Q1
mcq
1 mark
If A = [[5,0,0],[0,5,0],[0,0,5]], then A³ is:
- A. 3 * [[5,0,0],[0,5,0],[0,0,5]]
- B. [[125,0,0],[0,125,0],[0,0,125]]
- C. [[15,0,0],[0,15,0],[0,0,15]]
- D. [[5³,0,0],[0,5³,0],[0,0,5³]]
Solution hidden
Log in to view solution →
Q2
mcq
1 mark
If P(A ∪ B) = 0.9 and P(A ∩ B) = 0.4, then P(A') + P(B') is:
- A. 0.3
- B. 1
- C. 1.3
- D. 0.7
Solution hidden
Log in to view solution →
Q3
mcq
1 mark
If A = [[1,2,3],[4,3,7]] and B = [[4,3],[1,2],[0,5]], then the correct statement is:
- A. Only AB is defined.
- B. Only BA is defined.
- C. AB and BA, both are defined.
- D. AB and BA, both are not defined.
Solution hidden
Log in to view solution →
Q4
mcq
1 mark
If |2x 5; 6 5| / |12 x; 4 3| = 0 (determinant equation: [[2x,5],[12,x]] = [[6,5],[4,3]]), then the value of x is:
- A. 3
- B. 7
- C. -7
- D. -3
Solution hidden
Log in to view solution →
Q5
mcq
1 mark
If f(x) = sin²(ax)/x² for x ≠ 0, and f(x) = 1 for x = 0, is continuous at x = 0, then the value of a is:
- A. 1
- B. -1
- C. ±1
Solution hidden
Log in to view solution →
Q6
mcq
1 mark
If A = [aᵢⱼ] is a 3×3 diagonal matrix such that a₁₁ = 1, a₂₂ = 5 and a₃₃ = –2, then |A| is:
- B. -10
- C. 10
- D. 1
Solution hidden
Log in to view solution →
Q7
mcq
1 mark
The principal value of cot⁻¹(–1/√3) is:
- A. -π/3
- B. -2π/3
- C. π/3
- D. 2π/3
Solution hidden
Log in to view solution →
Q8
mcq
1 mark
If [[4,x+1],[x-2,3]] is a singular matrix, then the value of x is:
- B. 1
- C. -2
- D. -4
Solution hidden
Log in to view solution →
Q9
mcq
1 mark
If f(x) = {[x], x ∈ R} is the greatest integer function, then the correct statement is:
- A. f is continuous but not differentiable at x = 2.
- B. f is neither continuous nor differentiable at x = 2.
- C. f is continuous as well as differentiable at x = 2.
- D. f is not continuous but differentiable at x = 2.
Solution hidden
Log in to view solution →
Q10
mcq
1 mark
The slope of the curve y = –x³ + 3x² + 8x – 20 is maximum at:
- A. (1, –10)
- B. (1, 10)
- C. (10, 1)
- D. (–10, 1)
Solution hidden
Log in to view solution →
Q11
mcq
1 mark
∫√(1 + sin x) dx is equal to:
- A. 2(sin(x/2) – cos(x/2)) + C
- B. 2(sin(x/2) + cos(x/2)) + C
- C. –2(sin(x/2) – cos(x/2)) + C
- D. 2(sin(x/2) + cos(x/2)) + C
Solution hidden
Log in to view solution →
Q12
mcq
1 mark
∫₀^(π/2) cos x · e^(sin x) dx is equal to:
- B. 1 – e
- C. e – 1
- D. e
Solution hidden
Log in to view solution →
Q13
mcq
1 mark
The area of the region enclosed between the curve y = x|x|, x-axis, x = –2 and x = 2 is:
- A. 8/3
- B. 16/3
- D. 8
Solution hidden
Log in to view solution →
Q14
mcq
1 mark
The integrating factor of the differential equation (2x/e^y – 1/x)(dx/dy) = 1 (or equivalently as given) is:
- A. e^(–1/x)
- B. e^(2/x)
- C. e^(2x)
- D. e^(–2x)
Solution hidden
Log in to view solution →
Q15
mcq
1 mark
The sum of the order and degree of the differential equation [1 + (dy/dx)²]^(3/2) = d²y/dx² is:
- A. 2
- B. 5/2
- C. 3
- D. 4
Solution hidden
Log in to view solution →
Q16
mcq
1 mark
For a Linear Programming Problem (LPP), the given objective function Z = 3x + 2y is subject to constraints: x + 2y ≤ 10, 3x + y ≤ 15, x, y ≥ 0. The correct feasible region is:
- A. ABC
- B. AOEC
- C. CED
- D. Open unbounded region BCD
Solution hidden
Log in to view solution →
Q17
mcq
1 mark
Let vector a be a position vector whose tip is the point (2, –3). If AB⃗ = a⃗, where coordinates of A are (–4, 5), then the coordinates of B are:
- A. (–2, –2)
- B. (2, –2)
- C. (–2, 2)
- D. (2, 2)
Solution hidden
Log in to view solution →
Q18
mcq
1 mark
The respective values of |a⃗| and |b⃗|, if (a⃗ – b⃗)·(a⃗ + b⃗) = 512 and |a⃗| = 3|b⃗|, are:
- A. 48 and 16
- B. 3 and 1
- C. 24 and 8
- D. 6 and 2
Solution hidden
Log in to view solution →
Q19
mcq
1 mark
Assertion: The shaded portion of the graph represents the feasible region for the given Linear Programming Problem (LPP). Min Z = 50x + 70y subject to constraints 2x + y ≥ 8, 2y ≥ 10, x, y ≥ 0. Z = 50x + 70y has a minimum value = 380 at B(2, 4). Reason: The region representing 50x + 70y < 380 does not have any point common with 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.
