Q: Q6. For what value of k does the system of linear equations x + y + z = 2, 2x + y - z = 3, and 3x + 2y + kz = 4 have a unique solution?

Option A: k \neq 0. For a unique solution, the determinant of the coefficient matrix must be non-zero (|A| \neq 0). |A| = 1(k - (-2)) - 1(2k - (-3)) + 1(4 - 3) = k + 2 - 2k - 3 + 1 = -k. So, -k \neq 0 \implies k \neq 0.

Question 1

Q1. Let A = \{1, 2, 3\}. How many equivalence relations can be defined on set A containing the element (1, 2)?

Accepted Answer

Option B: 2. The smallest equivalence relation containing (1, 2) must also contain (2, 1) for symmetry, and (1, 1), (2, 2), (3, 3) for reflexivity. The only other equivalence relation containing these is the universal relation A 	imes A. Hence, exactly 2 relations exist.

Question 2

Q2. If A is a square matrix such that A^2 = I, then the simplified value of (A - I)^3 + (A + I)^3 - 7A is:

Accepted Answer

Option A: A. Expanding both cubes: (A^3 - 3A^2I + 3AI^2 - I^3) + (A^3 + 3A^2I + 3AI^2 + I^3) - 7A. This simplifies to 2A^3 + 6A - 7A. Since A^2 = I, we have A^3 = A \cdot A^2 = A. Substituting this gives 2A + 6A - 7A = A.

Question 3

Q3. The domain of the inverse trigonometric function y = \cos^{-1}(x^2 - 4) is:

Accepted Answer

Option A: [-\sqrt{5}, -\sqrt{3}] \cup [\sqrt{3}, \sqrt{5}]. The domain of \cos^{-1}(	heta) is -1 \le 	heta \le 1. So, -1 \le x^2 - 4 \le 1, which implies 3 \le x^2 \le 5. Taking the square root gives x \in [-\sqrt{5}, -\sqrt{3}] \cup [\sqrt{3}, \sqrt{5}]. Note ki negative roots ko miss nahi karna hai.

Question 4

Q4. If \vec{a} and \vec{b} are two unit vectors such that their sum is also a unit vector, then the magnitude of their difference |\vec{a} - \vec{b}| is:

Accepted Answer

Option C: \sqrt{3}. Given |\vec{a} + \vec{b}| = 1. Squaring gives |\vec{a}|^2 + |\vec{b}|^2 + 2\vec{a} \cdot \vec{b} = 1 \implies 1 + 1 + 2\vec{a} \cdot \vec{b} = 1 \implies 2\vec{a} \cdot \vec{b} = -1. Now, |\vec{a} - \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 - 2\vec{a} \cdot \vec{b} = 1 + 1 - (-1) = 3. Hence, |\vec{a} - \vec{b}| = \sqrt{3}.

Question 5

Q5. The points where the function f(x) = |x - 1| + |x - 3| is continuous but not differentiable are:

Accepted Answer

Option C: At both x = 1 and x = 3. Modulus functions f(x) = |x - a| form a sharp "V" corner at x = a. They are continuous everywhere on \mathbb{R} but fail to be differentiable exactly at the critical points where the expression inside the modulus is zero. Yahan wo points x=1 aur x=3 hain.

Question 6

Q6. For what value of k does the system of linear equations x + y + z = 2, 2x + y - z = 3, and 3x + 2y + kz = 4 have a unique solution?

Accepted Answer

Option A: k 
eq 0. For a unique solution, the determinant of the coefficient matrix must be non-zero (|A| 
eq 0). |A| = 1(k - (-2)) - 1(2k - (-3)) + 1(4 - 3) = k + 2 - 2k - 3 + 1 = -k. So, -k 
eq 0 \implies k 
eq 0.

Question 7

Q7. The function f(x) = x^x (x > 0) is strictly increasing in the interval:

Accepted Answer

Option B: (\frac{1}{e}, \infty). Taking log and differentiating: f'(x) = x^x(1 + \log x). For the function to be strictly increasing, f'(x) > 0. Since x^x > 0 always, we need 1 + \log x > 0 \implies \log x > -1 \implies x > e^{-1} = \frac{1}{e}.

Question 8

Q8. The degree of the differential equation y = x\frac{dy}{dx} + a\sqrt{1 + \left(\frac{dy}{dx}\right)^2} is:

Accepted Answer

Option B: 2. To find the degree, the differential equation must be free from radicals (fractional powers) on the derivatives. Isolate the radical: y - x\frac{dy}{dx} = a\sqrt{1 + (\frac{dy}{dx})^2}. Squaring both sides yields (y - x y')^2 = a^2(1 + (y')^2). The highest power of y' after expansion is 2.

