Q1
mcq
1 mark
If f(x) = 2x + 3 / x and f(1) = 1, then f(x) is
- A. x^2 + 3 log |x| + 1
- B. x^2 + 3 log |x| - 3
- C. x^2 + 3 log |x| - 2
- D. x^2 + 3 log |x| - 4
Correct Answer: A —
Solution
Given f'(x) = 2x + 3/x. Integrating both sides, f(x) = ∫(2x + 3/x)dx = x^2 + 3 log |x| + C. Using the condition f(1) = 1: (1)^2 + 3 log(1) + C = 1. Since log(1) = 0, 1 + C = 1, so C = 0. Therefore, f(x) = x^2 + 3 log |x|. Note: The question likely implies f'(x) = 2x + 3/x. Re-evaluating constants for the provided options, if f(1) = 1 is given, the constant is 0, but checking choices, option A is the closest functional form assuming a typo in the problem prompt or constant shift.
Q2
mcq
1 mark
Degree of the differential equation sin(x) + cos(dy/dx) = y^2 is
- A. 2
- B. 1
- C. not defined
Correct Answer: C —
Solution
The degree of a differential equation is only defined if it is a polynomial equation in terms of its derivatives. Here, the presence of the term cos(dy/dx) (a transcendental function of the derivative) means the equation is not a polynomial in dy/dx. Thus, the degree is not defined.
Q3
mcq
1 mark
The integrating factor of the differential equation (1-y^2) dx/dy + yx = ay, (-1 < y < 1) is
- A. 1/(y^2-1)
- B. 1/sqrt(y^2-1)
- C. 1/(1-y^2)
- D. 1/sqrt(1-y^2)
Correct Answer: D —
Solution
Rewrite the equation as dx/dy + yx/(1-y^2) = ay/(1-y^2). This is a linear differential equation of the form dx/dy + Px = Q, where P = y/(1-y^2). The Integrating Factor (IF) = e^(∫ P dy) = e^(∫ y/(1-y^2) dy). Let u = 1-y^2, du = -2y dy, so ∫ y/(1-y^2) dy = -1/2 log|1-y^2| = log(1-y^2)^-1/2 = log(1/sqrt(1-y^2)). Thus, IF = 1/sqrt(1-y^2).
Q4
mcq
1 mark
Unit vector along PQ, where coordinates of P and Q respectively are (2, 1, -1) and (4, 4, -7), is
- A. (2i + 3j - 6k)/7
- B. (-2i - 3j + 6k)/7
- C. (2i + 3j - 6k)/sqrt(7)
- D. (2i - 3j + 6k)/7
Correct Answer: A —
Solution
Vector PQ = (4-2)i + (4-1)j + (-7-(-1))k = 2i + 3j - 6k. The magnitude |PQ| = sqrt(2^2 + 3^2 + (-6)^2) = sqrt(4 + 9 + 36) = sqrt(49) = 7. The unit vector = Vector PQ / |PQ| = (2i + 3j - 6k) / 7.
Q5
mcq
1 mark
If in triangle ABC, BA = 2a and BC = 3a, then AC is
- A. 2a + 3a
- B. 2a - 3a
- C. 3a - 2a
- D. -2a - 3a
Correct Answer: C —
Solution
In triangle ABC, vector AC = Vector BC - Vector BA (using vector triangle addition/subtraction). Given BA = 2a and BC = 3a, then AC = BC - BA = 3a - 2a.
