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VARDAAN LEARNING INSTITUTE

Mathematics Half-Yearly Mock Test 02 (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 \( R \) be a relation on the set of integers \( \mathbb{Z} \) defined by \( (a, b) \in R \iff a - b \) is an even integer. Then \( R \) is:
(a) Reflexive only
(b) Symmetric only
(c) Equivalence relation
(d) Not transitive
2.
The function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = 3 - 4x \) is:
(a) One-one only
(b) Onto only
(c) Bijective
(d) Neither one-one nor onto
3.
The principal value of \( \cos^{-1}\left(\cos\left(\frac{7\pi}{6}\right)\right) \) is:
(a) \( \frac{7\pi}{6} \)
(b) \( \frac{5\pi}{6} \)
(c) \( \frac{\pi}{6} \)
(d) \( \frac{\pi}{3} \)
4.
If \( A \) is a square matrix of order \( 3 \times 3 \) such that \( |A| = 4 \), then the value of \( |2A| \) is:
(a) 8
(b) 16
(c) 32
(d) 64
5.
If \( A \) and \( B \) are symmetric matrices of the same order, then \( AB - BA \) is a:
(a) Skew-symmetric matrix
(b) Symmetric matrix
(c) Zero matrix
(d) Identity matrix
6.
If \( \begin{vmatrix} 2 & 4 \\ 5 & 1 \end{vmatrix} = \begin{vmatrix} 2x & 4 \\ 6 & x \end{vmatrix} \), then the value(s) of \( x \) is/are:
(a) \( \pm 3 \)
(b) \( \pm \sqrt{3} \)
(c) \( 3 \)
(d) \( \pm \sqrt{6} \)
7.
If \( A \) is an invertible matrix of order 3 and \( |A| = 5 \), then \( |\text{adj } A| \) is:
(a) 5
(b) 25
(c) 125
(d) \( \frac{1}{5} \)
8.
The function \( f(x) = \begin{cases} \frac{k\cos x}{\pi - 2x}, & x \ne \frac{\pi}{2} \\ 3, & x = \frac{\pi}{2} \end{cases} \) is continuous at \( x = \frac{\pi}{2} \). The value of \( k \) is:
(a) 3
(b) 6
(c) -6
(d) 2
9.
If \( y = \log(\cos(e^x)) \), then \( \frac{dy}{dx} \) is equal to:
(a) \( -e^x \tan(e^x) \)
(b) \( e^x \tan(e^x) \)
(c) \( -\tan(e^x) \)
(d) \( e^x \cot(e^x) \)
10.
The rate of change of the volume of a sphere with respect to its surface area when the radius is \( r = 4\text{ cm} \) is:
(a) \( 2\text{ cm} \)
(b) \( 4\text{ cm} \)
(c) \( 8\text{ cm} \)
(d) \( 1\text{ cm} \)
11.
The function \( f(x) = 2x^3 - 9x^2 + 12x + 15 \) is strictly decreasing in the interval:
(a) \( (-\infty, 1) \)
(b) \( (1, 2) \)
(c) \( (2, \infty) \)
(d) \( [1, 2] \)
12.
A unit vector in the direction of the vector \( \vec{a} = 2\hat{i} + 3\hat{j} + \hat{k} \) is:
(a) \( \frac{2\hat{i} + 3\hat{j} + \hat{k}}{\sqrt{14}} \)
(b) \( \frac{2\hat{i} + 3\hat{j} + \hat{k}}{14} \)
(c) \( \frac{\hat{i} + \hat{j} + \hat{k}}{\sqrt{3}} \)
(d) \( \frac{2\hat{i} + 3\hat{j} + \hat{k}}{6} \)
13.
If \( \vec{a} = \hat{i} + 2\hat{j} + 3\hat{k} \) and \( \vec{b} = 2\hat{i} - \hat{j} + 4\hat{k} \), then \( \vec{a} \cdot \vec{b} \) is:
(a) 12
(b) 14
(c) 10
(d) 16
14.
The direction cosines of a line which makes equal angles with the coordinate axes are:
(a) \( \left(\pm 1, \pm 1, \pm 1\right) \)
(b) \( \left(\pm \frac{1}{\sqrt{3}}, \pm \frac{1}{\sqrt{3}}, \pm \frac{1}{\sqrt{3}}\right) \)
(c) \( \left(\pm \frac{1}{3}, \pm \frac{1}{3}, \pm \frac{1}{3}\right) \)
(d) \( \left(\pm \frac{1}{\sqrt{2}}, \pm \frac{1}{\sqrt{2}}, 0\right) \)
15.
The lines \( \frac{x-1}{-3} = \frac{y-2}{2k} = \frac{z-3}{2} \) and \( \frac{x-1}{3k} = \frac{y-1}{1} = \frac{z-6}{-5} \) are perpendicular if \( k = \):
(a) \( -\frac{10}{7} \)
(b) \( \frac{10}{7} \)
(c) \( -\frac{7}{10} \)
(d) \( \frac{7}{10} \)
16.
The corner points of the feasible region for an LPP are \( (0, 2), (3, 0), (6, 0), (6, 8) \) and \( (0, 5) \). Let \( Z = 4x + 6y \). The maximum value of \( Z \) occurs at:
(a) \( (0, 5) \) only
(b) \( (6, 8) \) only
(c) Every point on the line joining \( (0, 5) \) and \( (6, 8) \)
(d) \( (6, 0) \) only
17.
If \( P(A) = \frac{1}{2} \), \( P(B) = 0 \), then \( P(A|B) \) is:
(a) 0
(b) \( \frac{1}{2} \)
(c) Not defined
(d) 1
18.
If \( A \) and \( B \) are two independent events with \( P(A) = 0.3 \) and \( P(B) = 0.4 \), then \( P(A' \cap B') \) is:
(a) 0.12
(b) 0.42
(c) 0.58
(d) 0.70
19.
Assertion (A): The function \( f(x) = x^3 - 3x^2 + 3x - 100 \) has neither a local maximum nor a local minimum at \( x = 1 \).
