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VARDAAN LEARNING INSTITUTE

Mathematics Half-Yearly Mock Test 03 (DAV Zone-L Pattern)
Class: 12 CBSE Subject: Mathematics Time: 2 Hours Max. Marks: 80
General Instructions:
SECTION A (20 Marks)(Multiple Choice Questions - 1 Mark Each)
1.
Let \( A = \{1, 2, 3\} \). The number of equivalence relations containing \( (1, 2) \) is:
(a) 1
(b) 2
(c) 3
(d) 4
2.
If \( f: [0, \infty) \to [0, \infty) \) is defined by \( f(x) = x^2 \), then \( f \) is:
(a) One-one and onto
(b) One-one but not onto
(c) Onto but not one-one
(d) Neither one-one nor onto
3.
The value of \( \tan^{-1}\sqrt{3} - \sec^{-1}(-2) \) is:
(a) \( \frac{\pi}{3} \)
(b) \( -\frac{\pi}{3} \)
(c) \( \frac{2\pi}{3} \)
(d) \( \pi \)
4.
If \( A \) is a matrix of order \( 2 \times 3 \) and \( B \) is a matrix of order \( 3 \times 2 \), then the order of \( (AB)' \) is:
(a) \( 2 \times 2 \)
(b) \( 3 \times 3 \)
(c) \( 2 \times 3 \)
(d) \( 3 \times 2 \)
5.
If \( A = \begin{bmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{bmatrix} \), then \( A + A' = I \), if the value of \( \alpha \) is:
(a) \( \frac{\pi}{6} \)
(b) \( \frac{\pi}{3} \)
(c) \( \pi \)
(d) \( \frac{3\pi}{2} \)
6.
If area of triangle is 35 sq units with vertices \( (2, -6), (5, 4) \) and \( (k, 4) \), then \( k \) is:
(a) 12
(b) -2
(c) -12, -2
(d) 12, -2
7.
If \( A \) is a non-singular square matrix of order 3 such that \( A^2 = 3A \), then the value of \( |A| \) is:
(a) 3
(b) 9
(c) 27
(d) 81
8.
The function \( f(x) = |x| + |x-1| \) is:
(a) Continuous everywhere
(b) Differentiable everywhere
(c) Not continuous at 0 and 1
(d) Differentiable at 0 and 1
9.
If \( y = e^{3\log x} \), then \( \frac{dy}{dx} \) is:
(a) \( 3x^2 \)
(b) \( 3e^{3\log x} \)
(c) \( \frac{3}{x} \)
(d) \( x^3 \)
10.
The side of an equilateral triangle is increasing at the rate of \( 2\text{ cm/s} \). The rate at which its area increases when side is \( 10\text{ cm} \) is:
(a) \( 10\sqrt{3}\text{ cm}^2/\text{s} \)
(b) \( \sqrt{3}\text{ cm}^2/\text{s} \)
(c) \( 10\text{ cm}^2/\text{s} \)
(d) \( \frac{10}{\sqrt{3}}\text{ cm}^2/\text{s} \)
11.
The slope of the normal to the curve \( y = 2x^2 + 3\sin x \) at \( x = 0 \) is:
(a) 3
(b) \( \frac{1}{3} \)
(c) -3
(d) \( -\frac{1}{3} \)
12.
If \( |\vec{a} \times \vec{b}| = \vec{a} \cdot \vec{b} \), then the angle between \( \vec{a} \) and \( \vec{b} \) is:
(a) 0
(b) \( \frac{\pi}{4} \)
(c) \( \frac{\pi}{2} \)
(d) \( \pi \)
13.
The projection of the vector \( \vec{a} = 2\hat{i} - \hat{j} + \hat{k} \) on the vector \( \vec{b} = \hat{i} + 2\hat{j} + 2\hat{k} \) is:
(a) \( \frac{2}{3} \)
(b) \( \frac{1}{3} \)
(c) 2
(d) \( \frac{\sqrt{6}}{3} \)
14.
If a line has direction ratios \( 2, -1, -2 \), then its direction cosines are:
(a) \( \frac{2}{3}, -\frac{1}{3}, -\frac{2}{3} \)
(b) \( \frac{2}{9}, -\frac{1}{9}, -\frac{2}{9} \)
(c) \( \frac{2}{\sqrt{5}}, -\frac{1}{\sqrt{5}}, -\frac{2}{\sqrt{5}} \)
(d) \( -\frac{2}{3}, \frac{1}{3}, \frac{2}{3} \)
15.
The distance of the point \( (2, 3, 4) \) from the \( x \)-axis is:
(a) 2
(b) \( \sqrt{13} \)
(c) 5
(d) \( \sqrt{29} \)
16.
The feasible region for an LPP is shown to be unbounded. If \( M \) is the maximum value of objective function \( Z = ax + by \) evaluated at corner points, then \( M \) is the maximum value of \( Z \) if:
(a) The open half plane \( ax + by > M \) has no point in common with feasible region
(b) The open half plane \( ax + by < M \) has no point in common with feasible region
(c) Feasible region is convex
(d) Maximum value does not exist always
17.
If \( A \) and \( B \) are two events such that \( P(A) = 0.4 \), \( P(B) = 0.8 \) and \( P(B|A) = 0.6 \), then \( P(A \cup B) \) is:
(a) 0.96
(b) 0.24
(c) 0.56
(d) 0.48
18.
If \( P(A \cap B) = \frac{1}{6} \), \( P(A) = \frac{1}{2} \), and \( A \) and \( B \) are independent events, then \( P(B) \) is:
(a) \( \frac{1}{12} \)
(b) \( \frac{1}{3} \)
(c) \( \frac{2}{3} \)
(d) \( \frac{1}{4} \)
19.
Assertion (A): If \( A \) is an invertible symmetric matrix, then \( A^{-1} \) is also symmetric.
Reason (R): For any invertible matrix \( A \), \( (A^{-1})' = (A')^{-1} \).
(a) Both A and R are true, R is correct explanation of A
(b) Both A and R are true, R is NOT correct explanation of A
