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GATE (Graduate Aptitude Test in Engineering) Instrumentation Engineering is an online examination conducted to get admission in M-Tech (Masters in Technology), ME (Master of Engineering ) and PhD at IITs (Indian Institutes of Technology), NITs (National Institutes of Technology), IIITs (Indian Institutes of Information Technology) and other universities in India. GATE exam is jointly conducted by IIT Bombay, IIT Delhi, IIT Guwahati, IIT Kanpur, IIT Kharagpur, IIT Madras, IIT Roorkee and Indian Institute for Science, Bangalore (IISC Bangalore). The exam is divided into two different parts General Aptitude (GA) and Candidate’s Selected Subject (Core Subjects). GATE Instrumentation Engineering online mock tests help candidates in preparing for the GATE exam. By attempting GATE online mock tests candidates get familiar with the pattern of actual question paper. BrPaper provides all GATE question papers completely free. Here you can practice free online mock tests of complete exam of GATE Instrumentation Engineering. We advise candidates to solve the online GATE Instrumentation Engineering mock tests to get good marks in the exam. Start practicing InstrumentationGATE 2020 Mock tests here online for free

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Category: GATE Instrumentation Engineering 2020 General Aptitude

Question 1:

He is known for his unscrupulous ways. He always sheds ____________ tears to deceive people.

fox\'s

crocodile\'s

crocodile

fox

Category: GATE Instrumentation Engineering 2020 General Aptitude

Question 2:

Jofra Archer, the England fast bowler, is ___________ than accurate.

more fast

faster

less faster

more faster

Category: GATE Instrumentation Engineering 2020 General Aptitude

Question 3:

Select the word that fits the analogy: Build: Building: :Grow : __________

Grown

Grew

Growth

Growed

Category: GATE Instrumentation Engineering 2020 General Aptitude

Question 4:

I do not think you know the case well enough to have opinions. Having said that, I agree with your other point. What does that phrase "having said that" mean in the given text?

as opposed to what I have said

despite what I have said

in addition to what I have said

contrary to what I have said

Category: GATE Instrumentation Engineering 2020 General Aptitude

Question 5:

Define [x] as the greatest integer less than or equal to x, for each x

\in (- \infty, \infty)

. If y=[x], then area under y for x

\in

[1, 4] is _________.

Category: GATE Instrumentation Engineering 2020 General Aptitude

Question 6:

Crowdfunding deals with mobilization of funds for a project from a large number of people, who would be willing to invest smaller amounts through web-based platforms in the project. Based on the above paragraph, which of the following is correct about crowd funding?

Funds raised through unwilling contributions on web-based platforms.

Funds raised through large contributions on web-based platforms.

Funds raised through coerced contributions on web-based platforms.

Funds raised through voluntary contributions on web-based platforms.

Category: GATE Instrumentation Engineering 2020 General Aptitude

Question 7:

P, Q, R, and S are to be uniquely coded using

α and β

. If P is coded as

αα and Q as αβ

, then R and S, respectively, can be coded as _________.

A	$βα and αβ$	B	$ββ and αα$
C	$αβ and ββ$	D	$βα and ββ$

Category: GATE Instrumentation Engineering 2020 General Aptitude

Question 8:

The sum of the first n terms in the sequence 8, 88, 888, 8888,... is _____________.

A	$\frac{81}{80} (10^{n} - 1) + \frac{9}{8} n$	B	$\frac{81}{80} (10^{n} - 1) - \frac{9}{8} n$
C	$\frac{81}{80} (10^{n} - 1) + \frac{8}{9} n$	D	$\frac{81}{80} (10^{n} - 1) - \frac{8}{9} n$

Category: GATE Instrumentation Engineering 2020 General Aptitude

Question 9:

Select the graph that schematically represents BOTH

y = x^{m} and y = x^{1 / m}

properly in the interval

0 \leq x \leq 1,

for integer values m, where

m > 1 .

Category: GATE Instrumentation Engineering 2020 General Aptitude

Question 10:

The bar graph shows the data of the students who appeared and passed in an examination for four schools P, Q, R and S. The average of success rates (in percentage) of these four schools is ___________.

58.5%

58.8%

59.0%

59.3%

Category: GATE Mathematics 2020

Question 11:

Suppose that

I_{1} and I_{2}

are topologies on X induced by metrics

d_{1} and d_{2}

, respectively,

I_{1} \underline{\subset} I_{2}

. Then which of the following is TRUE?

A	If a sequence converges in $(X, d_{2})$ then it convergers in $(X, d_{1})$
B	If a sequence converges in $(X, d_{1})$ then it convergers in $(X, d_{2})$
C	Every open ball in $(X, d_{1})$ is an open ball in $(X, d_{2})$
D	The map $x \mapsto x from (X, d_{1}) to (X, d_{2})$ is continuous

Category: GATE Mathematics 2020

Question 12:

Let

D = [- 1, 1] \times [- 1, 1] .

If the function

f : D \to ℝ

is defined by

f (x, y) = \{\begin{cases} \frac{x^{2} - y^{2}}{{(x^{2} + y^{2})}^{2}}, (x, y) \neq (0, 0) \\ 0, (x, y) = (0, 0) \end{cases},

A	f is continuous at (0, 0)
B	both the first-order partial derivatives of f exist at (0, 0)
C	$\int \int_{D} {\|f (x, y)\|}^{\frac{1}{2}} dx dy$ is finite
D	$\int \int_{D} \|f (x, y)\| dx dy$ is finite

Category: GATE Mathematics 2020

Question 13:

The initial value problem

y' = y^{\frac{3}{5}}, y (0) = b

has

a unique solution if b=0

no solution if b=1

infinitely many solutions if b=2

a unique solution if b=1

Category: GATE Mathematics 2020

Question 14:

Consider the following statements: I:

\log (|z|)

is harmonic on

ℂ \ \{0\}

II.

\log (|z|)

has a harmonic conjugate on

ℂ \ \{0\}

Then

both I and II are true

I is the true but II is false

I is the false but II is true

both I and II are false

Category: GATE Mathematics 2020

Question 15:

Let G and H be defined by

G = ℂ \ \{z = x + iy \in ℂ : x ⩽ 0, y = 0\}, H = ℂ \ \{z = x + iy \in ℂ : x \in ℤ, x \leq 0, y = 0\} .

Suppose

f : G \to ℂ and g : H \to ℂ

are analytic functions. Consider the following statements: I:

\int_{y} f dz

is independent of paths y in G joining -i and i II:

\int_{y} g dz

is independent of paths y in H joining -i and i Then

both I and II are true

I is true but II is false

I is false but II is true

both I and II are false

Category: GATE Mathematics 2020

Question 16:

Let

f (z) = e^{1 / z}, z \in ℂ \ \{0\}

and let, for

n \in ℕ,

R_{n} = \{z = x + iy \in ℂ : |x| < \frac{1}{n}, |y| < \frac{1}{n}\} \ \{0\} .

