Question 1

Q1. Let R be a relation on the set of integers \mathbb{Z} given by R = \{(a, b) : |a - b| 	ext{ is a multiple of } 4\}. The equivalence class of 1, denoted by [1], is:

Accepted Answer

Option B: \{\dots, -7, -3, 1, 5, 9, \dots\}. The equivalence class [1] contains all integers x such that |x - 1| is a multiple of 4. This means x - 1 = \dots, -8, -4, 0, 4, 8, \dots, which gives x = \dots, -7, -3, 1, 5, 9, \dots.

Question 2

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

Accepted Answer

Option B: \frac{5\pi}{6}. The range of the principal value branch of \cos^{-1}x is [0, \pi]. Since \frac{7\pi}{6} does not lie in this interval, we rewrite \cos(\frac{7\pi}{6}) as \cos(2\pi - \frac{5\pi}{6}) = \cos(\frac{5\pi}{6}). Thus, \cos^{-1}(\cos \frac{5\pi}{6}) = \frac{5\pi}{6}.

Question 3

Q3. If A is a square matrix of order 3 and |A| = -4, then the value of |2A^{-1}| is:

Accepted Answer

Option A: -2. For a matrix of order n, |kA| = k^n|A|. Here, n=3, so |2A^{-1}| = 2^3 |A^{-1}|. Also, |A^{- 1}| = \frac{1}{|A|}. Therefore, 8 	imes \left(\frac{1}{-4}ight) = -2.

Question 4

Q4. The integrating factor of the differential equation x \frac{dy}{dx} - y = 2x^2 is:

Accepted Answer

Option C: \frac{1}{x}. Dividing by x, the equation becomes \frac{dy}{dx} - \frac{1}{x}y = 2x. This is a linear differential equation of the form \frac{dy}{dx} + Py = Q, where P = -\frac{1}{x}. The integrating factor is e^{\int P dx} = e^{\int -\frac{1}{x} dx} = e^{-\log x} = e^{\log(x^{-1})} = \frac{1}{x}.

Question 5

Q5. If the projection of vector \vec{a} = \hat{i} - 2\hat{j} + 3\hat{k} on vector \vec{b} = 2\hat{i} - \hat{j} + \lambda\hat{k} is zero, then the value of \lambda is:

Accepted Answer

Option C: -\frac{4}{3}. The projection of \vec{a} on \vec{b} is given by \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}. If this is zero, then \vec{a} \cdot \vec{b} = 0. Taking the dot product: (1)(2) + (-2)(-1) + (3)(\lambda) = 0 \Rightarrow 2 + 2 + 3\lambda = 0 \Rightarrow \lambda = -\frac{4}{3}.

Question 6

Q6. The maximum value of Z = 4x + 3y subject to constraints x \ge 0, y \ge 0, 2x + 3y \le 18, and x + y \ge 10 is:

Accepted Answer

Option D: No feasible region exists. The line 2x + 3y = 18 has intercepts (9, 0) and (0, 6). The region 2x + 3y \le 18 lies below this line. The line x + y = 10 has intercepts (10, 0) and (0, 10), and x + y \ge 10 lies above it. These two shaded regions do not overlap in the first quadrant, so there is no feasible solution.

Question 7

Q7. What is the order and degree of the differential equation \left(\frac{d^2y}{dx^2}\right)^3 + \cos\left(\frac{dy}{dx}\right) = 0?

Accepted Answer

Option C: Order 2, Degree not defined. The highest order derivative is \frac{d^2y}{dx^2}, so the order is 2. However, the differential equation is not a polynomial equation in its derivatives because of the term \cos\left(\frac{dy}{dx}\right). Therefore, its degree is not defined.

Question 8

Q8. Evaluate the definite integral: \int_{-2}^{2} x|x| \, dx

Accepted Answer

Option B: 0. Let f(x) = x|x|. Then f(-x) = -x|-x| = -x|x| = -f(x). Since f(x) is an odd function, the integral of f(x) from -a to a is directly 0.

Question 9

Q9. If \vec{a}, \vec{b}, \vec{c} are unit vectors such that \vec{a} + \vec{b} + \vec{c} = \vec{0}, find the value of \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}.

Accepted Answer

Option C: -\frac{3}{2}. Given |\vec{a}+\vec{b}+\vec{c}|^2 = 0. Expanding this gives |\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 + 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = 0. Since they are unit vectors, 1 + 1 + 1 + 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = 0. Solving gives -\frac{3}{2}.

