Commerce (English Medium)
Science (English Medium)
Arts (English Medium)
Academic Year: 2021-2022
Date & Time: 6th December 2021, 11:30 am
Duration: 1h30m
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General Instructions:
- This question paper comprises of 50 questions out of which 40 questions are to be attempted as per instructions. All questions carry equal marks,
- The question paper consists of three Sections - Section A, B and C.
- Section - A contains 20 questions. Attempt any 16 questions from Q. No. 1 to 20.
- Section - B also contains 20 questions. Attempt any 16 questions from Q. No. 21 to 40.
- Section - C contains 10 questions including one Case Study. Attempt any 8 questions from Q. No. 41 to 50.
- There is only one correct option for every Multiple Choice Question (MCQ). Marks will not be awarded for answering more than one option.
- There is no negative marking.
Differential of log [log (log x5)] w.r.t x is ______.
`5/(xlog(x^5)log(logx^5)`
`5/(xlog(logx^5)`
`(5x^4)/(log(x^5)log(logx^5)`
`(5x^4)/(logx^5log(logx^5)`
Chapter:
The number of all possible matrices of order 2 × 3 with each entry 1 or 2 is ______.
16
6
64
24
Chapter:
A function f : R `rightarrow` R is defined as f(x) = x3 + 1. Then the function has ______.
no minimum value
no maximum value
both maximum and minimum values
neither maximum value nor minimum value
Chapter:
If sin y = x cos(a + y), then `dx/dy` is ______.
`cosa/(cos^2(a + y))`
`(-cosa)/(cos^2(a + y))`
`cosa/(sin^2y)`
`(-cosa)/(sin^2y)`
Chapter:
The points on the curve `x^2/9 + y^2/25 = 1`, where tangent is parallel to X-axis are ______.
(±5, 0)
(0, ±5)
(0, ±3)
(±3, 0)
Chapter:
Three points P(2x, x + 3), Q(0, x) and R(x + 3, x + 6) are collinear, then x is equal to ______.
0
2
3
1
Chapter:
The principal value of `cos^-1(1/2) + sin^-1(-1/sqrt(2))` is ______.
`π/12`
π
`π/3`
`π/6`
Chapter:
If (x2 + y2)2 = xy, then `dy/dx` is ______.
`(y + 4x(x^2 + y^2))/(4y(x^2 + y^2) - x)`
`(y - 4x(x^2 + y^2))/(x + 4(x^2 + y^2))`
`(y - 4x(x^2 + y^2))/(4y(x^2 + y^2) - x)`
`(4y(x^2 + y^2) - x)/(y - 4x(x^2 + y^2))`
Chapter:
If a matrix A is both symmetric and skew symmetric, then A is necessarily a ______.
Diagonal matrix
Zero square matrix
Square matrix
Identity matrix
Chapter:
Let set X = {1, 2, 3} and a relation R is defined in X as: R = {(1, 3), (2, 2), (3, 2)}, then minimum ordered pairs which should be added in relation R to make it reflexive and symmetric are ______.
{(1, 1), (2, 3), (1, 2)}
{(3, 3), (3, 1), (1, 2)}
{(1, 1), (3, 3), (3, 1), (2, 3)}
{(1, 1), (3, 3), (3, 1), (1, 2)}
Chapter:
A Linear Programming Problem is as follows:
Minimise z = 2x + y
Subject to the constraints x ≥ 3, x ≤ 9, y ≥ 0
x – y ≥ 0, x + y ≤ 14
The feasible region has
5 corner points including (0, 0) and (9, 5)
5 corner points including (7, 7) and (3, 3)
5 corner points including (14, 0) and (9, 0)
5 corner points including (3, 6) and (9, 5)
Chapter:
The function `f(x) = {{:((e^(3x) - e^(-5x))/x",", if x ≠ 0),(k, if x = 0):}` is continuous at x = 0 for the value of k, as ______.
3
5
2
8
Chapter:
If Cij denotes the cofactor of element Pij of the matrix P = `[(1, -1, 2),(0, 2, -3),(3, 2, 4)]`, then the value of C31.C23 is ______.
