State Eligibility Test 2017 Mathematical Sciences (Final ......State Eligibility Test – 2017...

State Eligibility Test – 2017

Mathematical Sciences

(Final Answer Key)

Part-A

Q.No: 1 Among the following, choose the odd one ऱ म म ऱ , य ।

A The axiom of choice ऑफ

B Well - ordering principle -

C Pigeon - hole principle -

D Zorn's lemma ( )

Q.No: 2

If denotes the null set, then among the following

the number of true statements is

य र म य र ऱ म य य :

A one

B Two

C Three

D Four

Q.No: 3 f(x) is an non-negative function in [0, [ such that f'(x) 2f(x) and f(0) = 0. Then

f(x) पर [0, [ ए र फऱ इ र f'(x) 2f(x)

f(0) = 0 ब

A f(x) is always a constant function f(x) अ फ ।

B f(x) is increasing function f(x) फ ।

C f(x) is decreasing function f(x) फ ।

D f'(x) changes sign f'(x) ।

Q.No: 4 The value of म

A is

B can be

C does not exist अ

D both (is ) and (does not exist) are true ( ) ( अ ) ।

Q.No: 5 Which of the following, subsets of 2 is Compact 2 प म य म

A {(x , y) : |x| 1, |y| 2} {(x , y) : |x| 1, |y| 2}

B {(x , y) : x2 y2 + 5} {(x , y) : x2 y2 + 5}

C {(x , y) : x = y} {(x , y) : x = y}

D {(x , y) : |x| 1, |y|2 3} {(x , y) : |x| 1, |y|2 3}

Q.No: 6 For each j = 1,2,3,..., let Aj be a finite set containing at least two distinct elements. Then which of the following is true?

य j = 1,2,3,..., ऱय म य Aj ए प र म म य म म म भ अवयव । ब ऱ म

य ?

A is uncountable अ

B is uncountable अ

C is uncountable अ

D All of the these

Q.No: 7 The value of is म

A 0 0

B 2 2

C

D /2 /2

Q.No: 8

"If f:[a,b] is a continuous function and f(a) . f(b) < 0, then there is x, a < x < b such that f(x) = 0". This result of the consequence of:

''य f:[a,b] ए फऱ f(a) . f(b) < 0 ए x, a < x < b f(x) = 0 ''

य ए

A Boundedness of [a,b] [a,b]

B Completeness of [a,b] [a,b]

C Compactness of [a,b] [a,b]

D Connectness of [a,b] [a,b]

Q.No: 9 Which of the following statements are not true? ऱ म य ?

A Every convergent sequence is bounded अ अ

B Every bounded sequence is convergent अ अ ।

C Every Cauchy sequence of real numbers is

convergent अ अ ।

D Every convergent sequence is a Cauchy sequence अ अ अ ।

Q.No: 10

Let f(x) and g(x) be differentiable functions for 0 x 2 such that f(0) = 4, f(2) = 8 , g(0) = 0 and f'(x) = g'(x) for all x in [0,2] then the value of g(2) must be

म ऱ य f(x)और g(x), 0 x 2 ऱय अव ऱ य फऱ f(0) = 4, f(2) = 8 , g(0) = 0 f'(x) = g'(x),[0,2] म x भ म ऱय , ब g(2) म

:

A 2 2

B 4 4

C -4 -4

D -2 -2

Q.No: 11 Let be a function such म य फऱ इ र

that and exist. Then at (0,0): अ व ब (0,0)पर

A f must be continuous but not differentiable f अ अ ।

B f must be differentiable f अ अ ।

C f need not to be continuous f ।

D f'(x) is continuous f'(x) ।

Q.No: 12 The integral dx, p > 0 converges absolutely

if म ऱ dx, p > 0 र प य अ भ र य

A p > 1 p > 1

B p = 1 p = 1

C p < 1 p < 1

D p > 0 p > 0

Q.No: 13 Which of the following statements is correct? ऱ म य ?

A f is measurable | f | is measurable f | f |

B | f | is measurable f is measurable | f | f

C | f | is measurable f + | f | is measurable | f | f + | f |

D None of these are correct

Q.No: 14 Let A be a 5 X 4 matrix with real entries such that AX = 0 if and only if X = 0, where X is a 4 X 1 matrix

and 0 is null matrix. Then the rank of A is:

य A ए व व व य 5 X 4 य AX

= 0 य और वऱ य X = 0 X ए 4 X 1 य 0 य य । ब A :

A 1 1

B 2 2

C 3 3

D 4 4

Q.No: 15 Let A be a real 3 X 4 matrix of rank 2. Then rank of A*A, where A* denotes the transpose of A, is

म य A 2 ए व व 3 X 4 य । ब A*A

, A*, प रव A र , :

A exactly 2 2

B exactly 3 3

C exactly 4 4

D at most 2 but not necessarily 2 अ अ 2 2 ।

Q.No: 16 If A is a 5 X 5 real matrix with trace 14 and if 2 and 3 are eigen values of A, each with algebraic multiplicity 2, then the determinant of A is equal to

य A ए 5 X 5 व व य अ र 14 य 2 और 3, A अ भऱ म , य ब य ब 2 A र म

A 180 180

B 144 144

C 24 24

D 0 0

Q.No: 17 If , then A50 is य ब A50

A

B

C

D

Q.No: 18

Let V1 and V2 be subspaces of a vector space V. Consider the following statements P & Q : P: V1 V2 is a subspace of V

Q: V1 + V2 = {x+y : x V1, y V2} is not subspace of V. Then

य V1 V2 ए म V प म य ऱ P Q पर व र र P : V1 V2 ए प म V Q: V1 + V2 = {x+y : x V1, y V2}, V ए प म

