Question 1

Q1. Let A and B be symmetric matrices of the same order. Then the matrix (AB - BA) is a:

Accepted Answer

Option B: Skew-symmetric matrix. (AB - BA)' = (AB)' - (BA)' = B'A' - A'B'. Since A and B are symmetric (A'=A, B'=B), this becomes BA - AB = -(AB - BA). A matrix whose transpose is its negative is skew-symmetric.

Question 2

Q2. Let R be a relation on the set N of natural numbers defined by n R m if n divides m. Then relation R is:

Accepted Answer

Option D: Reflexive, transitive but not symmetric. Every number divides itself, so n R n (Reflexive). If n divides m and m divides p, then n divides p (Transitive). However, 2 divides 4, but 4 does not divide 2, so it is not symmetric.

Question 3

Q3. The domain of the function f(x) = \sin^{-1}(\sqrt{x - 1}) is:

Accepted Answer

Option A: [1, 2]. For \sin^{-1}(y), the domain is -1 \le y \le 1. So, -1 \le \sqrt{x - 1} \le 1. Since a principal square root is always non-negative, 0 \le \sqrt{x - 1} \le 1. Squaring gives 0 \le x - 1 \le 1, which simplifies to 1 \le x \le 2.

Question 4

Q4. If A is a square matrix of order 3 such that |A| = 5, then the value of |adj(adj A)| is:

Accepted Answer

Option C: 625. The standard formula is |adj(adj A)| = |A|^{(n-1)^2}. Here, the order n=3. So, the value is |A|^{(3-1)^2} = |A|^4 = 5^4 = 625.

Question 5

Q5. If \vec{a} and \vec{b} are two unit vectors such that \vec{a} + \vec{b} is also a unit vector, then the angle between \vec{a} and \vec{b} is:

Accepted Answer

Option C: \frac{2\pi}{3}. Given |\vec{a}+\vec{b}| = 1. Squaring both sides: |\vec{a}|^2 + |\vec{b}|^2 + 2|\vec{a}||\vec{b}|\cos	heta = 1. Substituting unit vector values (1): 1 + 1 + 2(1)(1)\cos	heta = 1 \implies \cos	heta = -\frac{1}{2} \implies 	heta = \frac{2\pi}{3}.

Question 6

Q6. A water tank has the shape of an inverted right circular cone. Water is poured into it at a constant rate. How does the rate of change of the water level behave as the tank fills?

Accepted Answer

Option C: It decreases continuously.. Volume V = \frac{1}{3}\pi r^2 h. As the height h increases, the cross-sectional area gets wider. To maintain a constant volume rate (\frac{dV}{dt}), the rate at which the height increases (\frac{dh}{dt}) must decrease because more water is needed to raise the level by the same height.

Question 7

Q7. Three schools want to award their students for honesty, punctuality, and obedience. If we represent the prize amounts as a 3 	imes 1 column matrix X and the number of students per category as a 3 	imes 3 matrix P, then the product PX represents:

Accepted Answer

Option B: Total amount spent by each school on all awards.. Matrix P (Schools 	imes Categories) multiplied by Matrix X (Categories 	imes 1) yields a new matrix of order (Schools 	imes 1). This resulting column matrix explicitly shows the total money spent by each individual school.

Question 8

Q8. The population of a city grows at a rate proportional to its current population. If this scenario is modeled mathematically by a differential equation, what is the order and degree of this equation?

Accepted Answer

Option A: Order 1, Degree 1. The mathematical model is \frac{dP}{dt} = kP. This involves only the first derivative with a power of one. Therefore, it is a first-order, first-degree linear ordinary differential equation.

Question 9

Q9. Geometrically, the definite integral \int_0^2 |x - 1| dx represents:

Accepted Answer

Option A: The algebraic sum of areas of two triangles above the x-axis.. The modulus function |x - 1| is strictly non-negative. Its graph forms two identical V-shaped triangles resting on the x-axis from x=0 to 1, and x=1 to 2. The integral calculates the combined positive area of these two triangles.

Question 10

Q10. A medical test for a rare disease is 99% accurate. However, only 0.1% of the population actually has the disease. If a person tests positive, the probability they actually have the disease is surprisingly low. Based on Bayes' Theorem logic, why does this happen?

