GATE 2018 Mathematics Main exam Practice Online with solution

Category: GATE Mathematics 2018

Question 1:

The principal value of

{(- 1)}^{(- 2 i / π)}

A	$e^{2}$	B	$e^{2 i}$
C	$e^{- 2 i}$	D	$e^{- 2}$

Category: GATE Mathematics 2018

Question 2:

Let f : C → C be an entire function with f(0) = 1, f(1) = 2 and f' (0) = 0. If there existsM > 0 such that |

|f'' (z)| \leq m

for all

z \in C

, then f(2) =

2+ 5i

5+ 2i

Category: GATE Mathematics 2018

Question 3:

In the Laurent series expansion of

f (z) = \frac{1}{z (z - 1)}

valid for

|z - 1| > 1

, the coefficient of

\frac{1}{z - 1}

-2

-1

Category: GATE Mathematics 2018

Question 4:

Let X and Y be metric spaces, and let f: X → Y be a continuous map. For any subset S of X, which one of the following statements is true?

If S is open, then f(S) is open

If S is connected, then f(S) is connected

If S is closed, then f(S) is closed

If S is bounded, then f(S) is bounded

Category: GATE Mathematics 2018

Question 5:

Let

p_{n}

(x) be the polynomial solution of the differential equation

\frac{d}{d x} [(1 - x^{2}) y'] + n (n + 1) y = 0

with $p_{n}$ (1) = 1 for n = 1, 2, 3, . . . . If

\frac{d}{d x} [p_{n + 2} (x) - p_{n} (x)] = a_{n} p_{n + 1} (x),

then

a_{n}

2n + 1

2n + 2

2n + 3

Category: GATE Mathematics 2018

Question 6:

In the permutation group

S_{6}

, the number of elements of order 8 is

Category: GATE Mathematics 2018

Question 7:

Let R be a commutative ring with 1 (unity) which is not a field. Let I ⊂ R be a proper ideal such that every element of R not in I is invertible in R. Then the number of maximal ideals of R is

infinite

Category: GATE Mathematics 2018

Question 8:

Let f: R → R be a twice continuously differentiable function. The order of convergence of the secant method for finding the root of the equation f(x) = 0 is

A	$\frac{1 + \sqrt{5}}{2}$	B	$\frac{2}{1 + \sqrt{5}}$
C	$\frac{1 + \sqrt{5}}{3}$	D	$\frac{3}{1 + \sqrt{5}}$

Category: GATE Mathematics 2018

Question 9:

The Cauchy problem

υ υ_{x} + y u_{y} = x

, with

u (x, 1) = 2 x

when solved using its characteristic equations with an independent variable t, is found to admit of a solution in the form

x = \frac{3}{2} s e^{t} - \frac{1}{2} s e^{- t}, y = e^{t}, u = f (s, t) .

Then f(s, t) =

A	$\frac{3}{2} s e^{t} + \frac{1}{2} s e^{- t}$	B	$\frac{1}{2} s e^{t} + \frac{3}{2} s e^{- t}$
C	$\frac{1}{2} s e^{t} - \frac{3}{2} s e^{- t}$	D	$\frac{3}{2} s e^{t} - \frac{1}{2} s e^{- t}$

Category: GATE Mathematics 2018

Question 10:

An urn contains four balls, each ball having equal probability of being white or black. Three black balls are added to the urn. The probability that five balls in the urn are black is

2/7

3/8

1/2

5/7

Category: GATE Mathematics 2018

Question 11:

The general solution of the differential equation

x y' = y + \sqrt{x^{2} + y^{2}} f o r x > 0

is given by (with an arbitrary positive constant k)

A	$k y^{2} = x + \sqrt{x^{2} + y^{2}}$	B	$k x^{2} = x + \sqrt{x^{2} + y^{2}}$
C	$k x^{2} = y + \sqrt{x^{2} + y^{2}}$	D	$k y^{2} = y + \sqrt{x^{2} + y^{2}}$

Category: GATE Mathematics 2018

Question 12:

For a linear programming problem, which one of the following statements is FALSE?