A P is true but Q is false P Q अ

B P and Q both are true P Q ।

C P is false while Q is true P अ Q

D P and Q both are false P Q अ ।

Q.No: 19

Consider the six vector spaces Q(Q), R(Q), C(Q),

R(R), C(R) and C(C), where Q, R and C denotes the fields of rational numbers, real numbers and complex numbers respectively. Here the number of finite

dimensional vector spaces is

छ: र म य Q(Q), R(Q), C(Q), R(R), C(R) C(C)पर व र य , Q, R C म : प रम य

य , व व य म य र । य प र म वम र म य य :

A Three

B Four

C Five

D Six छ:

Q.No: 20

Let V be the vector space of all real polynomials of degree upto 2 and let S = {1, x2 + x, x2 - x} and T = {x, x2 + 1, 3x2 - 2x + 3}

Choose your answer as

म य V भ व व ब प र

म म य

S = {1, x2 + x, x2 - x}

T = {x, x2 + 1, 3x2 - 2x + 3}

अप र य

A S and T both are bases of V S T, V ।

B None of S and T is a basis for V S T V ।

C S is a basis, but T is not a basis for V S T, V ।

D S is not a basis but T is a basis for V S T, V ।

Q.No: 21

Let V denote the vector space of all infinitely differentiable functions and let be the differential operator. Consider the sets: S1 = {1, x, x2} S2 = {x, x2, x3}

S3 = {1+x, x+x2, 1+x2}

म V भ अ य अव ऱ य फऱ र म र अव ऱ य र म य : S1 = {1, x, x2} S2 = {x, x2, x3} S3 = {1+x, x+x2, 1+x2}

पर व र य । व म य , म य D3(f) = 0 ऱ

The sets which span the solution space of the problem D3(f) = 0 are

म प व र र ,

A All the three S1, S2 and S3 S1, S2 and S3

B S1 and S2 S1 S2

C S2 and S3 S2 S3

D S3 and S1 S3 S1

Q.No: 22

If V be a vector space over the field of real numbers

and a linear transformation is defined

by , x V. Then the nullity of T

य V व व य पर ब र म

र प र , x V प रभ । ब T य

A is zero

B is one

C is infinite अ

D Cannot be determined unless V is known V

Q.No: 23 Let x1 = 31/7 + 91/7 and x2 = 51/7 + 91/7. Then which of the following is true statement?

म x1 = 31/7 + 91/7 x2 = 51/7 + 91/7 ब म य

A x1 < x2 x1 < x2

B x2 < x1 x2 < x1

C f(x) = (6-x)1/7 + (6+x)1/7 is an increasing function in [1,3]

f(x) = (6-x)1/7 + (6+x)1/7 अ [1,3]

फ ।


Q.No: 24

If f(z) is analytic throughout the complex plane and its modulus is uniformly bounded, then f'(z) is identically

zero The above result is known by the name of

य f(z) प म ऱ म व ऱ इ म प

ए म प रब f'(z) व म य । पर प र म म

A Morera

B Cauchy

C Liouville

D Rouche

Q.No: 25 Let f(z) be an analytic function whose value lie on a straight line in the complex plane. Then

म व ऱ फऱ f(z) म म ऱ म ए

रऱ र पर । ब

A f(z) is identically equal to zero f(z) ।

B f(z) is constant f(z) अ ।

C f(z) is a linear map f(z) फ ।

D f(z) is a bilinear transformation f(z) ।

Q.No: 26 The image of line x = c under the conformal transformation w = z1/2 is a

अ प र w = z1/2 अ र x = c ब ब

A Circle

B Hyperbola अ

C Parabola

D None of these are correct ।

Q.No: 27 Under what assumptions on f does Cauchy's theorem

imply that where c is the unit circle?

प र प अ f पर म य य

ऱ c ए इ व ?

A f is analytic outside c f, c ।

B f is analytic at every point of c c f ।

C f is analytic in a region containing the closed

unit disc f उ , अ , ।

D f is analytic in the annulus containing c c , f ।

Q.No: 28 Let f:C C be an entire function such that, f(1/n) = 1/n2, for all positive integers n. Then

म f:C C इ र व फऱ भ म

प n ऱए f(1/n) = 1/n2 , ब

A Such a function does not exist फ अ

B f should be bounded f

C f has a pole at 0 f, 0 ।

D f(z) = z2 for all z in C C z , f(z) = z2

Q.No: 29

Let G be a group of order 99. Consider the following three statements:

(i) G has a normal sub group of order 3 (ii) G has a subgroup of order 11 (iii) the group G is abelian. Here the number of true statements is

म G 99 ए म । ऱ पर व र य G 3 ए म य प म र । G 11 ए प म र । G ए ब ऱ म । य य य ।

A zero

B one

C two

D three

Q.No: 30 " Every finite group G is isomorphic to a permutation group". This statement is

'' य प र म म G म य म य र '' य

-

A Cayley's theorem

B Lagrange's theorem

C Lioville's theorem


Q.No: 31 Let F be a field of 32 elements and A = { x F : x31 = 1 and xk 1 for all natural numbers k < 31}. Then the number of elements in A is

म F , 32 अवयव ए A = { x F : x31 =

1 एव भ य k < 31 ऱए xk 1}. ब A म अवयव य :

A 1

B 2

C 30

D 6 छ:

Q.No: 32 Let G be a group of order 36 and H be its subgroup of order 4. If Z(G) denotes the centre of G. Then

म G 36 ए म H इ 4

प म । य Z(G), G र ब

A H Z(G) H Z(G)