A	$f (R_{n + 1}) \neq f (R_{n})$	B	$f (R_{n}) \ f (R_{n + 1}) = f (R_{n} \ R_{n + 1})$
C	$f (\cap_{n = 1}^{\infty} R_{n}) = \cap_{n = 1}^{\infty} f (R_{n})$	D	$f (R_{n}) = f (R_{n + 1})$

A and B

B and C

C and D

D and A

Category: GATE Mathematics 2020

Question 17:

Suppose that

U = ℝ^{2} \ \{(x, y) \in ℝ^{2} : x, y \in ℚ\}, U = ℝ^{2} \ \{(x, y) \in ℝ^{2} : x > 0, y \frac{1}{x}\}

Then with respect to the Euclidean metric on

ℝ^{2},

both U and V are disconnected

u is disconnected but V is connected

u is connected but V is disconnected

both U and V are connected

Category: GATE Mathematics 2020

Question 18:

If (D1) and (D2) denote the dual problems of the linear programming problems (P1) and (P2), respectively, where (P1): minimize

x_{1} - 2 x_{2} subject to - x_{1} + x_{2} = 10, x_{1}, x_{2} \geq 0,

(P2): minimize

x_{1} - 2 x_{2} subject to - x_{1} + x_{2} = 10, x_{1} - x_{2} = 10, x_{1}, x_{2} \geq 0,

then

both (D1) and (D2) are infeasible

(P2) is infeasible and (D2) is feasible

(D1) is infeasible and (D2) is feasible but unbounded

(P1) is feasible but unbounded and (D1) is feasible

Category: GATE Mathematics 2020

Question 19:

(4, 0) and (0, - \frac{1}{2})

are critical points of the function

f (x, y) = 5 - (α - β) x^{2} + {βy}^{2} + (α + 1) y^{3} + x^{3},

where

α, β \in ℝ,

then

A	$(4, - \frac{1}{2})$ is a point of local maxima of f
B	$(4, - \frac{1}{2})$ is a saddle point of f
D	$α = 4, β = 2$
C	$(4, - \frac{1}{2})$ is a point of local maxima of f

Category: GATE Mathematics 2020

Question 20:

Consider the iterative scheme

x_{n} = \frac{x_{n - 1}}{2} + \frac{3}{x_{n - 1}}, n \geq 1 .

with initial point

x_{0} > 0 .

Then the sequence

\{x_{n}\}

A	converges only if $x_{0} > 1$
B	converges only if $x_{0} < 3$
C	converges for any $x_{0}$
D	does not converge for any $x_{0}$

Category: GATE Mathematics 2020

Question 21:

Let C[0, 1] denote the space of all real-valued continuous functions on [0,1] equipped with the supremum norm

| | . | |_{\infty} .

Let

T : C [0, 1] \to C [0, 1]

be the linear operator defined by

T (f) (x) = \int_{0}^{x} e^{- y} f (y) dy .

Then

A	$\| \| T \| \| = 1$	B	I-T is not invertible
C	T is subjective	D	$\| \| I + T \| \| = 1 + \| \| T \| \|$

Category: GATE Mathematics 2020

Question 22:

Suppose that M is a

5 \times 5

matrix with real entries and p(x) = diet (xI-M). Then

A	$p (0) = \det (M)$
B	every eigenvalue of M is real if p(1) + p(2)= 0 = p(2) + p(3)
C	$M^{- 1}$ is necessarily a polynomial in M of degree 4 if M is invertible
D	M is not invertible if $M^{2} - 2 M = 0$

Category: GATE Mathematics 2020

Question 23:

Let C [0, 1] denote the space of all real-valued continuous function on [0, 1] equipped with the supremum norm

| | . | |_{\infty} .

Let

f \in C [0, 1]

be such that

|f (x) - f (y)| \leq M |x - y|, for all x, y \in [0, 1] and for some M > 0 .

For

n \in ℕ, let f_{n} (x) = f (x^{1 + \frac{1}{n}}) . If S = \{f_{n} : n \in ℕ\},

then

the closure of S is compact

S is closed and bounded

S is bounded but not totally bounded

S is compact

Category: GATE Mathematics 2020

Question 24:

Let

K : ℝ \times (0, \infty) \to ℝ

be a function such that the solution of the initial value problem

\frac{\partial u}{\partial t} = \frac{\partial^{2} u}{\partial x^{2}}, u (x, 0) = f (x), x \in ℝ, t > 0,

is given by

u (x, t) = \int_{ℝ} k (x - y, t) f (y) dy

for all bounded continuous functions f. Then the value of

\int_{ℝ} k (x, t) dx

is ____________

0 to 0

1 to 1

2 to 2

3 to 3

Category: GATE Mathematics 2020

Question 25:

The number of cyclic subgroups of the quaternion group

Q_{8} = < a, b | a^{4} = 1, a^{2} = b^{2}, ba = a^{3} b >

is ______________.

1 to 1

3 to 3

5 to 5

7 to 7

Category: GATE Mathematics 2020

Question 26:

The number of elements of order 3 in the symmetric group

S_{6}

is __________

80 to 80

81 to 81

82 to 82

83 to 83

Category: GATE Mathematics 2020

Question 27:

Let F be the field with 4096 elements. The number of proper subfields of F is ____________

5 to 5

5 to 6

5 to 7

5 to 8

Category: GATE Mathematics 2020

Question 28:

(x_{1}^{*}, x_{2}^{*})

is an optimal solution of the linear programming problem,

minimize x_{1} + 2 x_{2} subject to 4 x_{1} - x_{2} \geq 8 2 x_{1} + x_{2} \geq 10 - x_{1} + x_{2} \geq 0

and

(λ_{1}^{*}, λ_{2}^{*}, λ_{3}^{*})

is an optimal solution of its dual problem, then

\sum_{i = 1}^{2} x_{i}^{*} + \sum_{j = 1}^{3} λ_{j}^{*}

is equal to ____________ (correct up to one decimal place)

3.5 to 3.5

4.5 to 4.5

5.5 to 5.5

6.5 to 6.5

Category: GATE Mathematics 2020

Question 29:

Let a, b, c,

\in ℝ

be such that the quadrature rule

\int_{- 1}^{1} f (x) dx \approx af (- 1) + bf (0) + cf' (1)

is exact for all polynomials of degree less than or equal to 2. Then b is equal to __________ (rounded off to two decimal places)

1.40 to 1.50

1.50 to 1.60

1.60 to 1.70

1.70 to 1.80

Category: GATE Mathematics 2020

Question 30:

Let

f (x) = x^{4}

and let p(x) be the interpolating polynomial of f at nodes 1, 2, and 3. Then p(0) is equal to _______

35 to 35

36 to 36

37 to 37

38 to 38

Category: GATE Mathematics 2020

Question 31:

For

n \geq 2,

define the sequence

\{x_{n}\}

x_{n} = \frac{1}{2 π} \int_{0}^{\frac{π}{2}} \tan^{\frac{1}{n}} t dt .

Then the sequence

\{x_{n}\}

converges to _______________ (correct up to two decimal places)

0.25 to 0.25

0.45 to 0.45

0.65 to 0.65

0.85 to 0.85

Category: GATE Mathematics 2020

Question 32:

Let

L^{2} [0, 10] = \{f : [0, 10] \to ℝ : f is Lebesgue measurable and \int_{0}^{10} f^{2} dx < \infty\}

equipped with the norm

| | f | | = {(\int_{0}^{10} f^{2} dx)}^{\frac{1}{2}}

and let T be the linear functional on

L^{2} [0, 10]

given by

T (f) = \int_{0}^{2} f (x) dx - \int_{3}^{10} f (x) dx .