(c) A is true, but R is false
(d) A is false, but R is true
20.
Assertion (A): The angle between the lines \( \vec{r} = \lambda(\hat{i} + \hat{j} + \hat{k}) \) and \( \vec{r} = \mu(\hat{i} - \hat{j} + \hat{k}) \) is \( \cos^{-1}\left(\frac{1}{3}\right) \).
Reason (R): The angle \( \theta \) between two lines with direction vectors \( \vec{b}_1 \) and \( \vec{b}_2 \) is given by \( \cos\theta = \frac{|\vec{b}_1 \cdot \vec{b}_2|}{|\vec{b}_1||\vec{b}_2|} \).
(a) Both A and R are true, R is correct explanation of A
(b) Both A and R are true, R is NOT correct explanation of A
(c) A is true, but R is false
(d) A is false, but R is true
SECTION B (10 Marks)(Very Short Answer Questions - 2 Marks Each)
21.
Find the principal value of \( \tan^{-1}\left(\tan\left(\frac{3\pi}{4}\right)\right) + \cos^{-1}\left(\cos\left(\frac{2\pi}{3}\right)\right) \). [2]
OR
Express \( \tan^{-1}\left(\frac{\cos x - \sin x}{\cos x + \sin x}\right) \), where \( -\frac{\pi}{4} < x < \frac{3\pi}{4} \), in the simplest form. [2]
22.
If \( A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix} \), show that \( A^2 - 5A + 7I = O \). Hence find \( A^{-1} \). [2]
23.
Find the value of \( k \) so that the function \( f(x) = \begin{cases} kx + 1, & \text{if } x \le 5 \\ 3x - 5, & \text{if } x > 5 \end{cases} \) is continuous at \( x = 5 \). [2]
24.
Find the area of a parallelogram whose adjacent sides are determined by the vectors \( \vec{a} = \hat{i} - \hat{j} + 3\hat{k} \) and \( \vec{b} = 2\hat{i} - 7\hat{j} + \hat{k} \). [2]
25.
Find the vector equation of a line passing through the point \( (2, -1, 4) \) and parallel to the vector \( \hat{i} + 2\hat{j} - \hat{k} \). Write its Cartesian form. [2]
OR
A die is thrown three times. Let event \( A \) be "4 on the third throw" and event \( B \) be "6 on the first and 5 on the second throw". Find \( P(A|B) \). [2]
SECTION C (18 Marks)(Short Answer Questions - 3 Marks Each)
26.
Check whether the relation \( R \) in \( \mathbb{R} \) defined by \( R = \{(a, b) : a \le b^3\} \) is reflexive, symmetric, or transitive. [3]
27.
If \( y = x^x + (\sin x)^{\cos x} \), find \( \frac{dy}{dx} \). [3]
OR
If \( x = a(\cos t + t\sin t) \) and \( y = a(\sin t - t\cos t) \), find \( \frac{d^2y}{dx^2} \) at \( t = \frac{\pi}{4} \). [3]
28.
A ladder \( 5\text{ m} \) long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of \( 2\text{ cm/s} \). How fast is its height on the wall decreasing when the foot of the ladder is \( 4\text{ m} \) away from the wall? [3]
29.
Find the adjoint of matrix \( A = \begin{bmatrix} 1 & -1 & 2 \\ 2 & 3 & 5 \\ -2 & 0 & 1 \end{bmatrix} \) and verify that \( A(\text{adj } A) = (\text{adj } A)A = |A|I \). [3]
30.
Solve the following Linear Programming Problem graphically:
Maximise: \( Z = 5x + 3y \)
Subject to: \[ 3x + 5y \le 15 \] \[ 5x + 2y \le 10 \] \[ x \ge 0, \quad y \ge 0 \] [3]
31.
An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of an accident is 0.01, 0.03 and 0.15 respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver? [3]
SECTION D (20 Marks)(Long Answer Questions - 5 Marks Each)
32.
Solve the following system of linear equations using the matrix inversion method: \[ 2x + 3y + 3z = 5 \] \[ x - 2y + z = -4 \] \[ 3x - y - 2z = 3 \] [5]
OR
Given matrices \( A = \begin{bmatrix} 1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2 \end{bmatrix} \) and \( B = \begin{bmatrix} 2 & 2 & -4 \\ -4 & 2 & -4 \\ 2 & -1 & 5 \end{bmatrix} \), find \( AB \). Hence use this result to solve the system of linear equations: \[ x - y = 3 \] \[ 2x + 3y + 4z = 17 \] \[ y + 2z = 7 \] [5]
33.
A wire of length \( 28\text{ m} \) is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum? [5]
OR
Show that the height of the closed cylinder of given volume and least surface area is equal to its diameter. [5]
34.
Find the shortest distance between the lines whose vector equations are: \[ \vec{r} = (\hat{i} + \hat{j}) + \lambda(2\hat{i} - \hat{j} + \hat{k}) \] \[ \vec{r} = (2\hat{i} + \hat{j} - \hat{k}) + \mu(3\hat{i} - 5\hat{j} + 2\hat{k}) \] [5]
35.
If \( y = (\tan^{-1} x)^2 \), show that: \[ (x^2 + 1)^2 \frac{d^2y}{dx^2} + 2x(x^2 + 1)\frac{dy}{dx} = 2 \] [5]
SECTION E (12 Marks)(Case-Based Integrated Assessment - 4 Marks Each)
36.