If a constraint is an equality, then the corresponding dual variable is unrestricted in sign

Both primal and its dual can be infeasible

If primal is unbounded, then its dual is infeasible

Even if both primal and dual are feasible, the optimal values of the primal and the dual can differ

Category: GATE Mathematics 2018

Question 13:

Let

A = [\begin{matrix} a & 2 f & 0 \\ 2 f & b & 3 f \\ 0 & 3 f & c \end{matrix}]

, where a, b, c, f are real numbers and

f \neq 0

. The geometric multiplicity of the largest eigenvalue of A equals _________________.

0 to 1

1 to 1

1 to 2

2 to 2

Category: GATE Mathematics 2018

Question 14:

Consider the subspaces

W_{1} = \{(x_{1}, x_{2}, x_{3}) \in R^{3} : x_{1} = x_{2} + 2 x_{3}\} W_{2} = \{(x_{1}, x_{2}, x_{3}) \in R^{3} : x_{1} = 3 x_{2} + 2 x_{3}\}

R^{3}

. Then the dimension of

W_{1} + W_{2}

equals ____________ .

1 to 1

2 to 2

3 to 3

4 to 4

Category: GATE Mathematics 2018

Question 15:

Let V be the real vector space of all polynomials of degree less than or equal to 2 with real coefficients. Let T : V → V be the linear transformation given by

T = (p) = 2 q + p' f o r p \in V

where p' is the derivative of p. Then the number of nonzero entries in the Jordan canonical form of a matrix of T equals.

0 to 0

1 to 1

3 to 3

5 to 5

Category: GATE Mathematics 2018

Question 16:

Let I = [2, 3), J be the set of all rational numbers in the interval [4, 6], K be the Cantor (ternary) set, and let L = {7 + x : x ∈ K}. Then the Lebesgue measure of the set I ∪ J ∪ L equals ___________.

1 to 1

1 to 2

2 to 2

2 to 3

Category: GATE Mathematics 2018

Question 17:

Let

u (x, y, z) = x^{2} - 2 y + 4 z^{2}

for

(x, y, z) \in R^{3} .

. Then the directional derivative of u in the direction

\frac{3}{5} \hat{i} - \frac{4}{5} \hat{k}

at the point (5, 1, 0) is _________________.

4 to 4

5 to 5

6 to 6

7 to 7

Category: GATE Mathematics 2018

Question 18:

If the Laplace transform of y(t) is given by Y (s) = L(y(t)) =

\frac{5}{2 (s - 1)} - \frac{2}{s - 2} + \frac{1}{2 (s - 3)}

, then

y (0) + y' (0) =

______________ .

0 to 0

-1 to -1

-1 to 1

1 to 1

Category: GATE Mathematics 2018

Question 19:

The number of regular singular points of the differential equation

[{(x - 1)}^{2} \sin z] y'' + [\cos x \sin (x - 1)] y' + (x - 1) y = 0

in the interval

[0, \frac{π}{2}]

is equal to _______________.

1 to 1

2 to 2

3 to 3

4 to 4

Category: GATE Mathematics 2018

Question 20:

Let F be a field with

7^{6}

elements and let K be a subfield of F with 49 elements. Then the dimension of F as a vector space over K is__________________.

1 to 1

2 to 2

3 to 3

4 to 4

Category: GATE Mathematics 2018

Question 21:

Let C([0, 1]) be the real vector space of all continuous real valued functions on [0, 1], and let T be the linear operator on C([0, 1]) given by

(T f) (x) = \int_{0}^{1} \sin (x + y) f (y) d y, x \in [0, 1] .

Then the dimension of the range space of T equals ___________________.