Then ||T|| is equal to ____________

0 to 0

1 to 1

2 to 2

3 to 3

Category: GATE Mathematics 2020

Question 33:

\{x_{13}, x_{22}, x_{23} = 10, x_{31}, x_{32}, x_{34}\}

is the set of basic variables of a balanced transportation problem seeking to minimize the cost of transportation from origins to destinations, where the cost matrix is,

and

λ, μ \in ℝ,

then

x_{32}

is equal to _________

0 to 5

1 to 5

3 to 5

5 to 5

Category: GATE Mathematics 2020

Question 34:

Let

ℤ_{225}

be the ring of integers modulo 225. If x is the number of prime and y is the number of nontrivial units in

ℤ_{225}

, then x + y is equal to ____

121 to 121

122 to 122

123 to 123

124 to 124

Category: GATE Mathematics 2020

Question 35:

Let u(x, t) be the solution of

\frac{\partial^{2} u}{\partial t} - \frac{\partial^{2} u}{\partial x^{2}} = 0, u (x, 0) = f (x), \frac{\partial u}{\partial t} (x, 0) = 0, x \in ℝ, t > 0,

where is a twice continuously differentiable function. If f(-2) =4, f (0) =0, and u (2, 2) = 8, then value of u (1, 3) is _________

9 to 9

10 to 10

11 to 11

12 to 12

Category: GATE Mathematics 2020

Question 36:

Let

{[e_{n}]}_{n = 1}^{\infty}

be an orthonormal basis for a parable Hilbert space H with the inner product <. ,. > Define

f_{n} = e_{n} - \frac{1}{n + 1} e_{n + 1} for n \in ℕ .

Then

A	the closure of the span $\{f_{n} : n \in ℕ\}$	B	$f = 0 if < f, f_{n} > = < f, e_{n} > for all n \in ℕ$
C	${\{f_{n}\}}_{n = 1}^{\infty}$ is an orthogonal sunset of H	D	there does exist nonzero $f \in H such that < f, e_{2} > = < f, f_{2} >$

Category: GATE Mathematics 2020

Question 37:

Suppose V is a finite-dimensional non-zero vector space over

ℂ and T : V \to V

is a linear transformation such that Range (T)=Nullspace (T). Then when of the following statements is FALSE?

A	The dimension of V is even	B	0 is the only eigenvalue of T
C	Both 0 and 1 are eigenvalues of T	D	$T^{2} = 0$

Category: GATE Mathematics 2020

Question 38:

Let

P \in M_{m \times n} (ℝ) .

Consider the following statement: I: If

XPY = 0 for all X \in M_{1 \times m} (ℝ) and M_{n \times 1} (ℝ), then P = 0 .

II. If

m = n, P is symmetric and P^{2} = 0, then P = 0 .

Then

both I and II are true

I is true but II is false

I is false but ii is true

both I and II are false

Category: GATE Mathematics 2020

Question 39:

For

n \in ℕ, let T_{n} : (l^{1}, | | . | |) \to (l^{\infty}, | | . | |) and T : (l^{\infty}, | | . | |_{1}) \to (l^{\infty}, | | . | |_{\infty})

be that bounded linear operators defined by

T_{n} (x_{1}, x_{2}, . .) = (y_{1}, y_{2} . . .), where y_{j} = \{\begin{cases} x_{j}, j \geq n \\ x_{n}, j > n \end{cases}

and

T (x_{1}, x_{2} . . .) = (x_{1}, x_{2}, . . .) .

A	$\| \| T \| \| does not converge to \| \| T \| \| as n \to \infty$	B	$\| \| T_{n} - T \| converge to zero n \to \infty$
C	$for all x \in l^{1}, \| \| T_{n} (x) - T (x) \| \| converge to zero as n \to \infty$	D	for each non-zero $x \in l^{1},$ exists a continuous linear functional g on $l^{\infty}$ such that $g (T_{n} (x))$ does not $g (T (x)) ad n \to \infty$

Category: GATE Mathematics 2020

Question 40:

Let

P (ℝ)

denote the power set of

ℝ,

equipped with the metric

d (U, V) = \sup_{x \in ℝ} |x_{U} (x) - X_{V} (x)|,

where

X_{U} and X_{V}

denote the characteristic functions of the subsets U and V, respectively, of

ℝ .

The set

\{\{m\} : m \in ℤ\}

in the metric space

(P (ℝ), d)

bounded but not totally bounded

totally bounded but not compact

compact

not bounded

Category: GATE Mathematics 2020

Question 41:

Let

f : ℝ \to ℝ

be defined by

f (x) = \sum_{n = 0}^{\infty} \frac{1}{2^{n}} X_{(n, n + 1]} (x),

Where

X_{(n, n + 1]}

is the characteristic function of the interval

(n, n + 1] . For α \in ℝ, let S_{α} = \{x \in ℝ : f (x) > α\} . Then

A	$S_{\frac{1}{2}}$ is open	B	$S_{\frac{\sqrt{5}}{2}}$ is not measurable
C	$S_{0}$ is closed	D	$S_{\frac{1}{\sqrt{2}}}$ is measurable

Category: GATE Mathematics 2020

Question 42:

For

n, \in ℕ, let f_{n}, g_{n} : (0, 1) \to ℝ

be, a function defined by

f_{n} (x) = x^{n} and g_{n} (x) = x^{n} (1 - x) .

Then

A	$\{f_{n}\}$ converges uniformly but $\{g_{n}\}$ does not converge uniformly	B	$\{g_{n}\}$ converges uniformly but $\{f_{n}\}$ does not converge uniformly
C	both $\{f_{n}\} and \{g_{n}\}$ converge uniformly	D	neither $\{f_{n}\} and \{g_{n}\}$ converge uniformly

Category: GATE Mathematics 2020

Question 43:

Let U be a solution of the differential equation

y' + xy = 0 and let ϕ = uψ

be a solution of the differential equation

y'' + 2 xy' + (x^{2} + 2) y = 0 satisfying ϕ (0) = 1 and ϕ' (0) = 0 . Then ϕ (x) is

A	$(\cos^{2} x) e^{- \frac{x^{2}}{2}}$	B	$(\cos x) e^{- \frac{x^{2}}{2}}$
C	$(1 + x^{2}) e^{- \frac{x^{2}}{2}}$	D	$(\cos x) e^{- x^{2}}$

Category: GATE Mathematics 2020

Question 44:

For

n \in ℕ \{0\}, let y_{n}

be a solution of the differential equation

xy'' + (1 - x) y' + n y = 0

Satisfying

y_{n} (0) = 1 .

For which of the following functions w(x), the integral

\int_{0}^{\infty} y_{p} (x) y_{q} (x) w (x) dx, (p \neq q)

is equal to zero?

A	$e^{- x^{2}}$	B	$e^{- x}$
C	${xe}^{- x^{2}}$	D	${xe}^{- x}$

Category: GATE Mathematics 2020

Question 45:

Suppose that

X = \{(0, 0)\} U \{(x, \sin \frac{1}{2}) : x \in ℝ \ \{0\}\}

and

Y = \{(0, 0)\} U \{(x, x \sin \frac{1}{2}) : x \in ℝ \ \{0\}\}

are metric spaces with metrics induced by the Euclidean metric of

ℝ^{2} .

Let

B_{X}

and

B_{Y}

be the open unit balls around (0, 0) in X and Y, respectively. Consider the following statement: I: The closure of

B_{X}

in X is compact II. The closure of

B_{y}

in Y is compact Then

both I and II are true

I is true but II is false

I is false but II is true

both I and II are false

Category: GATE Mathematics 2020

Question 46:

f : ℂ \ \{0\} \to ℂ

is a function such that

f (z) = f (\frac{z}{|z|})

and its restriction to the unit circle is continuous, then

A	f is continuous but not necessarily analytic
B	f is continuous but not necessarily a constant function
C	f is a constant function
D	$\lim_{z \to 0} f (z)$ exists

Category: GATE Mathematics 2020

Question 47:

For a subset S of a topological space, let Int (S) and

S

denote the interior and closure of S, respectively. Then which of the following statement is TRUE?