Question 10

Q10. Let A and B be two events such that P(A) = 0.4, P(B) = 0.8, and P(B|A) = 0.6. The value of P(A \cup B) is:

Accepted Answer

Option A: 0.96. We know P(B|A) = \frac{P(A \cap B)}{P(A)} \Rightarrow 0.6 = \frac{P(A \cap B)}{0.4} \Rightarrow P(A \cap B) = 0.24. Now, P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.4 + 0.8 - 0.24 = 0.96.

Question 11

Q11. A firm manufactures two products, X and Y. The total cost C(x) associated with the production of x units of item X is given by C(x) = 0.005x^3 - 0.02x^2 + 30x + 5000. The marginal cost when 3 units are produced is:

Accepted Answer

Option A: 30.015. Marginal cost (MC) is the derivative of C(x) with respect to x. MC = \frac{dC}{dx} = 0.015x^2 - 0.04x + 30. Substituting x = 3, MC = 0.015(9) - 0.04(3) + 30 = 0.135 - 0.12 + 30 = 30.015.

Question 12

Q12. 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. The mathematical concept best suited to find the probability that he is a scooter driver is:

Accepted Answer

Option B: Bayes' Theorem. We are given that an event (the accident) has already occurred, and we need to find the probability of a specific cause (it being a scooter driver) that led to this event. Reversing the condition to find the probability of the cause is the exact application of Bayes' Theorem.

Question 13

Q13. A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is fixed at 10 meters. To admit maximum light through the whole opening, what mathematical condition must be met?

Accepted Answer

Option B: The area function must have its first derivative equal to zero and second derivative negative.. To maximize the light, we must maximize the area. For a function to have a local maximum, the first derivative rule states that A'(x) = 0, and the second derivative test requires A''(x) < 0.

Question 14

Q14. Assertion (A): If A is a skew-symmetric matrix of odd order, then |A| = 0. Reason (R): For any square matrix A, |A| = |-A|.

Accepted Answer

Option C: A is true but R is false. Assertion A is true (a known property: determinant of odd order skew-symmetric matrix is 0). Reason R is false because |-A| = (-1)^n |A|, where n is the order. They are only equal if n is even.

Question 15

Q15. Assertion (A): Two independent events A and B (with P(A) 
eq 0, P(B) 
eq 0) cannot be mutually exclusive. Reason (R): If A and B are mutually exclusive, P(A \cap B) = 0, but for independent events, P(A \cap B) = P(A) \cdot P(B) 
eq 0.

Accepted Answer

Option A: Both A and R are true and R is the correct explanation of A. If events are independent, their intersection probability is the product of their individual probabilities. Since neither is zero, the product isn't zero. Hence, they cannot be mutually exclusive (where intersection is zero). Reason perfectly explains the Assertion.

Question 16

Q16. Case Study Passage (Questions 16 & 17): A logistics company uses a drone to deliver packages. The drone is programmed to follow a path in 3D space. At any time \(t\) seconds, the position vector of the drone is given by \(\vec{r} = (3t)\hat{i} + (4t)\hat{j} + (5t^2)\hat{k}\), taking the launch pad as the origin \((0,0,0)\).

What is the straight-line distance of the drone from the launch pad exactly at \(t = 1\) second?

Accepted Answer

Option B: 5\sqrt{2} units. At t = 1, the position vector is \vec{r} = 3\hat{i} + 4\hat{j} + 5\hat{k}. The distance from the origin is the magnitude |\vec{r}| = \sqrt{3^2 + 4^2 + 5^2} = \sqrt{9 + 16 + 25} = \sqrt{50} = 5\sqrt{2}.

Question 17

Q17. Case Study Passage (Questions 16 & 17): A logistics company uses a drone to deliver packages. The drone is programmed to follow a path in 3D space. At any time \(t\) seconds, the position vector of the drone is given by \(\vec{r} = (3t)\hat{i} + (4t)\hat{j} + (5t^2)\hat{k}\), taking the launch pad as the origin \((0,0,0)\).

The direction ratios of the drone's velocity vector at \(t = 2\) seconds are proportional to:

Accepted Answer

Option B: 3, 4, 20. Velocity vector \vec{v} = \frac{d\vec{r}}{dt} = 3\hat{i} + 4\hat{j} + 10t\hat{k}. At t = 2, \vec{v} = 3\hat{i} + 4\hat{j} + 20\hat{k}. Thus, the direction ratios are proportional to 3, 4, 20.

