Q: Q5. If A and B are two independent events such that P(A) = 0.3 and P(B) = 0.4, find the probability that exactly one of the events occurs.

Option A: 0.46. P(\text{exactly one}) = P(A) + P(B) - 2P(A \cap B). For independent events, P(A \cap B) = P(A) \times P(B) = 0.3 \times 0.4 = 0.12. Substitute values: 0.3 + 0.4 - 2(0.12) = 0.7 - 0.24 = 0.46.

Q: Q9. The piecewise function f(x) is defined as f(x) = \frac{k\cos x}{\pi - 2x} if x \neq \frac{\pi}{2}, and f(x) = 3 if x = \frac{\pi}{2}. If f(x) is continuous at x = \frac{\pi}{2}, the value of the constant k is:

Option B: 6. To find the limit as x \to \frac{\pi}{2}, substitute x = \frac{\pi}{2} - h. The limit becomes \lim_{h \to 0} \frac{k\cos(\pi/2 - h)}{\pi - 2(\pi/2 - h)} = \lim_{h \to 0} \frac{k\sin h}{2h} = \frac{k}{2}. For continuity, \frac{k}{2} = 3, meaning k = 6.

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

Option C: 12, -2. Area = \frac{1}{2} |\text{det}| = 35, meaning the determinant equals \pm 70. Expanding the determinant: 2(4-4) - (-6)(5-k) + 1(20-4k) = 0 + 30 - 6k + 20 - 4k = 50 - 10k. Setting 50 - 10k = \pm 70 yields k = -2 or k = 12.

Question 1

Q1. Let A = \{1, 2, 3\}. The number of equivalence relations containing the element (1, 2) is:

Accepted Answer

Option B: 2. The smallest equivalence relation containing (1, 2) is \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)\}. Adding even one more element like (1, 3) forces all remaining pairs to be included to maintain transitivity, forming the universal relation. So, only 2 such 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 the terms: (A-I)^3 + (A+I)^3 = (A^3 - 3A^2I + 3AI^2 - I^3) + (A^3 + 3A^2I + 3AI^2 + I^3) = 2A^3 + 6A. Since A^2 = I, we get A^3 = A^2 \cdot A = I \cdot A = A. So, 2A + 6A - 7A = A.

Question 3

Q3. The value of the definite integral \int_{-1}^{1} \log(\frac{2-x}{2+x}) dx is:

Accepted Answer

Option C: 0. Let f(x) = \log(\frac{2-x}{2+x}). Replacing x with -x gives f(-x) = \log(\frac{2+x}{2-x}) = \log(\frac{2-x}{2+x})^{-1} = -f(x). Since it is an odd function, the integral from -a to a is exactly 0.

Question 4

Q4. If \vec{a} and \vec{b} are two unit vectors and 	heta is the angle between them, then the value of \sin(\frac{	heta}{2}) is equal to:

Accepted Answer

Option A: \frac{1}{2}|\vec{a} - \vec{b}|. Consider |\vec{a} - \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 - 2|\vec{a}||\vec{b}|\cos	heta. Since they are unit vectors, this becomes 1 + 1 - 2\cos	heta = 2(1 - \cos	heta) = 4\sin^2(\frac{	heta}{2}). Taking the square root gives \frac{1}{2}|\vec{a} - \vec{b}|.

Question 5

Q5. If A and B are two independent events such that P(A) = 0.3 and P(B) = 0.4, find the probability that exactly one of the events occurs.

Accepted Answer

Option A: 0.46. P(	ext{exactly one}) = P(A) + P(B) - 2P(A \cap B). For independent events, P(A \cap B) = P(A) 	imes P(B) = 0.3 	imes 0.4 = 0.12. Substitute values: 0.3 + 0.4 - 2(0.12) = 0.7 - 0.24 = 0.46.

Question 6

Q6. A tech startup is producing a new gadget. The total cost C(x) (in ₹) associated with the production of x units is given by C(x) = 0.005x^3 - 0.02x^2 + 30x + 5000. What is the marginal cost when 3 units are produced?

Accepted Answer

Option B: ₹ 30.015. Marginal Cost MC = \frac{dC}{dx} = 0.015x^2 - 0.04x + 30. Substituting x = 3, we get MC = 0.015(9) - 0.04(3) + 30 = 0.135 - 0.12 + 30 = 30.015.

Question 7

Q7. The rate of growth of a specific bacteria in a culture is proportional to the number of bacteria currently present. If y(t) represents the bacteria count at time t, which differential equation accurately models this biological scenario?

Accepted Answer

Option C: \frac{dy}{dt} = ky. The phrase "proportional to the number present" translates mathematically to \frac{dy}{dt} \propto y. Removing the proportionality sign introduces a constant k, resulting in the linear differential equation \frac{dy}{dt} = ky.

Question 8

Q8. Two drones are flying along the straight line paths given by \vec{r} = (1+\lambda)\hat{i} + (2- \lambda)\hat{j} + (1+\lambda)\hat{k} and \vec{r} = 2\hat{i} + \hat{j} - \hat{k} + \mu(2\hat{i} + \hat{j} + 2\hat{k}). What mathematical condition must hold true to confirm their flight paths intersect?