B H = Z(G) H = Z(G)

C H is normal in G H, G

D H is ableian subgroup H उ

Q.No: 33 The number of non-trivial topologies on a three elements set X = {a,b,c} is

ए अवयव म य X = {a,b,c} पर अ छ

य य :

A 30 30

B 29 29

C 28 28

D 27 27

Q.No: 34 The cofinite topology on an infinite set is ए अ म य पर -प र म

A T0, compact but not T2 T0, T2

B T1, connected but not T2 T1, T2

C T2, but not T3 T2, T3

D T2, compact but not T4 T2, T4

Q.No: 35

Let be the real line and x be the plane as a topological space. Consider the map defined by f(x,y)=(x,0) . Then the map f is

म R व व र ऱ x ए म र म । फऱ . f(x,y)=(x,0)

व र प रभ ब फऱ f :

A Closed and continuous

B Continuous and open

C Open and closed

D Closed, open and continuous ,

Q.No: 36 Let g be a differentiable function म य g ए अव ऱ य फऱ भ x 0

satisfying for all x 0.

Then the value of is equal to

ऱय र ।

ब म :

A /6 /6

B /3 /3

C /4 /4

D /2 /2

Q.No: 37 If y =e-x is a solution of ODE, ,

then the correct relation between P and Q is :

य y =e-x . अ. म . , ए ऱ

P Q म य ब

A 1 + P + Q = 0 1 + P + Q = 0

B P = 1 + Q P = 1 + Q

C Q = 1 + P Q = 1 + P

D P + Q = 1 P + Q = 1

Q.No: 38 Given that y = x is one solution of ODE x2y" + xy' - y = 0, a second linearly independent solution will be:

y = x .अ. म . x2y" + xy' - y = 0 ए ऱ य य र र व ऱ

A

B

C

D

Q.No: 39

The region in which the

PDE is

hyperbolic, is

व म . अ.

म . अ परवऱ य

,

A xy > 1 xy > 1

B xy 1 xy 1

C xy 0 xy 0

D xy > 0 xy > 0

Q.No: 40 The differential equation is अव ऱ म र

A Linear and homogeneous

B non-linear and homogeneous अ

C linear and non-homogeneous अ

D Non-linear and non-homogeneous अ अ

Q.No: 41 The partial differential equation of the family of

surfaces z = (x+y) + Axy is प z = (x+y) + Axy ऱ अव ऱ म र

A xp - yq = 0 xp - yq = 0

B xp - yq = x - y xp - yq = x - y

C xp + yq = x + y xp + yq = x + y

D xp + yq = 0 xp + yq = 0

Q.No: 42

The function satisfying the integral

equation

म ऱ म र र व ऱ फऱ :

A

B

C

D

Q.No:43 The integral म ऱ

equation is equivalent to

म र

म य :

A y" - y = 0; y(0) = 0, y(1) = 0 y" - y = 0; y(0) = 0, y(1) = 0

B y" - y = 0; y(0) = 0, y'(1) = 0 y" - y = 0; y(0) = 0, y'(1) = 0

C y" + y = 0; y(0) = 0, y(1) = 0 y" + y = 0; y(0) = 0, y(1) = 0

D y" + y = 0; y(0) = 0, y'(0) = 0 y" + y = 0; y(0) = 0, y'(0) = 0

Q.No: 44

The variational problem of extremizing the

functional has

फऱ इ म र व र य म य

A a unique solution अ

B exactly two solutions

C an infinite number of solutions अ

D no solution

Q.No: 45

The Newton - Raphson iteration

formula can be used to compute the

य र फ प र व ,

ऱय य य

A square of A A

B reciprocal of A A

C square root of A A

D logarithm of A A

Q.No: 46

One bag contains 5 white balls and 7 black balls, another bag contains 3 white balls and 5 black balls. If a bag is chosen at random and a ball is drawn from it, the probability that it is a white ball is:

ए ब म 5 फ 7 ऱ , र ब म 3 फ 5 ऱ . य य छय ए ब म ए ऱ फ य

A 5/12 5/12

B 3/8 3/8

C 19/24 19/24

D 19/48 19/48

Q.No: 47 The function f(x) = ce-x, 0 x is a probability density function for

फऱ f(x) = ce-x, 0 x ए य व फऱ

A all values of c c

B all values of c > 1 c > 1

C all values of c < 1 c < 1

D c = 1 only c = 1

Q.No: 48

For a fixed t , consider the linear programming

problem maximize z = 3x + 4y, subject to x + y 100, x + 3y t and x,y 0 The maximum value of z is 400 for t equal to

ए t ऱय , र म म य अ म

र z = 3x + 4y,

ब x + y 100 ,

x + 3y t

x,y 0 ब t म ऱय z अ म म 400

:

A 300 300

B 200 200

C 100 100

D 50 50

Q.No: 49

Let X(t) = number of customers in the system at time t in an M/M/C queuing model with C = 3, arrival

rate > 0 and service rate > 0 .

Consider the following statements P and Q :

P: If X(t) has a stationary distribution then < 3 Q: The number of customers undergoing service at

time t is min{X(t),3}. Then

C = 3, म र > 0 व > 0 ए

M/M/C र म डऱ म मय t पर य म य X(t)

। ऱ P Q पर व र य P : य X(t) ए ब < 3 । Q : मय t पर व प व ऱ य य {X(t),3} । ब


B both P and Q are true P Q

C None of P and Q is true P Q

D Only Q is true Q

Q.No: 50

Consider the linear programming problem Maximize z = 2x1 + 3x2 - 4x3 + x4 subject to x1 + x2 + x3 = 2 x1 - x2 + x3 = 2

2x1 + 3x2 + 2x3 - x4 = 0 and x1, x2, x3, x4 > 0 Which of the following is true?