Solution hidden
Log in to view solution →
Q20
mcq
1 mark
Assertion: Let A = {x ∈ R : –1 ≤ x ≤ 1}. If f: A → A be defined as f(x) = x², then f is not an onto function. Reason: If y = –1 ∈ A, then x = √(–1) ∉ A.
- 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.
Solution hidden
Log in to view solution →
Q21
short answer
2 marks
Find the domain of the function f(x) = cos⁻¹(x² – 4).
Solution hidden
Log in to view solution →
Q22
short answer
2 marks
Surface area of a balloon (spherical), when air is blown into it, increases at a rate of 5 mm²/s. When the radius of the balloon is 8 mm, find the rate at which the volume of the balloon is increasing.
Solution hidden
Log in to view solution →
Q23
short answer
2 marks
(a) Differentiate sin x / cos x with respect to x.
OR
(b) If y = 5 cos x – 3 sin x, prove that d²y/dx² + y = 0.
Solution hidden
Log in to view solution →
Q24
short answer
2 marks
(a) Find a vector of magnitude 5 which is perpendicular to both the vectors 3î – 2ĵ + k̂ and 4î + 3ĵ – 2k̂.
OR
(b) Let a⃗, b⃗ and c⃗ be three vectors such that a⃗·b⃗ = a⃗·c⃗ and a⃗ × b⃗ = a⃗ × c⃗, a⃗ ≠ 0⃗. Show that b⃗ = c⃗.
Solution hidden
Log in to view solution →
Q25
short answer
2 marks
A man needs to hang two lanterns on a straight wire whose end points have coordinates A(4, 1, –2) and B(6, 2, –3). Find the coordinates of the points where he hangs the lanterns such that these points trisect the wire AB.
Solution hidden
Log in to view solution →
Q26
short answer
3 marks
Find the value of 'a' for which f(x) = √3 sin x – cos x – 2ax + 6 is decreasing in R.
Solution hidden
Log in to view solution →
Q27
short answer
3 marks
(a) Find: ∫ 2x / ((x² + 3)(x² – 5)) dx
OR
(b) Evaluate: ∫₁⁴ |x – 2| dx
Solution hidden
Log in to view solution →
Q28
short answer
2 marks
Differentiate sin x / cos x with respect to x.
Solution hidden
Log in to view solution →
Q28
short answer
3 marks
Find the particular solution of the differential equation x sin²(y/x) dx – x dy + y dx = 0 given that y = π/4, where x = 1.
Solution hidden
Log in to view solution →
Q29
short answer
2 marks
If y = 5 cos x – 3 sin x, prove that d²y/dx² + y = 0.
Solution hidden
Log in to view solution →
Q29
short answer
3 marks
In the Linear Programming Problem (LPP), find the point/points giving maximum value for Z = 5x + 10y subject to constraints:
x + 2y ≤ 120
x + y ≥ 60
x – 2y ≥ 0
x, y ≥ 0
Solution hidden
Log in to view solution →
Q30
short answer
2 marks
Find a vector of magnitude 5 which is perpendicular to both the vectors 3î – 2ĵ + k̂ and 4î + 3ĵ – 2k̂.
Solution hidden
Log in to view solution →
Q31
short answer
2 marks
Let a, b and c be three vectors such that a · b = a · c and a × b = a × c, a ≠ 0. Show that b = c.
Solution hidden
Log in to view solution →
Q32
short answer
3 marks
Find: ∫ 2x / ((x² + 3)(x² – 5)) dx
Solution hidden
Log in to view solution →
Q32
long answer
5 marks
Sketch a graph of y = x². Using integration, find the area of the region bounded by y = 9, x = 0 and y = x².
