KCET 2023 Mathematics Question Paper: Download Set A3 Question Paper with Answer Key PDF

KCET 2023 Mathematics Questions with Solutions

Let \( A = \{x, y, z, u\} \) and \( B = \{a, b\} \). A function \( f: A \to B \) is selected randomly. The probability that the function is an onto function is
 

• (A) \( \frac{5}{8} \)
• (B) \( \frac{1}{35} \)
• (C) \( \frac{7}{8} \)
• (D) \( \frac{1}{8} \)

The shaded region in the figure given is the solution of which of the inequalities?




 

• (A) \( x + y \geq 7, \ 2x - 3y + 6 \geq 0, \ x \geq 0, \ y \geq 0 \)
• (B) \( x + y \leq 7, \ 2x - 3y + 6 \leq 0, \ x \geq 0, \ y \geq 0 \)
• (C) \( x + y \leq 7, \ 2x - 3y + 6 \geq 0, \ x \geq 0, \ y \geq 0 \)
• (D) \( x + y \geq 7, \ 2x - 3y + 6 \leq 0, \ x \geq 0, \ y \geq 0 \)

If \( A \) and \( B \) are events such that \( P(A) = \frac{1}{4}, P(A/B) = \frac{1}{2} \) and \( P(B/A) = \frac{2}{3} \), then \( P(B) \) is
 

• (A) \( \frac{2}{3} \)
• (B) \( \frac{1}{2} \)
• (C) \( \frac{1}{6} \)
• (D) \( \frac{1}{3} \)

A bag contains \( 2n + 1 \) coins. It is known that \( n \) of these coins have heads on both sides, whereas the other \( n + 1 \) coins are fair. One coin is selected at random and tossed. If the probability that the toss results in heads is \( \frac{31}{42} \), then the value of \( n \) is
 

• (A) 8
• (B) 10
• (C) 5
• (D) 6

The value of



is
 

• (A) 1
• (B) 2
• (C) -1
• (D) 0

The modulus of the complex number \[ \frac{(1 + i)^2 (1 + 3i)}{(2 - 6i)(2 - 2i)} \]
is
 

• (A) \( \frac{1}{\sqrt{2}} \)
• (B) \( \frac{\sqrt{2}}{4} \)
• (C) \( \frac{4}{\sqrt{2}} \)
• (D) \( \frac{2}{\sqrt{2}} \)

Given that \( a, b \) and \( x \) are real numbers and \( a < b, \ x < 0 \), then
 

• (A) \( \frac{a}{x} < \frac{b}{x} \)
• (B) \( \frac{a}{x} \leq \frac{b}{x} \)
• (C) \( \frac{a}{x} > \frac{b}{x} \)
• (D) \( \frac{a}{x} \geq \frac{b}{x} \)

Ten chairs are numbered as 1 to 10. Three women and two men wish to occupy one chair each. First the women choose the chairs marked 1 to 6, then the men choose the chairs from the remaining. The number of possible ways is
 

• (A) \( 6C_3 \times 4P_2 \)
• (B) \( 6P_3 \times 4C_2 \)
• (C) \( 6C_3 \times 4C_2 \)
• (D) \( 6P_3 \times 4P_2 \)

Which of the following is an empty set?
 

• (A) \( \{ x : x^2 - 9 = 0, \ x \in \mathbb{R} \} \)
• (B) \( \{ x : x^2 = x + 2, \ x \in \mathbb{R} \} \)
• (C) \( \{ x : x^2 - 1 = 0, \ x \in \mathbb{R} \} \)
• (D) \( \{ x : x^2 + 1 = 0, \ x \in \mathbb{R} \} \)

If \( f(x) = ax + b \), where \( a \) and \( b \) are integers, \( f(-1) = -5 \) and \( f(3) = 3 \), then \( a \) and \( b \) are respectively
 