Accepted Answer

Option B: The very low base rate means the 1% false positives from healthy people vastly outnumber the true positives.. This is a classic application of conditional probability and the Base Rate Fallacy. Since 99.9% of people are healthy, a 1% error rate on that massive group produces far more false positives than the 99% accurate true positives found in the tiny 0.1% infected group.

Question 11

Q11. A factory makes tennis rackets (x) and cricket bats (y). A racket takes 1.5 hours of machine time, and a bat takes 3 hours. The factory has a maximum of 42 hours of machine time available. To maximize profit, this is formulated as an LPP . The correct machine time constraint is:

Accepted Answer

Option A: 1.5x + 3y \le 42. Total machine time consumed by x rackets and y bats is 1.5x + 3y. The word "maximum" implies the usage cannot exceed the available hours, forming the inequality 1.5x + 3y \le 42.

Question 12

Q12. A company designs a cylindrical can to hold 1 liter (1000 	ext{ cm}^3) of oil. To minimize manufacturing cost, they must minimize the total surface area. What relation between height (h) and radius (r) yields the most economical can?

Accepted Answer

Option B: h = 2r. Surface area S = 2\pi r^2 + 2\pi rh. Volume V = \pi r^2 h = 1000 \implies h = \frac{1000}{\pi r^2}. Substituting h into S and minimizing by setting \frac{dS}{dr}=0 yields 2\pi r^3 = 1000. Substituting this back gives h = 2r (height equals diameter).

Question 13

Q13. An insurance company insured 2000 scooter drivers, 4000 car drivers, and 6000 truck drivers. The probability of them having an accident is 0.01, 0.03, and 0.15 respectively. If an insured person meets with an accident, finding the probability that they are a scooter driver directly applies which concept?

Accepted Answer

Option C: Bayes' Theorem. We are given the final outcome (an accident has occurred) and need to calculate the "reverse probability" of a specific prior cause (it was a scooter driver). This scenario is the defining use-case of Bayes' Theorem.

Question 14

Q14. Assertion (A): The function f(x) = |x| is continuous at x = 0 but not differentiable at x = 0. Reason (R): A function is differentiable at a point only if it is continuous at that point, but the converse is not always true.

Accepted Answer

Option B: Both A and R are true but R is not the correct explanation of A. Both statements are mathematically true. However, R is just a general theorem. The specific reason A is true is because at x=0, the Left Hand Derivative is -1 and the Right Hand Derivative is 1 (LHD 
eq RHD), causing a sharp corner.

Question 15

Q15. Assertion (A): The lines \frac{x-1}{2} = \frac{y+1}{3} = \frac{z-1}{4} and \frac{x-3}{1} = \frac{y-k}{2} = \frac{z}{1} intersect if k = \frac{9}{2}. Reason (R): Two lines intersect if the shortest distance between them is zero.

Accepted Answer

Option A: Both A and R are true and R is the correct explanation of A. Two non-parallel lines in 3D space intersect if and only if their shortest distance is zero. Applying the coplanarity determinant (\vec{a_2} - \vec{a_1}) \cdot (\vec{b_1} 	imes \vec{b_2}) = 0 yields k = \frac{9}{2}. R perfectly explains the method to prove A.

Question 16

Q16. Case Study Passage (Questions 16 & 17) A drone is flying in a 3D coordinate space. Its position vector at any time \(t\) (in seconds) is given by \(\vec{r} = (2t)\hat{i} + (t^2)\hat{j} + (3)\hat{k}\). The control tower is situated at the origin \((0,0,0)\).

What is the mathematical path followed by the drone's shadow on the XY-plane?

Accepted Answer

Option C: A parabola. On the XY-plane, we only look at x and y components. x = 2t \implies t = \frac{x}{2}. The y- coordinate is y = t^2. Substituting t gives y = (\frac{x}{2})^2 = \frac{x^2}{4}, which represents a parabola.

Question 17

Q17. Case Study Passage (Questions 16 & 17) A drone is flying in a 3D coordinate space. Its position vector at any time \(t\) (in seconds) is given by \(\vec{r} = (2t)\hat{i} + (t^2)\hat{j} + (3)\hat{k}\). The control tower is situated at the origin \((0,0,0)\).