Accepted Answer

Option A: The shortest distance between the lines must be 0.. Two straight paths in 3D space intersect if and only if they are coplanar and not parallel. Mathematically, this means the shortest distance between the two skew-like lines must evaluate to exactly 0.

Question 9

Q9. The piecewise function f(x) is defined as f(x) = \frac{k\cos x}{\pi - 2x} if x 
eq \frac{\pi}{2}, and f(x) = 3 if x = \frac{\pi}{2}. If f(x) is continuous at x = \frac{\pi}{2}, the value of the constant k is:

Accepted Answer

Option B: 6. To find the limit as x 	o \frac{\pi}{2}, substitute x = \frac{\pi}{2} - h. The limit becomes \lim_{h 	o 0} \frac{k\cos(\pi/2 - h)}{\pi - 2(\pi/2 - h)} = \lim_{h 	o 0} \frac{k\sin h}{2h} = \frac{k}{2}. For continuity, \frac{k}{2} = 3, meaning k = 6.

Question 10

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

Accepted Answer

Option C: 12, -2. Area = \frac{1}{2} |	ext{det}| = 35, meaning the determinant equals \pm 70. Expanding the determinant: 2(4-4) - (-6)(5-k) + 1(20-4k) = 0 + 30 - 6k + 20 - 4k = 50 - 10k. Setting 50 - 10k = \pm 70 yields k = -2 or k = 12.

Question 11

Q11. A furniture dealer has ₹50,000 to invest and warehouse space to store at most 60 pieces. A table costs him ₹2500 and a chair costs ₹500. If he buys x tables and y chairs, the constraints representing this problem are:

Accepted Answer

Option A: x+y \le 60, 2500x + 500y \le 50000, x \ge 0, y \ge 0. The storage capacity means total items cannot exceed 60, so x+y \le 60. His budget restriction means total cost cannot exceed his capital, so 2500x + 500y \le 50000. Non-negative constraints apply because physical quantities cannot be negative.

Question 12

Q12. An AI diagnostic tool detects a disease 90\% of the time when it is present, but gives a false positive 5\% of the time for healthy patients. If 1\% of the population actually has the disease, what is the probability that a person has the disease given that the AI predicted positive? (Approximate to 3 decimal places)

Accepted Answer

Option A: 0.154. Using Bayes' Theorem: P(	ext{Disease}|	ext{Positive}) = \frac{P(	ext{Positive}|	ext{Disease})P(	ext{Disease})}{P(	ext{Positive}|	ext{Disease})P(	ext{Disea se}) + P(	ext{Positive}|	ext{Healthy})P(	ext{Healthy})}. Substituting the values: \frac{0.90 	imes 0.01}{(0.90 	imes 0.01) + (0.05 	imes 0.99)} = \frac{0.009}{0.009 + 0.0495} \approx 0.1538.

Question 13

Q13. A civil engineer is designing a parabolic arch for a small bridge. The shape of the arch is represented by y = 4 - x^2, where y is the height and the x-axis represents ground level. The total area enclosed by the arch and the ground is:

Accepted Answer

Option A: \frac{32}{3} sq. units. The curve intersects the ground (y=0) at x = -2 and x = 2. The area is \int_{-2}^{2} (4 - x^2) dx. Since it's an even function, 2 \int_{0}^{2} (4 - x^2) dx = 2 [4x - \frac{x^3}{3}]_{0}^{2} = 2(8 - \frac{8}{3}) = 2(\frac{16}{3}) = \frac{32}{3}.

Question 14

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

Accepted Answer

Option D: A is false but R is true. The assertion is mathematically false because \frac{2\pi}{3} falls outside the principal value branch [-\frac{\pi}{2}, \frac{\pi}{2}]. It must be converted to \sin^{-1}(\sin(\pi - \frac{\pi}{3})) = \frac{\pi}{3}. The Reason statement correctly defines the principal branch limits.

Question 15

Q15. Assertion (A): The function f(x) = |x - 2| is continuous but not differentiable at the point x = 2. Reason (R): The left-hand derivative and right-hand derivative of f(x) = |x - 2| at x = 2 are not equal to each other.

Accepted Answer

Option A: Both A and R are true and R is the correct explanation of A. Modulus functions form continuous V-shapes but have a "corner" at the critical point x=2. Evaluating limits shows LHD = -1 and RHD = 1. Since LHD 
eq RHD, the function lacks differentiability there, making R the exact explanation for A. Passage for Q16 & Q17: An architect is designing a cylindrical water tank with an open top to hold a constant volume V of water for a residential society. The cost of construction materials depends directly on the total surface area used. To optimize funds, the architect must minimize the surface area S.

Question 16

Q16. If the radius of the tank's base is r and its height is h, what is the expression for the total surface area S purely in terms of r and V?

Accepted Answer

Option A: S = \pi r^2 + \frac{2V}{r}. The volume is V = \pi r^2 h, which means h = \frac{V}{\pi r^2}. Since the tank has an open top, the surface area S = 	ext{base} + 	ext{curved surface} = \pi r^2 + 2\pi rh. Substituting h gives S = \pi r^2 + 2\pi r(\frac{V}{\pi r^2}) = \pi r^2 + \frac{2V}{r}.