र म म य अ म र z = 2x1 + 3x2 - 4x3 +

x4

ब

x1 + x2 + x3 = 2 x1 - x2 + x3 = 2

2x1 + 3x2 + 2x3 - x4 = 0

x1, x2, x3, x4 > 0 पर व र य । ब ऱ म य ?

A (1,0,1,4) is a basic feasible solution but (2,0,0,4) is not

(1,0,1,4) (2,0,0,4)

B Neither (1,0,1,4) nor (2,0,0,4) is a basic feasible

solution (1,0,1,4) (2,0,0,4)

C (1,0,1,4) in not a basic feasible solution but

(2,0,0,4) is (1,0,1,4) (2,0,0,4)

D Both (1,0,1,4) and (2,0,0,4) are basic feasible solutions

(1,0,1,4) (2,0,0,4)

Part-B

Q.No: 1 Let be a convergent series of positive terms.

Then म म प ए अ भ र , ब:

A is always divergent अ ।

B may be divergent अ

C may be divergent अ

D all the series are necessarily convergent अ

Q.No: 2 For a function which one of the following is true? फऱ ऱय ऱ म ए य

A is Riemann integral on [0,1] is Riemann integrable

[0,1] ।

B is Riemann intergable on [0,1] is [0,1] ।

Riemann integrable

C is Riemann integrabe on [0,1] is continuous

[0,1] ।

D None of these

Q.No: 3

The improper

integral converges, if अ म ऱ अ भ र , य :

A m > 0 and n < 0 m > 0 n < 0

B m > 0 and n > 0 m > 0 n > 0

C m < 0 and n > 1 m < 0 n > 1

D m < 0 and n < 0 m < 0 n < 0

Q.No: 4 Let Q be the set of all rational numbers and T be the set of all irrational numbers, then

म ऱ य Q भ प रम य य T भ अप रम य

य म य , ब

A Both Q and T are countable Q T

B Q is countable but T is Uncountable Q T

C T is countable but Q is Uncountable T Q

D Both Q and T are uncountable Q T

Q.No: 5 Which of the following is NOT true ? ऱ म अ य ?

A A bounded monotonic function is a function of bounded variation

फ फ

B A continuous function is necessarily a function of bounded variation

फ फ

C A function of bounded variation is necessarily bounded

फ

D The sum of two functions of bounded variation is

also of bounded variation फ

Q.No: 6 Let be a Riemann integrable function on [a, b] and

let . Then

म य अ र ऱ [a, b] पर, ए र म म ऱ य फऱ

य , ब:

A (x) is continuous on [a, b] फ (x) अ [a,b]

B (x) is continuous on [a, b] except at finite

number of points of [a,b]

[a,b] छ छ (x) अ

[a,b]

C (x) may be discontinuous at an infinite number of points in [a, b]

फ (x) अ [a,b] अ अ

D Nothing can be said about continuity of (x) फ (x) छ

Q.No: 7

Consider the following statements P and Q:

P: is convergent Series

Q: .

Then

P Q पर व र ए:

P: ए अ भ र

Q:

A P Q P Q

B Q P Q P

C P Q P Q

D None of these

Q.No: 8 For a subset A of a metric space, which of the following implies the other three?

ए र म प म य A ऱय , ऱ म

अ य र

A A is closed A

B A is bounded A

C Closure of B is compact for every B A B A , B ,

D A is compact A

Q.No: 9 For the series and the sequence < 1/n >, which of the following is true?

अ म < 1/n > ऱय , म य

A Both and < 1/n > are convergent < 1/n > अ

B is convergent but < 1/n > is not convergent अ < 1/n > अ

C is not convergent but < 1/n > is convergent अ < 1/n > अ

D as well as < 1/n > both are not convergent < 1/n > अ

Q.No: 10 The function

is both continuous and differentiable at x = 0 if

फऱ

x = 0 पर अव ऱ य , य

A - 1 e only X > e

D X e X e

Q.No: 12

Let < fn > be a sequence of real valued continuous

functions defined on [0, 1] and suppose

that . Then

म ऱ य < fn > अ र ऱ [0,1]पर प रभ व व

म फऱ अ म म ऱ य

ब:

A f is continuous f

B f is continuous provided < fn(x) > is a uniformly bounded sequence

f < fn(x) > अ

C f is continuous provided < fn(x) > is a decreasing sequence

f < fn(x) > अ

D f is continuous provided < fn(x) > is convergent for some x [0, 1]

f < fn(x) > छ x [0, 1] , अ

Q.No: 13 If , which of the

following theorems enables you to conclude that c0+c1x+⋯+cnx

n = 0 for some x ]0,1[ ?

य , म य म प ऱ म म र

c0+c1x+⋯+cnxn = 0, छ x ]0,1[ ऱए ?

A Rolle's Theorem

B Intermediate value Theorem

C Fundamental Theorem of algebra

D Mean value Theorem of integral calculus

Q.No: 14 Let X be a connected subset of real numbers. If every

element of X irrational, then the cardinality of X is

य X व व य ब प म य X य

अवयव अप रम य य X ड ऱ य :

A Infinite अ

B Countably infinite अ

C 2 2

D 1 1

Q.No: 15 Far all x ]0,1[, which of the following is true ? भ x ]0,1[ ऱय , ऱ म य ?