At \(t=1\), what is the straight-line distance of the drone from the control tower?

Accepted Answer

Option A: \sqrt{14} units. At t=1, substituting 1 into the position vector gives \vec{r} = 2\hat{i} + 1\hat{j} + 3\hat{k}. The distance from the origin (0,0,0) is the magnitude |\vec{r}| = \sqrt{2^2 + 1^2 + 3^2} = \sqrt{4 + 1 + 9} = \sqrt{14}.

Question 18

Q18. The value of \int_{-1}^1 (x^3 + x\cos x + 	an^5 x + 1) dx is:

Accepted Answer

Option B: 2. Let f(x) = x^3 + x\cos x + 	an^5 x. Note that f(-x) = -f(x), making it an odd function. Thus, \int_{- 1}^1 f(x) dx = 0. The integral simplifies to \int_{-1}^1 1 dx = [x]_{-1}^1 = 1 - (-1) = 2.

Question 19

Q19. The integrating factor (I.F.) of the differential equation x\frac{dy}{dx} + 2y = x^2 (x 
eq 0) is:

Accepted Answer

Option B: x^2. Divide the equation by x to get standard linear form: \frac{dy}{dx} + (\frac{2}{x})y = x. Here P = \frac{2}{x}. The Integrating Factor is e^{\int P dx} = e^{\int \frac{2}{x} dx} = e^{2\ln x} = e^{\ln(x^2)} = x^2.

Question 20

Q20. Two independent events A and B are such that P(A) = 0.3 and P(A \cup B) = 0.6. The value of P(B) is:

Accepted Answer

Option D: \frac{3}{7}. For independent events, P(A \cap B) = P(A)P(B). Using the addition theorem: P(A \cup B) = P(A) + P(B) - P(A \cap B). Substituting values: 0.6 = 0.3 + P(B) - 0.3P(B) \implies 0.3 = 0.7P(B) \implies P(B) = \frac{0.3}{0.7} = \frac{3}{7}. (Option A is a trap for mutually exclusive events).

Mock Test 09 Performance Solutions

Detailed Question Analysis

Q1. Let \(A\) and \(B\) be symmetric matrices of the same order. Then the matrix \((AB - BA)\) is a:

Q2. Let \(R\) be a relation on the set \(N\) of natural numbers defined by \(n R m\) if \(n\) divides \(m\). Then relation \(R\) is:

Q3. The domain of the function \(f(x) = \sin^{-1}(\sqrt{x - 1})\) is:

Q4. If \(A\) is a square matrix of order 3 such that \(|A| = 5\), then the value of \(|adj(adj A)|\) is:

Q5. If \(\vec{a}\) and \(\vec{b}\) are two unit vectors such that \(\vec{a} + \vec{b}\) is also a unit vector, then the angle between \(\vec{a}\) and \(\vec{b}\) is:

Q6. A water tank has the shape of an inverted right circular cone. Water is poured into it at a constant rate. How does the rate of change of the water level behave as the tank fills?

Q7. Three schools want to award their students for honesty, punctuality, and obedience. If we represent the prize amounts as a \(3 \times 1\) column matrix \(X\) and the number of students per category as a \(3 \times 3\) matrix \(P\), then the product \(PX\) represents:

Q8. The population of a city grows at a rate proportional to its current population. If this scenario is modeled mathematically by a differential equation, what is the order and degree of this equation?

Q9. Geometrically, the definite integral \(\int_0^2 |x - 1| dx\) represents:

Q10. A medical test for a rare disease is 99% accurate. However, only 0.1% of the population actually has the disease. If a person tests positive, the probability they actually have the disease is surprisingly low. Based on Bayes' Theorem logic, why does this happen?

Q11. A factory makes tennis rackets (\(x\)) and cricket bats (\(y\)). A racket takes 1.5 hours of machine time, and a bat takes 3 hours. The factory has a maximum of 42 hours of machine time available. To maximize profit, this is formulated as an LPP . The correct machine time constraint is:

Q12. A company designs a cylindrical can to hold 1 liter (\(1000 \text{ cm}^3\)) of oil. To minimize manufacturing cost, they must minimize the total surface area. What relation between height (\(h\)) and radius (\(r\)) yields the most economical can?

