KCET 2025 April 17 Mathematics Question Paper (Available): Download Question Paper with Answers PDF

Consider the following statements:
Statement-I: The set of all solution of the linear inequalities 3x + 8 \(<\) 17 and 2x + 8 \(>\) = 12 are x \(<\) 3 and x \(>\)= 2 respectively.
Statement-II: The common set of solution of linear inequalities 3x + 8 \(<\) 17 and 2x + 8 \(>\)= 12 is (2,3).

Which of the following is true?

• (1) Statement-I is false but Statement-II is true
• (2) Both the statements are true
• (3) Both the statements are false
• (4) Statement-I is true but Statement-II is false

The number of four digit even numbers that can be formed using the digits 0, 1, 2 and 3 without repetition is:

• (1) 10
• (2) 4
• (3) 6
• (4) 6

The number of diagonals that can be drawn in an octagon is:

• (1) 20
• (2) 28
• (3) 30
• (4) 15

If the number of terms in the binomial expansion of \((2x + 3)^n\) is 22, then the value of \(n\) is:

• (1) 6
• (2) 7
• (3) 9
• (4) 8

If the 4th, 10th, and 16th terms of a G.P. are \(x\), \(y\), and \(z\) respectively, then

• (1) \(y = \sqrt{xz}\)
• (2) \(x = \sqrt{yz}\)
• (3) \(y = \frac{x + z}{2}\)
• (4) \(z = \sqrt{xy}\)

If \(A\) is a square matrix such that \(A^2 = A\), then \((I - A)^3\) is:

• (1) \(I - A\)
• (2) \(I + A\)
• (3) \(I - A^3\)
• (4) \(I - A\)

If A and B are two matrices such that AB is an identity matrix and the order of matrix B is \(3 \times 4\), then the order of matrix A is:

• (1) \(3 \times 3\)
• (2) \(4 \times 3\)
• (3) \(4 \times 4\)
• (4) \(3 \times 4\)

Which of the following statements is not correct?

• (1) A diagonal matrix has all diagonal elements equal to zero.
• (2) A symmetric matrix \(A\) is a square matrix satisfying \(A' = A\).
• (3) A skew symmetric matrix has all diagonal elements equal to zero.
• (4) A row matrix has only one row.

If a matrix \( A = \begin{bmatrix} 1 & 1
1 & 1 \end{bmatrix} \) satisfies \( A^6 = kA' \), then the value of \( k \) is:

• (1) 1
• (2) \( \frac{1}{32} \)
• (3) 6
• (4) 32

If \( A = \begin{bmatrix} k & 2
2 & k \end{bmatrix} \) and \( |A^3| = 125 \), then the value of \( k \) is:

• (1) \( \pm 3 \)
• (2) -5
• (3) -4
• (4) \( \pm 2 \)

If \( A \) is a square matrix satisfying the equation \( A^2 - 5A + 7I = 0 \), where \( I \) is the identity matrix and \( 0 \) is the null matrix of the same order, then \( A^{-1} \) is:

• (1) \( \frac{1}{7} (A - 5I) \)
• (2) \( 7(5I - A) \)
• (3) \( \frac{1}{5}(7I - A) \)
• (4) \( \frac{1}{7}(5I - A) \)

If \( A \) is a square matrix of order \( 3 \times 3 \), \( \det A = 3 \), then the value of \( \det(3A^{-1}) \) is:

• (1) 3
• (2) 27
• (3) 9
• (4) \( \frac{1}{3} \)

If \( B = \begin{bmatrix} 1 & 3
2 & \alpha \end{bmatrix} \) is the adjoint of a matrix \( A \) and \( |A| = 2 \), then the value of \( \alpha \) is:

• (1) 5
• (2) 2
• (3) 3
• (4) 4

The system of equations \( 4x + 6y = 5 \) and \( 8x + 12y = 10 \) has:

• (1) Infinitely many solutions.
• (2) A unique solution.
• (3) Only two solutions.
• (4) No solution.

