Q: Q7. If f(x) = \begin{cases} \frac{k \cos x}{\pi - 2x}, & x \neq \frac{\pi}{2} \\ 3, & x = \frac{\pi}{2} \end{cases} is continuous at x = \frac{\pi}{2}, then the value of k is:

Option B: 6. For continuity, \lim_{x \to \pi/2} f(x) = f(\pi/2) = 3. Evaluating the limit: \lim_{x \to \pi/2} \frac{k \cos x}{\pi - 2x}. Using L'Hopital's rule (0/0 form), derivative of numerator is -k \sin x, derivative of denominator is -2. Limit is \frac{-k \sin(\pi/2)}{-2} = \frac{k}{2}. Equating to 3 gives \frac{k}{2} = 3 \implies k = 6.

Question 1

Q1. Let R be a relation on the set \mathbb{Z} of integers defined by aRb \iff (a-b) is divisible by 5. The equivalence class of 2, denoted by [2], is:

Accepted Answer

Option B: \{\dots, -8, -3, 2, 7, 12, \dots\}. The equivalence class [2] contains all integers x such that x - 2 = 5k for some integer k. Thus, x = 5k + 2. Substituting k = \dots, -2, -1, 0, 1, 2, \dots gives the set \{\dots, -8, -3, 2, 7, 12, \dots\}.

Question 2

Q2. If A = \begin{pmatrix} \alpha & \beta \ \gamma & -\alpha \end{pmatrix} and A^2 = I (where I is the identity matrix of order 2), then which of the following is true?

Accepted Answer

Option C: 1 - \alpha^2 - \beta\gamma = 0. Calculating A^2 = \begin{pmatrix} \alpha & \beta \ \gamma & -\alpha \end{pmatrix} \begin{pmatrix} \alpha & \beta \ \gamma & -\alpha \end{pmatrix} = \begin{pmatrix} \alpha^2 + \beta\gamma & 0 \ 0 & \beta\gamma + \alpha^2 \end{pmatrix}. Equating this to I = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}, we get \alpha^2 + \beta\gamma = 1, which rearranges to 1 - \alpha^2 - \beta\gamma = 0.

Question 3

Q3. If a straight line makes equal acute angles with the coordinate axes, then its direction cosines are:

Accepted Answer

Option B: \left\langle \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} ightangle. Let the line make angle 	heta with all three axes. The direction cosines are l = m = n = \cos	heta. Since l^2 + m^2 + n^2 = 1, we have 3\cos^2	heta = 1 \implies \cos	heta = \frac{1}{\sqrt{3}} (taking the positive root for acute angles).

Question 4

Q4. Evaluate: \int e^x \left(\frac{1}{x} - \frac{1}{x^2}\right) dx

Accepted Answer

Option C: \frac{e^x}{x} + C. This integral is of the standard form \int e^x [f(x) + f'(x)] dx = e^x f(x) + C. Here, f(x) = \frac{1}{x}, and its derivative is f'(x) = -\frac{1}{x^2}. Therefore, the result is e^x \left(\frac{1}{x}\right) + C.

Question 5

Q5. The shortest distance of the point (2, 3, 4) from the xy-plane is:

Accepted Answer

Option C: 4 units. The xy-plane has the equation z = 0. The shortest distance of any point (x_1, y_1, z_1) from the xy-plane is simply the absolute value of its z-coordinate, which is |4| = 4 units.

Question 6

Q6. A die is thrown twice and the sum of the numbers appearing is observed to be 6. What is the conditional probability that the number 4 has appeared at least once?

Accepted Answer

Option B: \frac{2}{5}. Let E be the event "sum is 6": E = \{(1,5), (2,4), (3,3), (4,2), (5,1)\}, so n(E) = 5. Let F be the event "4 appears at least once": F = \{(4,1), (4,2), \dots, (1,4), (2,4), \dots\}. The intersection E \cap F = \{(2,4), (4,2)\}. Therefore, P(F|E) = \frac{n(E \cap F)}{n(E)} = \frac{2}{5}.

Question 7

Q7. If f(x) = \begin{cases} \frac{k \cos x}{\pi - 2x}, & x 
eq \frac{\pi}{2} \ 3, & x = \frac{\pi}{2} \end{cases} is continuous at x = \frac{\pi}{2}, then the value of k is:

Accepted Answer

Option B: 6. For continuity, \lim_{x 	o \pi/2} f(x) = f(\pi/2) = 3. Evaluating the limit: \lim_{x 	o \pi/2} \frac{k \cos x}{\pi - 2x}. Using L'Hopital's rule (0/0 form), derivative of numerator is -k \sin x, derivative of denominator is -2. Limit is \frac{-k \sin(\pi/2)}{-2} = \frac{k}{2}. Equating to 3 gives \frac{k}{2} = 3 \implies k = 6.