Q14. Assertion (A): The function \(f(x) = |x|\) is continuous at \(x = 0\) but not differentiable at \(x = 0\). Reason (R): A function is differentiable at a point only if it is continuous at that point, but the converse is not always true.

Q15. Assertion (A): The lines \(\frac{x-1}{2} = \frac{y+1}{3} = \frac{z-1}{4}\) and \(\frac{x-3}{1} = \frac{y-k}{2} = \frac{z}{1}\) intersect if \(k = \frac{9}{2}\). Reason (R): Two lines intersect if the shortest distance between them is zero.

Q18. The value of \(\int_{-1}^1 (x^3 + x\cos x + \tan^5 x + 1) dx\) is:

Q19. The integrating factor (I.F.) of the differential equation \(x\frac{dy}{dx} + 2y = x^2\) (\(x \neq 0\)) is:

Q20. Two independent events \(A\) and \(B\) are such that \(P(A) = 0.3\) and \(P(A \cup B) = 0.6\). The value of \(P(B)\) is:

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Mock Test 09 Performance Solutions

Detailed Question Analysis

Q1. Let \(A\) and \(B\) be symmetric matrices of the same order. Then the matrix \((AB - BA)\) is a:

Q2. Let \(R\) be a relation on the set \(N\) of natural numbers defined by \(n R m\) if \(n\) divides \(m\). Then relation \(R\) is:

Q3. The domain of the function \(f(x) = \sin^{-1}(\sqrt{x - 1})\) is:

Q4. If \(A\) is a square matrix of order 3 such that \(|A| = 5\), then the value of \(|adj(adj A)|\) is:

Q5. If \(\vec{a}\) and \(\vec{b}\) are two unit vectors such that \(\vec{a} + \vec{b}\) is also a unit vector, then the angle between \(\vec{a}\) and \(\vec{b}\) is:

Q6. A water tank has the shape of an inverted right circular cone. Water is poured into it at a constant rate. How does the rate of change of the water level behave as the tank fills?

Q7. Three schools want to award their students for honesty, punctuality, and obedience. If we represent the prize amounts as a \(3 \times 1\) column matrix \(X\) and the number of students per category as a \(3 \times 3\) matrix \(P\), then the product \(PX\) represents:

Q8. The population of a city grows at a rate proportional to its current population. If this scenario is modeled mathematically by a differential equation, what is the order and degree of this equation?

Q9. Geometrically, the definite integral \(\int_0^2 |x - 1| dx\) represents:

Q10. A medical test for a rare disease is 99% accurate. However, only 0.1% of the population actually has the disease. If a person tests positive, the probability they actually have the disease is surprisingly low. Based on Bayes' Theorem logic, why does this happen?

Q11. A factory makes tennis rackets (\(x\)) and cricket bats (\(y\)). A racket takes 1.5 hours of machine time, and a bat takes 3 hours. The factory has a maximum of 42 hours of machine time available. To maximize profit, this is formulated as an LPP . The correct machine time constraint is:

Q12. A company designs a cylindrical can to hold 1 liter (\(1000 \text{ cm}^3\)) of oil. To minimize manufacturing cost, they must minimize the total surface area. What relation between height (\(h\)) and radius (\(r\)) yields the most economical can?

Q14. Assertion (A): The function \(f(x) = |x|\) is continuous at \(x = 0\) but not differentiable at \(x = 0\). Reason (R): A function is differentiable at a point only if it is continuous at that point, but the converse is not always true.

Q15. Assertion (A): The lines \(\frac{x-1}{2} = \frac{y+1}{3} = \frac{z-1}{4}\) and \(\frac{x-3}{1} = \frac{y-k}{2} = \frac{z}{1}\) intersect if \(k = \frac{9}{2}\). Reason (R): Two lines intersect if the shortest distance between them is zero.

Q18. The value of \(\int_{-1}^1 (x^3 + x\cos x + \tan^5 x + 1) dx\) is:

Q19. The integrating factor (I.F.) of the differential equation \(x\frac{dy}{dx} + 2y = x^2\) (\(x \neq 0\)) is:

Q20. Two independent events \(A\) and \(B\) are such that \(P(A) = 0.3\) and \(P(A \cup B) = 0.6\). The value of \(P(B)\) is:

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