If \( \vec{a} = \hat{i} + 2\hat{j} + \hat{k} \), \( \vec{b} = \hat{i} - \hat{j} + 4\hat{k} \), and \( \vec{c} = \hat{i} + \hat{j} + \hat{k} \) are such that \( \vec{a} + \lambda \vec{b} \) is perpendicular to \( \vec{c} \), then the value of \( \lambda \) is:

• (1) \( \pm 1 \)
• (2) 3
• (3) 0
• (4) \( -1 \)

If \( |\vec{a}| = 10, |\vec{b}| = 2 \) and \( \vec{a} \cdot \vec{b} = 12 \), then the value of \( |\vec{a} \times \vec{b}| \) is:

• (1) 10
• (2) 14
• (3) 16
• (4) 5

Consider the following statements:


% Statement
Statement (I): If either \( |\vec{a}| = 0 \) or \( |\vec{b}| = 0 \), then \( \vec{a} \cdot \vec{b} = 0 \).


% Statement
Statement (II): If \( \vec{a} \times \vec{b} = 0 \), then \( \vec{a} \) is perpendicular to \( \vec{b} \).


Which of the following is correct?

• (1) Statement (I) is false but Statement (II) is true
• (2) Both Statement (I) and Statement (II) are true
• (3) Both Statement (I) and Statement (II) are false
• (4) Statement (I) is true but Statement (II) is false

If a line makes angles \( 90^\circ, 60^\circ \) and \( \theta \) with \( x, y \) and \( z \) axes respectively, where \( \theta \) is acute, then the value of \( \theta \) is:

• (1) \( \dfrac{\pi}{4} \)
• (2) \( \dfrac{\pi}{3} \)
• (3) \( \dfrac{\pi}{2} \)
• (4) \( \dfrac{\pi}{6} \)

The equation of the line through the point \( (0, 1, 2) \) and perpendicular to the line \[ \frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 1}{-2} \]
is:

• (1) \( \frac{x}{-3} = \frac{y - 1}{4} = \frac{z - 2}{-4} \)
• (2) \( \frac{x}{-3} = \frac{y - 1}{4} = \frac{z - 2}{3} \)
• (3) \( \frac{x}{-4} = \frac{y - 1}{-4} = \frac{z - 2}{-3} \)
• (4) \( \frac{x}{3} = \frac{y - 1}{4} = \frac{z - 2}{-3} \)

A line passes through \( (-1, -3) \) and is perpendicular to \( x + 6y = 5 \). Its x-intercept is:

• (1) \( -\frac{1}{2} \)
• (2) \( -2 \)
• (3) \( 2 \)
• (4) \( \frac{1}{2} \)

The length of the latus rectum of \( x^2 + 3y^2 = 12 \) is:

• (1) \( \frac{1}{3} \) units
• (2) \( \frac{4}{\sqrt{3}} \) units
• (3) 24 units
• (4) \( \frac{2}{3} \) units

The value of \[ \lim_{x \to 1} \frac{x^4 - \sqrt{x}}{\sqrt{x} - 1} \]
is:

• (1) \( 7 \)
• (2) does not exist
• (3) \( 1 \)
• (4) \( 0 \)

If \[ y = \frac{\cos x}{1 + \sin x} \]
then:

• (a) \( \frac{dy}{dx} = \frac{-1}{1 + \sin x} \)
• (b) \( \frac{dy}{dx} = \frac{1}{1 + \sin x} \)
• (c) \( \frac{dy}{dx} = -\frac{1}{2} \sec^2 \left( \frac{\pi}{4} - \frac{x}{2} \right) \)
• (d) \( \frac{dy}{dx} = -\frac{1}{2} \sec^2 \left( \frac{\pi}{4} - \frac{x}{2} \right) \)

Match the following:

In the following, \( [x] \) denotes the greatest integer less than or equal to \( x \).