A	If S is open, then S = Int $(S)$	B	If the boundary of S is empty, then S is open
C	If the boundary of S is empty, then S is not closed	D	If $S \ S$ is a proper subset of the boundary of S, then S is open

Category: GATE Mathematics 2020

Question 48:

Suppose

I_{1}, I_{2} and I_{3}

are the smallest topologies on

ℝ containing S_{1}, S_{2} and S_{3}, respectively, where

S_{1} = \{(α, α + \frac{π}{n}) : α \in ℚ, n \in ℕ\}, S_{2} = \{(a, b) : a < b, a, b \in ℚ\}, S_{3} = \{(a, b) : a < b, a, b \in ℝ\} .

A	$I_{3} \underline{\supset} I_{1}$	B	$I_{3} \underline{\supset} I_{2}$
C	$I_{1} = I_{2}$	D	$I_{1} \underline{\supset} I_{2}$

Category: GATE Mathematics 2020

Question 49:

Let

M = [\begin{matrix} α & 3 & 0 \\ β & 3 & 1 \\ 0 & 1 & 2 \end{matrix}]

. Consider the following statements: I: There exists a lower triangular matrix L such that M =

{LL}^{t}, where L^{t}

denotes the transpose of L. II: Gauss-Seidel method for

Mx = b (b \in ℝ^{3})

converges for any initial choice

x_{0} \in ℝ^{3} .

Then

A	I is not true when $α > \frac{9}{2}, β = 3$	B	II is not true when $α > \frac{9}{2}, β = - 1$
C	II is not true when $α = 4, β = \frac{3}{2}$	D	$α = 5, β = 3$

Category: GATE Mathematics 2020

Question 50:

Let I and J be the ideals generated by

\{5 . \sqrt{10}\} and \{4, \sqrt{10}\}

in the ring

ℤ [\sqrt{10}] = \{a + b \sqrt{10} | α, b \in ℤ\},

respectively. Then

both I and J are maximal ideals

I is a maximal ideal but J is not a prime ideal

I is not a maximal ideal but J is not a prime ideal

neither I and J are maximal ideals

Category: GATE Mathematics 2020

Question 51:

Suppose V is finite-dimensional vector space over

ℝ . If W_{1}, W_{2}, and W_{3}

are subspaces of V, then which of the following statements is TRUE?

A	If $W_{1} + W_{2} + W_{3} = V then span (W_{1} \cup W_{2}) \cup span (W_{2} \cup W_{3}) \cup span (W_{3} \cup W_{1}) = V$
B	$If W_{1} \cap W_{2} = \{0\} and W_{1} \cap W_{3} = \{0\}, then W_{1} \cap (W_{2} + W_{3}) = \{0\}$
D	If $W_{1} + W_{2} = W_{1} + W_{3}, then W_{2} = W_{3}$
C	$If W_{1} \neq V, then span (V \ W_{1}) = V$

Category: GATE Mathematics 2020

Question 52:

Let

α, β \in ℝ, α \neq 0 . The system

x_{1} - 2 x_{2} + {αx}_{3} = 8 x_{1} - x_{2} + x_{4} = β x_{1}, x_{2}, x_{3}, x_{4} \geq 0

has NO basic feasible solution if

A	$α < 0, β > 8$	B	$α > 0, 0 < β < 8$
C	$α > 0, β < 0$	D	$α < 0, β < 8$

Category: GATE Mathematics 2020

Question 53:

Let

0 < p < 1 and let

X = \{f : ℝ \to ℝ is continuous and \int_{ℝ} {|f (x)|}^{p} dx < \infty\} .

For

f \in X,

define

{|f|}_{p} = {(\int_{ℝ} {|f (x)|}^{p} dx)}^{\frac{1}{p}}

Then

A	${\|.\|}_{p}$ defines a norm on X	B	${\|f + g\|}_{p} \leq {\|f\|}_{p} + {\|g\|}_{p} for all f, g \in X$
C	${\|f + g\|}_{p}^{p} \leq {\|f\|}_{p}^{p} + {\|g\|}_{p}^{p} for all f, g \in X$	D	$if f_{n} converges to f ponitwise on ℝ, then \lim_{n \to \infty} \|f_{n}\| = {\|f\|}_{p}$

Category: GATE Mathematics 2020

Question 54:

Suppose that

ϕ and ϕ_{2}

are linearly independent solutions of the differential equation

2 x^{2} y'' - (x + x^{2}) y' + (x^{2} - 2) y = 0,

and

ϕ (0) = 0 .

Then the smallest positive integer n such that

\lim_{x \to 0} x^{n} \frac{ϕ_{2} (x)}{ϕ_{1} (x)} = 0

is ___________.

1 to 1

2 to 2

3 to 3

4 to 4

Category: GATE Mathematics 2020

Question 55:

Suppose that

f (z) = \prod_{n = 1}^{17} (z - \frac{π}{n}), z \in ℂ and y (t) = e^{2 it}, t \in [0, 2 π] . If

\int_{Y} \frac{f' (z)}{f (z)} dz = απi,

then the value

α

is equal to _______

55 to 55

56 to 56

57 to 57

58 to 58

Category: GATE Mathematics 2020

Question 56:

y (t) = \frac{1}{2} e^{3 πi t}, t \in [0, 2] and

\int_{y} \frac{1}{z^{2} (e^{z} - 1)} dz = βπi,

then

β

is equal to __________ (correct up to one decimal place)

0.5 to 0.5

0.6 to 0.6

0.7 to 0.7

0.8 to 0.8

Category: GATE Mathematics 2020

Question 57:

Let

K = ℚ (\sqrt{3 + 2 \sqrt{2}}, ω), where ω

is a primitive cube root of unity. Then the degree of extension of K over

ℚ

is ___________

1 to 1

2 to 2

3 to 3

4 to 4

Category: GATE Mathematics 2020

Question 58:

Let

α \in ℝ . If (3, 0, 0, β)

is an optimal solution to the linear programming problem

minimize x_{1} + x_{2} + x_{3} - {αx}_{4} subject to 2 x_{1} - x_{2} + x_{3} = 6 - x_{1} + x_{2} + x_{4} = 3 x_{1}, x_{2}, x_{3}, x_{4} \geq 0

then the maximum value of

β - α

is _______________

3 to 3

5 to 5

7 to 7

9 to 9

Category: GATE Mathematics 2020

Question 59:

Suppose that

T : ℝ^{4} \to ℝ [x]

is a linear transformation over

ℝ

satisfying

T (- 1, 1, 1, 1) = x^{2} + 2 x^{4}, T (1, 2, 3, 4) = 1 - x^{2}, T (2, - 1, - 1, 0) = x^{3} - x^{4} .

Then the coefficient of

x^{4} in T (- 3, 5, 6, 6)

is _________________

1 to 1

3 to 3

5 to 5

7 to 7

Category: GATE Mathematics 2020

Question 60:

Let

\vec{F} (x, y, z) = (2 x - 2 y \cos x) \hat{i} + (2 y - y^{2} \sin x) \hat{j} + 4 z \hat{k}

and let S be the surface of the tetrahedron bounded by the planes

x = 0, y = 0, z = 0 and x + y + z = 1 .