A loge (1+x) < x loge (1+x) < x

B ex < 1+x ex < 1+x

C x < sinx x < sinx

D x < logex x < loge

x

Q.No: 16 Which of the following sets has a non-zero measure? ऱ म य म म प (म र) अ य

A Set of all rational numbers

B Set of all irrational numbers अ

C Set of all integers

D Set with finite number of elements अ

Q.No: 17 Let f : X X such that f(f(x)) = x all x X. Then म f : X X इ र f(f(x)) = x, भ x X ऱय । ब

A f is one-one and onto f छ

B f is one-one but not onto f , छ

C f is onto but not one-one f छ

D f is neither one-one nor onto f औ छ

Q.No: 18 Let d1, d2 and d3 be metrics on a set X with at least

two elements. Which of the following is NOT a metric on X?

य d1,d2 d3 म म अवयव म य X पर म ऱ म ए X पर म ?

A min{d1, 2} {d1, 2}

B max{d2, 2} अ {d2, 2}

C

D

Q.No: 19 Let f : be a monotone function. Then य f : ए ए फऱ , ब

A f is always continuous f

B f has only finitely many discontinuities f : अ

C f can have at most countably many discontinuities

f अ अ : अ

D f can have uncountably many discontinuities f अ : अ |

Q.No: 20 If A and B are 5x5 matrices of rank 4 and 2 respectively, then rank of AB is:

य A B, 5x5 य म : 4 2

ब AB :

A 5 5

B 4 4

C 3 3

D 2 2

Q.No: 21

The system of equations

-2x + y + z = a

x – 2y + z = b x + y -2 z = c

is consistent if

म र य

-2x + y + z = a

x – 2y + z = b x + y -2 z = c

, य

A a + b + c = 0 a + b + c = 0

B a + b - c = 0 a + b - c = 0

C a - b + c = 0 a - b + c = 0

D a + b + c 0 a + b + c 0

Q.No: 22 A is a square matrix of odd order and At is its transpose then A - At is always:

य A ए व म व य At इ प रव A-At व ए :

A Singular symmetric matrix

B Singular Skew-Symmetric matrix

C Non singular Skew-Symmetric matrix अ

D Non-singular matrix अ

Q.No: 23

If ,

then the trace of A102 is

य A102 अ र :

A 0 0

B 1 1

C 2 2

D 3 3

Q.No: 24

Let A be a non-zero linear transformation on a real

vector space V of dimension n. Let the subspace V0 V be the image of V under A. Let K = dim V0 < n and suppose that for some

, A2 = A. Then

म A ए n वम य व व र म V पर अ य

र प र । म प म V0 V, A अ V

ब व । म K = dim V0 < n एव म

छ ऱय A2 = A । ब

A = 1 = 1

B detA = n A = n

C is the only Eigen value of A ,A अ

D There is a non-trivial subspace V1 V such that Ax = 0 all x V1

अ छ उ V1 V x V1 Ax = 0

Q.No: 25

Let A be a non-zero 3x3 matrix with the property A2 = 0. Consider the following three statements I. A is not similar to a diagonal matrix

II. A has one non-zero Eigen vector III. A can have at most two linearly independent Eigen vectors Then, the number of true statements is

म ऱ य A ए 3x3 य र य म

A2 = 0 । ऱ पर व र य

I. A ए व य म प

II. A ए य र अ भऱ

III. A अ अ र व अ भऱ र

ब, य य :

A 1 1

B 2 2

C 3 3

D 0 0

Q.No: 26 The Eigen vector of the matrix are य अ भऱ :

A

B

C

D None of these

Q.No: 27 Let be a linear transformation with T(1,-

1,1) = (1,0) and T(1,1,1) = (0,1). Then T(a,-b,a) is given by

म य ए र प र T(1-,1,1)

= (1,0) T(1,1,1) = (0,1) ब T(a,-b,a) म :

A (a+b, a-b) (a+b, a-b)

B

C

D None of these

Q.No: 28 Let T : 3 be defined by T(x1,x2,x3) =

x1+2x2+3x3. Then Ker(T) is a

म य T : 3 इ र प रभ T(x1,x2,x3)

= x1+2x2+3x3 ब अ (T) :

A Plane

B Circle

C Straight line

D Cone

Q.No: 29 The rank of the linear transformation T : 3 3 defined by T(x1,x2,x3) = (x2,0,x3) is

T(x1,x2,x3) = (x2,0,x3) व र प रभ र प र T

: 3 3 :

A 3 3

B 2 2

C 1 1

D 0 0

Q.No: 30 Choose the correct statement from the following: ऱ म य य :

A Dimension of a subspace of a finite dimensional vector space divides the dimension of the vector space

उ उ

B The trace of 5x5 identity matrix with entries in

5 is non-zero 5 अ 5x5 अ अ

C For real square matrices A and B, the matrix AB-

BA can never be the identity matrix A B , AB-BA ।

D Every complex square matrix can be

diagonalised

Q.No: 31

For an analytic function f(z) in a domain D, consider the following three conditions

P: Re(f(z)) is constant

Q:Im(f(z)) is constant R:Arg (f(z)) is constant Which of these conditions imply that f(z) is constant?

ए ड म D म व ऱ फऱ f(z) ऱय ,

य पर व र य :

P: व व (f(z)) अ र

Q: प (f(z)) अ र

R: (f(z)) अ र

इ म - य र f(z) अ र ?

A P and Q but not R P Q R

B P and R but not Q P R Q

C Q and R but not P Q R P

D All P, Q and R P, Q R

Q.No: 32

Consider the following two statements:

P: if f(z) and are analytic in a domain D then f is

constant.

Q: is analytic then so is .

Then which of the following is correct?