• (A) 0, 2
• (B) 2, 3
• (C) -3, -1
• (D) 2, -3

The value of \[ e^{\log_{10} \tan 1^\circ + \log_{10} \tan 2^\circ + \log_{10} \tan 3^\circ + \cdots + \log_{10} \tan 89^\circ} \]
is
 

• (A) \( \frac{1}{e} \)
• (B) 1
• (C) 0
• (D) 3

A line passes through \( (2, 2) \) and is perpendicular to the line \( 3x + y = 3 \). Its y-intercept is
 

• (A) 1
• (B) \( \frac{4}{3} \)
• (C) \( \frac{1}{3} \)
• (D) \( \frac{2}{3} \)

The distance between the foci of a hyperbola is 16 and its eccentricity is \( \sqrt{2} \). Its equation is
 

• (A) \( 2x^2 - 3y^2 = 7 \)
• (B) \( y^2 - x^2 = 32 \)
• (C) \( x^2 - y^2 = 32 \)
• (D) \( \frac{x^2}{4} - \frac{y^2}{9} = 1 \)

If \[ \lim_{x \to 0} \frac{\sin(2 + x) - \sin(2 - x)}{x} = A \cos B \]
then the values of \( A \) and \( B \) respectively are
 

• (A) 2, 1
• (B) 1, 1
• (C) 2, 2
• (D) 1, 2

If \( n \) is even and the middle term in the expansion of \( \left( x^2 + \frac{1}{x} \right)^n \) is \( 924x^6 \), then \( n \) is equal to
 

• (A) 12
• (B) 8
• (C) 10
• (D) 14

The \(n^{th}\) term of the series \[ 1 + \frac{3}{7} + \frac{5}{7^2} + \frac{7}{7^3} + \dots \]
is
 

• (A) \( \frac{2n - 1}{7^n} \)
• (B) \( \frac{2n + 1}{7^{n-1}} \)
• (C) \( \frac{2n - 1}{7^{n-1}} \)
• (D) \( \frac{2n + 1}{7^n} \)

If \( p \left( \frac{1}{q} + \frac{1}{r} \right), q \left( \frac{1}{r} + \frac{1}{p} \right), r \left( \frac{1}{p} + \frac{1}{q} \right) \) are in A.P., then \( p, q, r \) are:
 

• (A) are in A.P.
• (B) are not in G.P.
• (C) are not in A.P.
• (D) are in G.P.

Let \( f : \mathbb{R} \to \mathbb{R} \) be defined by \( f(x) = 3x^2 - 5 \) and \( g : \mathbb{R} \to \mathbb{R} \) by \( g(x) = \frac{x}{x^2 + 1} \). Then \( g \circ f \) is
 

• (A) \( \frac{3x^2}{x^4 + 2x^2 - 4} \)
• (B) \( \frac{3x^2}{9x^4 + 30x^2 - 2} \)
• (C) \( \frac{3x^2 - 5}{9x^4 - 30x^2 + 26} \)
• (D) \( \frac{3x^2 - 5}{9x^4 - 6x^2 + 26} \)

Let the relation \( R \) be defined in \( \mathbb{N} \) by \( aRb \) if \( 3a + 2b = 27 \). Then \( R \) is
 

• (A) \( \{(1, 12), (3, 9), (5, 6), (7, 3), (9, 0)\} \)
• (B) \( \{(2, 1), (9, 3), (6, 5), (3, 7)\} \)
• (C) \( \{(1, 12), (3, 9), (5, 6), (7, 3)\} \)
• (D) \( \{(0, \frac{27}{2}), (1, 12), (3, 9), (5, 6), (7, 3)\} \)

Let \( f(x) = \sin 2x + \cos 2x \) and \( g(x) = x^2 - 1 \), then \( g(f(x)) \) is invertible in the domain
 

• (A) \( x \in \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \)
• (B) \( x \in \left[ 0, \frac{\pi}{4} \right] \)
• (C) \( x \in \left[ -\frac{\pi}{4}, \frac{\pi}{4} \right] \)
• (D) \( x \in \left[ -\frac{\pi}{8}, \frac{\pi}{8} \right] \)

The contrapositive of the statement

\textit{"If two lines do not intersect in the same plane then they are parallel." is
 

• (A) If two lines are not parallel then they do not intersect in the same plane.
• (B) If two lines are parallel then they do not intersect in the same plane.
• (C) If two lines are not parallel then they intersect in the same plane.
• (D) If two lines are parallel then they intersect in the same plane.