Solution hidden
Log in to view solution →
Q33
short answer
3 marks
Evaluate: ∫₁⁴ (|x – 2| + |x – 4|) dx
Solution hidden
Log in to view solution →
Q33
long answer
5 marks
A furniture workshop produces three types of furniture - chairs, tables and beds each day. On a particular day the total number of furniture pieces produced is 45. It was also found that production of beds exceeds that of chairs by 8, while the total production of beds and chairs together is twice the production of tables. Determine the units produced of each type of furniture, using matrix method.
Solution hidden
Log in to view solution →
Q36
short answer
3 marks
If a + b + c = 0 such that |a| = 3, |b| = 5, |c| = 7, then find the angle between a and b.
Solution hidden
Log in to view solution →
Q36
long answer
(iii)(a) Find a relation between x and y such that the surface area (S) is minimum. OR (iii)(b) If surface area (S) is constant, the volume V = (1/4)(Sx - 2x³), x being the edge of base. Show that volume (V) is maximum for x = √(S/6).
Solution hidden
Log in to view solution →
Q37
short answer
3 marks
If a and b are unit vectors inclined with each other at an angle θ, then prove that |a – b| / 2 = sin(θ/2).
Solution hidden
Log in to view solution →
Q37
long answer
(iii)(a) Let R be a relation defined by the teacher to plan the seating arrangement of students in pairs, where R = {(x, y) : x, y are Roll Numbers of students such that y = 3x}. List the elements of R. Is the relation R reflexive, symmetric and transitive? Justify your answer. OR (iii)(b) Let R be a relation defined by R = {(x, y): x, y are Roll Numbers of students such that y = x³}. List the elements of R. Is R a function? Justify your answer.
Solution hidden
Log in to view solution →
Q38
short answer
3 marks
The probability that a student buys a colouring book is 0.7 and that she buys a box of colours is 0.2. The probability that she buys a colouring book, given that she buys a box of colours, is 0.3. Find the probability that the student:
(i) Buys both the colouring book and the box of colours.
(ii) Buys a box of colours given that she buys the colouring book.
Solution hidden
Log in to view solution →
Q38
short answer
Case Study-3: A gardener wanted to plant vegetables in his garden. Hence he bought 10 seeds of brinjal plant, 12 seeds of cabbage plant and 8 seeds of radish plant. The shopkeeper assured him of germination probabilities of brinjal, cabbage and radish to be 25%, 35% and 40% respectively. But before he could plant the seeds, they got mixed up in the bag and he had to sow them randomly. (i) Calculate the probability of a randomly chosen seed to germinate.
Solution hidden
Log in to view solution →
Q38
short answer
(ii) What is the probability that it is a cabbage seed, given that the chosen seed germinates?
Solution hidden
Log in to view solution →
Q39
short answer
3 marks
A person has a fruit box that contains 6 apples and 4 oranges. He picks out a fruit three times, one after the other, after replacing the previous one in the box. Find:
(i) The probability distribution of the number of oranges he draws.
(ii) The expectation of the random variable (number of oranges).
Solution hidden
Log in to view solution →
Q42
long answer
5 marks
For a positive constant 'a', differentiate a^(t + 1/t) with respect to a^(t + 1/t), where t is a non-zero real number.
Solution hidden
Log in to view solution →
Q43
long answer
5 marks
Find dy/dx if y^x + x^y + x^x = a^b, where a and b are constants.
Solution hidden
Log in to view solution →
Q44
long answer
5 marks
Find the foot of the perpendicular drawn from the point (1, 1, 4) on the line (x + 2)/5 = (y + 1)/2 = (z – 4)/–3 + 4.
Solution hidden
Log in to view solution →
Q45
long answer
5 marks
Find the point on the line (x – 1)/3 = (y + 1)/2 = (z – 4)/3 at a distance of 2√2 units from the point (–1, –1, 2).
Solution hidden
Log in to view solution →
Q46
short answer
Taking length = breadth = x m and height = y m, express the surface area (S) of the box in terms of x and its volume (V), which is constant.
Solution hidden
Log in to view solution →
Q48
short answer
Find a relation between x and y such that the surface area (S) is minimum.
Solution hidden
Log in to view solution →
Q49
short answer
If surface area (S) is constant, the volume V = (1/4)(Sx – 2x³), x being the edge of base. Show that volume (V) is maximum for x = √(S/6).
Solution hidden
Log in to view solution →
Q50
short answer
Let A be the set of 30 students of class XII in a school. Let f: A → N, N is a set of natural numbers such that function f(x) = Roll Number of student x. Is f a bijective function?
Solution hidden
Log in to view solution →
Q51
short answer
Give reasons to support your answer to (i).
Solution hidden
Log in to view solution →
Q52
short answer
Let R be a relation defined by the teacher to plan the seating arrangement of students in pairs, where R = {(x, y) : x, y are Roll Numbers of students such that y = 3x}. List the elements of R. Is the relation R reflexive, symmetric and transitive? Justify your answer.
Solution hidden
Log in to view solution →