Question 18

Q18. Evaluate: \int_{0}^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, dx

Accepted Answer

Option C: \frac{\pi}{4}. Let I = \int_{0}^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, dx -- (1). Using the property \int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx, we get I = \int_{0}^{\pi/2} \frac{\sqrt{\cos x}}{\sqrt{\cos x} + \sqrt{\sin x}} \, dx -- (2). Adding (1) and (2) gives 2I = \int_{0}^{\pi/2} 1 \, dx = [x]_{0}^{\pi/2} = \frac{\pi}{2}. Therefore, I = \frac{\pi}{4}.

Question 19

Q19. Let A = \begin{bmatrix} 1 & \sin 	heta \ -\sin 	heta & 1 \end{bmatrix}, where 0 \le 	heta \le 2\pi. Then the determinant |A| lies in the interval:

Accepted Answer

Option C: [1, 2]. |A| = (1)(1) - (-\sin 	heta)(\sin 	heta) = 1 + \sin^2 	heta. We know that for all real 	heta, 0 \le \sin^2 	heta \le 1. Adding 1 throughout gives 1 \le 1 + \sin^2 	heta \le 2. Therefore, |A| \in [1, 2].

Question 20

Q20. If the system of equations 2x + 5y = 0 and 8x + ky = 0 has a non-trivial (infinite) solution, then the value of k must make the coefficient matrix:

Accepted Answer

Option B: Singular. A homogeneous system of linear equations AX = 0 has non-trivial solutions only if the determinant of the coefficient matrix A is zero, which means A must be a singular matrix. (For the record, \begin{vmatrix} 2 & 5 \ 8 & k \end{vmatrix} = 0 \Rightarrow 2k - 40 = 0 \Rightarrow k = 20).

Mock Test 07 Performance Solutions

Detailed Question Analysis

Q1. Let \(R\) be a relation on the set of integers \(\mathbb{Z}\) given by \(R = \{(a, b) : |a - b| \text{ is a multiple of } 4\}\). The equivalence class of \(1\), denoted by \([1]\), is:

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

Q3. If \(A\) is a square matrix of order \(3\) and \(|A| = -4\), then the value of \(|2A^{-1}|\) is:

Q4. The integrating factor of the differential equation \(x \frac{dy}{dx} - y = 2x^2\) is:

Q5. If the projection of vector \(\vec{a} = \hat{i} - 2\hat{j} + 3\hat{k}\) on vector \(\vec{b} = 2\hat{i} - \hat{j} + \lambda\hat{k}\) is zero, then the value of \(\lambda\) is:

Q6. The maximum value of \(Z = 4x + 3y\) subject to constraints \(x \ge 0, y \ge 0, 2x + 3y \le 18\), and \(x + y \ge 10\) is:

Q7. What is the order and degree of the differential equation \(\left(\frac{d^2y}{dx^2}\right)^3 + \cos\left(\frac{dy}{dx}\right) = 0\)?

Q8. Evaluate the definite integral: \(\int_{-2}^{2} x|x| \, dx\)

Q9. If \(\vec{a}\), \(\vec{b}\), \(\vec{c}\) are unit vectors such that \(\vec{a} + \vec{b} + \vec{c} = \vec{0}\), find the value of \(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}\).

Q10. Let \(A\) and \(B\) be two events such that \(P(A) = 0.4\), \(P(B) = 0.8\), and \(P(B|A) = 0.6\). The value of \(P(A \cup B)\) is:

Q11. A firm manufactures two products, \(X\) and \(Y\). The total cost \(C(x)\) associated with the production of \(x\) units of item \(X\) is given by \(C(x) = 0.005x^3 - 0.02x^2 + 30x + 5000\). The marginal cost when \(3\) units are produced is:

Q13. A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is fixed at \(10\) meters. To admit maximum light through the whole opening, what mathematical condition must be met?

Q14. Assertion (A): If \(A\) is a skew-symmetric matrix of odd order, then \(|A| = 0\). Reason (R): For any square matrix \(A\), \(|A| = |-A|\).

Q15. Assertion (A): Two independent events \(A\) and \(B\) (with \(P(A) \neq 0, P(B) \neq 0\)) cannot be mutually exclusive. Reason (R): If \(A\) and \(B\) are mutually exclusive, \(P(A \cap B) = 0\), but for independent events, \(P(A \cap B) = P(A) \cdot P(B) \neq 0\).

Q18. Evaluate: \(\int_{0}^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, dx\)

Q19. Let \(A = \begin{bmatrix} 1 & \sin \theta \\ -\sin \theta & 1 \end{bmatrix}\), where \(0 \le \theta \le 2\pi\). Then the determinant \(|A|\) lies in the interval:

Q20. If the system of equations \(2x + 5y = 0\) and \(8x + ky = 0\) has a non-trivial (infinite) solution, then the value of \(k\) must make the coefficient matrix:

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