Math - 2019 Past Year Paper


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Math - 2019 Past Year Paper - Question 1

Let a1 = b1 = 0, and for each n ≥ 2, let an and bn be real numbers given by

Then which one of the following is TRUE about the sequences {an} and {bn}? 

Math - 2019 Past Year Paper - Question 2

Let  Let V be the subspace of  defined by 

Then the dimension of V is

Math - 2019 Past Year Paper - Question 3

Let be a twice differentiable function. Define

f(x,y,z) = g(x2 + y2 - 2z2).

Detailed Solution for Math - 2019 Past Year Paper - Question 3

 Correct Answer :- A

Explanation : f(x,y,z) = g(x2 + y2 - 2z2).

df'/dx = g'(x2 + y2 - 2z2) (2x)

df"/dx” = g"(x2 + y2 - 2z2) (4x2) + g'(x2 + y2 - 2z2)*2.........(1)

df/dy = g'(x2 + y2 - 2z2) (2y)

df"/dy” = g"(x2 + y2 - 2z2) (4y2) + g'(x2 + y2 - 2z2)*2.........(2)

df'/dz = g'(x2 + y2 - 2z2) (2y)

df"/dz” = g"(x2 + y2 - 2z2) (4z2) + g'(x2 + y2 - 2z2)*2.........(3)

Adding (1), (2) and (3)

g"(x2 + y2 - 2z2)(4x2 + 4y2 + 16z2) + g'(x2+ y2 - 2z2) (2 + 2 - 4)

= 4(x2 + y2 + 4z2) g"(x2 + y2 - 2z2)

Math - 2019 Past Year Paper - Question 4

be sequences of positive real numbers such that nan < bn < n2an for
all n > 2. If the radius of convergence of the power series then the power series

Detailed Solution for Math - 2019 Past Year Paper - Question 4

If the radius of convergenceis 4, Then

Math - 2019 Past Year Paper - Question 5

Let S be the set of all limit points of the set  be the set of all positive 
rational numbers. Then

Math - 2019 Past Year Paper - Question 6

If xhyk is an integrating factor of the differential equation y(1 + xy) dx + x(1 — xy) dy = 0, then the ordered pair (h, k) is equal to

Detailed Solution for Math - 2019 Past Year Paper - Question 6

If xh yk is an I.F. of differential equation, Then given equation become exact differential equation.
xh uk+1 (1 + xy)dx + xh +1 yk (1 – xy) dy = 0
So

=> (k + 1)ykxh + (k + 2)xh + 1yk + 1
= (h + 1)xhyk - (h + 1)xk + 1yk + 1
Comparing coefficients of both the sides, we have
h – k = 0
h + k – 4
⇒ h = –2, k = – 2

Math - 2019 Past Year Paper - Question 7

If y(x) = λe2x + eβx, β ≠ 2, is a solution of the differential equation

satisfying dy/dx (0) = 5, then y(0) is equal to

Detailed Solution for Math - 2019 Past Year Paper - Question 7

Math - 2019 Past Year Paper - Question 8

The equation of the tangent plane to the surface  at the point (2, 0, 1) is

Detailed Solution for Math - 2019 Past Year Paper - Question 8

equation of the tangent plane at point (2, 0, 1) is (x – 2) fx(2, 0, 1) + (y – 0) fy(2, 0, 1) + (z – 1) fz(0, 0, 1) = 0

Here fx(2,0,1) = 3, fy(2,0,1) =0, fz(2,0,1) = 4
so, we have
(x – 2).3 + (y – 0).0 + (z – 1)4 = 0
⇒ 3x + 4z = 10
which is required tangent plane.

Math - 2019 Past Year Paper - Question 9

The value of the integral is

Detailed Solution for Math - 2019 Past Year Paper - Question 9

By the change of order of integration



Let t = (1 – x)2
dt = –2(1 – x) dx

Math - 2019 Past Year Paper - Question 10

The area of the surface generated by rotating the curve x = y3, 0 ≤ y ≤ 1, about the y-axis, is

Math - 2019 Past Year Paper - Question 11

Let H and K be subgroups of  If the order of H is 24 and the order of K is 36, then the order of the subgroup H ∩ K is

Math - 2019 Past Year Paper - Question 12

Let P be a 4 × 4 matrix with entries from the set of rational numbers. If  with  is a root of the characteristic polynomial of P and I is the 4 × 4 identity matrix, then

