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VARDAAN LEARNING INSTITUTE

Mathematics Half-Yearly Mock Test 01 (DAV Zone-L Pattern)
Class: 12 CBSE Subject: Mathematics Time: 3 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 \( N \) given by \( R = \{(a, b) : a = b - 2, b > 6\} \). Choose the correct answer:
(a) \((2, 4) \in R\)
(b) \((3, 8) \in R\)
(c) \((6, 8) \in R\)
(d) \((8, 7) \in R\)
2.
The principal value of \(\sin^{-1}\left(-\frac{1}{2}\right)\) is:
(a) \(\frac{\pi}{3}\)
(b) \(-\frac{\pi}{3}\)
(c) \(-\frac{\pi}{6}\)
(d) \(\frac{\pi}{6}\)
3.
If a matrix \( A \) is both symmetric and skew-symmetric, then \( A \) is a:
(a) Diagonal matrix
(b) Zero matrix
(c) Square matrix
(d) Identity matrix
4.
If \( A \) is a square matrix of order \( 3 \times 3 \) such that \( | \text{adj } A | = 64 \), then \( |A| \) is equal to:
(a) \(\pm 8\)
(b) \(8\)
(c) \(\pm 4\)
(d) \(64\)
5.
The function \( f(x) = |x| \) is:
(a) Continuous everywhere
(b) Differentiable everywhere
(c) Not continuous at \( x = 0 \)
(d) Continuous but not differentiable at \( x = 0 \)
6.
The derivative of \( \sin(x^2) \) with respect to \( x \) is:
(a) \(\cos(x^2)\)
(b) \(2x \cos(x^2)\)
(c) \(-2x \sin(x^2)\)
(d) \(x^2 \cos(x)\)
7.
The function \( f(x) = x^3 - 3x^2 + 3x - 100 \) is:
(a) Decreasing on \(\mathbb{R}\)
(b) Increasing on \(\mathbb{R}\)
(c) Neither increasing nor decreasing
(d) Strictly decreasing
8.
If \(\vec{a}\) and \(\vec{b}\) are unit vectors such that \( |\vec{a} + \vec{b}| = 1 \), then \( |\vec{a} - \vec{b}| \) is:
(a) \(1\)
(b) \(\sqrt{3}\)
(c) \(\sqrt{2}\)
(d) \(2\)
9.
The projection of the vector \(\hat{i} + 3\hat{j} + 7\hat{k}\) on the vector \(7\hat{i} - \hat{j} + 8\hat{k}\) is:
(a) \(\frac{60}{\sqrt{114}}\)
(b) \(\frac{50}{\sqrt{114}}\)
(c) \(\frac{48}{\sqrt{114}}\)
(d) \(\frac{60}{114}\)
10.
The angle between two lines whose direction ratios are \(2, 2, -1\) and \(10, 2, -11\) is:
(a) \(0^\circ\)
(b) \(\frac{\pi}{4}\)
(c) \(\frac{\pi}{3}\)
(d) \(\frac{\pi}{2}\)
11.
The corner points of the feasible region determined by a set of linear constraints are \((0, 10), (2, 2)\) and \((4, 0)\). Let \( Z = 3x + 5y \). The minimum value of \( Z \) occurs at:
(a) \((0, 10)\)
(b) \((2, 2)\)
(c) \((4, 0)\)
(d) Both (b) and (c)
12.
If \( P(A) = 0.6 \), \( P(B) = 0.3 \), and \( P(A \cap B) = 0.2 \), then \( P(A | B') \) is equal to:
(a) \(\frac{4}{7}\)
(b) \(\frac{2}{7}\)
(c) \(\frac{1}{7}\)
(d) \(\frac{3}{7}\)
13.
The domain of the function \(\cos^{-1}(2x - 1)\) is:
(a) \([-1, 1]\)
(b) \([0, 1]\)
(c) \([-1, 0]\)
(d) \([0, \pi]\)
14.