0 to 2

-2 to 2

-2 to -2

2 to 2

Category: GATE Mathematics 2018

Question 22:

Let a ∈ (−1, 1) be such that the quadrature rule

\int_{- 1}^{1} f (x) d x ≃ f (- a) + f (a)

is exact for all polynomials of degree less than or equal to 3. Then

3 a^{2}

= ____________ .

1 to 1

2 to 2

3 to 3

4 to 4

Category: GATE Mathematics 2018

Question 23:

Let X and Y have joint probability density function given by

f x, y (x . y) = \{\begin{cases} 2, 0 \leq x \leq 1 - y, 0 \leq y \leq 1 \\ 0, o t h e r w i s e . \end{cases}

If fY denotes the marginal probability density function of Y, then fY (1/2) = ______________.

0 to 0

1 to 1

2 to 2

3 to 3

Category: GATE Mathematics 2018

Question 24:

Let the cumulative distribution function of the random variable X be given by

F_{x} (x) = \{\begin{cases} 0, x < 0, \\ x, 0 \leq x < 1 / 2, \\ (1 + x) / 2, 1 / 2 \leq x < 1, \\ 1, x \geq 1 . \end{cases}

Then P(X = 1/2) = _________.

0.00 to 0.00

0.25 to 0.25

0.50 to 0.50

0.75 to 0.75

Category: GATE Mathematics 2018

Question 25:

Let

\{X_{j}\}

be a sequence of independent Bernoulli random variables with

P (X_{j} = 1) 1 / 4

and let

Y_{n} = \frac{1}{n} \sum_{j = 1}^{n} X_{j}^{2}

. Then

Y_{n}

converges, in probability, to ____________.

0.05 to 0.05

0.15 to 0.15

0.25 to 0.25

0.35 to 0.35

Category: GATE Mathematics 2018

Question 26:

Let Γ be the circle given by

z = 4 e^{i θ}

, where θ varies from 0 to 2π. Then

\oint_{r} \frac{e^{z}}{z^{2} - 2 z} d z =

A	$2 πi (e^{2} - 1)$	B	$πi (1 - e^{2})$
C	$πi (e^{2} - 1)$	D	$2 πi (1 - e^{2})$

Category: GATE Mathematics 2018

Question 27:

The image of the half plane Re(z) + Im(z) > 0 under the map

w = \frac{z - 1}{z + i}

is given by

A	$R e (w) > 0$	B	$I m (w) > 0$
C	$\|w\| > 1$	D	$\|w\| < 1$

Category: GATE Mathematics 2018

Question 28:

Let

D \subset R^{2}

denote the closed disc with the center at the origin and radius 2. Then

\int \int_{D} e^{- (x^{2} + y^{2})} d x d y =

A	$π (1 - e^{- 4})$	B	$\frac{π}{2} (1 - e^{- 4})$
C	$π (1 - e^{- 2})$	D	$\frac{π}{2} (1 - e^{- 2})$

Category: GATE Mathematics 2018

Question 29:

Consider the polynomial

p (X) = X^{4} + 4

in the ring Q[X] of polynomials in the variable X with coefficients in the field Q of rational numbers. Then

the set of zeros of p(X) in C forms a group under multiplication

p(X) is reducible in the ring Q[X]

the splitting field of p(X) has degree 3 over Q

the splitting field of p(X) has degree 4 over Q

Category: GATE Mathematics 2018

Question 30:

Which one of the following statements is true?