Choose the correct answer from the options given below:

• (1) a - iv, b - iii, c - i, d - ii
• (2) a - iii, b - ii, c - iv, d - i
• (3) a - iii, b - ii, c - i, d - iii
• (4) a - ii, b - iv, c - i, d - iii

The function \( f(x) = \begin{cases} e^x + ax, & x < 0
b(x-1)^2, & x \geq 0 \end{cases} is differentiable at x = 0. Then,

• (1) \( a = 3, b = 1 \)
• (2) \( a = -3, b = 1 \)
• (3) \( a = 3, b = -1 \)
• (4) \( a = -3, b = -1 \)

A function \( f(x) = \begin{cases} \frac{1}{e^x - 1}, & if x \neq 0
\frac{1}{e^x + 1}, & if x = 0 \end{cases} is given. Then, which of the following is true?

• (1) not continuous at x = 0
• (2) differentiable at x = 0
• (3) differentiable at x = 0, but not continuous at x = 0
• (4) continuous at x = 0

If \( y = a \sin^3 t \), \( x = a \cos^3 t \), then \( \frac{dy}{dx} \) at \( t = \frac{3\pi}{4} \) is:

• (1) \( \frac{1}{\sqrt{3}} \)
• (2) \( -\sqrt{3} \)
• (3) \( 1 \)
• (4) \( -1 \)

The derivative of \( \sin x \) with respect to \( \log x \) is:

• (1) \( x \cos x \)
• (2) \( \cos x \log x \)
• (3) \( \cos x \)
• (4) \( \cos x \)

The minimum value of \( 1 - \sin x \) is:

• (1) \( -1 \)
• (2) \( 1 \)
• (3) \( 2 \)
• (4) \( 0 \)

The function \( f(x) = \tan x - x \)

• (1) always decreases
• (2) never increases
• (3) neither increases nor decreases
• (4) always increases

The value of \( \int \frac{dx}{(x+1)(x+2)} \) is:

• (1) \( \log \left| \frac{x-1}{x-2} \right| + c \)
• (2) \( \log \left| \frac{x+2}{x+1} \right| + c \)
• (3) \( \log \left| \frac{x+1}{x+2} \right| + c \)
• (4) \( \log \left| \frac{x-1}{x+2} \right| + c \)

The value of \( \int_{-1}^1 \sin^5 x \cos^4 x \, dx \) is:

• (1) \( \pi \)
• (2) \( \frac{\pi}{2} \)
• (3) \( 0 \)
• (4) \( -\pi \)

The value of \( \int_0^{\frac{2\pi}{0}} \left( 1 + \sin \left( \frac{x}{2} \right) \right) \, dx \) is:

• (1) 4
• (2) 2
• (3) 0
• (4) 8

The integral \[ \int \frac{dx}{x^2 \left( x^4 + 1 \right)^{3/4}} \]
equals:

• (1) \( \left( x^4 + 1 \right)^{1/4} + c \)
• (2) \( - \left( x^4 + 1 \right)^{1/4} + c \)
• (3) \( - \frac{(x^4 + 1)^{1/4}}{x^4} + c \)
• (4) \( \left( \frac{x^4 + 1}{x^4} \right)^{1/4} + c \)

The value of the integral \[ \int_0^1 \log(1 - x) \, dx \]
is:

• (1) 0
• (2) \( \log(2) \)
• (3) \( \log \left( \frac{1}{2} \right) \)
• (4) 1

The area bounded by the curve \[ y = \sin\left(\frac{x}{3}\right), \quad x axis, \quad the lines x = 0 and x = 3\pi \]
is:

• (1) 1 sq. units
• (2) 6 sq. units
• (3) 3 sq. units
• (4) 9 sq. units

The area of the region bounded by the curve \[ y = x^2 \quad and the line \quad y = 16 \quad is: \]

• (1) \( \frac{256}{3} \) sq. units
• (2) 64 sq. units
• (3) \( \frac{128}{3} \) sq. units
• (4) \( \frac{32}{3} \) sq. units

General solution of the differential equation \[ \frac{dy}{dx} + y \tan x = \sec x \quad is: \]

• (1) \( y \tan x = \sec x + c \)
• (2) \( \cos x = y \tan x + c \)
• (3) \( y \sec x = \tan x + c \)
• (4) \( y \sec x = \sec x \, \int \sec x \, dx + c \)