ऱ पर व र य :

P: य f(z) ड म D म व ऱ f अ र

Q: व ऱ भ

ब म


B P is false but Q is true P अ Q

C Both P and Q are true P Q

D Both P and Q are false P Q अ

Q.No: 33 The value of the integral

Where c is positively oriented circle | z | = 2, is

म ऱ c म अ भ व य व |

z | = 2 , म :

A

B 0 0

C

D

Q.No: 34 The mobius transformation maps | z | <

2 onto :

म बय प र , | z | < 2 छ

र :

A It self अ

B | w |> 2 | w |> 2

C | w |> ½ | w |> 1/2

D | w |> 4 | w |> 4

Q.No: 35 The value of where c is oriented circle | z | = 2, is

, c म अ भ व य व | z

| = 2 , म :

A 0 0

B i i

C

D

Q.No: 36 The singularity of f(z) = esinz at z = is z = पर f(z) = esinz व :

A a pole

B a removable singularity अ

C a non-isolated essential singularity अ अ

D an isolated essential singularity अ

Q.No: 37 The function f(z) = x2 + y2 + ixy, where z = x + iy, is analytic:

फऱ f(z) = x2 + y2 + ixy, z = x + iy व ऱ

A at z = 0 only z = 0

B at infinity many points :

C every where

D Nowhere

Q.No: 38 , defined for z C. Then which of the

following is false?

, z C ऱय प रभ ब म

अ य

A f is entire function f

B only singularities of f are poles f अ ( ) ?

C f has infinitely many poles on the imaginary axis अ अ f अ : अ

D each pole of f is simple f अ

Q.No: 39

Let f : C C be a meromorphic function analytic at 0

satisfying for n 1. Then which of

the following is wrong?

म f : C C ए अ फऱ 0 पर व ऱ

n 1 ऱय र । ब

म अ य ( ऱ ) ?

A f(0) = ½ f(0) = 1/2

B f has a simple pole at z = -2 f z = -2 अ

C f(2) = ¼ f(2) = 1/4

D no such meromorphic function exists अ फ अ

Q.No: 40

Let G be a group of order 49. Consider the statements P & Q:

P: The normalize of an element x( e) of G is whole group G Q: The centre of G is whole Of G

Then which of the following is correct?

म य G ए 49 म , P Q पर व र य :

P: G ए अवयव x( e) म य प म G

Q: G प म G

म व प

A Both P and Q are true P Q

B P is true but Q is false P , Q अ

C Q is true but P is false Q , P अ


Q.No: 41

Let Q be the field of rational numbers and let Q(a) be

the field obtained by adjoining a Q to Q. Consider the statements P and R:

are field isomorphic. Choose your answer as:

म Q ए प रम य य म Q(a) a Q म ऱ र P Q पर व र य :

य र अप र य :

A If P and R are both true P R

B If P is true, R is false P , R अ

C If P is false, R is true P अ , R

D If P and T are both false P R अ

Q.No: 42

Consider the following statements:

1. Every finite integral domain is a field 2. Every finite division ring (Skew field) is a field 3.Every integral domain which is also a division ring is

a field Here the number of true statements is

पर व र य : 1. य प र म प य ए 2. य प र म वभ वऱय ( ब म ऱ य ) ए 3. य प य ए वभ वऱय भ , ए

य य य :

A 3 3

B 2 2

C 1 1

D 0 0

Q.No: 43 The number of group homomorphisms from the symmetric group S3 to the additive group Z/6Z is.

म म म S3 म म Z/6Z पर म म र य :

A 1 1

B 2 2

C 3 3

D 0 0

Q.No: 44 Let G be a simple group of order 60. Then म G 60 ए रऱ म । ब

A G has six sylow-5 subgroups G छ: -5 उ

B G has four sylow-3 subgroups G -3 उ

C G has cyclic subgroup of order 6 G 6 उ

D G has a unique element of order 2 G 2 अ अ

Q.No: 45 The total number of non- isomorphic groups of order 122 is

122 अ य र म ऱ य

A 1 1

B 2 2

C 61 61

D 4 4

Q.No: 46

For a field F of integers modulo 11, consider the following three statements P, Q and R :

P: x2+1 and x2+x+4 are irreducible over F

Q: and are isomorphic fields

R: Number of elements in is 121 Then choose the correct option below:

प म डय ऱ 11

F ऱय , P, Q R पर व र य : P: F पर x2+1 x2+x+4 ऱ र य

Q: य र

R: म अवयव य 121 ब म म व प :

A only one of the statements is true

B only two statements are true

C All P, Q and R are true statements P, Q R

D P and Q are true but R is false P Q R अ

Q.No: 47 Let X = {a, b, c} and = { ,{a},{b},{a,b}, X} be a topology on X. Then

म X = {a, b, c} = { ,{a},{b},{a,b},X}, X

पर । ब

A (X, ) is T2 and regular (X, ), T2

B (X, ) is regular but not normal (X, )

C (X, ) is not normal but connected (x, )

D (X, ) is not connected (X, )

Q.No: 48 Which of the following property is not hereditary in a

topological space

य म म म व

A Regularity

B Normality

C Hausdorff property उ फ

D None of these

Q.No: 49 Consider usual topology and lower limit topology '

on the set of real numbers. Then choose the correct

option below:

व व य म य पर र म ' पर व र य । एव म म व प य :

A Identity mapping is

continuous फ

B Identity mapping is continuous फ

C

Both (Identity mapping is

continuous & Identity

mapping is continuous) are true

( फ )

( फ )

D None of these

Q.No: 50 Let be defined by Then

म , व र प रभ । ब

A f is one – one, continuous but not onto f , छ

B f is onto, continuous but not one – one f छ ,

C f is one – one,onto and continuous f , छ

D f is one – one, onto but f-1 is not continuous f छ f-1

Q.No: 51

Let P be a continuous function on and W , the Wronskian of two linearly independent solutions y1 and y2 of the ODE

Let W(1) = a, W(2) = b and W(3) = c, then

म P, पर ए फऱ W .अ. म .