If \( A = \begin{bmatrix} \alpha & \beta \\ \gamma & -\alpha \end{bmatrix} \) is such that \( A^2 = I \), then:
(a) \( 1 + \alpha^2 + \beta\gamma = 0 \)
(b) \( 1 - \alpha^2 + \beta\gamma = 0 \)
(c) \( 1 - \alpha^2 - \beta\gamma = 0 \)
(d) \( \alpha^2 + \beta\gamma + 1 = 0 \)
15.
Area of triangle with vertices \((-2, 3), (3, 2)\) and \((-1, -8)\) is:
(a) 15 sq. units
(b) 25 sq. units
(c) 30 sq. units
(d) 45 sq. units
16.
If \( x = e^t \cos t \), \( y = e^t \sin t \), then \(\frac{dy}{dx}\) at \( t = \frac{\pi}{4} \) is:
(a) 1
(b) \(-1\)
(c) \(\sqrt{2}\)
(d) \(0\)
17.
The maximum value of \(\sin x \cdot \cos x\) is:
(a) \(1\)
(b) \(\frac{1}{2}\)
(c) \(\frac{1}{4}\)
(d) \(\sqrt{2}\)
18.
If \(\vec{a} \times \vec{b} = \vec{0}\) and \(\vec{a} \cdot \vec{b} = 0\), then what can be concluded about vectors \(\vec{a}\) and \(\vec{b}\)?
(a) \(\vec{a}\) is perpendicular to \(\vec{b}\)
(b) \(\vec{a}\) is parallel to \(\vec{b}\)
(c) Either \(\vec{a} = \vec{0}\) or \(\vec{b} = \vec{0}\)
(d) \(\vec{a}\) and \(\vec{b}\) are unit vectors
DIRECTION for Q19 and Q20: In the following questions, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option:
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is NOT the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
19.
Assertion (A): The function \( f(x) = \tan x \) is discontinuous at \( x = \frac{\pi}{2} \).
Reason (R): \(\tan \frac{\pi}{2}\) is not defined.
20.
Assertion (A): If \( A \) is a \( 3 \times 3 \) invertible matrix, then \( |\text{adj } A| = |A|^2 \).
Reason (R): For any square matrix \( A \) of order \( n \), \( |\text{adj } A| = |A|^{n-1} \).
SECTION B (10 Marks)(Very Short Answer Type - 2 Marks Each)
21.
Check the injectivity and surjectivity of the function \( f: \mathbb{N} \to \mathbb{N} \) given by \( f(x) = x^2 \).
22.
Find the value of \(\tan^{-1}\left(\sqrt{3}\right) - \cot^{-1}\left(-\sqrt{3}\right)\).
23.
If \( y = \sin(\log x) \), prove that \( x^2 \frac{d^2y}{dx^2} + x \frac{dy}{dx} + y = 0 \).
24.
Find the intervals in which the function \( f(x) = 2x^3 - 3x^2 - 36x + 7 \) is strictly increasing.
25.
Find a vector of magnitude 5 units which is perpendicular to both the vectors \(\vec{a} = 2\hat{i} + \hat{j} + 3\hat{k}\) and \(\vec{b} = 3\hat{i} - \hat{j} + 2\hat{k}\).
SECTION C (18 Marks)(Short Answer Type - 3 Marks Each)
26.
Show that the relation \( R \) in the set \( A = \{x \in \mathbb{Z} : 0 \le x \le 12\} \) given by \( R = \{(a, b) : |a - b| \text{ is a multiple of } 4\} \) is an equivalence relation.
27.
Express the matrix \( A = \begin{bmatrix} 3 & -2 \\ -1 & 4 \end{bmatrix} \) as the sum of a symmetric and a skew-symmetric matrix.
28.
Find \(\frac{dy}{dx}\) if \( x^y = y^x \).
29.
Find the points on the curve \( \frac{x^2}{4} + \frac{y^2}{25} = 1 \) at which the tangents are parallel to the x-axis.
30.