\hat{n}

is the unit outward normal to the tetrahedron, then the value of

\int \int_{S} \vec{F} . \hat{n} dS

is ______________ (rounded off to two decimal places)

1.00 to 1.10

1.10 to 1.20

1.20 to 1.30

1.30 to 1.40

Category: GATE Mathematics 2020

Question 61:

Let

\vec{F} = (x + 2 y) e^{z} \hat{i} + ({ye}^{z} + x^{2}) \hat{j} + y^{2} z \hat{k}

and let S be the surface

x^{2} + y^{2} + z = 1, z \geq 0 .

\hat{n}

is a unit normal to S and

|\int \int_{S} (\nabla \times \vec{F}) . \hat{n} dS| = απ

Then

α is equal to_______

2 to 2

4 to 4

6 to 6

8 to 8

Category: GATE Mathematics 2020

Question 62:

Let G be a non-cyclic group of order 57. Then the number of elements of order 3 is G is _______________

37 to 37

38 to 38

39 to 39

40 to 40

Category: GATE Mathematics 2020

Question 63:

The coefficient of

{(x - 1)}^{5}

in the Taylor expansion about x = 1 of the function

$F (x) = \int_{1}^{x} \frac{\log_{e} t}{t - 1} dt, 0 < x < 2$

is ___________ (correct uo to two decimal places)

0.01 to 0.04

0.02 to 0.04

0.03 to 0.04

0.04 to 0.04

Category: GATE Mathematics 2020

Question 64:

Let u(x, y) be the solution to the initial value problem

\frac{\partial u}{\partial x} + (\sqrt{u}) \frac{\partial u}{\partial y} = 0, u (x, 0) = 1 + x^{2}

Then the value u(0, 1) is ___________ (rounded off to three decimal places)

1.610 to 1.625

1.620 to 1.635

1.630 to 1.645

1.640 to 1.655

Category: GATE Mathematics 2020

Question 65:

The value of

\lim_{n \to \infty} \int_{0}^{1} {nx}^{n} {ex}^{n} e^{x^{2}} dx

is ____________ (rounded off to three decimal places)

2.700 to 2.715

2.710 to 2.725

2.720 to 2.735

2.730 to 2.745

⮜ Back Next ⮞

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Answered & Marked for Review (will be considered for evaluation)

Question 1:

He is known for his unscrupulous ways. He always sheds ____________ tears to deceive people.

Correct Answer : crocodile

Question 2:

Jofra Archer, the England fast bowler, is ___________ than accurate.

Correct Answer : more fast

Question 3:

Select the word that fits the analogy: Build: Building: :Grow : __________

Correct Answer : Growth

Question 4:

I do not think you know the case well enough to have opinions. Having said that, I agree with your other point. What does that phrase "having said that" mean in the given text?

Correct Answer : despite what I have said

Question 5:

Define [x] as the greatest integer less than or equal to x, for each x

\in (- \infty, \infty)

. If y=[x], then area under y for x

\in

[1, 4] is _________.

Correct Answer : 6

Question 6:

Correct Answer : Funds raised through voluntary contributions on web-based platforms.

Question 7:

P, Q, R, and S are to be uniquely coded using

α and β

. If P is coded as

αα and Q as αβ

, then R and S, respectively, can be coded as _________.

A	$βα and αβ$	B	$ββ and αα$
C	$αβ and ββ$	D	$βα and ββ$

Correct Answer : D

Question 8:

The sum of the first n terms in the sequence 8, 88, 888, 8888,... is _____________.

A	$\frac{81}{80} (10^{n} - 1) + \frac{9}{8} n$	B	$\frac{81}{80} (10^{n} - 1) - \frac{9}{8} n$
C	$\frac{81}{80} (10^{n} - 1) + \frac{8}{9} n$	D	$\frac{81}{80} (10^{n} - 1) - \frac{8}{9} n$

Correct Answer : D

Question 9:

Select the graph that schematically represents BOTH

y = x^{m} and y = x^{1 / m}

properly in the interval

0 \leq x \leq 1,

for integer values m, where

m > 1 .

Correct Answer : A

Question 10:

Correct Answer : 59.0%

Question 11:

Suppose that

I_{1} and I_{2}

are topologies on X induced by metrics

d_{1} and d_{2}

, respectively,

I_{1} \underline{\subset} I_{2}

. Then which of the following is TRUE?

A	If a sequence converges in $(X, d_{2})$ then it convergers in $(X, d_{1})$
B	If a sequence converges in $(X, d_{1})$ then it convergers in $(X, d_{2})$
C	Every open ball in $(X, d_{1})$ is an open ball in $(X, d_{2})$
D	The map $x \mapsto x from (X, d_{1}) to (X, d_{2})$ is continuous

Correct Answer : A

Question 12:

Let

D = [- 1, 1] \times [- 1, 1] .

If the function

f : D \to ℝ

is defined by

f (x, y) = \{\begin{cases} \frac{x^{2} - y^{2}}{{(x^{2} + y^{2})}^{2}}, (x, y) \neq (0, 0) \\ 0, (x, y) = (0, 0) \end{cases},

A	f is continuous at (0, 0)
B	both the first-order partial derivatives of f exist at (0, 0)
C	$\int \int_{D} {\|f (x, y)\|}^{\frac{1}{2}} dx dy$ is finite
D	$\int \int_{D} \|f (x, y)\| dx dy$ is finite

Correct Answer : C

Question 13:

The initial value problem

y' = y^{\frac{3}{5}}, y (0) = b

has

Correct Answer : a unique solution if b=1

Question 14:

Consider the following statements: I:

\log (|z|)

is harmonic on

ℂ \ \{0\}

II.

\log (|z|)

has a harmonic conjugate on

ℂ \ \{0\}

Then

Correct Answer : I is the true but II is false

Question 15:

Let G and H be defined by

G = ℂ \ \{z = x + iy \in ℂ : x ⩽ 0, y = 0\}, H = ℂ \ \{z = x + iy \in ℂ : x \in ℤ, x \leq 0, y = 0\} .

Suppose

f : G \to ℂ and g : H \to ℂ

are analytic functions. Consider the following statements: I:

\int_{y} f dz

is independent of paths y in G joining -i and i II:

\int_{y} g dz

is independent of paths y in H joining -i and i Then

Correct Answer : I is true but II is false

Question 16:

Let

f (z) = e^{1 / z}, z \in ℂ \ \{0\}

and let, for

n \in ℕ,

R_{n} = \{z = x + iy \in ℂ : |x| < \frac{1}{n}, |y| < \frac{1}{n}\} \ \{0\} .

A	$f (R_{n + 1}) \neq f (R_{n})$	B	$f (R_{n}) \ f (R_{n + 1}) = f (R_{n} \ R_{n + 1})$
C	$f (\cap_{n = 1}^{\infty} R_{n}) = \cap_{n = 1}^{\infty} f (R_{n})$	D	$f (R_{n}) = f (R_{n + 1})$

Correct Answer : D and A

Question 17:

Suppose that

U = ℝ^{2} \ \{(x, y) \in ℝ^{2} : x, y \in ℚ\}, U = ℝ^{2} \ \{(x, y) \in ℝ^{2} : x > 0, y \frac{1}{x}\}

Then with respect to the Euclidean metric on

ℝ^{2},

Correct Answer : u is connected but V is disconnected

Question 18:

If (D1) and (D2) denote the dual problems of the linear programming problems (P1) and (P2), respectively, where (P1): minimize

x_{1} - 2 x_{2} subject to - x_{1} + x_{2} = 10, x_{1}, x_{2} \geq 0,

(P2): minimize

x_{1} - 2 x_{2} subject to - x_{1} + x_{2} = 10, x_{1} - x_{2} = 10, x_{1}, x_{2} \geq 0,

then

Correct Answer : both (D1) and (D2) are infeasible

Question 19:

(4, 0) and (0, - \frac{1}{2})

are critical points of the function

f (x, y) = 5 - (α - β) x^{2} + {βy}^{2} + (α + 1) y^{3} + x^{3},

where

α, β \in ℝ,

then

A	$(4, - \frac{1}{2})$ is a point of local maxima of f
B	$(4, - \frac{1}{2})$ is a saddle point of f
D	$α = 4, β = 2$
C	$(4, - \frac{1}{2})$ is a point of local maxima of f

Correct Answer : B

Question 20:

Consider the iterative scheme

x_{n} = \frac{x_{n - 1}}{2} + \frac{3}{x_{n - 1}}, n \geq 1 .

with initial point

x_{0} > 0 .