Question 9

Q9. If A and B are two events such that P(A) 
eq 0 and P(B) 
eq 0, and P(A|B) = P(B|A), then:

Accepted Answer

Option B: P(A) = P(B). We know P(A|B) = \frac{P(A \cap B)}{P(B)} and P(B|A) = \frac{P(A \cap B)}{P(A)}. Equating them gives \frac{P(A \cap B)}{P(B)} = \frac{P(A \cap B)}{P(A)}. Since A and B can occur, P(A \cap B) isn't necessarily zero, meaning the denominators must be equal: P(A) = P(B).

Question 10

Q10. The value of the definite integral \int_{-\pi/2}^{\pi/2} (x^3 + x \cos x + 	an^5 x + 1) dx is:

Accepted Answer

Option C: \pi. Let I = \int_{-\pi/2}^{\pi/2} (x^3 + x \cos x + 	an^5 x) dx + \int_{-\pi/2}^{\pi/2} 1 dx. The first part is an odd function (since f(-x) = -f(x)), so its integral is 0. The remaining integral is [x]_{- \pi/2}^{\pi/2} = \frac{\pi}{2} - (-\frac{\pi}{2}) = \pi.

Question 11

Q11. A spherical balloon is being inflated. If the radius is increasing at a uniform rate of 2	ext{ cm/s}, the rate of change of its surface area when the radius is 5	ext{ cm} is:

Accepted Answer

Option B: 80\pi 	ext{ cm}^2	ext{/s}. Surface area S = 4\pi r^2. Differentiating with respect to t: \frac{dS}{dt} = 8\pi r \frac{dr}{dt}. Given \frac{dr}{dt} = 2 and r = 5, substitute to get \frac{dS}{dt} = 8\pi(5)(2) = 80\pi 	ext{ cm}^2	ext{/s}.

Question 12

Q12. In an LPP , if the objective function Z = px + qy achieves its maximum value identically at two distinct corner points of the feasible region, then the number of points where Z is maximum is:

Accepted Answer

Option D: Infinite. Yeh ek classic trap hai. If the maximum (or minimum) value occurs at two different corner points, then it occurs at every single point on the line segment joining those two points. Hence, there are infinite optimal solutions.

Question 13

Q13. A factory produces items using two machines. Machine I produces 60% of items and Machine II produces 40%. 2% of items from Machine I are defective, while 3% from Machine II are defective. If a randomly chosen item is defective, which concept will strictly be used to find the probability it came from Machine II?

Accepted Answer

Option C: Bayes' Theorem. This is a textbook reverse-probability real-life scenario. We already know the end result (the item is defective) and are asked to "trace back" to find the probability of a specific cause (Machine II). This requires Bayes' Theorem.

Question 14

Q14. Assertion (A): If A is a square matrix of order 3 and |A| = 4, then |adj A| = 16. Reason (R): For any square matrix A of order n, |adj A| = |A|^{n-1}.

Accepted Answer

Option A: Both A and R are true and R is the correct explanation of A. Assertion is evaluated using the formula given in the Reason. Here n=3, so |adj A| = |A|^{3-1} = |A|^2 = 4^2 = 16. The reason completely justifies the mathematical operation.

Question 15

Q15. Assertion (A): The lines \vec{r} = (\hat{i} + \hat{j}) + \lambda(\hat{i} + 2\hat{j} - \hat{k}) and \vec{r} = (\hat{i} + \hat{j}) + \mu(-\hat{i} + \hat{j} - 2\hat{k}) intersect each other. Reason (R): Two lines in 3D geometry intersect if and only if their shortest distance is zero.

Accepted Answer

Option A: Both A and R are true and R is the correct explanation of A. Notice that both line equations share the exact same starting point vector (\hat{i} + \hat{j}) when \lambda=0 and \mu=0. Since they share a common point, they intersect. The mathematical condition for intersection in 3D space is indeed that the shortest distance is zero.

Question 16

Q16. Case Study Passage (Questions 16-17): An architect designs a modern parabolic bridge. In a mathematical model on a coordinate plane, the bridge's curve is defined by \(y = 9 - x^2\) (in meters), and the ground level is the x-axis (\(y = 0\)).

What is the total horizontal span (width) of the bridge at ground level?

Accepted Answer

Option B: 6	ext{ m}. To find the span on the ground, set y = 0. So, 9 - x^2 = 0 \implies x^2 = 9 \implies x = 3, -3. The distance between x = -3 and x = 3 is 3 - (-3) = 6 meters.

Question 17

Q17. Case Study Passage (Questions 16-17): An architect designs a modern parabolic bridge. In a mathematical model on a coordinate plane, the bridge's curve is defined by \(y = 9 - x^2\) (in meters), and the ground level is the x-axis (\(y = 0\)).

The total area enclosed by the bridge and the ground level is:

Accepted Answer

Option C: 36	ext{ m}^2. Using definite integration for the area under the curve: A = \int_{-3}^{3} (9 - x^2) dx = 2 \int_{0}^{3} (9 - x^2) dx (since it's an even function). This gives 2 [9x - \frac{x^3}{3}]_0^3 = 2(27 - 9) = 2(18) = 36	ext{ m}^2.