The mean of 100 observations is 50 and their standard deviation is 5. Then the sum of squares of all observations is
 

• (A) 250000
• (B) 255000
• (C) 50000
• (D) 252500

Let \( f : \mathbb{R} \to \mathbb{R} \) and \( g : [0, \infty) \to \mathbb{R} \) be defined by \( f(x) = x^2 \) and \( g(x) = \sqrt{x} \). Which one of the following is not true?
 

• (A) \( (fo g)(2) = 2 \)
• (B) \( (go f)(-2) = 2 \)
• (C) \( (go f)(4) = 4 \)
• (D) \( (fo g)(-4) = 4 \)

If \( A \) and \( B \) are two matrices such that \( AB = B \) and \( BA = A \), then \( A^2 + B^2 = \)
 

• (A) \( AB \)
• (B) \( 2BA \)
• (C) \( A + B \)
• (D) \( 2AB \)

If \( A = \begin{bmatrix} 2-k & 2
1 & 3-k \end{bmatrix} \) is a singular matrix, then the value of \( 5k - k^2 \) is equal to
 

• (A) -4
• (B) 6
• (C) 4
• (D) -6

The area of a triangle with vertices \( (-3, 0) \), \( (3, 0) \), and \( (0, k) \) is 9 sq. units. The value of \( k \) is
 

• (A) 6
• (B) 3
• (C) 9
• (D) -9

If \( \Delta = \begin{vmatrix} 1 & a & a^2
1 & b & b^2
1 & c & c^2 \end{vmatrix} \) and \( \Delta_1 = \begin{vmatrix} 1 & 1 & 1
bc & ca & ab
a & b & c \end{vmatrix} \), then
 

• (A) \( \Delta_1 \neq \Delta \)
• (B) \( \Delta_1 = -\Delta \)
• (C) \( \Delta_1 = \Delta \)
• (D) \( \Delta_1 = 3\Delta \)

If \[ \sin^{-1}\left( \frac{2a}{1 + a^2} \right) + \cos^{-1}\left( \frac{1 - a^2}{1 + a^2} \right) = \tan^{-1}\left( \frac{2x}{1 - x^2} \right) \]
where \( a, x \in (0, 1) \), then the value of \( x \) is
 

• (A) \( \frac{2a}{1 + a^2} \)
• (B) \( \frac{2a}{1 - a^2} \)
• (C) 0
• (D) \( \frac{a}{2} \)

The value of \[ \cot^{-1}\left[ \frac{\sqrt{1 - \sin x} + \sqrt{1 + \sin x}}{\sqrt{1 - \sin x} - \sqrt{1 + \sin x}} \right] \]
where \( x \in \left( 0, \frac{\pi}{4} \right) \) is
 

• (A) \( \pi - \frac{x}{3} \)
• (B) \( \pi - \frac{x}{2} \)
• (C) \( \frac{x}{2} \)
• (D) \( \frac{x}{2} - \pi \)

If \[ x \begin{bmatrix} 3
2 \end{bmatrix} + y \begin{bmatrix} 1
-1 \end{bmatrix} = \begin{bmatrix} 15
5 \end{bmatrix} \]
then the value of \( x \) and \( y \) are
 

• (A) \( x = -4, y = -3 \)
• (B) \( x = -4, y = 3 \)
• (C) \( x = 4, y = 3 \)
• (D) \( x = 4, y = -3 \)