Question 8

Q8. In a Linear Programming Problem, the objective function is Z = 3x + 5y. The corner points of the bounded feasible region are (0, 0), (5, 0), (4, 3), and (0, 6). The maximum value of Z occurs at:

Accepted Answer

Option C: (0, 6). Evaluate Z at each corner point: At (0,0), Z=0. At (5,0), Z=15. At (4,3), Z=3(4) + 5(3) = 12 + 15 = 27. At (0,6), Z=3(0) + 5(6) = 30. The maximum value is 30, which occurs at (0,6).

Question 9

Q9. The general solution of the differential equation \frac{dy}{dx} = \frac{y}{x} represents a family of:

Accepted Answer

Option C: Straight lines passing through the origin. Separating variables, we get \frac{dy}{y} = \frac{dx}{x}. Integrating both sides gives \ln|y| = \ln|x| + \ln|C| \implies \ln|y| = \ln|Cx| \implies y = Cx. This is the standard equation for a family of straight lines passing through the origin.

Question 10

Q10. If the area of a triangle with vertices (2, -6), (5, 4), and (k, 4) is 35 sq. units, then k is:

Accepted Answer

Option C: 12 or -2. Area = \frac{1}{2} \begin{vmatrix} 2 & -6 & 1 \ 5 & 4 & 1 \ k & 4 & 1 \end{vmatrix} = \pm 35. Expanding the determinant: \frac{1}{2} [2(4-4) + 6(5-k) + 1(20-4k)] = \pm 35. Simplifying: \frac{1}{2} [30 - 6k + 20 - 4k] = \pm 35 \implies 50 - 10k = \pm 70. Solving yields k = -2 or k = 12.

Question 11

Q11. The total revenue R(x) received from the sale of x units of a product is given by R(x) = 3x^2 + 36x + 5. The marginal revenue when x = 15 is:

Accepted Answer

Option D: 126. Marginal Revenue (MR) is the rate of change of total revenue, MR = \frac{dR}{dx} = 6x + 36. Substituting x = 15, we get MR = 6(15) + 36 = 90 + 36 = 126.

Question 12

Q12. Using integration, the area of the region bounded by the curve y = \sqrt{x} and the lines x = 1, x = 4 and the x-axis is:

Accepted Answer

Option B: \frac{14}{3} sq. units. The required area is \int_{1}^{4} \sqrt{x} dx = \int_{1}^{4} x^{1/2} dx. The antiderivative is \frac{2}{3}x^{3/2}. Evaluating the limits: \frac{2}{3}[4^{3/2} - 1^{3/2}] = \frac{2}{3}[8 - 1] = \frac{2}{3}(7) = \frac{14}{3} sq. units.

Question 13

Q13. A factory has three machines A, B, and C producing 50%, 30%, and 20% of its items, respectively. The percentage of defective items produced by them is 1%, 2%, and 3%. If an item is drawn at random and found defective, what is the probability it was produced by machine A?

Accepted Answer

Option A: \frac{5}{17}. Let D be the event of drawing a defective item. Using Bayes' Theorem: P(A|D) = \frac{P(A)P(D|A)}{P(A)P(D|A) + P(B)P(D|B) + P(C)P(D|C)}. P(A|D) = \frac{0.5 	imes 0.01}{(0.5 	imes 0.01) + (0.3 	imes 0.02) + (0.2 	imes 0.03)} = \frac{0.005}{0.005 + 0.006 + 0.006} = \frac{0.005}{0.017} = \frac{5}{17}.

Question 14

Q14. Assertion (A): The principal value of \sin^{-1}\left(\sin \frac{2\pi}{3}\right) is \frac{2\pi}{3}. Reason (R): The range of the principal value branch of \sin^{-1}x is [-\frac{\pi}{2}, \frac{\pi}{2}].

Accepted Answer

Option D: A is false but R is true. Assertion A is false because \frac{2\pi}{3} lies outside the principal range [-\frac{\pi}{2}, \frac{\pi}{2}]. The correct principal value is \sin^{-1}(\sin(\pi - \frac{\pi}{3})) = \sin^{-1}(\sin \frac{\pi}{3}) = \frac{\pi}{3}. Reason R states the correct standard range.

Question 15

Q15. Assertion (A): The function f(x) = e^x is strictly increasing on \mathbb{R}. Reason (R): For a strictly increasing function f(x) on an interval I, f'(x) > 0 for all x \in I.

Accepted Answer

Option A: Both A and R are true and R is the correct explanation of A. The derivative of f(x) = e^x is f'(x) = e^x. Since e^x > 0 for all real numbers x, the function is strictly increasing. The reason correctly outlines the first derivative test for strictly increasing functions.