Detailed Solution for Math - 2019 Past Year Paper - Question 12

Given P is a 4 × 4 matrix with rational entries.
Let characteristic polynomial of P is
hp(x) = x4 + ax3 + bx2 + cx + d,   ...(i)
where a, b, c and d are rational.
Since √2 + i is a root of (1), Then √2 − i is also a root of (1)
⇒ (x2 − 22x + 3) is a factor of (1)
Since x2 + 3 − 2√2x is factor of (1), Then
x2 + 3 + 2√2x is also a factor of (1)
⇒ (x2 + 3 − 2√2x) (x2 + 3 + 2√2x) is a factor of (1)
x4 + 6x2 + 9 – 8x2.
= x4 – 2x2 + 9 ...(ii)
From (1) & (2), we have
P4 – 2P2 + 9I = 0
⇒ P4 = 2P2 – 9I

Math - 2019 Past Year Paper - Question 13

The set  as a subset of

Math - 2019 Past Year Paper - Question 14

The set as a subset of

Math - 2019 Past Year Paper - Question 15

For −1 < x < 1, the sum of the power series 

Math - 2019 Past Year Paper - Question 16

Let f(x) = (ln x)2 , x > 0. Then

Math - 2019 Past Year Paper - Question 17

Let be a differentiable function such that f'(x) > f(x) for all and f(0) = 1. Then f( 1) lies in the interval

Detailed Solution for Math - 2019 Past Year Paper - Question 17

Let be defined as
f(x) = eax, a > 1 x ← R
f′(x) = aeax.
f(o) = aeao = 1 & f′(x) = aeax > eax = f(x) ∀ x ∈ R.
hence f(1) = ea
.
e < ea < ∞
⇒ f(1). lies in the interval (e, ∞)

Math - 2019 Past Year Paper - Question 18

For which one of the following values of k, the equation 2x3 + 3x2 − 12x − k = 0 has three distinct real roots?

Detailed Solution for Math - 2019 Past Year Paper - Question 18

Let g(x) = 2x3 + 3x2 – 12x.
For the roots of g(x) = 0 x(2x2 + 3x – 12) = 0

for the graph of g(x):
g′(x) = 6x2 + 6x – 12
g′(x) = 0 ⇒ x = 1, –2
function has maximum at x = –2 & minimum at x = 1 in between the roots of g(x) = 0. The graph of function shown as.
g(–2) = 2(–2)3 + 3(–2)2 – 12(–2) = 20
If y = K intersect at three distinct points with g(x), then K should be less than 20, So K = 16

Math - 2019 Past Year Paper - Question 19

Which one of the following series is divergent?

Detailed Solution for Math - 2019 Past Year Paper - Question 19

For option (b);

Let vn = 1 / n3
Then,

Asis convergent, sois convergent. 
For option (c);

Let vn = 1 / n3
Then
Thenis also convergent.
For option (d)





Sois also convergent.

Hence option (a) is divergent
(By Cauchy-condensation test)

Math - 2019 Past Year Paper - Question 20

Let S be the family of orthogonal trajectories of the family of curves 2x2 + y2 = k, for  and k > 0. If   passes through the point (1, 2), then passes through

Detailed Solution for Math - 2019 Past Year Paper - Question 20

For the orthogonal trajectories of the family of curves,

integrating, we have
loge y = loge√x + log c1 ⇒ y = c1√x
The curve passes through the point (1, 2), then c1 = 2
Now, orthogonal trajectories in y = 2√x
At point (2, 2√2) satisfy the given condition.
 

Math - 2019 Past Year Paper - Question 21

Let x, x + ex and 1 + x + ex be solutions of a linear second order ordinary differential equation with constant coefficients. If y(x) is the solution of the same equation satisfying y(0) = 3 and y'(0) = 4, then y(1) is equal to

Detailed Solution for Math - 2019 Past Year Paper - Question 21

Here x, x + ex are two linear independent
solution. so general solution can be written as
y = c1x + c2 (x + ex)

= (c1+ c2) x + c2ex

y′= (c1+ c2) + c2ex
.
y(0) = c2 = 3
y′ (0) = 4 = c1+ 2c2

⇒ c1 = 4 – 6 = –2
so, y(x) (–2 + 3) x + 3e= 3ex+ x

y(1) = 3e + 1

Math - 2019 Past Year Paper - Question 22

The function f(x,y) = x3 + 2xy + y3 has a saddle point at

Detailed Solution for Math - 2019 Past Year Paper - Question 22

Here
fx = 3x2 + 2y ⇒ fxx = 6x, fxy = 2
fy = 2x + 3y2 ⇒  fyy = 6y
then fxx fyy – (fxy)2 = 36xy – 4
so at the point (0, 0) fxx fyy – (fxy)2 < 0
⇒ (0, 0) is a saddle point.