If the function is \( f(x) = \frac{1}{x+2} \), then the point of discontinuity of the composite function \( y = f(f(x)) \) is
 

• (A) \( \frac{2}{5} \)
• (B) \( \frac{1}{2} \)
• (C) \( -\frac{5}{2} \)
• (D) \( \frac{5}{2} \)

If \( y = a \sin x + b \cos x \), then \( y^2 + \left( \frac{dy}{dx} \right)^2 \) is a
 

• (A) function of \( x \) and \( y \)
• (B) constant
• (C) function of \( x \)
• (D) function of \( y \)

If \[ f(x) = 1 + nx + \frac{n(n-1)}{2} x^2 + \frac{n(n-1)(n-2)}{6} x^3 + \dots + x^n \]
then \( f''(1) \) is
 

• (A) \( n(n-1) 2^n \)
• (B) \( 2^{n-1} \)
• (C) \( (n-1)2^{n-1} \)
• (D) \( n(n-1)2^{n-2} \)

If \[ A = \begin{bmatrix} 1 & -\tan \frac{\alpha}{2}
\tan \frac{\alpha}{2} & 1 \end{bmatrix} \]
and \( AB = I \), then \( B \) =:
 

• (A) \( \cos^2 \frac{\alpha}{2} \cdot I \)
• (B) \( \sin^2 \frac{\alpha}{2} \cdot A \)
• (C) \( \cos^2 \frac{\alpha}{2} \cdot A^T \)
• (D) \( \cos^2 \frac{\alpha}{2} \cdot A \)

If \( u = \sin^{-1}\left( \frac{2x}{1 + x^2} \right) \) and \( v = \tan^{-1}\left( \frac{2x}{1 - x^2} \right) \), then \( \frac{du}{dv} \) is
 

• (A) \( \frac{1 - x^2}{1 + x^2} \)
• (B) 1
• (C) \( \frac{1}{2} \)
• (D) 2

The function \( f(x) = \cot x \) is discontinuous on every point of the set
 

• (A) \( \left\{ x = \left( 2n + 1 \right) \frac{\pi}{2}, n \in \mathbb{Z} \right\} \)
• (B) \( \left\{ x = \frac{n\pi}{2}, n \in \mathbb{Z} \right\} \)
• (C) \( \left\{ x = n\pi, n \in \mathbb{Z} \right\} \)
• (D) \( \left\{ x = 2n\pi, n \in \mathbb{Z} \right\} \)

A particle moves along the curve \( \frac{x^2}{16} + \frac{y^2}{4} = 1 \). When the rate of change of abscissa is 4 times that of its ordinate, then the quadrant in which the particle lies is
 

• (A) III or IV
• (B) II or III
• (C) I or III
• (D) II or IV

An enemy fighter jet is flying along the curve given by \( y = x^2 + 2 \). A soldier is placed at \( (3, 2) \) and wants to shoot down the jet when it is nearest to him. Then the nearest distance is
 

• (A) 2 units
• (B) \( \sqrt{5} \) units
• (C) \( \sqrt{3} \) units
• (D) \( \sqrt{6} \) units

Evaluate the integral ∫₂⁸ (5√(10-x))/(5√x + 5√(10-x)) dx:

• (A) 4
• (B) 3
• (C) 5
• (D) 6

Evaluate the integral \[ \int \sqrt{\csc x - \sin x} \, dx = \]
 

• (A) \( 2 \sqrt{\sin x} + C \)
• (B) \( \frac{2}{\sqrt{\sin x}} + C \)
• (C) \( \sqrt{\sin x} + C \)
• (D) \( \frac{\sqrt{\sin x}}{2} + C \)

If \( f(x) \) and \( g(x) \) are two functions with \( g(x) = x - \frac{1}{x} \) and \( f \circ g (x) = x^3 - \frac{1}{x^3} \), then \( f'(x) \) =:
 