र य व ऱ y1 y2 र य , य W(1) = a, W(2) = b and W(3) = c,

A a < 0 and b > 0 a < 0 b > 0

B a < 0, b < 0 and c > 0 a < 0, b < 0 c > 0

C a b > c a b > c

D 0 < a c > 0 0 < a c > 0

Q.No: 52 Solution of ODE ysin2x dx + (-y2 – cos2x)dy = 0 is .अ. म . ysin2x dx + (-y2 – cos2x)dy = 0 ऱ :

A 3ycos2x + y3 = c, c is constant 3ycos2x + y3 = c, c अ

B 3ycos2x + 2y3 = c, c is constant 3ycos2x + 2y3 , c अ

C 3y2sin2x -2y3 = c, c is constant 3y2sin2x -2y3 , cअ

D 3ycos2x + y3 = c, c is constant 3ycos2x + y3 = c, c अ

Q.No: 53 If y1(x) and y2(x) are solution of ODE y’’+ xy’ + (1-x2)y = sinx then which of the following is also its solution:

य y1(x) y2(x) .अ. म .ODE y’’+ xy’ + (1-x2)y =

sinx ऱ म भ इ ऱ :

A y1(x) + y2(x) y1(x) + y2(x)

B y1(x) - y2(x) y1(x) - y2(x)

C 2y1(x) - y2(x) 2y1(x) - y2(x)

D y1(x) - 2y2(x) y1(x) - 2y2(x)

Q.No: 54 If y1(x) and y2(x) are solutions of y" + x2y' + (1-x)y = 0 such that y1(0) = 0, y1'(0) = -1 and y2(0) = -1, y2'(0) = 1, then the Wronskian W(y1,y2) on

य y" + x2y' + (1-x)y = 0 ऱ y1(x) y2(x) इ

र y1(0) = 0, y1'(0) = -1 y2(0) = -1, y2'(0)

= 1 ब र य W(y1,y2) , पर

A is never zero

B is identically zero

C is zero only at finite number of points :

D is zero at countably infinite number of points अ

Q.No: 55 The PDE y3uxx – (x2 + 1)uyy = 0 is .अ. म . y3uxx – (x2 + 1)uyy = 0 :

A Parabolic in {(x,y): x < 0} {(x,y): x < 0}

B hyperbolic in {(x,y): y > 0} {(x,y): y > 0} अ

C elliptic in 2 2

D parabolic in {(x,y): x > 0} {(x,y): x > 0}

Q.No: 56 If (x) be a solution of ODE, xy" + y' + xy = 0, then which of the following differential equation is satisfied

by the function x1/2 (x):

य (x) .अ. म . xy" + y' + xy = 0 ऱ फऱ

x1/2 (x), म म अव ऱ म र र .

A x2y" + (4x2 + 1)y = 0 x2y" + (4x2 + 1)y = 0

B x2y" + (4x2 - 1)y = 0 x2y" + (4x2 - 1)y = 0

C x2y" + (x2 + 1/4)y = 0 x2y" + (x2 + 1/4)y = 0

D x2y" + (x2 – 1/4)y = 0 x2y" + (x2 – 1/4)y = 0

Q.No: 57

A particular solution of the

PDE is .अ. म ए व ऱ :

A

B

C

D

Q.No: 58

Using Euler's method taking step size = 0.1, the approximate value of y obtained corresponding to x =

0.2 for the initial value problem dy/dx = x2 + y2 , y(0) = 1 is

प प = 0.1 ऱ र यऱर व य र र भ म

म य dy/dx = x2 + y2, y(0) = 1 ऱय x = 0.2 y म

A 1.322 1.322

B 1.122 1.122

C 1.222 1.222

D 1.110 1.110

Q.No: 59 A function y(x) is given as y(1) = 3.24, y(6) = 8.43 and y(10) = 24.6. Using Lagrange’s interpolation, the approximate value of y(4) is

फऱ y(x) ऱय य y(1) = 3.24, y(6) = 8.43 y(10) = 24.6, ब ऱ अ व य र पर y(4) म :

A 4.35 4.35

B 4.82 4.82

C 5.65 5.65

D 6.12 6.12

Q.No: 60

A function f(x) is given as f(1.0) = 2.32, f(1.2) = 4.68

and f(1.4) = 6.48. Using Simpson's rule, the value

of is

ए फऱ f(x) ऱय , f(1.0) = 2.32, f(1.2) = 4.68 f(1.4) = 6.48 य य । प यम य

र म :

A 1.81 1.81

B 1.83 1.83

C 1.85 1.85

D 1.89 1.89

Q.No: 61 Consider the functional I , y(0) = 1, where y C2([0,1]). If y extremizes I, then

फऱ I , y(0) = 1, y C2([0,1]) पर व र ए,य I y र म र , ब

A

B

C

D

Q.No: 62

If 1, 2 be the characteristic numbers and f1 and f2 the corresponding Eigen functions for the homogeneous integral

equation , consider the following statement:

Then

य म म ऱ

म र अ भऱ म 1एव 2 f1 एव f2 इ अ भऱ

फऱ पर व र य :

A both P and Q are true P Q

B P is true, Q is false P , Q अ

C P is false, Q is true P अ , Q


Q.No: 63

The integral

equation has

म ऱ

म र र :