Question 16

Q16. Case Study Passage (Questions 16 & 17): An architect is designing a decorative arch for a modern art museum. Using a coordinate geometry software, he models the shape of the arch as a continuous curve given by the equation \(y = 9 - x^2\), where \(y\) represents the height in meters and \(x\) represents the horizontal distance from the center line of the arch. The ground is represented by the x-axis.

What is the total horizontal width of the arch at the ground level?

Accepted Answer

Option B: 6 meters. The arch meets the ground where height y = 0. Setting 9 - x^2 = 0 gives x^2 = 9, so x = 3 and x = -3. The total width is the distance between -3 and 3, which is 3 - (-3) = 6 meters.

Question 17

Q17. Case Study Passage (Questions 16 & 17): An architect is designing a decorative arch for a modern art museum. Using a coordinate geometry software, he models the shape of the arch as a continuous curve given by the equation \(y = 9 - x^2\), where \(y\) represents the height in meters and \(x\) represents the horizontal distance from the center line of the arch. The ground is represented by the x-axis.

The total cross-sectional area enclosed by the arch and the ground is:

Accepted Answer

Option C: 36 sq. meters. The area is given by the integral of the curve from x = -3 to x = 3. Area = \int_{-3}^{3} (9 - x^2) dx. Since it's an even function, 2 \int_{0}^{3} (9 - x^2) dx = 2 [9x - \frac{x^3}{3}]_{0}^{3} = 2 [27 - 9] = 2(18) = 36 sq. meters.

Question 18

Q18. Evaluate the expression: \sin\left(\frac{1}{2} \cos^{-1} \frac{4}{5}\right)

Accepted Answer

Option B: \frac{1}{\sqrt{10}}. Let \cos^{-1}\left(\frac{4}{5}ight) = 	heta \implies \cos	heta = \frac{4}{5}. We need to find \sin(\frac{	heta}{2}). Using the half-angle formula: 1 - 2\sin^2(\frac{	heta}{2}) = \cos	heta \implies 1 - 2\sin^2(\frac{	heta}{2}) = \frac{4}{5} \implies 2\sin^2(\frac{	heta}{2}) = \frac{1}{5} \implies \sin^2(\frac{	heta}{2}) = \frac{1}{10}. Since 	heta is in quadrant I, \sin(\frac{	heta}{2}) is positive, so \frac{1}{\sqrt{10}}.

Question 19

Q19. The degree of the differential equation e^{\frac{dy}{dx}} = x is:

Accepted Answer

Option A: 1. While exponential forms of derivatives usually make the degree "not defined," this equation can be algebraically manipulated to isolate the derivative. Taking the natural logarithm of both sides gives \frac{dy}{dx} = \ln x. Here, the highest order derivative has a power of 1, and the equation is a polynomial in its derivatives. So, the degree is 1.

Question 20

Q20. If A is a non-singular square matrix of order 3 such that |adj A| = 64, then the value of |A| is:

Accepted Answer

Option C: \pm 8. The property relating a matrix and its adjoint's determinant is |adj A| = |A|^{n-1}, where n is the order of the matrix. Here n = 3, so |adj A| = |A|^{3-1} = |A|^2. We are given |A|^2 = 64, which means |A| = \pm 8.

Mock Test 02 Performance Solutions

Detailed Question Analysis

Q1. Let \(R\) be a relation on the set \(\mathbb{Z}\) of integers defined by \(aRb \iff (a-b)\) is divisible by 5. The equivalence class of \(2\), denoted by \([2]\), is:

Q2. If \(A = \begin{pmatrix} \alpha & \beta \\ \gamma & -\alpha \end{pmatrix}\) and \(A^2 = I\) (where \(I\) is the identity matrix of order 2), then which of the following is true?

Q3. If a straight line makes equal acute angles with the coordinate axes, then its direction cosines are:

Q4. Evaluate: \(\int e^x \left(\frac{1}{x} - \frac{1}{x^2}\right) dx\)

Q5. The shortest distance of the point \((2, 3, 4)\) from the xy-plane is:

Q6. A die is thrown twice and the sum of the numbers appearing is observed to be 6. What is the conditional probability that the number 4 has appeared at least once?

Q7. If \(f(x) = \begin{cases} \frac{k \cos x}{\pi - 2x}, & x \neq \frac{\pi}{2} \\ 3, & x = \frac{\pi}{2} \end{cases}\) is continuous at \(x = \frac{\pi}{2}\), then the value of \(k\) is:

Q8. In a Linear Programming Problem, the objective function is \(Z = 3x + 5y\). The corner points of the bounded feasible region are \((0, 0), (5, 0), (4, 3)\), and \((0, 6)\). The maximum value of \(Z\) occurs at:

Q9. The general solution of the differential equation \(\frac{dy}{dx} = \frac{y}{x}\) represents a family of:

Q10. If the area of a triangle with vertices \((2, -6), (5, 4)\), and \((k, 4)\) is 35 sq. units, then \(k\) is:

Q11. The total revenue \(R(x)\) received from the sale of \(x\) units of a product is given by \(R(x) = 3x^2 + 36x + 5\). The marginal revenue when \(x = 15\) is:

Q12. Using integration, the area of the region bounded by the curve \(y = \sqrt{x}\) and the lines \(x = 1, x = 4\) and the x-axis is:

Q13. A factory has three machines A, B, and C producing 50%, 30%, and 20% of its items, respectively. The percentage of defective items produced by them is 1%, 2%, and 3%. If an item is drawn at random and found defective, what is the probability it was produced by machine A?