• (A) \( x^2 - \frac{1}{x^2} \)
• (B) \( 1 - \frac{1}{x^2} \)
• (C) \( 3x^2 + 3 \)
• (D) \( 3x^2 + \frac{3}{x^4} \)

A circular plate of radius 5 cm is heated. Due to expansion, its radius increases at the rate of 0.05 cm/sec. The rate at which its area is increasing when the radius is 5.2 cm is:
 

• (A) \( 5.05 \pi \, cm^2/sec \)
• (B) \( 0.52 \pi \, cm^2/sec \)
• (C) \( 5.2 \pi \, cm^2/sec \)
• (D) \( 27.4 \pi \, cm^2/sec \)

The distance \( s \) in meters travelled by a particle in \( t \) seconds is given by \[ s = \frac{2t^3}{3} - 18t + \frac{5}{3} \]
The acceleration when the particle comes to rest is:
 

• (A) \( 12 \, m^2/sec \)
• (B) \( 18 \, m^2/sec \)
• (C) \( 3 \, m^2/sec \)
• (D) \( 10 \, m^2/sec \)

Evaluate the integral \[ \int_{0}^{\pi} \frac{x \tan x}{\sec x - \csc x} \, dx = \]
 

• (A) \( \frac{\pi}{2} \)
• (B) \( \frac{\pi^2}{2} \)
• (C) \( \frac{\pi}{4} \)
• (D) \( \frac{\pi^2}{4} \)

Evaluate the integral \[ \int \sqrt{5 - 2x + x^2} \, dx = \]
 

• (A) \( \frac{x}{2} \sqrt{5 + 2x + x^2} + 2 \log |x - 1| + \sqrt{5 + 2x + x^2} + C \)
• (B) \( \frac{x - 1}{2} \sqrt{5 - 2x + x^2} + 2 \log |x - 1| + \sqrt{5 - 2x + x^2} + C \)
• (C) \( \frac{x - 1}{2} \sqrt{5 - 2x + x^2} + 2 \log |x + 1| + \sqrt{5 - 2x + x^2} + C \)
• (D) \( \frac{x}{2} \sqrt{5 - 2x + x^2} + 4 \log |x + 1| + \sqrt{x^2 - 2x + 5} + C \)

Evaluate the integral \[ \int \frac{1}{1 + 3 \sin^2 x + 8 \cos^2 x} \, dx = \]
 

• (A) \( \frac{1}{6} \tan^{-1} \left( \frac{2 \tan x}{3} \right) + C \)
• (B) \( 6 \tan^{-1} \left( \frac{2 \tan x}{3} \right) + C \)
• (C) \( \frac{1}{6} \tan^{-1} (2 \tan x) + C \)
• (D) \( \tan^{-1} \left( \frac{2 \tan x}{3} \right) + C \)

Evaluate the integral: ∫₋₂⁰ (x³ + 3x² + 3x + 3) cos(x + 1) dx

• (A) 4
• (B) 1
• (C) 0
• (D) 3

The degree of the differential equation \[ 1 + \left( \frac{dy}{dx} \right)^2 + \left( \frac{d^2y}{dx^2} \right)^2 = \sqrt[3]{\frac{d^3y}{dx^3} + 1 is } \]
 

• (A) 1
• (B) 2
• (C) 6
• (D) 3

If \( |\vec{a} + \vec{b}| = |\vec{a} - \vec{b}| \), then:
 

• (A) \( \vec{a} \) and \( \vec{b} \) are coincident.
• (B) \( \vec{a} \) and \( \vec{b} \) are inclined to each other at \( 60^\circ \).
• (C) \( \vec{a} \) and \( \vec{b} \) are perpendicular.
• (D) \( \vec{a} \) and \( \vec{b} \) are parallel.