5
24
–24
–5
Chapter:
The function y = x2e–x is decreasing in the interval
(0, 2)
(2, ∞)
(–∞, 0)
(–∞, 0) ∪ (2, ∞)
Chapter:
If R = {(x, y) : x, y ∈ Z, x2 + y2 ≤ 4} is a relation on Z, then the domain of R is ______.
{0, 1, 2}
{0, −1, −2}
{−2, −1, 0, 1, 2}
{−1, 0, 1}
None of these
Chapter:
The system of linear equations
5x + ky = 5,
3x + 3y = 5;
will be consistent if
k ≠ – 3
k = –5
k = 5
k ≠ 5
Chapter:
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The equation of tangent to the curve y(1 + x2) = 2 – x, where it crosses x-axis is ______.
x + 5y = 2
x – 5y = 2
5x – y = 2
5x + y = 2
Chapter: [0.06] Applications of Derivatives
`[(3c + 6, a - d),(a + d, 2 - 3b)] = [(12, 2),(-8, -4)]` are equal, then value of ab – cd is ______.
4
16
–4
–16
Chapter:
The principal value of `tan^-1(tan (9π)/8)` is ______.
`π/8`
`(3π)/8`
`- π/8`
`- (3π)/8`
Chapter:
For two matrices P = `[(3, 4),(-1, 2),(0, 1)]` and QT = `[(-1, 2, 1),(1, 2, 3)]` P – Q is ______.
`[(2, 3),(-3, 0),(0, -3)]`
`[(4, 3),(-3, 0),(-1, -2)]`
`[(4, 3),(-0, -3),(-1, -2)]`
`[(2, 3),(0, -3),(0, -3)]`
Chapter:
The function f(x) = 2x3 – 15x2 + 36x + 6 is increasing in the interval
(–∞, 2) ∪ (3, ∞)
(–∞, 2)
(–∞, 2] ∪ [3, ∞)
[3, ∞)
Chapter:
If x = 2 cos θ – cos 2θ and y = 2 sin θ – sin 2θ, then `dy/dx` is ______.
`(cosθ + cos2θ)/(sinθ - sin2θ_`
`(cosθ - cos2θ)/(sin2θ - sinθ)`
`(cosθ - cos2θ)/(sinθ - sin2θ)`
`(cos2θ - cosθ)/(sin2θ + sinθ)`
Chapter:
What is the domain of the function cos–1 (2x – 3)?
[–1, 1]
(1, 2)
(–1, 1)
[1, 2]
Chapter:
A matrix A = [aij]3 × 3 is defined by
`a_(ij) = {{:(2i + 3j",", i < j),(5",", i = j),(3i - 2j",", i > j):}`
The number of elements in A which are more than 5, is ______.
3
4
5
6
Chapter:
If a function f defined by
`f(x) = {{:((k cos x)/(π - 2x)",", if x ≠π/2),(3, if x = π/2):}`
is continuous at `x = π/2`, then the value of k is ______.
2
3
6
–6
Chapter:
For the matrix `X = [(0, 1, 1),(1, 0, 1),(1, 1, 0)], (X^2 - X)` is ______.
2I
3I
I
5I
Chapter:
Let X = {x2 : x ∈ N} and the function f : N `rightarrow` X is defined by f(x) = x2, x ∈ N. Then this function is ______.
injective only
not bijective
surjective only
bijective
Chapter:
The corner points of the feasible region for a Linear Programming problem are P(0, 5), Q(1, 5), R(4, 2) and S(12, 0). The minimum value of the objective function Z = 2x + 5y is at the point ______.
P
Q
R
S
Chapter:
The equation of the normal to the curve ay2 = x3 at the point (am2, am3) is ______.
2y – 3mx + am3 = 0
2x + 3my 3am4 – am2 = 0
2x + 3my + 3am4 – 2am = 0
2x + 3my – 3am4 – 2am2 = 0
Chapter:
If A is a square matrix of order 3 and |A| = –5, then |adj A| is ______.