Math - 2019 Past Year Paper - Question 23

The area of the part of the surface of the paraboloid x+ y2 + z = 8 lying inside the cylinder x2 + y2 = 4 is

Detailed Solution for Math - 2019 Past Year Paper - Question 23

Math - 2019 Past Year Paper - Question 24

be the circle (x − 1)2 + y2 = 1, oriented counterclockwise. Then the value of the line integral

 is

Detailed Solution for Math - 2019 Past Year Paper - Question 24

By Green's Theorem,




Now by using polar x = rcosθ, y = r sinθ. r = 0 to 2 cosθ, θ = 0 to 2θ.
we obtain the solution.

Math - 2019 Past Year Paper - Question 25

 be the curve of intersection of the plane x + y + z = 1 and the cylinder x2 + y2 = 1. Then the value of

 is

Detailed Solution for Math - 2019 Past Year Paper - Question 25

By stoke theorem,

Here








 

Math - 2019 Past Year Paper - Question 26

The tangent line to the curve of intersection of the surface x2 + y2 − z = 0 and the plane x + y = 3 at the point (1, 1, 2) passes through

Math - 2019 Past Year Paper - Question 27

The set of eigenvalues of which one of the following matrices is NOT equal to the set of eigenvalues of 

Math - 2019 Past Year Paper - Question 28

For , define 

Then, at (0, 0), the function f is

Math - 2019 Past Year Paper - Question 29

Let {an} be a sequence of positive real numbers such that a1 = 1,  for all n ≥ 1.

Then the sum of the series  lies in the interval

Math - 2019 Past Year Paper - Question 30

Let {an} be a sequence of positive real numbers. The series  converges if the series

Detailed Solution for Math - 2019 Past Year Paper - Question 30

Here, {an} is a sequence of a positive real number.
The series converges if the series converges.
 

*Multiple options can be correct
Math - 2019 Past Year Paper - Question 31

Let G be a noncyclic group of order 4. Consider the statements I and II:
I. There is NO injective (one-one) homomorphism from
II. There is NO surjective (onto) homomorphism from
Then

*Multiple options can be correct
Math - 2019 Past Year Paper - Question 32

Let G be a nonabelian group, y ∈ G, and let the maps f, g, h from G to itself be defined by  f(x) = yxy-1, g(x) = x-1 and h = g ° g.
Then

*Multiple options can be correct
Math - 2019 Past Year Paper - Question 33

Let S and T be linear transformations from a finite dimensional vector space V to itself such that S(T(v)) = 0 for all v ∈ V. Then

*Multiple options can be correct
Math - 2019 Past Year Paper - Question 34

Let  be differentiable vector fields and let g be a differentiable scalar function. Then

*Multiple options can be correct
Math - 2019 Past Year Paper - Question 35

Consider the intervals S = (0, 2] and T = [1, 3). Let S° and T° be the sets of interior points of S and T, respectively. Then the set of interior points of S\T is equal to

Math - 2019 Past Year Paper - Question 36

Let {an} be the sequence given by

Then which of the following statements is/are TRUE about the subsequences {a6n−1} and {a6n+4}?

*Multiple options can be correct
Math - 2019 Past Year Paper - Question 37

Let  f(x) = cos(|π - x|) + (π - n) sin |x| and g(x) = x2  If h(x) = f(g(x)), then

*Multiple options can be correct
Math - 2019 Past Year Paper - Question 38

 be given by f(x) = (sin x)π − π sin x + π.

Then which of the following statements is/are TRUE?

*Multiple options can be correct
Math - 2019 Past Year Paper - Question 39

Let 

Then at (0, 0),

*Multiple options can be correct
Math - 2019 Past Year Paper - Question 40

Let {an} be the sequence of real numbers such that a1 = 1 and an+1 = an + a2n for all n > 1.