A no solution

B unique solution अ

C infinitely many solution अ :

D exactly two solutions

Q.No: 64 If L and V are respectively the Lagrangian function and potential energy of a conservative holonomic dynamical system, then

य र ऱ म य ऱ फऱ म : L V

A L = constant L = अ

B L + V = constant L + V = अ

C L + 2V = constant L + 2V = अ

D L – V = constant L – V = अ

Q.No: 65

Consider the following three statements :

i. Hamilton's principle follows from the D'Alembert's principle ii. Hamilton's principle follows from Lagrange's equations.

iii. Newton's second law of motion follows from the Hamilton's principle Then the number of true statements is

पर व र य :

i. म डएऱ व अ म

ii. म ऱ म र अ म

iii. य व य यम म अ म

ब य य

A 1 1

B 2 2

C 3 3

D 0 0

Q.No: 66

For a simple pendulum of length l and mass m, consider the following statements

i. The Lagrangian function

is

ii. The equation of motion is

here, choose

ऱ ब l यम m ए र ऱ ऱय

पर व र य :

i. इ ऱ य फऱ

।

ii. इ म र । य य :

A both (i) and (ii) are true (i) (ii)

B (i) is true but (ii) is false (i) (ii) अ

C (i) is false but (ii) is true (i) अ (ii)

D (i) and (ii) are both false (i) (ii) अ

Q.No: 67 If x is a Poisson random variate with mean 3, then P

(|x – 3|<1) will be

य x म य 3 ए व य छ व र , P (|x –

3|<1)

A

B

C

D

Q.No: 68

The mean and variance of the number of defective items drawn randomly one by one with replacement from a lot are found to be 10 and 6 respectively. Then distribution of the number of defective items is

य ऱ य छ : प : प र य ए -ए र ऱ य य म य म य व र म :

10 व 6 य म य ब :

A Poisson with mean 10 10

B Binomial with n = 25 and p = 0.4 n = 25 p = 0.4

C Normal with mean 10 and variance 6 10 6

D None of these

Q.No: 69

From the six letters A, B, C, D, E and F, three letters are chosen at random with replacement. What is the probability that either the word BAD or the word CAD can be formed from the chosen letters?

छ: अ र A, B, C, D, E F म य छ : अ र प :

प । य अ र BAD य

CAD र र य य

A 3/216 3/216

B 6/216 6/216

C 9/216 9/216

D 12/216 12/216

Q.No: 70

At a doctor's clinic patients arrive at an average rate of 10 per hour. The consultancy time per patient is exponentially distributed with an average of 6 minutes per patient. The doctor does not admit any patient if at any time 10 patients are waiting. Then at the steady state of this M|M|1|R queque the expected

number of patients waiting is

ए ड र ऱ म 10 म य र र य म । र पर म ऱ म य 6 म र य

र : ब । भ मय य 10 र र र ड र अ य र व अ म , र अव म इ M|M|1|R र म य र य य :

A 0 0

B 5 5

C 9 9

D 10 10

Q.No: 71 Suppose X has density ,

where > 0 is unknown. Define Y as follows: Y= k if k X < k + 1, k = 0, 1, 2,. . . , then the distribution of Y is

म X व । > 0 अ । Y प रभ व र : Y= k if

k X < k + 1, k = 0, 1, 2,. . . ,

Y ब

A Normal

B Binomial

C Poisson

D Geometric

Q.No: 72 If a primal linear programming problem has degenerate optimal solution then its dual problem has

य ए य र म म य अप इ म ऱ र व म य ऱ :

A Alternative optima solution

B degenerate optimal solution अ

C no feasible solution

D no optimal solution

Q.No: 73

If (1, 0, 3) is an optimal solution of LPP: Maximize z = c1 x + c2 y + c3 z Subject to

x + y + z 4,

य र म म य अ म र z = c1 x + c2 y + c3 z

ब

x + y + z 4,

x 2, z 3, 3x + z 7, x, y, z 0 . Then

x 2, z 3, 3x + z 7,

x, y, z 0 (1, 0, 3) ए इ म ऱ

A c1 c2 c3 c1 c2 c3

B c3 c2 c1 c3 c2 c1

C c2 c1 c3 c2 c1 c3

D c2 c3 c1 c2 c3 c1

Q.No: 74

Consider the LPP Maximize : x1 + x2

Subject to x1 - 2x2 12 , x2 - 2x1 12 , x1 , x2 0. Then

र म म य अ म र : x1 + x2

ब : x1 - 2x2 12 ,

x2 - 2x1 12 ,

x1 , x2 0 . ब

A The LPP admits an optimal solution , (ऑ )

B The LPP admits no feasible solution ,

C The LPP is unbounded अ

D None of these

Q.No: 75

The optimal table for the primal LPP Maximize z = 6x1 + 12x2 + 12x3 - 6x4

Subject to x1 + x2 + x3 = 4, x1 + 4x2 + x4 = 8 and x1, x2, x3, x4 0 is

If y1 and y2 are the dual variables corresponding to

the first and second constraints then their values in the optimal solution of the dual problem are respectively

य र म म य

अ म र : z = 6x1 + 12x2 + 12x3 - 6x4

ब : x1 + x2 + x3 = 4,

x1 + 4x2 + x4 = 8 x1, x2, x3, x4 0 इ म

ऱ -

य म व य य ब व म य

र y1 y2 व म य इ म ऱ म म

म : :

A 0 and 6 0 6

B 12 and 0 12 0

C 6 and 3 6 3

D 4 and 4 4 4

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