Then the sequence

\{x_{n}\}

A	converges only if $x_{0} > 1$
B	converges only if $x_{0} < 3$
C	converges for any $x_{0}$
D	does not converge for any $x_{0}$

Correct Answer : C

Question 21:

Let C[0, 1] denote the space of all real-valued continuous functions on [0,1] equipped with the supremum norm

| | . | |_{\infty} .

Let

T : C [0, 1] \to C [0, 1]

be the linear operator defined by

T (f) (x) = \int_{0}^{x} e^{- y} f (y) dy .

Then

A	$\| \| T \| \| = 1$	B	I-T is not invertible
C	T is subjective	D	$\| \| I + T \| \| = 1 + \| \| T \| \|$

Correct Answer : D

Question 22:

Suppose that M is a

5 \times 5

matrix with real entries and p(x) = diet (xI-M). Then

A	$p (0) = \det (M)$
B	every eigenvalue of M is real if p(1) + p(2)= 0 = p(2) + p(3)
C	$M^{- 1}$ is necessarily a polynomial in M of degree 4 if M is invertible
D	M is not invertible if $M^{2} - 2 M = 0$

Correct Answer : C

Question 23:

Let C [0, 1] denote the space of all real-valued continuous function on [0, 1] equipped with the supremum norm

| | . | |_{\infty} .

Let

f \in C [0, 1]

be such that

|f (x) - f (y)| \leq M |x - y|, for all x, y \in [0, 1] and for some M > 0 .

For

n \in ℕ, let f_{n} (x) = f (x^{1 + \frac{1}{n}}) . If S = \{f_{n} : n \in ℕ\},

then

Correct Answer : the closure of S is compact

Question 24:

Let

K : ℝ \times (0, \infty) \to ℝ

be a function such that the solution of the initial value problem

\frac{\partial u}{\partial t} = \frac{\partial^{2} u}{\partial x^{2}}, u (x, 0) = f (x), x \in ℝ, t > 0,

is given by

u (x, t) = \int_{ℝ} k (x - y, t) f (y) dy

for all bounded continuous functions f. Then the value of

\int_{ℝ} k (x, t) dx

is ____________

Correct Answer : 1 to 1

Question 25:

The number of cyclic subgroups of the quaternion group

Q_{8} = < a, b | a^{4} = 1, a^{2} = b^{2}, ba = a^{3} b >

is ______________.

Correct Answer : 5 to 5

Question 26:

The number of elements of order 3 in the symmetric group

S_{6}

is __________

Correct Answer : 80 to 80

Question 27:

Let F be the field with 4096 elements. The number of proper subfields of F is ____________

Correct Answer : 5 to 5

Question 28:

(x_{1}^{*}, x_{2}^{*})

is an optimal solution of the linear programming problem,

minimize x_{1} + 2 x_{2} subject to 4 x_{1} - x_{2} \geq 8 2 x_{1} + x_{2} \geq 10 - x_{1} + x_{2} \geq 0

and

(λ_{1}^{*}, λ_{2}^{*}, λ_{3}^{*})

is an optimal solution of its dual problem, then

\sum_{i = 1}^{2} x_{i}^{*} + \sum_{j = 1}^{3} λ_{j}^{*}

is equal to ____________ (correct up to one decimal place)

Correct Answer : 5.5 to 5.5

Question 29:

Let a, b, c,

\in ℝ

be such that the quadrature rule

\int_{- 1}^{1} f (x) dx \approx af (- 1) + bf (0) + cf' (1)

is exact for all polynomials of degree less than or equal to 2. Then b is equal to __________ (rounded off to two decimal places)

Correct Answer : 1.70 to 1.80

Question 30:

Let

f (x) = x^{4}

and let p(x) be the interpolating polynomial of f at nodes 1, 2, and 3. Then p(0) is equal to _______

Correct Answer : 36 to 36

Question 31:

For

n \geq 2,

define the sequence

\{x_{n}\}

x_{n} = \frac{1}{2 π} \int_{0}^{\frac{π}{2}} \tan^{\frac{1}{n}} t dt .

Then the sequence

\{x_{n}\}

converges to _______________ (correct up to two decimal places)

Correct Answer : 0.25 to 0.25

Question 32:

Let

L^{2} [0, 10] = \{f : [0, 10] \to ℝ : f is Lebesgue measurable and \int_{0}^{10} f^{2} dx < \infty\}

equipped with the norm

| | f | | = {(\int_{0}^{10} f^{2} dx)}^{\frac{1}{2}}

and let T be the linear functional on

L^{2} [0, 10]

given by

T (f) = \int_{0}^{2} f (x) dx - \int_{3}^{10} f (x) dx .

Then ||T|| is equal to ____________

Correct Answer : 3 to 3

Question 33:

\{x_{13}, x_{22}, x_{23} = 10, x_{31}, x_{32}, x_{34}\}

is the set of basic variables of a balanced transportation problem seeking to minimize the cost of transportation from origins to destinations, where the cost matrix is,

and

λ, μ \in ℝ,

then

x_{32}

is equal to _________

Correct Answer : 5 to 5

Question 34:

Let

ℤ_{225}

be the ring of integers modulo 225. If x is the number of prime and y is the number of nontrivial units in

ℤ_{225}

, then x + y is equal to ____

Correct Answer : 121 to 121

Question 35:

Let u(x, t) be the solution of

\frac{\partial^{2} u}{\partial t} - \frac{\partial^{2} u}{\partial x^{2}} = 0, u (x, 0) = f (x), \frac{\partial u}{\partial t} (x, 0) = 0, x \in ℝ, t > 0,

where is a twice continuously differentiable function. If f(-2) =4, f (0) =0, and u (2, 2) = 8, then value of u (1, 3) is _________

Correct Answer : 10 to 10

Question 36:

Let

{[e_{n}]}_{n = 1}^{\infty}

be an orthonormal basis for a parable Hilbert space H with the inner product <. ,. > Define

f_{n} = e_{n} - \frac{1}{n + 1} e_{n + 1} for n \in ℕ .

Then

A	the closure of the span $\{f_{n} : n \in ℕ\}$	B	$f = 0 if < f, f_{n} > = < f, e_{n} > for all n \in ℕ$
C	${\{f_{n}\}}_{n = 1}^{\infty}$ is an orthogonal sunset of H	D	there does exist nonzero $f \in H such that < f, e_{2} > = < f, f_{2} >$

Correct Answer : A

Question 37:

Suppose V is a finite-dimensional non-zero vector space over

ℂ and T : V \to V

is a linear transformation such that Range (T)=Nullspace (T). Then when of the following statements is FALSE?