The component of \( \hat{i} \) in the direction of the vector \( \hat{i} + \hat{j} + 2 \hat{k} \) is:
 

• (A) \( 6\sqrt{6} \)
• (B) \( \frac{\sqrt{6}}{6} \)
• (C) \( \sqrt{6} \)
• (D) \( 6 \)

In the interval \((0, \frac{\pi}{2})\), the area lying between the curves \( y = \tan x \) and \( y = \cot x \) and the X-axis is:

 

• (A) \( 4 \log 2 \) sq. units
• (B) \( \log 2 \) sq. units
• (C) \( 3 \log 2 \) sq. units
• (D) \( 2 \log 2 \) sq. units

The area of the region bounded by the line \( y = x + 1 \), and the lines \( x = 3 \) and \( x = 5 \) is:

 

• (A) \( \frac{11}{2} \) sq. units
• (B) \( 7 \) sq. units
• (C) \( 10 \) sq. units
• (D) \( \frac{7}{2} \) sq. units

If a curve passes through the point \( (1, 1) \) and at any point \( (x, y) \) on the curve, the product of its slope and the x-coordinate of the point is equal to the y-coordinate of the point, then the curve also passes through the point:

 

• (A) \( (-1, 2) \)
• (B) \( (\sqrt{3}, 0) \)
• (C) \( (2, 2) \)
• (D) \( (3, 0) \)

The length of the perpendicular drawn from the point \( (3, -1, 11) \) to the line \( \frac{x /2} = \frac{y - 2}{3} = \frac{z - 3}{4} \) is:

 

• (A) \( \sqrt{33} \)
• (B) \( \sqrt{53} \)
• (C) \( \sqrt{66} \)
• (D) \( \sqrt{29} \)

The equation of the plane through the points \( (2, 1, 0) \), \( (3, 2, -2) \), and \( (3, 1, 7) \) is:

 

• (A) \( 6x - 3y + 2z - 7 = 0 \)
• (B) \( 7x - 9y - z - 5 = 0 \)
• (C) \( 3x - 2y + 6z - 27 = 0 \)
• (D) \( 2x - 3y + 4z - 27 = 0 \)

The point of intersection of the line \( x + 1 = \frac{y + 3}{3} = \frac{-z + 2}{2} \) with the plane \( 3x + 4y + 5z = 10 \) is:

 

• (A) \( (2, 6, -4) \)
• (B) \( (2, 6, 4) \)
• (C) \( (-2, 6, -4) \)
• (D) \( (2, -6, -4) \)

If \( (2, 3, -1) \) is the foot of the perpendicular from \( (4, 2, 1) \) to a plane, then the equation of the plane is:

 

• (A) \( 2x - y + 2z = 0 \)
• (B) \( 2x + y + 2z - 5 = 0 \)
• (C) \( 2x - y + 2z + 1 = 0 \)
• (D) \( 2x + y + 2z - 1 = 0 \)

If \(\left|\mathbf{a \times \mathbf{b\right|^2 + \left|\mathbf{a \cdot \mathbf{b\right|^2 = 144 \text{ and \left|\mathbf{a\right| = 4 \text{, then \left|\mathbf{b\right|\text{ is equal to \text{?

 

• (A) 8
• (B) 4
• (C) 12
• (D) 3

If \( \mathbf{a} + 2 \mathbf{b} + 3 \mathbf{c} = \mathbf{0} \) and \[ ( \mathbf{a} \times \mathbf{b}) + (\mathbf{b} \times \mathbf{c}) + (\mathbf{c} \times \mathbf{a}) = \lambda (\mathbf{b} \times \mathbf{c}), \]
then the value of \( \lambda \) is equal to:

 

• (A) 4
• (B) 6
• (C) 2
• (D) 3

If a line makes an angle of \( \frac{\pi}{3} \) with each X and Y axis, then the acute angle made by the Z-axis is:

 

• (A) \( \frac{\pi}{2} \)
• (B) \( \frac{\pi}{4} \)
• (C) \( \frac{\pi}{6} \)
• (D) \( \frac{\pi}{3} \)