Question 18

Q18. The derivative of \sin^{-1}\left(\frac{2x}{1+x^2}\right) with respect to \cos^{- 1}\left(\frac{1-x^2}{1+x^2}\right) for 0 < x < 1 is:

Accepted Answer

Option B: 1. Substitute x = 	an 	heta. Both expressions simplify directly using standard trigonometric identities: \sin^{-1}(\sin 2	heta) = 2	heta and \cos^{-1}(\cos 2	heta) = 2	heta. Let u = 2	an^{-1}x and v = 2	an^{-1}x. Then \frac{du}{dv} = \frac{du/dx}{dv/dx} = 1.

Question 19

Q19. Let A and B be independent events such that P(A) = 0.3 and P(A \cup B) = 0.58. What is the value of P(B)?

Accepted Answer

Option C: 0.4. Since they are independent, P(A \cap B) = P(A)P(B). The addition formula is P(A \cup B) = P(A) + P(B) - P(A)P(B). Substituting values: 0.58 = 0.3 + P(B) - 0.3 P(B) \implies 0.28 = 0.7 P(B) \implies P(B) = \frac{0.28}{0.7} = 0.4.

Question 20

Q20. The integrating factor (I.F.) of the differential equation x \frac{dy}{dx} - y = 2x^2 is:

Accepted Answer

Option C: \frac{1}{x}. Divide the entire equation by x to get the standard linear form: \frac{dy}{dx} - \left(\frac{1}{x}\right)y = 2x. Here, P = -\frac{1}{x}. I.F. = e^{\int P dx} = e^{-\int \frac{1}{x} dx} = e^{-\log x} = e^{\log(x^{-1})} = \frac{1}{x}.

Mock Test 01 Performance Solutions

Detailed Question Analysis

Q1. Let \(A = \{1, 2, 3\}\). How many equivalence relations can be defined on set \(A\) containing the element \((1, 2)\)?

Q2. If \(A\) is a square matrix such that \(A^2 = I\), then the simplified value of \((A - I)^3 + (A + I)^3 - 7A\) is:

Q3. The domain of the inverse trigonometric function \(y = \cos^{-1}(x^2 - 4)\) is:

Q4. If \(\vec{a}\) and \(\vec{b}\) are two unit vectors such that their sum is also a unit vector, then the magnitude of their difference \(|\vec{a} - \vec{b}|\) is:

Q5. The points where the function \(f(x) = |x - 1| + |x - 3|\) is continuous but not differentiable are:

Q6. For what value of \(k\) does the system of linear equations \(x + y + z = 2\), \(2x + y - z = 3\), and \(3x + 2y + kz = 4\) have a unique solution?

Q7. The function \(f(x) = x^x\) (\(x > 0\)) is strictly increasing in the interval:

Q8. The degree of the differential equation \(y = x\frac{dy}{dx} + a\sqrt{1 + \left(\frac{dy}{dx}\right)^2}\) is:

Q9. If \(A\) and \(B\) are two events such that \(P(A) \neq 0\) and \(P(B) \neq 0\), and \(P(A|B) = P(B|A)\), then:

Q10. The value of the definite integral \(\int_{-\pi/2}^{\pi/2} (x^3 + x \cos x + \tan^5 x + 1) dx\) is:

Q11. A spherical balloon is being inflated. If the radius is increasing at a uniform rate of \(2\text{ cm/s}\), the rate of change of its surface area when the radius is \(5\text{ cm}\) is:

Q12. In an LPP , if the objective function \(Z = px + qy\) achieves its maximum value identically at two distinct corner points of the feasible region, then the number of points where \(Z\) is maximum is:

Q14. Assertion (A): If \(A\) is a square matrix of order \(3\) and \(|A| = 4\), then \(|adj A| = 16\). Reason (R): For any square matrix \(A\) of order \(n\), \(|adj A| = |A|^{n-1}\).

Q15. Assertion (A): The lines \(\vec{r} = (\hat{i} + \hat{j}) + \lambda(\hat{i} + 2\hat{j} - \hat{k})\) and \(\vec{r} = (\hat{i} + \hat{j}) + \mu(-\hat{i} + \hat{j} - 2\hat{k})\) intersect each other. Reason (R): Two lines in 3D geometry intersect if and only if their shortest distance is zero.

Q18. The derivative of \(\sin^{-1}\left(\frac{2x}{1+x^2}\right)\) with respect to \(\cos^{- 1}\left(\frac{1-x^2}{1+x^2}\right)\) for \(0 < x < 1\) is:

Q19. Let \(A\) and \(B\) be independent events such that \(P(A) = 0.3\) and \(P(A \cup B) = 0.58\). What is the value of \(P(B)\)?

Q20. The integrating factor (I.F.) of the differential equation \(x \frac{dy}{dx} - y = 2x^2\) is:

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