A	The dimension of V is even	B	0 is the only eigenvalue of T
C	Both 0 and 1 are eigenvalues of T	D	$T^{2} = 0$

Correct Answer : C

Question 38:

Let

P \in M_{m \times n} (ℝ) .

Consider the following statement: I: If

XPY = 0 for all X \in M_{1 \times m} (ℝ) and M_{n \times 1} (ℝ), then P = 0 .

II. If

m = n, P is symmetric and P^{2} = 0, then P = 0 .

Then

Correct Answer : both I and II are true

Question 39:

For

n \in ℕ, let T_{n} : (l^{1}, | | . | |) \to (l^{\infty}, | | . | |) and T : (l^{\infty}, | | . | |_{1}) \to (l^{\infty}, | | . | |_{\infty})

be that bounded linear operators defined by

T_{n} (x_{1}, x_{2}, . .) = (y_{1}, y_{2} . . .), where y_{j} = \{\begin{cases} x_{j}, j \geq n \\ x_{n}, j > n \end{cases}

and

T (x_{1}, x_{2} . . .) = (x_{1}, x_{2}, . . .) .

A	$\| \| T \| \| does not converge to \| \| T \| \| as n \to \infty$	B	$\| \| T_{n} - T \| converge to zero n \to \infty$
C	$for all x \in l^{1}, \| \| T_{n} (x) - T (x) \| \| converge to zero as n \to \infty$	D	for each non-zero $x \in l^{1},$ exists a continuous linear functional g on $l^{\infty}$ such that $g (T_{n} (x))$ does not $g (T (x)) ad n \to \infty$

Correct Answer : C

Question 40:

Let

P (ℝ)

denote the power set of

ℝ,

equipped with the metric

d (U, V) = \sup_{x \in ℝ} |x_{U} (x) - X_{V} (x)|,

where

X_{U} and X_{V}

denote the characteristic functions of the subsets U and V, respectively, of

ℝ .

The set

\{\{m\} : m \in ℤ\}

in the metric space

(P (ℝ), d)

Correct Answer : bounded but not totally bounded

Question 41:

Let

f : ℝ \to ℝ

be defined by

f (x) = \sum_{n = 0}^{\infty} \frac{1}{2^{n}} X_{(n, n + 1]} (x),

Where

X_{(n, n + 1]}

is the characteristic function of the interval

(n, n + 1] . For α \in ℝ, let S_{α} = \{x \in ℝ : f (x) > α\} . Then

A	$S_{\frac{1}{2}}$ is open	B	$S_{\frac{\sqrt{5}}{2}}$ is not measurable
C	$S_{0}$ is closed	D	$S_{\frac{1}{\sqrt{2}}}$ is measurable

Correct Answer : D

Question 42:

For

n, \in ℕ, let f_{n}, g_{n} : (0, 1) \to ℝ

be, a function defined by

f_{n} (x) = x^{n} and g_{n} (x) = x^{n} (1 - x) .

Then

A	$\{f_{n}\}$ converges uniformly but $\{g_{n}\}$ does not converge uniformly	B	$\{g_{n}\}$ converges uniformly but $\{f_{n}\}$ does not converge uniformly
C	both $\{f_{n}\} and \{g_{n}\}$ converge uniformly	D	neither $\{f_{n}\} and \{g_{n}\}$ converge uniformly

Correct Answer : B

Question 43:

Let U be a solution of the differential equation

y' + xy = 0 and let ϕ = uψ

be a solution of the differential equation

y'' + 2 xy' + (x^{2} + 2) y = 0 satisfying ϕ (0) = 1 and ϕ' (0) = 0 . Then ϕ (x) is

A	$(\cos^{2} x) e^{- \frac{x^{2}}{2}}$	B	$(\cos x) e^{- \frac{x^{2}}{2}}$
C	$(1 + x^{2}) e^{- \frac{x^{2}}{2}}$	D	$(\cos x) e^{- x^{2}}$

Correct Answer : B

Question 44:

For

n \in ℕ \{0\}, let y_{n}

be a solution of the differential equation

xy'' + (1 - x) y' + n y = 0

Satisfying

y_{n} (0) = 1 .

For which of the following functions w(x), the integral

\int_{0}^{\infty} y_{p} (x) y_{q} (x) w (x) dx, (p \neq q)

is equal to zero?

A	$e^{- x^{2}}$	B	$e^{- x}$
C	${xe}^{- x^{2}}$	D	${xe}^{- x}$

Correct Answer : B

Question 45:

Suppose that

X = \{(0, 0)\} U \{(x, \sin \frac{1}{2}) : x \in ℝ \ \{0\}\}

and

Y = \{(0, 0)\} U \{(x, x \sin \frac{1}{2}) : x \in ℝ \ \{0\}\}

are metric spaces with metrics induced by the Euclidean metric of

ℝ^{2} .

Let

B_{X}

and

B_{Y}

be the open unit balls around (0, 0) in X and Y, respectively. Consider the following statement: I: The closure of

B_{X}

in X is compact II. The closure of

B_{y}

in Y is compact Then

Correct Answer : I is false but II is true

Question 46:

f : ℂ \ \{0\} \to ℂ

is a function such that

f (z) = f (\frac{z}{|z|})

and its restriction to the unit circle is continuous, then

A	f is continuous but not necessarily analytic
B	f is continuous but not necessarily a constant function
C	f is a constant function
D	$\lim_{z \to 0} f (z)$ exists

Correct Answer : A

Question 47:

For a subset S of a topological space, let Int (S) and

S

denote the interior and closure of S, respectively. Then which of the following statement is TRUE?

A	If S is open, then S = Int $(S)$	B	If the boundary of S is empty, then S is open
C	If the boundary of S is empty, then S is not closed	D	If $S \ S$ is a proper subset of the boundary of S, then S is open

Correct Answer : B

Question 48:

Suppose

I_{1}, I_{2} and I_{3}

are the smallest topologies on

ℝ containing S_{1}, S_{2} and S_{3}, respectively, where

S_{1} = \{(α, α + \frac{π}{n}) : α \in ℚ, n \in ℕ\}, S_{2} = \{(a, b) : a < b, a, b \in ℚ\}, S_{3} = \{(a, b) : a < b, a, b \in ℝ\} .

A	$I_{3} \underline{\supset} I_{1}$	B	$I_{3} \underline{\supset} I_{2}$
C	$I_{1} = I_{2}$	D	$I_{1} \underline{\supset} I_{2}$

Correct Answer : C

Question 49:

Let

M = [\begin{matrix} α & 3 & 0 \\ β & 3 & 1 \\ 0 & 1 & 2 \end{matrix}]

. Consider the following statements: I: There exists a lower triangular matrix L such that M =

{LL}^{t}, where L^{t}

denotes the transpose of L. II: Gauss-Seidel method for

Mx = b (b \in ℝ^{3})

converges for any initial choice

x_{0} \in ℝ^{3} .

Then

A	I is not true when $α > \frac{9}{2}, β = 3$	B	II is not true when $α > \frac{9}{2}, β = - 1$
C	II is not true when $α = 4, β = \frac{3}{2}$	D	$α = 5, β = 3$

Correct Answer : D

Question 50:

Let I and J be the ideals generated by

\{5 . \sqrt{10}\} and \{4, \sqrt{10}\}

in the ring

ℤ [\sqrt{10}] = \{a + b \sqrt{10} | α, b \in ℤ\},

respectively. Then

Correct Answer : I is a maximal ideal but J is not a prime ideal

Question 51:

Suppose V is finite-dimensional vector space over

ℝ . If W_{1}, W_{2}, and W_{3}

are subspaces of V, then which of the following statements is TRUE?