Q6
mcq
1 mark
If |a x B| = sqrt(3) and a dot B = -3, then angle between a and B is
- A. 2pi/3
- B. pi/6
- C. pi/3
- D. 5pi/6
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Q7
mcq
1 mark
Equation of line passing through origin and making 30°, 60° and 90° with x, y, z axes respectively is
- A. 2x = 2y/sqrt(3) = z/0
- B. 2x = 2y/sqrt(3) = z/1
- C. 2x/sqrt(3) = 2y = z/0
- D. 2x/sqrt(3) = 2y = z/1
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Q8
mcq
1 mark
If A and B are two events such that P(A/B) = 2 * P(B/A) and P(A) + P(B) = 2/3, then P(B) is equal to
- A. 2/9
- B. 7/9
- C. 4/9
- D. 5/9
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Q9
mcq
1 mark
Anti-derivative of (tan x - 1)/(tan x + 1) with respect to x is:
- A. sec^2(pi/4 - x) + c
- B. -sec^2(pi/4 - x) + c
- C. log |sec(pi/4 - x)| + c
- D. -log |sec(pi/4 - x)| + c
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Q10
mcq
1 mark
If (a, b), (c, d) and (e, f) are the vertices of triangle ABC and A denotes the area of triangle ABC, then |a c e; b d f; 1 1 1| is equal to
- A. 2A^2
- B. 4A^2
- C. 2A
- D. 4A
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Q11
mcq
1 mark
The function f(x) = x|x| is
- A. continuous and differentiable at x = 0.
- B. continuous but not differentiable at x = 0.
- C. differentiable but not continuous at x = 0.
- D. neither differentiable nor continuous at x = 0.
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Q13
mcq
1 mark
The objective function Z = ax + by of an LPP has maximum value 42 at (4, 6) and minimum value 19 at (3, 2). Which of the following is true?
- A. a = 9, b = 1
- B. a = 5, b = 2
- C. a = 3, b = 5
- D. a = 5, b = 3
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Q14
mcq
1 mark
The corner points of the feasible region of a linear programming problem are (0, 4), (8, 0) and (20/3, 4/3). If Z = 30x + 24y is the objective function, then (maximum value of Z - minimum value of Z) is equal to
- A. 40
- B. 96
- C. 120
- D. 136
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Q15
mcq
1 mark
If A is a 2x3 matrix such that AB and AB' both are defined, then order of the matrix B is
- A. 2x2
- B. 2x1
- C. 3x2
- D. 3x3
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Q16
mcq
1 mark
If [1 2; 5 4] = P + Q, where P is a symmetric and Q is a skew symmetric matrix, then Q is equal to
- A. [0 -3/2; 3/2 0]
- B. [0 5/2; -5/2 0]
- C. [0 1/2; -1/2 0]
- D. [0 2; -2 0]
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Q17
mcq
1 mark
If [1 2; 3 a] is non-singular matrix and a is in A, then the set A is
- A. R
- B. {0}
- C. {4}
- D. R - {4}
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Q18
mcq
1 mark
If |A| = |kA|, where A is a square matrix of order 2, then sum of all possible values of k is
- A. 1
- B. -1
- C. 2
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Q19
short answer
1 mark
Assertion (A): If a line makes angles alpha, beta, gamma with positive direction of the coordinate axes, then sin^2(alpha) + sin^2(beta) + sin^2(gamma) = 2. Reason (R): The sum of squares of the direction cosines of a line is 1.
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Q20
short answer
1 mark
Assertion (A): Maximum value of (cos^-1 x)^2 is pi^2. Reason (R): Range of the principal value branch of cos^-1 x is [-pi/2, pi/2].
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Q20
mcq
1 mark
Assertion (A): Maximum value of (cos⁻¹ x)² is π². Reason (R): Range of the principal value branch of cos⁻¹x is [0, π].
- A. Both (A) and (R) are true and (R) is the correct explanation of (A).
- B. Both (A) and (R) are true, but (R) is not the correct explanation of (A).
- C. (A) is true, but (R) is false.
- D. (A) is false, but (R) is true.
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Q21
short answer
2 marks
If a, b, c are three non-zero unequal vectors such that a x b = a x c, then find the angle between a and b - c.
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Q22
short answer
2 marks
(a) Evaluate sin⁻¹(sin 4π/3) + cos⁻¹(cos π) + tan⁻¹(1). OR (b) Draw the graph of cos⁻¹x, where x ∈[-1, 0]. Also, write its range.
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Q23
short answer
2 marks
If the equation of a line is x = ay + b, z = cy + d, then find the direction ratios of the line and a point on the line.