Reason (R): If \( f'(c) = 0 \) and \( f''(c) = 0 \), then \( x = c \) is necessarily a point of inflection.
(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): For any two non-zero vectors \( \vec{a} \) and \( \vec{b} \), \( (\vec{a} + \vec{b}) \times (\vec{a} - \vec{b}) = -2(\vec{a} \times \vec{b}) \).
Reason (R): The vector cross product is anti-commutative, i.e., \( \vec{a} \times \vec{b} = -(\vec{b} \times \vec{a}) \).
(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 domain and principal value of \( f(x) = \sin^{-1}(2x - 3) \). [2]
OR
Evaluate: \( \tan^{-1}\left[ 2\cos\left(2\sin^{-1}\frac{1}{2}\right) \right] \). [2]
22.
Find a matrix \( X \) such that \( 2A + B + X = O \), where \( A = \begin{bmatrix} -1 & 2 \\ 3 & 4 \end{bmatrix} \) and \( B = \begin{bmatrix} 3 & -2 \\ 1 & 5 \end{bmatrix} \). [2]
23.
If \( x = a(\theta - \sin\theta) \) and \( y = a(1 + \cos\theta) \), find \( \frac{dy}{dx} \) at \( \theta = \frac{\pi}{2} \). [2]
24.
Find a vector of magnitude 6 units which is parallel to the vector \( 2\hat{i} - \hat{j} + 2\hat{k} \). [2]
25.
Find the vector and Cartesian equations of the line passing through the points \( A(1, -1, 2) \) and \( B(3, 4, -2) \). [2]
OR
Two cards are drawn successively without replacement from a well-shuffled pack of 52 cards. Find the probability that both cards are aces. [2]
SECTION C (18 Marks)(Short Answer Questions - 3 Marks Each)
26.
Show that the relation \( R \) in the set \( \mathbb{Z} \) of integers defined by \( R = \{(a, b) : 3 \text{ divides } (a - b)\} \) is an equivalence relation. Also write the equivalence class \( [0] \). [3]
27.
Differentiate the function \( y = (\sin x)^x + x^{\sin x} \) with respect to \( x \). [3]
OR
If \( x\sqrt{1+y} + y\sqrt{1+x} = 0 \) for \( -1 < x < 1 \), prove that \( \frac{dy}{dx} = -\frac{1}{(1+x)^2} \). [3]
28.
Find the intervals in which the function \( f(x) = \sin x + \cos x \), where \( 0 \le x \le 2\pi \), is:
(a) strictly increasing,
(b) strictly decreasing. [3]
29.
For the matrix \( A = \begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix} \), verify that \( A^2 - 4A + 3I = O \). Hence, find \( A^{-1} \). [3]
30.
Solve the following Linear Programming Problem graphically:
Minimise: \( Z = 200x + 500y \)
Subject to the constraints: \[ x + 2y \ge 10 \] \[ 3x + 4y \le 24 \] \[ x \ge 0, \quad y \ge 0 \] [3]
31.
A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black balls. One of the two bags is selected at random and a ball is drawn from the bag which is found to be red. Find the probability that the ball was drawn from the first bag. [3]
SECTION D (20 Marks)(Long Answer Questions - 5 Marks Each)
32.
Solve the following system of linear equations using matrix method: \[ x - y + 2z = 7 \] \[ 3x + 4y - 5z = -5 \] \[ 2x - y + 3z = 12 \] [5]
OR
If \( A = \begin{bmatrix} 1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1 \end{bmatrix} \), find \( A^{-1} \). Using \( A^{-1} \), solve the system of linear equations: \[ x + 2y + z = 4 \] \[ -x + y + z = 0 \] \[ x - 3y + z = 2 \] [5]
33.
An open topped box is to be constructed by removing equal squares of side \( x \) from each corner of a rectangular sheet of tin of dimensions \( 24\text{ cm} \times 9\text{ cm} \) and folding up the sides. Find the dimensions of the box so that the volume is maximum. What is the maximum volume? [5]
OR
Show that the semi-vertical angle of a right circular cone of given surface area and maximum volume is \( \sin^{-1}\left(\frac{1}{3}\right) \). [5]
34.
Find the shortest distance between the following pairs of skew lines: \[ \vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(\hat{i} - 3\hat{j} + 2\hat{k}) \] \[ \vec{r} = (4\hat{i} + 5\hat{j} + 6\hat{k}) + \mu(2\hat{i} + 3\hat{j} + \hat{k}) \] Hence, state whether the two lines intersect. [5]
35.
If \( y = (x + \sqrt{x^2 + 1})^m \), prove that: \[ (x^2 + 1)\frac{d^2y}{dx^2} + x\frac{dy}{dx} - m^2y = 0 \] [5]
SECTION E (12 Marks)(Case-Based Integrated Assessment - 4 Marks Each)
36.