Question 17

Q17. For the total surface area S to be absolutely minimum, what geometric relationship must exist between the radius r and height h?

Accepted Answer

Option B: h = r. To minimize, set the first derivative to zero: \frac{dS}{dr} = 2\pi r - \frac{2V}{r^2} = 0, which yields \pi r^3 = V. Replacing V back with \pi r^2 h, we get \pi r^3 = \pi r^2 h. Dividing by \pi r^2 gives r = h.

Question 18

Q18. Evaluate the tricky integral: \int e^x (\frac{1 - x}{1 + x^2})^2 dx

Accepted Answer

Option A: \frac{e^x}{1+x^2} + C. Expand the square inside: (\frac{1-x}{1+x^2})^2 = \frac{1+x^2-2x}{(1+x^2)^2} = \frac{1}{1+x^2} + \frac{-2x}{(1+x^2)^2}. This perfectly matches the form \int e^x [f(x) + f'(x)] dx where f(x) = \frac{1}{1+x^2}. The standard result is e^x f(x) + C.

Question 19

Q19. If A is a 3 	imes 3 square matrix such that |A| = 4, then what is the value of the determinant |adj(adj A)|?

Accepted Answer

Option D: 256. The standard property for the determinant of an adjoint of an adjoint matrix is |adj(adj A)| = |A|^{(n-1)^2}, where n is the order. Here n = 3, so the power is (3-1)^2 = 4. Therefore, 4^4 = 256.

Question 20

Q20. A vector \vec{r} makes equal angles with the positive directions of the x, y, and z-axes. The direction cosines of the vector \vec{r} are:

Accepted Answer

Option B: (\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}). Let the angles made with the axes be \alpha, \beta, \gamma. Given they are equal, \alpha = \beta = \gamma. The fundamental identity for direction cosines is \cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1. This becomes 3\cos^2\alpha = 1, giving \cos\alpha = \frac{1}{\sqrt{3}}.

Mock Test 10 Performance Solutions

Detailed Question Analysis

Q1. Let \(A = \{1, 2, 3\}\). The number of equivalence relations containing the element \((1, 2)\) is:

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 value of the definite integral \(\int_{-1}^{1} \log(\frac{2-x}{2+x}) dx\) is:

Q4. If \(\vec{a}\) and \(\vec{b}\) are two unit vectors and \(\theta\) is the angle between them, then the value of \(\sin(\frac{\theta}{2})\) is equal to:

Q5. If \(A\) and \(B\) are two independent events such that \(P(A) = 0.3\) and \(P(B) = 0.4\), find the probability that exactly one of the events occurs.

Q6. A tech startup is producing a new gadget. The total cost \(C(x)\) (in ₹) associated with the production of \(x\) units is given by \(C(x) = 0.005x^3 - 0.02x^2 + 30x + 5000\). What is the marginal cost when \(3\) units are produced?

Q7. The rate of growth of a specific bacteria in a culture is proportional to the number of bacteria currently present. If \(y(t)\) represents the bacteria count at time \(t\), which differential equation accurately models this biological scenario?

Q9. The piecewise function \(f(x)\) is defined as \(f(x) = \frac{k\cos x}{\pi - 2x}\) if \(x \neq \frac{\pi}{2}\), and \(f(x) = 3\) if \(x = \frac{\pi}{2}\). If \(f(x)\) is continuous at \(x = \frac{\pi}{2}\), the value of the constant \(k\) is:

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

Q11. A furniture dealer has ₹\(50,000\) to invest and warehouse space to store at most \(60\) pieces. A table costs him ₹\(2500\) and a chair costs ₹\(500\). If he buys \(x\) tables and \(y\) chairs, the constraints representing this problem are:

Q13. A civil engineer is designing a parabolic arch for a small bridge. The shape of the arch is represented by \(y = 4 - x^2\), where \(y\) is the height and the x-axis represents ground level. The total area enclosed by the arch and the ground is:

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

Q15. Assertion (A): The function \(f(x) = |x - 2|\) is continuous but not differentiable at the point \(x = 2\). Reason (R): The left-hand derivative and right-hand derivative of \(f(x) = |x - 2|\) at \(x = 2\) are not equal to each other.

Q16. If the radius of the tank's base is \(r\) and its height is \(h\), what is the expression for the total surface area \(S\) purely in terms of \(r\) and \(V\)?

Q17. For the total surface area \(S\) to be absolutely minimum, what geometric relationship must exist between the radius \(r\) and height \(h\)?

Q18. Evaluate the tricky integral: \(\int e^x (\frac{1 - x}{1 + x^2})^2 dx\)

Q19. If \(A\) is a \(3 \times 3\) square matrix such that \(|A| = 4\), then what is the value of the determinant \(|adj(adj A)|\)?

Q20. A vector \(\vec{r}\) makes equal angles with the positive directions of the x, y, and z-axes. The direction cosines of the vector \(\vec{r}\) are:

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