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QUESTION: 1

Suppose N is a normal subgroup of a group G. Which one of the following is true?

Solution:

QUESTION: 2

Let y(x) = u(x) sin sin x + v(x) cos x be a solution of the differential equation y” + y = sec x.

Then u(x) is

Solution:

QUESTION: 3

Let a, b, c, d be distinct non- zero real numbers with a + b = c + d. Then an eigenvalue of the

matrix is

Solution:

QUESTION: 4

Let S be a nonempty subset of R. If S is a finite union of disjoint bounded intervals, then which one of the following is true?

Solution:

QUESTION: 5

Let {x_{n}} be a convergent sequence of real numbers.

for n ≥ 1, then which one of the following is the limit of this sequence?

Solution:

QUESTION: 6

The volume of the portion of the solid cylinder x^{2} + y^{2} ≤ 2 bounded above by the surface z = x^{2} + y^{2} and bounded below by the xy- plane is

Solution:

QUESTION: 7

Let f: R→R be a differentiable function with f(0) = 0. If for all x ∈ R, 1 < f^{'}(x) < 2, then which one of the following statements is true on (0, ∝)?

Solution:

QUESTION: 8

If an integral curve of the differential equation passes through (0, 0) and (α, 1),

then α is equal to

Solution:

QUESTION: 9

An integrating factor of the differential equation

Solution:

QUESTION: 10

Let A be a nonempty subset of R Let I(A) denote the set of interior points of A. Then I(A) can be

Solution:

QUESTION: 11

Let S_{3} be the group of permutations of three distinct symbols. The direct sum _{ }has an element of order

Solution:

QUESTION: 12

The orthogonal trajectories of the family of curves y = C_{1}x^{3} are

Solution:

QUESTION: 13

Let G be a nonabelian group. Let α ∈ G have order 4 and let β ∈ G have order 3. Then the order of the element αβ in G

Solution:

QUESTION: 14

Let S be the bounded surface of the cylinder x^{2} + y^{2} = 1 cut by the planes z = 0 and z = 1 + x. Then the value of the surface integral is equal to

Solution:

QUESTION: 15

Suppose that the dependent variables z and w are functions of the independent variables x and y, defined by the equations f(x, y, z, w) = 0 and g(x, y, z, w) = 0, where f_{z}g_{w} – f_{w}g_{z} = 1.

Which one of the following is correct

Solution:

QUESTION: 16

Solution:

QUESTION: 17

Solution:

QUESTION: 18

Let P_{2}(R) be the vector space of polynomials in x of degree at most 2 with real coefficients. Let M_{2}(R) be the vector space of 2 × 2 real matrices. If a linear transformation is defined as then

Solution:

QUESTION: 19

Let B_{1} = {(1, 2), (2, –1)} and B_{2} = {(1, 0), (0, 1} be ordered bases of R^{2} . If T : R^{2} → R^{2}^{ } is a linear transformation such that

then T(5, 5) is equal to

Solution:

QUESTION: 20

Which one of the following statements is FALSE?

Solution:

QUESTION: 21

Let f : R → R be a strictly increasing continuous function. If {a_{n}} is a sequence in [0, 1], then the sequence {f(a_{n})} is

Solution:

QUESTION: 22

Which one of the following statements is true for the series

Solution:

QUESTION: 23

Solution:

QUESTION: 24

If y(t) is a solution of the differential equation y” + 4y = 2e^{t}, then

is equal to

Solution:

QUESTION: 25

For what real values of x and y, does the integral attain its maximum?

Solution:

QUESTION: 26

The area of the planar region bounded by the curves x = 6y^{2} – 2 and x = 2y^{2} is

Solution:

QUESTION: 27

For n ≥ 2, let f_{n}: R → R be given by f_{n}(x) = x^{n} sin x. Then at x = 0, f_{n} has a

Solution:

QUESTION: 28

Solution:

QUESTION: 29

Let G and H be nonempty subsets of R , where G is connected and G U H is not connected.

Which one of the following statements is true for all such G and H ?

Solution:

QUESTION: 30

Then the value of

Solution:

*Multiple options can be correct

QUESTION: 31

Let f: R → R be a function defined by

In which of the following interval(s), f takes the value 1?

Solution:

*Multiple options can be correct

QUESTION: 32

Which of the following statements is (are) true?

Solution:

*Multiple options can be correct

QUESTION: 33

Which of the following conditions implies (imply) the convergence of a sequence {x_{n}} of real numbers?

Solution:

*Multiple options can be correct

QUESTION: 34

If C is the curve of intersection of the surfaces x2 + y2 = 1 and y + z = 2, then which of the following is (are) equal to

Solution:

*Multiple options can be correct

QUESTION: 35

Let V be the set of 2 × 2 matrices with complex entries such that a_{11} + a_{22} = 0. Let W be the set of matrices in V with . Then, under usual matrix addition and scalar multiplication, which of the following is (are) true?

