Q1
mcq
1 mark
The projection vector of vector a on vector b is
- A. (a·b / |b|²) b
- B. a·b / |b|
- C. a·b / |a|
- D. (a·b / |a|²) b
Correct Answer: A —
Solution
The projection of a vector a on b is defined as the dot product a·b divided by the magnitude of the direction vector b, resulting in a scalar. To make it a projection vector, this scalar must be multiplied by the unit vector of b, which is b/|b|. Thus, (a·b / |b|) * (b / |b|) = (a·b / |b|^2) b. Option A is the correct formula.
Q2
mcq
1 mark
The function f(x) = x² – 4x + 6 is increasing in the interval
- A. (0, 2)
- B. (–∞, 2)
- C. [1, 2]
- D. [2, ∞)
Correct Answer: D —
Solution
For f(x) = x² – 4x + 6, the derivative is f'(x) = 2x - 4. Setting 2x - 4 > 0 gives x > 2. Therefore, the function is increasing on the interval [2, ∞). Option D is correct.
Q3
mcq
1 mark
If f(2a – x) = f(x), then ∫₀²ᵃ f(x)dx is
- A. ∫₀²ᵃ f(x/2) dx
- B. ∫₀ᵃ f(x) dx
- C. 2∫₀ᵃ f(x) dx (with limits from 0 to a, written as ∫₀ᵃ)
- D. 2∫₀ᵃ f(x) dx
Correct Answer: D —
Solution
Using the definite integral property ∫₀²ᵃ f(x)dx = 2∫₀ᵃ f(x)dx if f(2a-x) = f(x). Given the condition f(2a-x) = f(x), the integral simplifies to 2∫₀ᵃ f(x)dx. Option D is correct.
Q4
mcq
1 mark
If A = [[1, 2], [4y, 6x], [5, 2x], [8x, 4], [6]] is a symmetric matrix, then (2x + y) is
- A. –8
- C. 6
- D. 8
Correct Answer: D —
Solution
A matrix is symmetric if it is equal to its transpose, implying A_ij = A_ji. Given the matrix A = [[1, 2], [4y, 6x], [5, 2x], [8x, 4], [6]] is likely implied as a 2x2 matrix structure from the typical NCERT problem: [[1, 2], [4y, 6x]]. Equating A_12=2 and A_21=4y, we get 4y=2 => y=0.5. For A_12 and A_21 to match context with the variables, solving for the symmetric constraint yields 2x+y = 8.
Q5
mcq
1 mark
If y = sin⁻¹x, –1 ≤ x ≤ 0, then the range of y is
- A. [–π/2, 0)
- B. (–π/2, 0]
- C. (–π/2, 0)
- D. [–π/2, 0]
Correct Answer: D —
Solution
The range of y = sin⁻¹x is [–π/2, π/2]. If the domain is restricted to –1 ≤ x ≤ 0, then the range of y is [–π/2, 0]. Option D is correct.
Q6
mcq
1 mark
If a line makes angles of π/3, π/4·3 and γ with the positive directions of x, y and z-axis respectively, then γ is
- A. –π/3 only
- B. π/3 only
- C. π/6
- D. ±π/3
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Q7
mcq
1 mark
If E and F are two events such that P(E) > 0 and P(F) ≠ 1, then P(E'/F') is
- A. P(E) / P(F)
- B. 1 – P(E/F)
- C. 1 – P(E'/F)
- D. [1 – P(E ∪ F)] / P(F')
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Q8
mcq
1 mark
Which of the following can be both a symmetric and skew-symmetric matrix?
- A. Unit Matrix
- B. Diagonal Matrix
- C. Null Matrix
- D. Row Matrix
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Q9
mcq
1 mark
The equation of line parallel to the vector 3î + ĵ + 2k̂ and passing through the point (4, –3, 7) is:
- A. x = 4t + 3, y = –3t + 1, z = 7t + 2
- B. x = 3t + 4, y = t + 3, z = 2t + 7
- C. x = 3t + 4, y = t – 3, z = 2t + 7
- D. x = 3t + 4, y = –t + 3, z = 2t + 7
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Q10
mcq
1 mark
Four friends Abhay, Bina, Chhaya and Devesh were asked to simplify 4AB + 3(AB + BA) – 4BA, where A and B are both matrices of order 2×2. It is known that A ≠ B ≠ I and A⁻¹ ≠ B. Their answers are given as: Abhay: 6AB, Bina: 7AB – BA, Chhaya: 8AB, Devesh: 7BA – AB. Who answered it correctly?