125
–25
25
±25
Chapter:
The simplest form of `tan^-1 [(sqrt(1 + x) - sqrt(1 - x))/(sqrt(1 + x) + sqrt(1 - x))]` is ______.
`π/4 - x/2`
`π/4 + x/2`
`π/4 - 1/2 cos^-1x`
`π/4 + 1/2 cos^-1x`
Chapter:
If for the matrix A = `[(α, -2),(-2, α)]`, |A3| = 125, then the value of α is ______.
±3
–3
±1
1
Chapter:
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If y = sin (msin–1x), then which one of the following equations is true?
`(1 - x^2) (d^2y)/(dx^2) + x dy/dx + m^2y = 0`
`(1 - x^2) (d^2y)/(dx^2) - x dy/dx + m^2y = 0`
`(1 + x^2) (d^2y)/(dx^2) - x dy/dx - m^2y = 0`
`(1 + x^2) (d^2y)/(dx^2) + x dy/dx - m^2x = 0`
Chapter:
The principal value of `[tan^-1sqrt(3) - cot^-1(-sqrt(3))]` is ______.
π
`-π/2`
0
`2sqrt(3)`
Chapter:
The maximum value of `(1/x)^x` is ______.
e
ex
`"e"^(1/"e")`
`(1/"e")^(1/"e")`
ee
Chapter: [0.06] Applications of Derivatives
Let matrix X = [xij] is given by X = `[(1, -1, 2),(3, 4, -5),(2, -1, 3)]`. Then the matrix Y = [mij], where mij = Minor of xij, is ______.
`[(7, -5, -3),(19, 1, -11),(-11, 1, 7)]`
`[(7, -19, 11),(5, -1, -1),(3, 11, 7)]`
`[(7, 19, -11),(-3, 11, 7),(-2, -1, -1)]`
`[(7, 19, -11),(-1, -1, 1),(-3, -11, 7)]`
Chapter:
A function f : R `rightarrow` R defined by f(x) = 2 + x2 is ______.
not one-one
one-one
not onto
neither one-one nor onto
Chapter:
A Linear Programming Problem is as follows:
Maximise / Minimise objective function
Z = 2x – y + 5
Subject to the constraints
3x + 4y ≤ 60
x + 3y ≤ 30
x ≥ 0, y ≥ 0
If the corner points of the feasible region are A(0, 10), B(12, 6), C(20, 0) and O(0, 0), then which of the following is true?
Maximum value of Z is 40
Minimum value of Z is –5
Difference of maximum and minimum values of Z is 35
At two corner points, value of Z are equal
Chapter:
If x = –4 is a root of `|(x, 2, 3),(1, x, 1),(3, 2, x)| = 0`, then the sum of the other two roots is ______.
4
–3
2
5
Chapter:
The absolute maximum value of the function `f(x) = 4x - 1/2 x^2` in the interval `[-2, 9/2]` is ______.
8
9
6
10
Chapter:
In a sphere of radius r, a right circular cone of height h having maximum curved surface area is inscribed. The expression for the square of curved surface of cone is ______.
2π2rh(2rh + h2)
π2hr(2rh + h2)
2π2r(2rh2 – h3)
2π2r2(2rh – h2)
Chapter:
The corner points of the feasible region determined by a set of constraints (linear inequalities) are P(0, 5), Q(3, 5), R(5, 0) and S(4, 1) and the objective function is Z = ax + 2by where a, b > 0. The condition on a and b such that the maximum Z occurs at Q and S is ______.
a – 5b = 0
a – 3b = 0
a – 2b = 0
a – 8b = 0
Chapter:
If curves y2 = 4x and xy = c cut at right angles, then the value of c is ______.
`4sqrt(2)`
8
`2sqrt(2)`
`-4sqrt(2)`
Chapter:
The inverse of the matrix X = `[(2, 0, 0),(0, 3, 0),(0, 0, 4)]` is ______.