If 'a' and 'b' are the order and degree respectively of the differentiable equation \[ \frac{d^2 y}{dx^2} + \left(\frac{dy}{dx}\right)^3 + x^4 = 0, \quad then \, a - b = \, \_ \_ \]

• (1) 2
• (2) -1
• (3) 0
• (4) 1

The distance of the point \( P(-3,4,5) \) from the yz-plane is:

• (1) 5 units
• (2) 3 units
• (3) 4 units
• (4) 3 units

If \( A = \{ x : x is an integer and x^2 - 9 \geq 0 \} \), \[ B = \{ x : x is a natural number and 2 \leq x \leq 5 \}, \quad C = \{ x : x is a prime number \leq 4 \} \]
Then \( (B - C) \cup A \) is:

• (1) \( \{2, 3, 4\} \)
• (2) \( \{3, 4, 5\} \)
• (3) \( \{2, 3, 5\} \)
• (4) \( \{-3, 3, 4\} \)

A and B are two sets having 3 and 6 elements respectively.
Consider the following statements:
- Statement (I): Minimum number of elements in \( A \cup B \) is 3
- Statement (II): Maximum number of elements in \( A \cap B \) is 3

Which of the following is correct?

• (1) Statement (I) is false, statement (II) is true.
• (2) Both statements (I) and (II) are true.
• (3) Both statements (I) and (II) are false.
• (4) Statement (I) is true, statement (II) is false.

Domain of the function \( f(x) = \frac{1}{(x-2)(x-5)} \) is:

• (1) \( (-\infty, 2) \cup (5, \infty) \)
• (2) \( (-\infty, 3] \cup (5, \infty) \)
• (3) \( (-\infty, 3) \cup (5, \infty) \)
• (4) \( (-\infty, 2] \cup [5, \infty) \)

If \( f(x) = \sin[\lfloor x^2 \rfloor] - \sin[\lfloor -x^2 \rfloor] \), where \( \lfloor x \rfloor \) denotes the greatest integer less than or equal to \( x \), then which of the following is not true?

• (1) \( f(\frac{\pi}{2}) = 1 \)
• (2) \( f(\frac{\pi}{4}) = 1 + \frac{1}{\sqrt{2}} \)
• (3) \( f(\pi) = -1 \)
• (4) \( f(0) = 0 \)

Which of the following is not correct?

• (1) \( \sin 2\pi = \sin(-2\pi) \)
• (2) \( \sin 4\pi = \sin 6\pi \)
• (3) \( \tan 45^\circ = \tan(-315^\circ) \)
• (4) \( \cos 5\pi = \cos 4\pi \)

If \( \cos x + \cos^2 x = 1 \), then the value of \( \sin^2 x + \sin^4 x \) is:

• (1) 1
• (2) 0
• (3) 2
• (4) -1

The mean deviation about the mean for the data \( 4, 7, 8, 9, 10, 12, 13, 17 \) is:

• (1) 3
• (2) 8.5
• (3) 4.03
• (4) 10

A random experiment has five outcomes \(w_1, w_2, w_3, w_4, w_5\). The probabilities of the occurrence of the outcomes \(w_1, w_2, w_4, w_5\) are respectively \( \frac{1}{6}, a, b, \frac{1}{12} \) such that \(12a + 12b - 1 = 0\). Then the probabilities of occurrence of the outcome \(w_3\) is:

• (1) \( \frac{1}{3} \)
• (2) \( \frac{1}{6} \)
• (3) \( \frac{1}{12} \)
• (4) \( \frac{2}{3} \)

A die has two faces each with number '1', three faces each with number '2' and one face with number '3'. If the die is rolled once, then \(P(1 or 3)\) is:

• (1) \( \frac{1}{2} \)
• (2) \( \frac{1}{3} \)
• (3) \( \frac{1}{6} \)
• (4) \( \frac{2}{3} \)

Let \( A = \{a, b, c\} \), then the number of equivalence relations on \( A \) containing \( (b, c) \) is:

• (1) 3
• (2) 2
• (3) 4
• (4) 1

Let the functions \( f \) and \( g \) be \[ f : [0, \frac{\pi}{2}] \to \mathbb{R} given by f(x) = \sin x and g(x) = \cos x, where R is the set of real numbers. \]
Consider the following statements:
Statement (I): \( f \) and \( g \) are one-to-one.
Statement (II): \( f + g \) is one-to-one.
Which of the following is correct?