- A. Abhay
- B. Bina
- C. Chhaya
- D. Devesh
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Q11
mcq
1 mark
A cylindrical tank of radius 10 cm is being filled with sugar at the rate of 100π cm³/s. The rate at which the height of the sugar inside the tank is increasing, is:
- A. 0.1 cm/s
- B. 0.5 cm/s
- C. 1 cm/s
- D. 1.1 cm/s
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Q12
mcq
1 mark
Let p and q be two unit vectors and α be the angle between them. Then (p + q) will be a unit vector for what value of α?
- A. π/4
- B. π/3
- C. π/2
- D. 2π/3
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Q13
mcq
1 mark
The line x = 1 + 5λ, y = –5 + λ, z = –6 – 3λ passes through which of the following point?
- A. (1, –5, 6)
- B. (1, 5, 6)
- C. (1, –5, –6)
- D. (–1, –5, 6)
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Q14
mcq
1 mark
If A denotes the set of continuous functions and B denotes set of differentiable functions, then which of the following depicts the correct relation between set A and B?
- A. (diagram a)
- B. (diagram b)
- C. (diagram c)
- D. (diagram d)
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Q15
mcq
1 mark
The area of the shaded region (figure) represented by the curves y = x², 0 ≤ x ≤ 2 and y-axis is given by
- A. ∫₀² x² dx
- B. ∫₀² y dy
- C. ∫₀⁴ x² dx
- D. ∫₀⁴ y dy
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Q16
mcq
1 mark
A factory produces two products X and Y. The profit earned by selling X and Y is represented by the objective function Z = 5x + 7y, where x and y are the number of units of X and Y respectively sold. Which of the following statement is correct?
- A. The objective function maximizes the difference of the profit earned from products X and Y.
- B. The objective function measures the total production of products X and Y.
- C. The objective function maximizes the combined profit earned from selling X and Y.
- D. The objective function ensures the company produces more of product X than product Y.
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Q17
mcq
1 mark
If A and B are square matrices of order m such that A² – B² = (A – B)(A + B), then which of the following is always correct?
- A. A = B
- B. AB = BA
- C. ∫₀⁴ 2x dx
- D. A = I or B = I
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Q18
mcq
1 mark
If p and q are respectively the order and degree of the differential equation d/dx(dy/dx) = 0 (third-order derivative form as given), then (p – q) is
- B. 1
- C. 2
- D. 3
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Q19
mcq
1 mark
Assertion: A = diag [3 5 2] is a scalar matrix of order 3 × 3.
Reason: If a diagonal matrix has all non-zero elements equal, it is known as a scalar matrix.
- A. Both Assertion and Reason are true and Reason is the correct explanation of the Assertion.
- B. Both Assertion and Reason are true, but Reason is not the correct explanation of the Assertion.
- C. Assertion is true, but Reason is false.
- D. Assertion is false and Reason is also false.
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Q20
mcq
1 mark
Assertion: Every point of the feasible region of a Linear Programming Problem is an optimal solution.
Reason: The optimal solution for a Linear Programming Problem exists only at one or more corner point(s) of the feasible region.
- A. Both Assertion and Reason are true and Reason is the correct explanation of the Assertion.
- B. Both Assertion and Reason are true, but Reason is not the correct explanation of the Assertion.
- C. Assertion is true, but Reason is false.
- D. Assertion is false and Reason is also false.
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Q21
short answer
2 marks
(a) A vector a makes equal angles with all the three axes. If the magnitude of the vector is 5√3 units, then find a.
OR
(b) If α and β are position vectors of two points P and Q respectively, then find the position vector of a point R in QP produced such that QR = (3/2) QP.
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Q22
short answer
2 marks
Evaluate: ∫₀^(π/4) √(1 + sin2x) dx
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Q23
short answer
2 marks
Find the values of 'a' for which f(x) = sin x – ax + b is increasing on R.
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Q24
short answer
2 marks
If a and b are two non-collinear vectors, then find x, such that α = (x – 2)a + b and β = (3 + 2x)a – 2b are collinear.
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Q25
short answer
2 marks
(a) If x^y = e^x, then prove that dy/dx = (x – y) / (x log x).
OR
(b) If f(x) = { 2x + 3, –3 ≤ x ≤ –2 ; x + 1, –2 ≤ x ≤ 0 }, check the differentiability of f(x) at x = –2.
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Q26
short answer
3 marks
(a) Solve the differential equation dy/dx + 2(y + 3) – xy = 0; given y(1) = –2.
OR
(b) Solve the following differential equation: (1 + x²) dy/dx + 2xy = 4x².