`24[(1//2, 0, 0),(0, 1//3, 0),(0, 0, 1//4)]`
`1/24[(1, 0, 0),(0, 1, 0),(0, 0, 1)]`
`1/24[(2, 0, 0),(0, 3, 0),(0, 0, 4)]`
`[(1//2, 0, 0),(0, 1//3, 0),(0, 0, 1//4)]`
Chapter:
For an L.P.P. the objective function is Z = 4x + 3y and the feasible region determined by a set of constraints (linear inequations) is shown in the graph.
Which one of the following statements is true?
Maximum value of Z is at R.
Maximum value of Z is at Q.
Value of Z at R is less than the value at P.
Value of Z at Q is less than the value at R.
Chapter:
|
Case Study In a residential society comprising of 100 houses, there were 60 children between the ages of 10-15 years. They were inspired by their teachers to start composting to ensure that biodegradable waste is recycled. For this purpose, instead of each child doing it for only his/her house, children convinced the Residents welfare association to do it as a society initiative. For this they identified a square area in the local park. Local authorities charged amount of ₹ 50 per square metre for space so that there is no misuse of the space and Resident welfare association takes it seriously. Association hired a labourer for digging out 250 m3 and he charged ₹ 400 X (depth)2. Association will like to have minimum cost.
|
Let side of square plot is x m and its depth is h metres, then cost C for the pit is
`50/h + 400h^2`
`12500/h + 400h^2`
`250/h + h^2`
`250/h + 400h^2`
Chapter:
|
Case Study In a residential society comprising of 100 houses, there were 60 children between the ages of 10-15 years. They were inspired by their teachers to start composting to ensure that biodegradable waste is recycled. For this purpose, instead of each child doing it for only his/her house, children convinced the Residents welfare association to do it as a society initiative. For this they identified a square area in the local park. Local authorities charged amount of ₹ 50 per square metre for space so that there is no misuse of the space and Resident welfare association takes it seriously. Association hired a labourer for digging out 250 m3 and he charged ₹ 400 X (depth)2. Association will like to have minimum cost.
|
Value of h (in m) for which `(dC)/(dh) = 0` is
1.5
2
2.5
3
Chapter:
|
Case Study In a residential society comprising of 100 houses, there were 60 children between the ages of 10-15 years. They were inspired by their teachers to start composting to ensure that biodegradable waste is recycled. For this purpose, instead of each child doing it for only his/her house, children convinced the Residents welfare association to do it as a society initiative. For this they identified a square area in the local park. Local authorities charged amount of ₹ 50 per square metre for space so that there is no misuse of the space and Resident welfare association takes it seriously. Association hired a labourer for digging out 250 m3 and he charged ₹ 400 X (depth)2. Association will like to have minimum cost.
|
`(d^2C)/(dh^2)` is given by
`25000/h^3 + 800`
`500/h^3 + 800`
`100/h^3 + 800`
`500/h^3 + 2`
Chapter:
|
Case Study In a residential society comprising of 100 houses, there were 60 children between the ages of 10-15 years. They were inspired by their teachers to start composting to ensure that biodegradable waste is recycled. For this purpose, instead of each child doing it for only his/her house, children convinced the Residents welfare association to do it as a society initiative. For this they identified a square area in the local park. Local authorities charged amount of ₹ 50 per square metre for space so that there is no misuse of the space and Resident welfare association takes it seriously. Association hired a labourer for digging out 250 m3 and he charged ₹ 400 X (depth)2. Association will like to have minimum cost.
|
Value of x (in m) for minimum cost is
5
`10sqrt(5/3)`
`5sqrt(5)`
10
Chapter:
|
Case Study In a residential society comprising of 100 houses, there were 60 children between the ages of 10-15 years. They were inspired by their teachers to start composting to ensure that biodegradable waste is recycled. For this purpose, instead of each child doing it for only his/her house, children convinced the Residents welfare association to do it as a society initiative. For this they identified a square area in the local park. Local authorities charged amount of ₹ 50 per square metre for space so that there is no misuse of the space and Resident welfare association takes it seriously. Association hired a labourer for digging out 250 m3 and he charged ₹ 400 X (depth)2. Association will like to have minimum cost.
|
Total minimum cost of digging the pit (in ₹) is
4,100
7,500
7,850
3,220
Chapter:
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