CASE STUDY 1: Fuel Efficiency & Speed Optimisation

The fuel cost per hour for running an electric high-speed train is directly proportional to the square of its speed in km/h. At a speed of \( 40\text{ km/h} \), the fuel cost is ₹800 per hour. Fixed operating charges (independent of speed) are ₹3,200 per hour. Let the speed of the train be \( v\text{ km/h} \), and the journey distance be \( 400\text{ km} \).
(i) Express the total cost of the journey \( C(v) \) as a function of the speed \( v \). (1 Mark)
(ii) Find \( \frac{dC}{dv} \) and determine the critical speed at which the total cost is minimized. (2 Marks)
OR
Using the second derivative test, prove that the total cost is strictly minimized at this critical speed. (2 Marks)
(iii) Calculate the minimum total cost for the entire \( 400\text{ km} \) journey. (1 Mark)
37.

CASE STUDY 2: Air Traffic Control Radar Tracking

An Air Traffic Control (ATC) radar station located at the origin \( O(0,0,0) \) tracks two civilian transport drones in 3D airspace. At a certain instant, Drone A is at point \( P(2, 3, 4) \) and Drone B is at point \( Q(4, 1, 2) \) (coordinates in kilometers).
(i) Find the position vector \( \vec{PQ} \) representing the relative displacement from Drone A to Drone B. (1 Mark)
(ii) Find the distance between the two drones at this instant. (1 Mark)
(iii) Drone A flies with constant velocity in direction \( \vec{d}_1 = \hat{i} + 2\hat{j} + 2\hat{k} \) while Drone B flies in direction \( \vec{d}_2 = 2\hat{i} - \hat{j} + 2\hat{k} \). Find the cosine of the angle between their flight paths. (2 Marks)
OR
Find a unit vector perpendicular to both flight path directions \( \vec{d}_1 \) and \( \vec{d}_2 \). (2 Marks)
38.

CASE STUDY 3: Diagnostic Testing for Rare Disease

A pathology lab is evaluating a rapid diagnostic kit for a rare disease which affects \( 0.2\% \) of a large metropolitan population. Medical trials show that the test kit is \( 99\% \) effective in detecting the disease when it is indeed present (sensitivity), but yields a false positive result for \( 0.5\% \) of healthy people tested. A person is selected at random from the population and given the test.
(i) What is the probability that a randomly chosen person is actually disease-free? (1 Mark)
(ii) Find the total probability that a randomly chosen person tests positive. (1 Mark)
(iii) Given that the person's test result is positive, find the probability that the person actually has the disease. (2 Marks)
OR
Given that the person's test result is negative, find the probability that the person is genuinely disease-free. (2 Marks)
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