CASE STUDY 1: Manufacturing Cylindrical Storage Cans

An engineering company designs closed cylindrical food storage cans of fixed volume \( V = 128\pi\text{ cm}^3 \). The company aims to minimize material costs by minimizing the total surface area \( S \) of the can. Let \( r \) be the base radius and \( h \) be the height of the cylinder.
(i) Express the height \( h \) in terms of the radius \( r \). (1 Mark)
(ii) Write the total surface area \( S(r) \) as a function of radius \( r \) alone, and find \( \frac{dS}{dr} \). (1 Mark)
(iii) Find the value of the radius \( r \) that minimizes the surface area and prove using the second derivative test that this critical radius gives minimum surface area. (2 Marks)
OR
Show that for minimum surface area, the height of the can must equal its base diameter, and calculate the minimum surface area. (2 Marks)
37.

CASE STUDY 2: Solar Energy Tracking Tower

A solar farm has an observation tower whose top is at coordinates \( T(1, 2, 3) \) in meters. A central solar collector panel is positioned at \( P(4, 6, 8) \). Sunrays hit the panel in the direction of the vector \( \vec{s} = 3\hat{i} + 4\hat{j} + 12\hat{k} \).
(i) Find the vector \( \vec{TP} \) connecting the tower top to the panel. (1 Mark)
(ii) Find the straight-line distance between the tower top and the solar collector panel. (1 Mark)
(iii) Find the angle \( \theta \) between the line \( TP \) and the direction of the incident sunrays \( \vec{s} \). (2 Marks)
OR
Find the scalar projection of the vector \( \vec{TP} \) onto the direction of the sunrays \( \vec{s} \). (2 Marks)
38.

CASE STUDY 3: Industrial Quality Assurance

In an automated bolt-manufacturing factory, machines \( M_1, M_2, M_3 \) produce \( 25\% \), \( 35\% \), and \( 40\% \) of total output bolts respectively. Historical quality records show that \( 5\% \) of bolts manufactured by \( M_1 \), \( 4\% \) by \( M_2 \), and \( 2\% \) by \( M_3 \) are defective. A bolt is drawn at random from a batch of mixed production.
(i) What is the probability that the randomly chosen bolt is manufactured by Machine \( M_2 \)? (1 Mark)
(ii) Find the total probability that a randomly chosen bolt is defective. (1 Mark)
(iii) If the drawn bolt is found to be defective, find the probability that it was manufactured by Machine \( M_3 \). (2 Marks)
OR
If the drawn bolt is found to be defective, find the probability that it was manufactured by Machine \( M_1 \). (2 Marks)
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