\textsf{₹}50$

(D) $\textsf{₹}60$

Question 12. A sum of $\textsf{₹}8400$ is borrowed at compound interest at the rate of 10% per annum for 2 years. Find the amount to be paid after 2 years.

(A) $\textsf{₹}10000$

(B) $\textsf{₹}10164$

(C) $\textsf{₹}10248$

(D) $\textsf{₹}9240$

Question 13. The denominator of a fraction is one more than twice the numerator. If the sum of the fraction and its reciprocal is $2 \frac{16}{21}$, find the fraction.

(A) $3/7$

(B) $7/3$

(C) $2/5$

(D) $5/2$

Question 14. If the speed of a boat in still water is 15 km/h, and it goes 30 km upstream and returns downstream to the same point in 4 hours 30 minutes, find the speed of the stream.

(A) 3 km/h

(B) 4 km/h

(C) 5 km/h

(D) 6 km/h

Question 15. A cistern can be filled by two pipes A and B in 12 and 15 minutes respectively. Both pipes are opened together, but after 3 minutes, pipe A is shut off. How much more time will pipe B take to fill the cistern?

(A) 8 minutes

(B) 8.25 minutes

(C) 9 minutes

(D) 7.5 minutes

Question 16. A mixture of 15 litres contains alcohol and water in the ratio 1:4. If 3 litres of water is added to the mixture, what is the new ratio of alcohol to water?

(A) 1:3

(B) 1:4

(C) 1:5

(D) 2:5

Question 17. The ratio of incomes of two persons is 9:7 and the ratio of their expenditures is 4:3. If each of them saves $\textsf{₹}2000$ per month, find their monthly incomes.

(A) $\textsf{₹}18000$ and $\textsf{₹}14000$

(B) $\textsf{₹}14000$ and $\textsf{₹}10000$

(C) $\textsf{₹}9000$ and $\textsf{₹}7000$

(D) $\textsf{₹}20000$ and $\textsf{₹}16000$

Question 18. The sum of the areas of two squares is 468 m$^2$. If the difference of their perimeters is 24 m, find the sides of the two squares.

(A) 18 m and 12 m

(B) 20 m and 14 m

(C) 15 m and 9 m

(D) 22 m and 16 m

Question 19. A motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.

(A) 4 km/h

(B) 5 km/h

(C) 6 km/h

(D) 8 km/h

Question 20. A person invested a sum of money at simple interest. The amount received after 3 years was $\textsf{₹}1120$ and after 5 years was $\textsf{₹}1280$. What was the original sum and the rate of interest?

(A) $\textsf{₹}800$, Rate = 10%

(B) $\textsf{₹}880$, Rate = $100/11\%$

(C) $\textsf{₹}900$, Rate = 9%

(D) $\textsf{₹}850$, Rate = 11%