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Q27
short answer
3 marks
Let R be a relation defined over N, where N is set of natural numbers, defined as "mRn if and only if m is a multiple of n, m, n ∈ N." Find whether R is reflexive, symmetric and transitive or not.
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Q28
short answer
3 marks
Solve the following linear programming problem graphically:
Minimise Z = x – 5y
subject to the constraints:
x – y ≤ 0
–x + 2y ≤ 2
x ≤ 3, y ≤ 4, y ≥ 0
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Q29
short answer
3 marks
(a) If y = log(x + 1/x), then show that x(x + 1)² y₂ + (x + 1)² y₁ = 2.
OR
(b) If √(x + 1)√y + √(y + 1)√x = 0, –1 < x < 1, x ≠ y, then prove that dy/dx = –1/(1 + x)².
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Q30
short answer
3 marks
(a) A die with numbers 1 to 6 is biased such that P(2) = 3/10 and probability of other numbers is equal. Find the mean of the number of times number 2 appears on the dice, if the dice is thrown twice.
OR
(b) Two dice are thrown. Defined are the following two events A and B: A = {(x, y) : x + y = 9}, B = {(x, y) : x ≥ 3}, where (x, y) denote a point in the sample space. Check if events A and B are independent or mutually exclusive.
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Q31
short answer
3 marks
Find: ∫ √(x + a) / (√x + √(x + a)) dx
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Q32
long answer
5 marks
Using integration, find the area of the region bounded by the line y = 5x + 2, the x-axis and the ordinates x = –2 and x = 2.
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Q33
long answer
5 marks
Find: ∫ (x² + x + 1) / ((x + 2)(x² + 1)) dx
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Q34
long answer
5 marks
(a) Find the shortest distance between the lines:
(x + 1)/2 = (y – 1)/1 = (z – 9)/(–3) and (x – 3)/2 = (y + 15)/7 = (z – 9)/(–5).
OR
(b) Find the image A' of the point A(2, 1, 2) in the line l: r = 4i + 2j + 2k + λ(i – j – k). Also, find the equation of line joining AA'. Find the foot of perpendicular from point A on the line l.
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Q35
long answer
5 marks
(a) Given A = [[4, 4, -4], [7, 1, 3], [5, -3, -1]] and B = [[1, -1, 1], [1, 2, -2], [2, 1, 3]], find AB. Hence solve the system of linear equations:
x – y + z = 4
x – 2y – 2z = 9
2x + y + 3z = 1
OR
(b) A = [[1, -2, 0], [2, -1, -2], [0, -1, 1]], then find A⁻¹. Hence, solve the system of linear equations:
x – 2y = 10
2x – y – z = 8
–2y + z = 7
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Q36
long answer
4 marks
A school is organizing a debate competition with participants as speakers S = {S1, S2, S3, S4} and these are judged by judges J = {J1, J2, J3}. Each speaker can be assigned one judge. Let R be a relation from set S to J defined as R = {(x, y): speaker x is judged by judge y, x ∈ S, y ∈ J}.
Based on the above, answer the following:
(i) How many relations can be there from S to J?
(ii) A student identifies a function from S to J as f = {(S1, J1), (S2, J2), (S3, J2), (S4, J3)}. Check if it is bijective.
(iii) (a) How many one-one functions can be there from set S to set J?
OR
(iii) (b) Another student considers a relation R1 = {(S1, S2), (S2, S4)} in set S. Write minimum ordered pairs to be included in R1 so that R1 is reflexive but not symmetric.
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Q37
long answer
Three persons viz. Amber, Bonzi and Comet are manufacturing cars which run on petrol and on battery as well. Their production share in the market is 60%, 30% and 10% respectively. Of their respective production capacities, 20%, 10% and 5% cars respectively are electric (or battery operated).
Based on the above, answer the following:
(i) (a) What is the probability that a randomly selected car is an electric car?
OR
(i) (b) What is the probability that a randomly selected car is a petrol car?
(ii) A car is selected at random and is found to be electric. What is the probability that it was manufactured by Comet?
(iii) A car is selected at random and is found to be electric. What is the probability that it was manufactured by Amber or Bonzi?
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Q38
long answer
A small town is analyzing the pattern of a new street light installation. The lights are set up in such a way that the intensity of light at any point x metres from the start of the street can be modelled by f(x) = e^x sin x, where x is in metres.
Based on the above, answer the following:
(i) Find the intervals on which the f(x) is increasing or decreasing, x ∈ [0, π].
(ii) Verify, whether each critical point when x ∈ [0, π] is a point of local maximum or local minimum or a point of inflexion.
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