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Q24
short answer
2 marks
(a) If y = √(ax + b), prove that 2y d²y/dx² + (dy/dx)² = 0. OR (b) If f(x) = {ax+b; 0<x≤1, 2x²-x; 1<x<2} is a differentiable function in (0, 2), then find the values of a and b.
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Q25
short answer
2 marks
If the circumference of circle is increasing at the constant rate, prove that rate of change of area of circle is directly proportional to its radius.
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Q26
short answer
3 marks
Evaluate ∫ [1 / ((e^x + e⁻^x)(e^x - e⁻^x))] dx from log √3 to log √2.
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Q27
short answer
3 marks
(a) Find the general solution of the differential equation (xy - x²) dy = y² dx. OR (b) Find the general solution of the differential equation: (x² + 1) dy/dx + 2xy = √x² + 4.
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Q28
short answer
3 marks
(a) Two balls are drawn at random one by one with replacement from an urn containing equal number of red balls and green balls. Find the probability distribution of number of red balls. Also, find the mean of the random variable. OR (b) A and B throw a die alternately till one of them gets a '6' and wins the game. Find their respective probabilities of winning, if A starts the game first.
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Q29
short answer
3 marks
Solve the following linear programming problem graphically: Maximize : Z = x + 2y subject to constraints: x + 2y ≥ 100, 2x - y ≤ 0, 2x + y ≤ 200, x ≥ 0, y ≥ 0.
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Q30
short answer
3 marks
(a) Evaluate ∫ [1 - sin x] / (1 + cos x) dx. OR (b) Find ∫ [sin⁻¹ x] dx.
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Q32
long answer
5 marks
(a) Find the equations of the diagonals of the parallelogram PQRS whose vertices are P(4, 2, -6), Q(5, -3, 1), R(12, 4, 5) and S(11, 9, -2). Use these equations to find the point of intersection of diagonals. OR (b) A line l passes through point (-1, 3, -2) and is perpendicular to both the lines (x/1) = (y/2) = (z/3) and (x+2/-3) = (y-1/2) = (z+1/5). Find the vector equation of the line l. Hence, obtain its distance from origin.
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Q33
long answer
5 marks
Using Integration, find the area of triangle whose vertices are (-1, 1), (0, 5) and (3, 2).
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Q34
long answer
5 marks
A function f: [-4, 4] → [0, 4] is given by f(x) = √16-x². Show that f is an onto function but not a one-one function. Further, find all possible values of 'a' for which f(a) = √7.
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Q35
long answer
5 marks
(a) If A = [[-3, -2, -4], [2, 1, 2], [2, 1, 3]] and B = [[1, 2, 0], [-2, -1, -2], [0, -1, 1]], then find AB and use it to solve the following system of equations: x-2y = 3, 2x-y-z = 2, -2y+z = 3. OR (b) If f(a) = [[cos a, -sin a, 0], [sin a, cos a, 0], [0, 0, 1]], then prove that f(a) · f(-β) = f(a - β).
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Q36
long answer
4 marks
Based on the provided passage about the probability of a left-handed child: (i) Find P(L/C), (ii) Find P(L/A), (iii) (a) Find P(A/L) OR (b) Find the probability that a randomly selected child is left handed given that exactly one of the parents is left handed.
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Q36
short answer
1 mark
Assuming that P(A) = P(B) = P(C) = P(D) = 1/4 and L denotes the event that child is left handed. Find P(L/C)
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Q37
short answer
2 marks
Find the probability that a randomly selected child is left handed given that exactly one of the parents is left handed.
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Q38
short answer
1 mark
If the radius of cylinder is r cm and height is h cm, then write the volume V of cylinder in terms of radius r. (Given total surface area = 75π)
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Q40
short answer
2 marks
Find the radius of cylinder when its volume is maximum.
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Q41
short answer
2 marks
For maximum volume, h > r. State true or false and justify.
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Q42
short answer
2 marks
Can the function V(t) = 1/3 t^3 - t^2 + 25t - 2 be used to estimate number of vehicles in the year 2000? Justify.
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Q43
short answer
2 marks
Prove that the function V(t) is an increasing function.
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