Q: Q4. The value of k for which f(x) = \frac{\tan 3x}{x} for x \neq 0 and f(0) = k is continuous at x=0 is:

Option A: 3. Continuity ke liye \lim_{x \to 0} \frac{\tan 3x}{x} = f(0). \lim_{x \to 0} 3 \cdot \frac{\tan 3x}{3x} = 3(1) = 3.

Question 1

Q1. For a relation R on a set A = \{1, 2, 3\} defined as R = \{(1,1), (2,2), (3,3), (1,2), (2,3)\}, which of the following is true?

Accepted Answer

Option C: R is reflexive but neither symmetric nor transitive.. Relation reflexive hai kyunki (1,1), (2,2), (3,3) \in R. (1,2) \in R par (2,1) 
otin R (not symmetric). (1,2) aur (2,3) \in R par (1,3) 
otin R (not transitive).

Question 2

Q2. The principal value of \cos^{-1}(\cos(7\pi/6)) is:

Accepted Answer

Option B: 5\pi/6. Principal value branch of \cos^{-1}x is [0, \pi]. 7\pi/6 iske bahar hai. We write \cos(2\pi - 5\pi/6) = \cos(5\pi/6). So, 5\pi/6 is the answer.

Question 3

Q3. If A is a square matrix such that A^2 = I, then A^{-1} is equal to:

Accepted Answer

Option B: A. Given A^2 = I. Multiply both sides by A^{-1}: A^{-1}A^2 = A^{-1}I \implies (A^{-1}A)A = A^{-1} \implies IA = A^{-1} \implies A = A^{-1}.

Question 4

Q4. The value of k for which f(x) = \frac{	an 3x}{x} for x 
eq 0 and f(0) = k is continuous at x=0 is:

Accepted Answer

Option A: 3. Continuity ke liye \lim_{x 	o 0} \frac{	an 3x}{x} = f(0). \lim_{x 	o 0} 3 \cdot \frac{	an 3x}{3x} = 3(1) = 3.

Question 5

Q5. The slope of the tangent to the curve y = x^3 - x + 1 at the point where x = 2 is:

Accepted Answer

Option B: 11. Slope is dy/dx. dy/dx = 3x^2 - 1. At x=2, 3(2^2) - 1 = 12 - 1 = 11.

Question 6

Q6. \int e^x (	an x + \log(\sec x)) \, dx equals:

Accepted Answer

Option B: e^x \log(\sec x) + C. Form \int e^x (f(x) + f'(x)) dx = e^x f(x) + C. Let f(x) = \log(\sec x), then f'(x) = \frac{1}{\sec x} \cdot \sec x 	an x = 	an x. Result matches option B.

Question 7

Q7. The degree of the differential equation \frac{d^2y}{dx^2} + \sqrt{1 + \left(\frac{dy}{dx}\right)^3} = 0 is:

Accepted Answer

Option B: 2. Pehle radical hatayenge: \frac{d^2y}{dx^2} = -\sqrt{1 + (dy/dx)^3}. Square both sides: (d^2y/dx^2)^2 = 1 + (dy/dx)^3. Highest order is 2, and its power is 2.

Question 8

Q8. A unit vector perpendicular to both vectors \vec{a} = 2\hat{i} + \hat{j} + \hat{k} and \vec{b} = \hat{i} - \hat{j} + 2\hat{k} is:

Accepted Answer

Option A: \frac{3\hat{i} - 3\hat{j} - 3\hat{k}}{\sqrt{27}}. \vec{n} = \vec{a} 	imes \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 2 & 1 & 1 \ 1 & -1 & 2 \end{vmatrix} = \hat{i}(2+1) - \hat{j}(4-1) + \hat{k}(-2-1) = 3\hat{i} - 3\hat{j} - 3\hat{k}. Unit vector = \frac{\vec{n}}{|\vec{n}|}.

Question 9

Q9. If a line makes angles \alpha, \beta, \gamma with axes, then \sin^2\alpha + \sin^2\beta + \sin^2\gamma is:

Accepted Answer

Option B: 2. \cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1. Convert to sine: (1-\sin^2\alpha) + (1-\sin^2\beta) + (1-\sin^2\gamma) = 1 \implies 3 - (\sin^2\alpha + \sin^2\beta + \sin^2\gamma) = 1 \implies 	ext{sum} = 2.

Question 10

Q10. The probability of getting a total of 7 when two dice are thrown is:

Accepted Answer

Option A: 1/6. Pairs for 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). Total 6 outcomes out of 36. 6/36 = 1/6.