• (1) Statement (I) is false, statement (II) is true.
• (2) Both statements (I) and (II) are true.
• (3) Both statements (I) and (II) are false.
• (4) Statement (I) is true, statement (II) is false.

Find \[ \sec^2 \left( \tan^{-1} 2 \right) + \csc^2 \left( \cot^{-1} 3 \right) = ? \]

• (1) 5
• (2) 15
• (3) 10
• (4) 1

The equation \[ 2 \cos^{-1} x = \sin^{-1} \left( 2 \sqrt{1 - x^2} \right) \]
\text{is valid for all values of \(x\) satisfying:

• (1) \( -1 \leq x \leq 1 \)
• (2) \( 0 \leq x \leq 1 \)
• (3) \( \frac{1}{\sqrt{2}} \leq x \leq 1 \)
• (4) \( 0 \leq x \leq \frac{1}{\sqrt{2}} \)

Consider the following statements:

Statement (I): In a LPP, the objective function is always linear.
Statement (II): In a LPP, the linear inequalities on variables are called constraints.
Which of the following is correct?

• (1) Statement (I) is true, Statement (II) is false.
• (2) Both Statements (I) and (II) are false.
• (3) Statement (I) is false, Statement (II) is true.
• (4) Both statements (I) and (II) are true.

The maximum value of \( z = 3x + 4y \), subject to the constraints \( x + y \leq 40, x + 2y \geq 60 \) and \( x, y \geq 0 \) is:

• (1) 120
• (2) 140
• (3) 40
• (4) 130

Consider the following statements.
Statement (I): If \( E \) and \( F \) are two independent events, then \( E' \) and \( F' \) are also independent.
Statement (II): Two mutually exclusive events with non-zero probabilities of occurrence cannot be independent.
Which of the following is correct?

• (1) Statement (I) is false and statement (II) is true.
• (2) Both the statements are true.
• (3) Both the statements are false.
• (4) Statement (I) is true and statement (II) is false.

If \( A \) and \( B \) are two non-mutually exclusive events such that \( P(A | B) = P(B | A) \), then:

• (1) \( A = B \)
• (2) \( A \cap B = \emptyset \)
• (3) \( P(A) = P(B) \)
• (4) \( A \subseteq B \) but \( A \neq B \)

If \( A \) and \( B \) are two events such that \( A \subseteq B \) and \( P(B) \neq 0 \), then which of the following is correct?

• (1) \( P(A) < P(B) \)
• (2) \( P(A | B) \geq P(A) \)
• (3) \( P(A) = P(B) \)
• (4) \( P(A | B) = \frac{P(A)}{P(B)} \)

Meera visits only one of the two temples A and B in her locality. Probability that she visits temple A is \( \frac{2}{5} \). If she visits temple A, the probability that she meets her friend is \( \frac{1}{3} \). The probability that she meets her friend, whereas it is \( \frac{2}{7} \) if she visits temple B. Meera met her friend at one of the two temples. The probability that she met her friend at temple B is:

• (1) \( \frac{5}{16} \)
• (2) \( \frac{3}{16} \)
• (3) \( \frac{9}{16} \)
• (4) \( \frac{7}{16} \)

If \( Z_1 \) and \( Z_2 \) are two non-zero complex numbers, then which of the following is not true?

• (1) \( |Z_1 Z_2| = |Z_1| |Z_2| \)
• (2) \( Z_1 Z_2 = Z_1 \cdot Z_2 \)
• (3) \( |Z_1 + Z_2| \geq |Z_1| + |Z_2| \)
• (4) \( Z_1 + Z_2 = Z_1 + Z_2 \)