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
Solution hidden
Log in to view solution →
Q2
mcq
1 mark
Degree of the differential equation sin(x) + cos(dy/dx) = y^2 is
- A. 2
- B. 1
- C. not defined
Solution hidden
Log in to view solution →
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)
Solution hidden
Log in to view solution →
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
Solution hidden
Log in to view solution →
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
Solution hidden
Log in to view solution →
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
Solution hidden
Log in to view solution →
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
Solution hidden
Log in to view solution →
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
Solution hidden
Log in to view solution →
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
Solution hidden
Log in to view solution →
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
Solution hidden
Log in to view solution →
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.
Solution hidden
Log in to view solution →
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
Solution hidden
Log in to view solution →
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
Solution hidden
Log in to view solution →
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
Solution hidden
Log in to view solution →
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]
Solution hidden
Log in to view solution →
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}
Solution hidden
Log in to view solution →
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
Solution hidden
Log in to view solution →
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.
Solution hidden
Log in to view solution →
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].
Solution hidden
Log in to view solution →
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.
Solution hidden
Log in to view solution →
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.
Solution hidden
Log in to view solution →
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.
Solution hidden
Log in to view solution →
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.
Solution hidden
Log in to view solution →
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.
Solution hidden
Log in to view solution →
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.
Solution hidden
Log in to view solution →
Q26
short answer
3 marks
Evaluate ∫ [1 / ((e^x + e⁻^x)(e^x - e⁻^x))] dx from log √3 to log √2.
Solution hidden
Log in to view solution →
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.
Solution hidden
Log in to view solution →
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.
Solution hidden
Log in to view solution →
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.
Solution hidden
Log in to view solution →
Q30
short answer
3 marks
(a) Evaluate ∫ [1 - sin x] / (1 + cos x) dx. OR (b) Find ∫ [sin⁻¹ x] dx.
Solution hidden
Log in to view solution →
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.
Solution hidden
Log in to view solution →
Q33
long answer
5 marks
Using Integration, find the area of triangle whose vertices are (-1, 1), (0, 5) and (3, 2).
Solution hidden
Log in to view solution →
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.
Solution hidden
Log in to view solution →
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).
Solution hidden
Log in to view solution →
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.
Solution hidden
Log in to view solution →
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)
Solution hidden
Log in to view solution →
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.
Solution hidden
Log in to view solution →
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)
Solution hidden
Log in to view solution →
Q40
short answer
2 marks
Find the radius of cylinder when its volume is maximum.
Solution hidden
Log in to view solution →
Q41
short answer
2 marks
For maximum volume, h > r. State true or false and justify.
Solution hidden
Log in to view solution →
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.
Solution hidden
Log in to view solution →
Q43
short answer
2 marks
Prove that the function V(t) is an increasing function.
Solution hidden
Log in to view solution →