Question 11

Q11. A wire of length 20m is to be cut into two pieces. One piece is made into a square and the other into a circle. To minimize the combined area, the side of the square is:

Accepted Answer

Option A: 20/(\pi+4). Let square side s and circle radius r. 4s + 2\pi r = 20 \implies r = (20-4s)/2\pi. Area A = s^2 + \pi r^2. Differentiate dA/ds = 0 to get s.

Question 12

Q12. The maximum value of Z = 3x + 4y subject to x+y \le 4, x \ge 0, y \ge 0 is:

Accepted Answer

Option B: 16. Corner points: (0,0), (4,0), (0,4). Z(0,0)=0, Z(4,0)=12, Z(0,4)=16. Max is 16.

Question 13

Q13. If P(A) = 0.4, P(B) = 0.8, P(B|A) = 0.6, then P(A|B) is:

Accepted Answer

Option A: 0.3. P(A \cap B) = P(B|A) \cdot P(A) = 0.6 	imes 0.4 = 0.24. P(A|B) = P(A \cap B) / P(B) = 0.24 / 0.8 = 0.3.

Question 14

Q14. Assertion (A): The function f(x) = |x| is not differentiable at x=0. Reason (R): A function is differentiable if its graph is a smooth curve.

Accepted Answer

Option B: Both A and R are true but R is not correct explanation.. A is true (sharp corner at x=0). R is also a general truth, but differentiability is defined by LHD=RHD, not just "smoothness."

Question 15

Q15. Assertion (A): |adj A| = |A|^{n-1} for a square matrix of order n. Reason (R): For any matrix A, A \cdot adj A = |A|I.

Accepted Answer

Option A: Both A and R are true and R is correct explanation of A.. A \cdot adj A = |A|I. Determinant lenge: |A \cdot adj A| = | |A|I | \implies |A| \cdot |adj A| = |A|^n \cdot |I| = |A|^n. Thus |adj A| = |A|^{n-1}.

Question 16

Q16. [Case Study] A box contains 5 red and 5 black balls. Two balls are drawn one by one without replacement. What is the probability that both balls are red?

Accepted Answer

Option C: 2/9. P(R_1) = 5/10 = 1/2. P(R_2|R_1) = 4/9. P(R_1 \cap R_2) = 1/2 	imes 4/9 = 2/9.

Question 17

Q17. [Case Study] Continuing from Q16, what is the probability that the second ball is red given the first was black?

Accepted Answer

Option B: 5/9. First ball black: 9 balls left, 5 red, 4 black. P(R_2|B_1) = 5/9.

Question 18

Q18. The number of all possible matrices of order 3 	imes 3 with each entry 0 or 1 is:

Accepted Answer

Option D: 512. 9 positions, each has 2 choices. 2^9 = 512.

Question 19

Q19. The area of the region bounded by y^2 = 4ax and x = a is:

Accepted Answer

Option A: 8/3 a^2. Area = 2 \int_0^a \sqrt{4ax} \, dx = 4\sqrt{a} \int_0^a x^{1/2} \, dx = 4\sqrt{a} [x^{3/2} / (3/2)]_0^a = 8/3 \sqrt{a} \cdot a^{3/2} = 8/3 a^2.

Question 20

Q20. If \int_0^a f(x) \, dx = \int_0^a f(a-x) \, dx, find \int_0^{\pi/2} \frac{\sin x}{\sin x + \cos x} \, dx.

Accepted Answer

Option B: \pi/4. Standard property. I = \int_0^{\pi/2} \frac{\cos x}{\cos x + \sin x} \, dx. 2I = \int_0^{\pi/2} 1 \, dx = \pi/2 \implies I = \pi/4.

Mock Test 08 Performance Solutions

Detailed Question Analysis

Q1. For a relation \(R\) on a set \(A = \{1, 2, 3\}\) defined as \(R = \{(1,1), (2,2), (3,3), (1,2), (2,3)\}\), which of the following is true?

Q2. The principal value of \(\cos^{-1}(\cos(7\pi/6))\) is:

Q3. If \(A\) is a square matrix such that \(A^2 = I\), then \(A^{-1}\) is equal to:

Q4. The value of \(k\) for which \(f(x) = \frac{\tan 3x}{x}\) for \(x \neq 0\) and \(f(0) = k\) is continuous at \(x=0\) is:

Q5. The slope of the tangent to the curve \(y = x^3 - x + 1\) at the point where \(x = 2\) is:

Q6. \(\int e^x (\tan x + \log(\sec x)) \, dx\) equals:

Q7. The degree of the differential equation \(\frac{d^2y}{dx^2} + \sqrt{1 + \left(\frac{dy}{dx}\right)^3} = 0\) is:

Q8. A unit vector perpendicular to both vectors \(\vec{a} = 2\hat{i} + \hat{j} + \hat{k}\) and \(\vec{b} = \hat{i} - \hat{j} + 2\hat{k}\) is:

Q9. If a line makes angles \(\alpha, \beta, \gamma\) with axes, then \(\sin^2\alpha + \sin^2\beta + \sin^2\gamma\) is:

Q10. The probability of getting a total of 7 when two dice are thrown is:

Q11. A wire of length 20m is to be cut into two pieces. One piece is made into a square and the other into a circle. To minimize the combined area, the side of the square is:

Q12. The maximum value of \(Z = 3x + 4y\) subject to \(x+y \le 4, x \ge 0, y \ge 0\) is:

Q13. If \(P(A) = 0.4, P(B) = 0.8, P(B|A) = 0.6\), then \(P(A|B)\) is:

Q14. Assertion (A): The function \(f(x) = |x|\) is not differentiable at \(x=0\). Reason (R): A function is differentiable if its graph is a smooth curve.

Q15. Assertion (A): \(|adj A| = |A|^{n-1}\) for a square matrix of order \(n\). Reason (R): For any matrix \(A\), \(A \cdot adj A = |A|I\).

Q16. [Case Study] A box contains 5 red and 5 black balls. Two balls are drawn one by one without replacement. What is the probability that both balls are red?

Q17. [Case Study] Continuing from Q16, what is the probability that the second ball is red given the first was black?

Q18. The number of all possible matrices of order \(3 \times 3\) with each entry 0 or 1 is:

Q19. The area of the region bounded by \(y^2 = 4ax\) and \(x = a\) is:

Q20. If \(\int_0^a f(x) \, dx = \int_0^a f(a-x) \, dx\), find \(\int_0^{\pi/2} \frac{\sin x}{\sin x + \cos x} \, dx\).

ExamSpark Quick Mathematics Revision Signals

Quick Links & ExamSpark Resources

Mock Test 08 Performance Solutions

Detailed Question Analysis

Q1. For a relation \(R\) on a set \(A = \{1, 2, 3\}\) defined as \(R = \{(1,1), (2,2), (3,3), (1,2), (2,3)\}\), which of the following is true?

Q2. The principal value of \(\cos^{-1}(\cos(7\pi/6))\) is:

Q3. If \(A\) is a square matrix such that \(A^2 = I\), then \(A^{-1}\) is equal to:

Q4. The value of \(k\) for which \(f(x) = \frac{\tan 3x}{x}\) for \(x \neq 0\) and \(f(0) = k\) is continuous at \(x=0\) is:

Q5. The slope of the tangent to the curve \(y = x^3 - x + 1\) at the point where \(x = 2\) is:

Q6. \(\int e^x (\tan x + \log(\sec x)) \, dx\) equals:

Q7. The degree of the differential equation \(\frac{d^2y}{dx^2} + \sqrt{1 + \left(\frac{dy}{dx}\right)^3} = 0\) is:

Q8. A unit vector perpendicular to both vectors \(\vec{a} = 2\hat{i} + \hat{j} + \hat{k}\) and \(\vec{b} = \hat{i} - \hat{j} + 2\hat{k}\) is:

Q9. If a line makes angles \(\alpha, \beta, \gamma\) with axes, then \(\sin^2\alpha + \sin^2\beta + \sin^2\gamma\) is:

Q10. The probability of getting a total of 7 when two dice are thrown is:

Q11. A wire of length 20m is to be cut into two pieces. One piece is made into a square and the other into a circle. To minimize the combined area, the side of the square is:

Q12. The maximum value of \(Z = 3x + 4y\) subject to \(x+y \le 4, x \ge 0, y \ge 0\) is:

Q13. If \(P(A) = 0.4, P(B) = 0.8, P(B|A) = 0.6\), then \(P(A|B)\) is:

Q14. Assertion (A): The function \(f(x) = |x|\) is not differentiable at \(x=0\). Reason (R): A function is differentiable if its graph is a smooth curve.

Q15. Assertion (A): \(|adj A| = |A|^{n-1}\) for a square matrix of order \(n\). Reason (R): For any matrix \(A\), \(A \cdot adj A = |A|I\).

Q16. [Case Study] A box contains 5 red and 5 black balls. Two balls are drawn one by one without replacement. What is the probability that both balls are red?

Q17. [Case Study] Continuing from Q16, what is the probability that the second ball is red given the first was black?

Q18. The number of all possible matrices of order \(3 \times 3\) with each entry 0 or 1 is:

Q19. The area of the region bounded by \(y^2 = 4ax\) and \(x = a\) is:

Q20. If \(\int_0^a f(x) \, dx = \int_0^a f(a-x) \, dx\), find \(\int_0^{\pi/2} \frac{\sin x}{\sin x + \cos x} \, dx\).

ExamSpark Quick Mathematics Revision Signals

Quick Links & ExamSpark Resources

Related Study Links