Every group of order 12 has a non-trivial proper normal subgroup

Some group of order 12 does not have a non-trivial proper normal subgroup

Every group of order 12 has a subgroup of order 6

Every group of order 12 has an element of order 12

Category: GATE Mathematics 2018

Question 31:

For an odd prime p, consider the ring

Z [\sqrt{- p}] = \{a + b \sqrt{- p} : a, b \in Z\} \subset C

. Then the element 2 in

Z [\sqrt{- p}]

a unit

a square

a prime

irreducible

Category: GATE Mathematics 2018

Question 32:

Consider the following two statements: P: The matrix

[\begin{matrix} 0 & 5 \\ 0 & 7 \end{matrix}]

has infinitely many LU factorizations, where L is lower triangular with each diagonal entry 1 and U is upper triangular. Q: The matrix

[\begin{matrix} 0 & 0 \\ 2 & 5 \end{matrix}]

has no LU factorization, where L is lower triangular with each diagonal entry 1 and U is upper triangular. Then which one of the following options is correct?

P is TRUE and Q is FALSE

Both P and Q are TRUE

P is FALSE and Q is TRUE

Both P and Q are FALSE

Category: GATE Mathematics 2018

Question 33:

If the characteristic curves of the partial differential equation

x u_{x x} + 2 x^{2} u_{x y} = u_{x} - 1

are

μ (x, y) = c_{1}

and

v (x, y) = c_{2}

, where

c_{1} a n d c_{2}

are constants, then

A	$μ (x, y) = x^{2} - y, v (x, y) = y$	B	$μ (x, y) = x^{2} + y, v (x, y) = y$
C	$μ (x, y) = x^{2} + y, v (x, y) = x^{2}$	D	$μ (x, y) = x^{2} - y, v (x, y) = x^{2}$

Category: GATE Mathematics 2018

Question 34:

Let f: X → Y be a continuous map from a Hausdorff topological space X to a metric space Y. Consider the following two statements: P: f is a closed map and the inverse image

f^{- 1} (y) = \{x \in X : f (x) = y\}

is compact for each y ∈ Y. Q: For every compact subset

K \subset Y

, the inverse image

f^{- 1} (K)

is a compact subset of X. Which one of the following is true?

Q implies P but P does NOT imply Q

P implies Q but Q does NOT imply P

P and Q are equivalent

neither P implies Q nor Q implies P

Category: GATE Mathematics 2018

Question 35:

Let X denote

R^{2}

endowed with the usual topology. Let Y denote R endowed with the co-finite topology. If Z is the product topological space Y × Y, then

the topology of X is the same as the topology of Z

the topology of X is strictly coarser (weaker) than that of Z

the topology of Z is strictly coarser (weaker) than that of X

the topology of X cannot be compared with that of Z

Category: GATE Mathematics 2018

Question 36:

Consider R n with the usual topology for n = 1, 2, 3. Each of the following options gives topological spaces X and Y with respective induced topologies. In which option is X homeomorphic to Y?

A	$X = \{(x, y, z) \in R^{3} : x^{2} + y^{2} = 1\}, Y = \{(x, y, z) \in R^{3} : z = 0, x^{2} + y^{2} \neq 0\},$
B	$X = \{(x, y) \in R^{2} : y = \sin (1 / x), 0 < x \leq 1\} \cup \{x, y\} \in R^{2} : x = 0, - 1 \emptyset y \emptyset 1, Y = [0, 1] \subset R$
C	$X = \{(x, y) \in R^{2} : y = x \sin (1 / x), 0 < x \leq 1\}, Y = [0, 1] \subset R$
D	$X = \{(x, y, z) \in R^{3} : x^{2} + y^{2} = 1\}, Y = \{(x, y, z) \in R^{3} : x^{2} + y^{2} = z^{2} \neq 0\}$

Category: GATE Mathematics 2018

Question 37:

Let

\{X_{i}\}

be a sequence of independent Poisson(λ) variables and let

W_{n} = \frac{1}{n} \sum_{i = 1}^{n} X_{i}

. Then the limiting distribution of

\sqrt{n} (W_{n} - λ)

is the normal distribution with zero mean and variance given by

A	1	B	$\sqrt{λ}$
C	$λ$	D	$λ^{2}$

Category: GATE Mathematics 2018

Question 38:

Let

X_{1}, X_{2}

, . . . ,

X_{n}

be independent and identically distributed random variables with probability density function given by

f x (x; θ) = \{\begin{cases} θ e^{- (x - 1)}, x \geq 1, \\ 0 o t h e r w i s e . \end{cases}

Also, let

X = \frac{1}{n} \sum_{i = 1}^{n} X_{1}

. Then the maximum likelihood estimator of θ is

A	$1 / x$	B	$(1 / x) - 1$
C	$1 / (x - 1)$	D	$X$

Category: GATE Mathematics 2018

Question 39:

9 Consider the Linear Programming Problem (LPP Maximize $α x_{1} + x_{2}$ Subject to $2 x_{1} + x_{2} \leq 6, - x_{1} + x_{2} \leq 1, x_{1} + x_{2} \leq 4, x_{1} \geq 0, x_{2} \geq 0,$ where α is a constant. If (3, 0) is the only optimal solution, then

A	$α < - 2$	B	$- 2 < α < 1$
C	$1 < α < 2$	D	$α < 2$

Category: GATE Mathematics 2018

Question 40:

Let

M_{2} R

be the vector space of all 2 × 2 real matrices over the field R. Define the linear transformation

S : M_{2} (R) \to M_{2} (R)

S (X) = 2 X + X^{T}

, where

X^{T}

denotes the transpose of the matrix X. Then the trace of S equals ____________.

5 to 5

10 to 10

15 to 15

20 to 20

Category: GATE Mathematics 2018

Question 41:

Consider R3 with the usual inner product. If d is the distance from (1, 1, 1) to the subspacespan{(1, 1, 0), (0, 1, 1)} of

R^{3}

, then

3 d^{2}

= __________.

0 to 0

1 to 1

2 to 2

3 to 3

Category: GATE Mathematics 2018

Question 42:

Consider the matrix

A = I_{g} - 2 u^{T} u

with

u = \frac{1}{3}

[1, 1, 1, 1, 1, 1, 1, 1, 1], where I9 is the 9 × 9 identity matrix and uT is the transpose of u. If λ and µ are two distinct eigenvalues of A, then |λ − µ| = _________.

0 to 1

1 to 1

0 to 2

2 to 2

Category: GATE Mathematics 2018

Question 43:

Let

f (z) = z^{3} e^{z^{2}}

for z ∈ C and let Γ be the circle

z = e^{i θ}

, where θ varies from 0 to 4π. Then

\frac{1}{2 πri} \oint_{r} \frac{f' (z)}{f (z)} d z

2 to 2

4 to 4

6 to 6

8 to 8

Category: GATE Mathematics 2018

Question 44:

Let S be the surface of the solid

V = \{(x, y, z) : 0 \leq x \leq 1, 0 \leq y \leq 2, 0 \leq z \leq 3\} .

Let nˆ denote the unit outward normal to S and let Then the surface integral

\int \int_{s} \vec{F} . \hat{n} d S

equals ____________.

17 to 17

18 to 18

19 to 19

20 to 20

Category: GATE Mathematics 2018

Question 45:

Let A be a 3 × 3 matrix with real entries. If three solutions of the linear system of differential equations x˙(t) = Ax(t) are given by

[\begin{matrix} e^{t} & - & e^{2 t} \\ - e^{t} & + & e^{2 t} \\ e^{t} & + & e^{2 t} \end{matrix}], [\begin{matrix} - e^{2 t} & - & e^{- t} \\ e^{2 t} & + & e^{- t} \\ e^{2 t} & + & e^{- t} \end{matrix}] a n d [\begin{matrix} e^{- t} & - & 2 e^{t} \\ e^{- t} & + & 2 e^{t} \\ - e^{- t} & + & 2 e^{t} \end{matrix}],

then the sum of the diagonal entries of A is equal to __________.

