Commerce (English Medium)
Science (English Medium)
Arts (English Medium)
Academic Year: 2024-2025
Date & Time: 8th March 2025, 10:30 am
Duration: 3h
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General Instructions:
Read the following instructions very carefully and strictly follow them:
- This Question paper contains 38 questions. All questions are compulsory.
- Question paper is divided into FIVE Sections - Section A, B, C, D and E.
- In Section A - Question Number 1 to 18 are Multiple Choice Questions (MCQs) and Question Number 19 & 20 are Assertion-Reason based questions of 1 mark each.
- In Section B - Question Number 21 to 25 are Very Short Answer (VSA) type questions, carrying 2 marks each.
- In Section C - Question Number 26 to 31 are Short Answer (SA) type questions, carrying 3 marks each.
- In Section D - Question Number 32 to 35 are Long Answer (LA) type questions, carrying 5 marks each.
- In Section E - Question Number 36 to 38 are case study based questions, carrying 4 marks each.
- There is no overall choice. However, an internal choice has been provided in 2 questions in Section - B, 3 questions in Section - C, 2 questions in Section - D and 2 questions in Section - E.
- Use of calculator is NOT allowed.
The projection vector of vector `veca` on vector `vecb` is ______.
`((veca . vecb)/|vecb|^2)vecb`
`(veca . vecb)/|vecb|`
`(veca . vecb)/|veca|`
`((veca . vecb)/|veca|^2)vecb`
Chapter:
The function f(x) = x2 – 4x + 6 is increasing in the interval:
(0, 2)
(–∞, 2]
[1, 2]
[2, ∞)
Chapter:
If f(2a – x) = f(x), then `int_0^(2a)f(x) dx` is ______.
`int_0^(2a)f(x/2) dx`
`int_0^a f(x) dx`
`2int_a^0 f(x) dx`
`2int_0^a f(x) dx`
Chapter:
If A = `[(1, 12, 4y), (6x, 5, 2x), (8x, 4, 6)]` is a symmetric matrix, then (2x + y) is ______.
−8
0
6
8
Chapter:
If y = sin−1x, −1 ≤ x ≤ 0, then thje range of y is ______.
`((-pi)/2, 0)`
`[(-pi)/2, 0]`
`[(-pi)/2, 0)`
`((-pi)/2, 0]`
Chapter:
If a line makes angles of `(3pi)/4, pi/3` and θ with the positive directions of x, y and z-axis respectively, then θ is ______.
`(-pi)/3 only`
`pi/3 only`
`pi/6`
`± pi/3`
Chapter:
If E and F are two events such that P(E) > 0 and P(F) ≠ 1, then `P(barE//barF)` is ______.
`(P(barE))/(P(barF))`
`1 - P(barE//F)`
1 − P(E/F)
`(1 - P(E ∪ F))/(P(barF)`
Chapter:
Which of the following can be both a symmetric and skew-symmetric matrix?
Unit Matrix
Diagonal Matrix
Null Matrix
Row Matrix
Chapter:
The equation of a line parallel to the vector `3hati + hatj + 2hatk`and passing through the point (4, −3, 7) is ______.
x = 4t + 3, y = −3t + 1, z = 7t + 2
x = 3t + 4, y = t + 3, z = 2t + 7
x = 3t + 4, y = t – 3, z = 2t + 7
x = 3t + 4, y = −t + 3, z = 2t + 7
Chapter:
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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: 8 AB
Devesh: 7BA − AB
Who answered it correctly?
Abhay
Bina
Chhaya
Devesh
Chapter:
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 ______.
0.1 cm/s
0.5 cm/s
1 cm/s
1.1 cm/s
Chapter:
Let `vecp and vecq` be two unit vectors and α be the angle between them. Then `(vecp + vecq)` will be a unit vector for what value of α?
`pi/4`
`pi/2`
`(pi)/3`
`(2pi)/3`
Chapter:
The line x = 1 + 5μ, y = −5 + μ, z = −6 − 3μ passes through which of the following point?
(1, −5, 6)
(1, 5, 6)
(1, −5, −6)
(−1, −5, 6)
Chapter:
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?
Chapter:
The area of the shaded region (figure) represented by the curves y = x2, 0 ≤ x ≤ 2 and y-axis is given by
`int_0^2 x^2 dx`
`int_0^2 sqrty dy`
`int_0^4 x^2 dx`
`int_0^4 sqrty dy`
Chapter:
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?
The objective function maximizes the difference of the profit earned from products X and Y.
The objective function measures the total production of products X and Y.
The objective function maximizes the combined profit earned from selling X and Y.
The objective function ensures the company produces more of product X than product Y.
Chapter:
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 = B
AB = BA
A = 0 or B = 0
A = I or B = I
Chapter:
If p and q are respectively the order and degree of the differential equation `d/dx((dy)/(dx))^3 = 0`, then (p − q) is ______.
0
1
2
3
Chapter:
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Assertion (A): A = diag [3 5 2] is a scalar matrix of order 3 × 3.
Reason (R): If a diagonal matrix has all non-zero elements equal, it is known as a scalar matrix.
Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A).
Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
Assertion (A) is true but Reason (R) is false.
Assertion (A) is false but Reason (R) is true.
Chapter:
Assertion (A): Every point of the feasible region of a Linear Programming Problem is an optimal solution.
Reason (R): The optimal solution for a Linear Programming Problem exists only at one or more corner point(s) of the feasible region.
Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A).
Explanation:
Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
Assertion (A) is true but Reason (R) is false.
Assertion (A) is false but Reason (R) is true.
Chapter:
A vector `veca` makes equal angles with all the three axes. If the magnitude of the vector is `5sqrt3` units, then find `veca`.
Chapter:
If `vecα and vecβ` 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.
Chapter:
Find the values of ‘a’ for which f(x) = sin x − ax + b is increasing on R.
Chapter:
If `veca and vecb` are two non-collinear vectors, then find x, such that `vecα = (x - 2)veca + vecb and vecβ = (3 + 2x)veca - 2vecb` are collinear.
Chapter:
If x = `e^(x/y)`, then prove that `dy/dx = (x - y)/(xlogx)`.
Chapter: [5] Continuity and Differentiability
If f(x) = `{(2x - 3",", -3 ≤ x ≤ -2), (x + 1",", -2 ≤ x ≤ 0):}`
Check the differentiability of f(x) at x = -2.
Chapter:
Solve:
`2(y + 3) - xy (dy)/(dx)` = 0, given that y(1) = – 2.
Chapter: [9] Differential Equations
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