Find the shortest distance between the lines given by vector equations:
\( \vec{r} = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k}) \) and \( \vec{r} = (2\hat{i} - \hat{j} - \hat{k}) + \mu(2\hat{i} + \hat{j} + 2\hat{k}) \).
31.
Probabilities of solving a specific problem independently by A and B are \(\frac{1}{2}\) and \(\frac{1}{3}\) respectively. If both try to solve the problem independently, find the probability that (i) the problem is solved, (ii) exactly one of them solves the problem.
SECTION D (20 Marks)(Long Answer Type - 5 Marks Each)
32.
Using matrices, solve the following system of linear equations:
\( x - y + 2z = 7 \)
\( 3x + 4y - 5z = -5 \)
\( 2x - y + 3z = 12 \)
33.
Discuss the continuity and differentiability of the function \( f(x) = |x| + |x - 1| \) at \( x = 0 \) and \( x = 1 \).
34.
An open box with a square base is to be made from a given quantity of sheet of area \( c^2 \). Show that the maximum volume of the box is \( \frac{c^3}{6\sqrt{3}} \).
35.
Find the equation of the plane passing through the point \((1, -1, 2)\) and perpendicular to the planes \( 2x + 3y - 2z = 5 \) and \( x + 2y - 3z = 8 \). Also, find the distance of this plane from the point \((2, 1, -1)\).
SECTION E (12 Marks)(Case-Based Questions - 4 Marks Each)
36.
Case Study 1: Inverse Trigonometric Functions in Architecture & Bridges
An architect is designing a suspension bridge where the angle of inclination of suspension cables is modeled using inverse trigonometric functions to ensure structural load distribution and stability. Let \( \theta = \cos^{-1}\left(\frac{3}{5}\right) + \tan^{-1}\left(\frac{1}{3}\right) \).
(i) Evaluate \(\sin\left(\cos^{-1}\frac{3}{5}\right)\). (1 Mark)
(ii) Convert \(\cos^{-1}\left(\frac{3}{5}\right)\) into an equivalent \(\tan^{-1}\) form. (1 Mark)
(iii) Find the simplified exact value of \(\tan \theta\). (2 Marks)
OR
Solve for \( x \): \(\tan^{-1}\left(\frac{1-x}{1+x}\right) = \frac{1}{2}\tan^{-1} x\) (where \( x > 0 \)). (2 Marks)
37.
Case Study 2: Linear Programming in Industrial Manufacturing
A factory manufactures two types of gadgets, A and B. Each gadget of type A requires 2 hours of machine time and 1 hour of craftsman time. Each gadget of type B requires 1 hour of machine time and 3 hours of craftsman time. The factory has available at most 12 hours of machine time and 18 hours of craftsman time per day. If the profit on each gadget of type A is ₹ 300 and on type B is ₹ 400.
(i) Formulate the objective function for maximizing the total profit. (1 Mark)
(ii) Write down all the linear inequality constraints representing the problem. (1 Mark)
(iii) Find the corner points of the feasible region and determine the maximum profit per day. (2 Marks)
OR
Can the factory produce 4 gadgets of type A and 5 gadgets of type B per day under the given constraints? Justify mathematically. (2 Marks)
38.
Case Study 3: Conditional Probability in Medical Diagnostics
In a certain medical testing facility, a rare disease is known to affect 2% of the population. A diagnostic test is available, which is 90% accurate for those who have the disease and 95% accurate for those who do not have the disease. A randomly chosen person is tested and found positive.
(i) Define appropriate events for a person having the disease and the test being positive, and state the prior probability \( P(D) \). (1 Mark)
(ii) Find the probability of a false positive result (test is positive given the person does not have the disease). (1 Mark)
(iii) Using Bayes' Theorem, find the exact probability that the person actually has the disease given that their test result is positive. (2 Marks)
OR
Find the total probability that a randomly chosen person's test will come out positive. (2 Marks)
Created by Team Vardaan | DAV Zone-L Half-Yearly Board Pattern