Then

*Answer can only contain numeric values
Math - 2019 Past Year Paper - Question 41

Let x be the 100-cycle (1 2 3 ⋯ 100) and let y be the transposition (49 50) in the permutation group S100. Then the order of xy is ______


Detailed Solution for Math - 2019 Past Year Paper - Question 41


= (1 2 ------ 48, 50, 51 ------ 100) So the order of xy is 99

*Answer can only contain numeric values
Math - 2019 Past Year Paper - Question 42

Let W1 and W2 be subspaces of the real vector space defined by
W1 = {(x1,x2, ...,x100) : xi = 0 if i is divisible by 4},
W2 = { (x1;x2, ....x100) : xi = 0 if i is divisible by 5}.
Then the dimension of W1 ∩ W2 is ____


*Answer can only contain numeric values
Math - 2019 Past Year Paper - Question 43

Consider the following system of three linear equations in four unknowns x1, x2, x3 and x4
x1 + x2 + x3 + x4 = 4,
x1 + 2x2 + 3x3 + 4x4 = 5,
x1 + 3x2 + 5x3 + kx4 = 5.
If the system has no solutions, then k = _____


*Answer can only contain numeric values
Math - 2019 Past Year Paper - Question 44

Let and  be the ellipse


oriented counter clockwise. Then the value of (round off to 2 decimal places) is_______________


*Answer can only contain numeric values
Math - 2019 Past Year Paper - Question 45

The coefficient of   in the Taylor series expansion of the function  about x = π/2, is ____________


*Answer can only contain numeric values
Math - 2019 Past Year Paper - Question 46

Let be given by

Then

max {f(x): x ∈ [0,1]} - min [f(x):x ∈ [0,1]}
is ___________


*Answer can only contain numeric values
Math - 2019 Past Year Paper - Question 47

If  then g′(1) = _______


*Answer can only contain numeric values
Math - 2019 Past Year Paper - Question 48

Let 

Then the directional derivative of f at (0, 0) in the direction of 


*Answer can only contain numeric values
Math - 2019 Past Year Paper - Question 49

The value of the integral  (round off to 2 decimal places) is ___________


*Answer can only contain numeric values
Math - 2019 Past Year Paper - Question 50

The volume of the solid bounded by the surfaces x = 1 - y2 and x = y2 - 1, and the planes z = 0 and z = 2 (round off to 2 decimal places) is    


*Answer can only contain numeric values
Math - 2019 Past Year Paper - Question 51

The volume of the solid of revolution of the loop of the curve y2 = x4 (x + 2) about the x-axis (round off to 2 decimal places) is ___________


*Answer can only contain numeric values
Math - 2019 Past Year Paper - Question 52

The greatest lower bound of the set  (round off to 2 decimal places) is ______________


*Answer can only contain numeric values
Math - 2019 Past Year Paper - Question 53

Let G =  : n < 55, gcd(n, 55) = 1} be the group under multiplication modulo 55. Let x ∈ G be such that x2 = 26 and x > 30. Then x is equal to _______


*Answer can only contain numeric values
Math - 2019 Past Year Paper - Question 54

The number of critical points of the function  is  ___________


*Answer can only contain numeric values
Math - 2019 Past Year Paper - Question 55

The number of elements in the set {x ∈ S3: x4 = e), where e is the identity element of the permutation group S3, is    


*Answer can only contain numeric values
Math - 2019 Past Year Paper - Question 56

If  is an eigenvector corresponding to a real eigenvalue of the matrix  then z − y is equal to__________


*Answer can only contain numeric values
Math - 2019 Past Year Paper - Question 57

Let M and N be any two 4 × 4 matrices with integer entries satisfying

Then the maximum value of det(M) + det(N) is ___________


*Answer can only contain numeric values
Math - 2019 Past Year Paper - Question 58

Let M be a 3 × 3 matrix with real entries such that M2 = M + 2I, where I denotes the 3 × 3 identity matrix. If α, β and γ are eigenvalues of M such that αβγ = -4, then α+β+γ is equal to ______


*Answer can only contain numeric values
Math - 2019 Past Year Paper - Question 59

Let y(x) = xv(x) be a solution of the differential equation If v(0) = 0 and v(1) = 1, then v(-2) is equal to ______


*Answer can only contain numeric values
Math - 2019 Past Year Paper - Question 60

If y(x) is the solution of the initial value problem  then y(ln 2) is (round off to 2 decimal places) equal to ____________


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