Solution:

*Multiple options can be correct

QUESTION: 36

The initial value problem has

Solution:

*Multiple options can be correct

QUESTION: 37

Which of the following statements is (are) true on the interval (0, π/2) ?

Solution:

*Multiple options can be correct

QUESTION: 38

Let f: R^{2} → R be defined by

At (0,0)

Solution:

*Multiple options can be correct

QUESTION: 39

Let f, g : [0, 1] → [0, 1] be functions. Let R(f) and R(g) be the ranges of f and g, respectively.

Which of the following statements is (are) true?

Solution:

*Multiple options can be correct

QUESTION: 40

Let f: (–1, 1) → R be the function defined by Then

Solution:

*Answer can only contain numeric values

QUESTION: 41

Let C be the straight line segment from P(0, π) to (4, π/2) in the xy- plane. Then the value of

Solution:

*Answer can only contain numeric values

QUESTION: 42

Let S be the portion of the surface bounded by the planes x = 0, x = 2, y = 0, and y = 3. The surface area of S, correct upto three decimal places, is ____________

Solution:

*Answer can only contain numeric values

QUESTION: 43

The number of distinct normal subgroups of S_{3} is _____

Solution:

*Answer can only contain numeric values

QUESTION: 44

If the directional derivative of f at (0, 0) exists along the direction cos αi sin α j , where sin a ≠ 0, then the value of cot a is _________

Solution:

*Answer can only contain numeric values

QUESTION: 45

The maximum rate of change of f at (π/4 , 0, 1) correct upto three decimal places, is ________

Solution:

*Answer can only contain numeric values

QUESTION: 46

If the power series

converges for |x| < c and diverges for |x| > c, then the value of c, correct upto three decimal places, is--

Solution:

*Answer can only contain numeric values

QUESTION: 47

If 5^{2015} ≡ n. ( mod 11) and n ∈ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, then n is equal to _________

Solution:

*Answer can only contain numeric values

QUESTION: 48

If the set is linearly dependent in the vector space of all 2 × 2 matrices with real entries, then x is equal to ___

Solution:

*Answer can only contain numeric values

QUESTION: 49

let f: R → R be defined by

The number of points at which f is continuous, is ______________

Solution:

*Answer can only contain numeric values

QUESTION: 50

Let f: (0, 1) → R be a continuously differentiable function such that f' has finitely many zeros in (0, 1) and f^{' }changes sign at exactly two of these points. Then for any y ∈ R , the maximum number of solutions to f(x) = y in (0, 1) is ______________

Solution:

*Answer can only contain numeric values

QUESTION: 51

Let R be the planar region bounded by the lines x = 0, y = 0 and the curve x^{2} + y^{2} = 4, in the first quadrant. Let C be the boundary of R, oriented counter- clockwise. Then the value of

is ----

Solution:

*Answer can only contain numeric values

QUESTION: 52

Suppose G is a cyclic group and σ, τ ∈ G are such that order(σ) = 12 and order (τ) = 21. Then the order of the smallest group containing σ and τ is ______________

Solution:

*Answer can only contain numeric values

QUESTION: 53

Solution:

*Answer can only contain numeric values

QUESTION: 54

Let M_{2}(R) be the vector space of 2 × 2 real matrices. Let V be a subspace of M_{2}(R) defined by

Then the dimension is ____

Solution:

*Answer can only contain numeric values

QUESTION: 55

correct upto three decimal places, is _________

Solution:

We now change the order of integration. Then the integral equals

*Answer can only contain numeric values

QUESTION: 56

The coefficient of in the Taylor series expansion of the function

about the point π/4 correct upto three decimal places, is ______

Solution:

*Answer can only contain numeric values

QUESTION: 57

as the power series expansion then a_{5 } correct upto three decimal places, is equal to _

Solution:

We have

a_{1}x + a_{2}x_{2} + a_{3}x_{3} + a_{4}x_{4} + a_{5}x_{5} +..

On differentiation, it gives, by Leibnitz rule on the left, and termwise differentiation on the right,

e_{x} = 1 +

cosx

Then

The coefficient of t^{4} in (*) is 5a_{5}.

The coefficient of x^{4} in (**) is 1/2 + 1/4!, i.e., 1/2 + 1/24 = 13/24.

Comparing these coefficients (note that symbols x and t play the same role), we get

13/24 = 5a_{5}

Which given a_{5} = 1/5. 13/24 ≈ 0.108.

*Answer can only contain numeric values

QUESTION: 58

Let ℓ be the length of the portion of the curve x = x(y) between the lines y = 1 and y = 3, where x(y) satisfies

The value of ℓ, correct upto three decimal places, is ___________

Solution:

*Answer can only contain numeric values

QUESTION: 59

Solution:

*Answer can only contain numeric values

QUESTION: 60

Let P and Q be two real matrices of size 4 × 6 and 5 × 4, respectively. If rank(Q) = 4 and rank(QP) = 2, then rank(P) is equal to ______

Solution:

### CBSE Past Year Paper Session (2015), Math Class 12

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