A	If $W_{1} + W_{2} + W_{3} = V then span (W_{1} \cup W_{2}) \cup span (W_{2} \cup W_{3}) \cup span (W_{3} \cup W_{1}) = V$
B	$If W_{1} \cap W_{2} = \{0\} and W_{1} \cap W_{3} = \{0\}, then W_{1} \cap (W_{2} + W_{3}) = \{0\}$
D	If $W_{1} + W_{2} = W_{1} + W_{3}, then W_{2} = W_{3}$
C	$If W_{1} \neq V, then span (V \ W_{1}) = V$

Correct Answer : D

Question 52:

Let

α, β \in ℝ, α \neq 0 . The system

x_{1} - 2 x_{2} + {αx}_{3} = 8 x_{1} - x_{2} + x_{4} = β x_{1}, x_{2}, x_{3}, x_{4} \geq 0

has NO basic feasible solution if

A	$α < 0, β > 8$	B	$α > 0, 0 < β < 8$
C	$α > 0, β < 0$	D	$α < 0, β < 8$

Correct Answer : D

Question 53:

Let

0 < p < 1 and let

X = \{f : ℝ \to ℝ is continuous and \int_{ℝ} {|f (x)|}^{p} dx < \infty\} .

For

f \in X,

define

{|f|}_{p} = {(\int_{ℝ} {|f (x)|}^{p} dx)}^{\frac{1}{p}}

Then

A	${\|.\|}_{p}$ defines a norm on X	B	${\|f + g\|}_{p} \leq {\|f\|}_{p} + {\|g\|}_{p} for all f, g \in X$
C	${\|f + g\|}_{p}^{p} \leq {\|f\|}_{p}^{p} + {\|g\|}_{p}^{p} for all f, g \in X$	D	$if f_{n} converges to f ponitwise on ℝ, then \lim_{n \to \infty} \|f_{n}\| = {\|f\|}_{p}$

Correct Answer : C

Question 54:

Suppose that

ϕ and ϕ_{2}

are linearly independent solutions of the differential equation

2 x^{2} y'' - (x + x^{2}) y' + (x^{2} - 2) y = 0,

and

ϕ (0) = 0 .

Then the smallest positive integer n such that

\lim_{x \to 0} x^{n} \frac{ϕ_{2} (x)}{ϕ_{1} (x)} = 0

is ___________.

Correct Answer : 3 to 3

Question 55:

Suppose that

f (z) = \prod_{n = 1}^{17} (z - \frac{π}{n}), z \in ℂ and y (t) = e^{2 it}, t \in [0, 2 π] . If

\int_{Y} \frac{f' (z)}{f (z)} dz = απi,

then the value

α

is equal to _______

Correct Answer : 56 to 56

Question 56:

y (t) = \frac{1}{2} e^{3 πi t}, t \in [0, 2] and

\int_{y} \frac{1}{z^{2} (e^{z} - 1)} dz = βπi,

then

β

is equal to __________ (correct up to one decimal place)

Correct Answer : 0.5 to 0.5

Question 57:

Let

K = ℚ (\sqrt{3 + 2 \sqrt{2}}, ω), where ω

is a primitive cube root of unity. Then the degree of extension of K over

ℚ

is ___________

Correct Answer : 4 to 4

Question 58:

Let

α \in ℝ . If (3, 0, 0, β)

is an optimal solution to the linear programming problem

minimize x_{1} + x_{2} + x_{3} - {αx}_{4} subject to 2 x_{1} - x_{2} + x_{3} = 6 - x_{1} + x_{2} + x_{4} = 3 x_{1}, x_{2}, x_{3}, x_{4} \geq 0

then the maximum value of

β - α

is _______________

Correct Answer : 7 to 7

Question 59:

Suppose that

T : ℝ^{4} \to ℝ [x]

is a linear transformation over

ℝ

satisfying

T (- 1, 1, 1, 1) = x^{2} + 2 x^{4}, T (1, 2, 3, 4) = 1 - x^{2}, T (2, - 1, - 1, 0) = x^{3} - x^{4} .

Then the coefficient of

x^{4} in T (- 3, 5, 6, 6)

is _________________

Correct Answer : 5 to 5

Question 60:

Let

\vec{F} (x, y, z) = (2 x - 2 y \cos x) \hat{i} + (2 y - y^{2} \sin x) \hat{j} + 4 z \hat{k}

and let S be the surface of the tetrahedron bounded by the planes

x = 0, y = 0, z = 0 and x + y + z = 1 .

\hat{n}

is the unit outward normal to the tetrahedron, then the value of

\int \int_{S} \vec{F} . \hat{n} dS

is ______________ (rounded off to two decimal places)

Correct Answer : 1.30 to 1.40

Question 61:

Let

\vec{F} = (x + 2 y) e^{z} \hat{i} + ({ye}^{z} + x^{2}) \hat{j} + y^{2} z \hat{k}

and let S be the surface

x^{2} + y^{2} + z = 1, z \geq 0 .

\hat{n}

is a unit normal to S and

|\int \int_{S} (\nabla \times \vec{F}) . \hat{n} dS| = απ

Then

α is equal to_______

Correct Answer : 2 to 2

Question 62:

Let G be a non-cyclic group of order 57. Then the number of elements of order 3 is G is _______________

Correct Answer : 38 to 38

Question 63:

The coefficient of

{(x - 1)}^{5}

in the Taylor expansion about x = 1 of the function

$F (x) = \int_{1}^{x} \frac{\log_{e} t}{t - 1} dt, 0 < x < 2$

is ___________ (correct uo to two decimal places)

Correct Answer : 0.04 to 0.04

Question 64:

Let u(x, y) be the solution to the initial value problem

\frac{\partial u}{\partial x} + (\sqrt{u}) \frac{\partial u}{\partial y} = 0, u (x, 0) = 1 + x^{2}

Then the value u(0, 1) is ___________ (rounded off to three decimal places)

Correct Answer : 1.610 to 1.625

Question 65:

The value of

\lim_{n \to \infty} \int_{0}^{1} {nx}^{n} {ex}^{n} e^{x^{2}} dx

is ____________ (rounded off to three decimal places)

Correct Answer : 2.710 to 2.725

Back
Scrore Card
Total Question		Total Attempted
Correct Answers		Incorrect Answers
Score

A	$\| \| T \| \| does not converge to \| \| T \| \| as n \to \infty$	B	$\| \| T_{n} - T \| converge to zero n \to \infty$
C	$for all x \in l^{1}, \| \| T_{n} (x) - T (x) \| \| converge to zero as n \to \infty$	D	for each non-zero $x \in l^{1},$ exists a continuous linear functional g on $l^{\infty}$ such that $g (T_{n} (x))$ does not $g (T (x)) ad n \to \infty$

A	${\|.\|}_{p}$ defines a norm on X	B	${\|f + g\|}_{p} \leq {\|f\|}_{p} + {\|g\|}_{p} for all f, g \in X$
C	${\|f + g\|}_{p}^{p} \leq {\|f\|}_{p}^{p} + {\|g\|}_{p}^{p} for all f, g \in X$	D	$if f_{n} converges to f ponitwise on ℝ, then \lim_{n \to \infty} \|f_{n}\| = {\|f\|}_{p}$

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