Q14. Assertion (A): The principal value of \(\sin^{-1}\left(\sin \frac{2\pi}{3}\right)\) is \(\frac{2\pi}{3}\). Reason (R): The range of the principal value branch of \(\sin^{-1}x\) is \([-\frac{\pi}{2}, \frac{\pi}{2}]\).

Q15. Assertion (A): The function \(f(x) = e^x\) is strictly increasing on \(\mathbb{R}\). Reason (R): For a strictly increasing function \(f(x)\) on an interval \(I\), \(f'(x) > 0\) for all \(x \in I\).

Q18. Evaluate the expression: \(\sin\left(\frac{1}{2} \cos^{-1} \frac{4}{5}\right)\)

Q19. The degree of the differential equation \(e^{\frac{dy}{dx}} = x\) is:

Q20. If \(A\) is a non-singular square matrix of order 3 such that \(|adj A| = 64\), then the value of \(|A|\) is:

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Mock Test 02 Performance Solutions

Detailed Question Analysis

Q1. Let \(R\) be a relation on the set \(\mathbb{Z}\) of integers defined by \(aRb \iff (a-b)\) is divisible by 5. The equivalence class of \(2\), denoted by \([2]\), is:

Q2. If \(A = \begin{pmatrix} \alpha & \beta \\ \gamma & -\alpha \end{pmatrix}\) and \(A^2 = I\) (where \(I\) is the identity matrix of order 2), then which of the following is true?

Q3. If a straight line makes equal acute angles with the coordinate axes, then its direction cosines are:

Q4. Evaluate: \(\int e^x \left(\frac{1}{x} - \frac{1}{x^2}\right) dx\)

Q5. The shortest distance of the point \((2, 3, 4)\) from the xy-plane is:

Q6. A die is thrown twice and the sum of the numbers appearing is observed to be 6. What is the conditional probability that the number 4 has appeared at least once?

Q7. If \(f(x) = \begin{cases} \frac{k \cos x}{\pi - 2x}, & x \neq \frac{\pi}{2} \\ 3, & x = \frac{\pi}{2} \end{cases}\) is continuous at \(x = \frac{\pi}{2}\), then the value of \(k\) is:

Q8. In a Linear Programming Problem, the objective function is \(Z = 3x + 5y\). The corner points of the bounded feasible region are \((0, 0), (5, 0), (4, 3)\), and \((0, 6)\). The maximum value of \(Z\) occurs at:

Q9. The general solution of the differential equation \(\frac{dy}{dx} = \frac{y}{x}\) represents a family of:

Q10. If the area of a triangle with vertices \((2, -6), (5, 4)\), and \((k, 4)\) is 35 sq. units, then \(k\) is:

Q11. The total revenue \(R(x)\) received from the sale of \(x\) units of a product is given by \(R(x) = 3x^2 + 36x + 5\). The marginal revenue when \(x = 15\) is:

Q12. Using integration, the area of the region bounded by the curve \(y = \sqrt{x}\) and the lines \(x = 1, x = 4\) and the x-axis is:

Q13. A factory has three machines A, B, and C producing 50%, 30%, and 20% of its items, respectively. The percentage of defective items produced by them is 1%, 2%, and 3%. If an item is drawn at random and found defective, what is the probability it was produced by machine A?

Q14. Assertion (A): The principal value of \(\sin^{-1}\left(\sin \frac{2\pi}{3}\right)\) is \(\frac{2\pi}{3}\). Reason (R): The range of the principal value branch of \(\sin^{-1}x\) is \([-\frac{\pi}{2}, \frac{\pi}{2}]\).

Q15. Assertion (A): The function \(f(x) = e^x\) is strictly increasing on \(\mathbb{R}\). Reason (R): For a strictly increasing function \(f(x)\) on an interval \(I\), \(f'(x) > 0\) for all \(x \in I\).

Q18. Evaluate the expression: \(\sin\left(\frac{1}{2} \cos^{-1} \frac{4}{5}\right)\)

Q19. The degree of the differential equation \(e^{\frac{dy}{dx}} = x\) is:

Q20. If \(A\) is a non-singular square matrix of order 3 such that \(|adj A| = 64\), then the value of \(|A|\) is:

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