1 to 1

2 to 2

3 to 3

4 to 4

Category: GATE Mathematics 2018

Question 46:

y_{1} (x) e^{- x^{2}}

y is a solution of the differential equation

x y'' + α y' + β x^{3} y = 0

for some real numbers α and β, then αβ = ________.

1 to 1

2 to 2

3 to 3

4 to 4

Category: GATE Mathematics 2018

Question 47:

Let

L^{2} ([0, 1])

be the Hilbert space of all real-valued square-integrable functions on [0, 1] with the usual inner product. Let φ be the linear functional on

L^{2} ([0, 1])

defined by

ϕ (f) = \int_{1 / 4}^{3 / 4} 3 \sqrt{2} f d μ,

where µ denotes the Lebesgue measure on [0, 1]. Then

||ϕ||

= ________________ .

0 to 3

1 to 3

2 to 3

3 to 3

Category: GATE Mathematics 2018

Question 48:

Let U be an orthonormal set in a Hilbert space H and let x ∈ H be such that

||x|| = 2

. Consider the set

E = \{u \in U : |<x, u>| \geq \frac{1}{4}\} .

Then the maximum possible number of elements in E is ___________ .

64 to 64

65 to 65

66 to 66

67 to 67

Category: GATE Mathematics 2018

Question 49:

If p(x) = 2 − (x + 1) + x(x + 1) − βx(x + 1)(x − α) interpolates the points (x, y) in the table then α + β = ________.

2 to 2

3 to 3

4 to 4

5 to 5

Category: GATE Mathematics 2018

Question 50:

If sin

\sin (πx) a_{0} + \sum_{n = 1}^{\infty} a_{n} \cos (n πx)

for 0 < x < 1, then

(a_{0} + a_{1}) π

= ________________.

0 to 0

1 to 1

2 to 2

3 to 3

Category: GATE Mathematics 2018

Question 51:

For n = 1, 2, . . . , let

f_{n} (x) = \frac{2 n x^{n - 1}}{1 + x}, x \in [0, 1]

. Then

\lim_{n \to \infty} \int_{0}^{1} f_{n} (x) d x

= _________.

0 to 0

-1 to 1

-1 to -1

1 to 1

Category: GATE Mathematics 2018

Question 52:

Let

X_{1}, X_{2}, X_{3}, X_{4}

be independent exponential random variables with mean 1, 1/2, 1/3, 1/4, respectively. Then Y = min

(X_{1}, X_{2}, X_{3}, X_{4})

has an exponential distribution with a mean equal to

0.1 to 0.1

0.1 to 0.2

0.1 to 0.3

0.1 to 0.4

Category: GATE Mathematics 2018

Question 53:

Let X be the number of heads in 4 tosses of a fair coin by Person 1 and let Y be the number of heads in 4 tosses of a fair coin by Person 2. Assume that all the tosses are independent. Then the value of P(X = Y ) correct up to three decimal places is

0.270 to 0.272

0.272 to 0.274

0.274 to 0.276

0.276 to 0.278

Category: GATE Mathematics 2018

Question 54:

Let

X_{1} a n d X_{2}

be independent geometric random variables with the same probability mass function given by

P (X = K) = p {(1 - p)}^{k - 1}, k = 1, 2

, . . .. Then the value of

P (X_{1} = 2 | X_{1} + X_{2} = 4)

correct up to three decimal places is _________.

0.330 to 0.334

0.331 to 0.334

0.332 to 0.334

0.333 to 0.334

Category: GATE Mathematics 2018

Question 55:

A certain commodity is produced by the manufacturing plants

P_{1} a n d P

whose capacities are 6 and 5 units, respectively. The commodity is shipped to markets

M_{1}, M_{2}, M_{3} a n d M_{3}

whose requirements are 1, 2, 3, and 5 units, respectively. The transportation cost per unit from plant Pi to market Mj is as follows: Then the optimal cost of transportation is__________.

54 to 54

55 to 55

56 to 56

57 to 57

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