## 13 May 50 Multiple Choice Questions

*Find the correct option:*

**1)** The units digit of the number is

A) 3

B) 1

C) 9

D) 2

**2)** If is a cyclic group of order 24 and where and Then the value of is

A) 4

B) 6

C) 8

D) 10

**3)** The ring is

A) a PID

B) a UFD

C) a PID and a UFD

D) none of these

**4)** Consider the matrix

then

A) is diagonalizable over

B) is diagonalizable over

C) is not diagonalizable.

D) none of these.

**5)** The system of linear equations

has

A) a unique solution

B) infinite solutions

C) no solution

D) two solutions

**6)** The functional form of the linear transformation from to itself whose matrix with respect to the basis is

A)

B)

C)

D)

**7)** If is defined by then the nullity of is

A) 0

B) 1

C) 2

D) 3

**8)** Let be an one-to-one map. Then which of the following is not correct?

A) may be a subset of

B) may be a subset of

C) should be equal to

D) cardinality of should be equal to cardinality of

**9)** Let and for all Then the interval in which is increasing is

A)

B)

C)

D) none of these

**10)** Consider the function

Choose the correct answer:

A) is continuous, but not differentiable at the origin.

B) is continuous and differentiable at the origin.

C) and exist, and

D) and are continuous at (0,0), and and do not exist.

**11)** Let

Choose the correct one

A)

B)

C)

D)

**12)** The sequence is

A) monotonically increasing

B) increasing and bounded

C) non-increasing and bounded

D) non-increasing, but not bounded

**13)** Consider the following two statements:

(I) A complex valued function is analytic in a region

(II) is such that and in a region

Choose the correct statement:

A) I and II are equivalent statements.

B) I does not imply II.

C) II implies I

D) II is necessary condition for I, but not sufficient in general.

**14)** Let be the line segment joining to Then the value of is

A)

B)

C)

D)

**15)** Detect the wrong statement:

A) Let be continuous in a simply connected an suppose that around every simple closed curve in Then is a constant function.

B) If and are any two points in then is independent of the path in a region joining and

C) Let be analytic in the region bounded by non-overlaping simple closed curves [where are inside ] and on these curves. Then

D) The following result is always true

**16)** The solution of the equation satisfying the condition is

A)

B)

C)

D)

**17)** The solution of the partial differential equation is given by

A) and

B) and

C) and

D) and

**18)** Given that is one solution of 0." />

Then the second linearly independent solution is

A)

B)

C)

D)

**19)** For the differential equation is a partial integral if

A)

B)

C)

D)

**20)** The solution of the partial differential equation is

A)

B)

C)

D)

**21)** The moment of inertia of a rectangle of mass and sides about a diagonal is

A)

B)

C)

D) none of these

**22)** The product of inertia of uniform rectangular lamina about a pair of axes at its C.G. parallel to its edge is

A)

B)

C)

D)

**23)** The moment of inertia of a cube of edge and mass about a line through its centre is

A)

B)

C)

D)

**24)** A system consisting of two particles moves on a plane. Then the degree of freedom is

A) 2

B) 3

C) 4

D) 6

**25)** For a conservative holonomic dynamical system, the Lagrangian kinetic energy and potential energy are connected by

A)

B)

C)

D)

**26)** Kinematics is concerned with

A) the physical causes of the motion.

B) the condition under which no motion is apparent

C) the geometry of the motion

D) none of these

**27)** The angular momentum and the external torque of a rigid body about a point is connected by

A)

B)

C)

D)

**28)** The directional derivative of at the point in the direction of is

A)

B)

C)

D) none of these

**29)** A unit vector normal to the surface at the point is

A)

B)

C)

D)

**30)** If then the value of at all points except is

A) 0

B) 1

C) 3

D) -2

**31)** The vector is solenoidal. Then the value of is

A) 2

B) 1

C) 0

D) -2

**32)** If where then equals

A)

B)

C)

D)

**33)** The directional derivative of at the point in the direction of is

A) -1

B) 0

C) 1

D) 2

**34)** The Fourier sin transform of is defined by

A)

B)

C)

D)

**35)** If and then is given by

A)

B)

C)

D)

**36)** equals

A)

B)

C)

D)

**37)** equals

A)

B)

C)

D) none of these

**38)** The firs iterated kernel of the kernel ; is given by

A)

B)

C)

D)

**39)** The necessary condition for an admissible function to have an extremum of are

A)

B) must be continuous

C) must be continuous

D) All of these

**40)** In Simpson’s rule we replace the graph of the given function by some

A) second degree polynomials

B) third degree polynomials

C) fourth degree polynomials

D) fifth degree polynomials

**41)** The basis of polynomial interpretation is

A) Taylor’s Theorem

B) Weierstrass Approximation Theorem

C) Rolle’s Theorem

D) Mean Value Theorem

**42)** “Mathematical Expectation of the product of two random variables is equal to the product of their expectations” is true for

A) any two random variables.

B) if the random variables are independent.

C) if the covariance between the random variables is non zero.

D) if the variance of the random variables are equal.

**43)** If a random variable follows normal distribution with mean and variance then the random variable follows normal distribution with

A) Mean = variance =

B) Mean = variance =

C) Mean = variance =

D) none of these

**44)** The number of real roots of the equation on equals the difference between the number of changes in sign Sturm sequence at and provided that

A)

B)

C)

D)

**45) **Let and Then

A)

B) since and are cyclic.

C) and are not isomorphic.

D) and are not isomorphic, since their identities are not equal.

*[Hint:- The group is cyclic.]*

**46) **Consider the group under addition modulo

(i) is the unique subgroup of of order

(ii) is the unique subgroup of of order

Then,

A) (i) is true, but (ii) is false

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

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

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

**47) **How many elements of order are there in ?

A) 5

B) 21

C) 35

D) 33

**48)** Let and be continuous and Then

A) for some

B)

C) for some

D)

*[Hint:- For all there exist such that ]*

**49) **Following are three statements:

(i) Any n-dimensional real vector space is isomorphic to

(ii) Any n-dimensional complex vector space is isomorphic to

(iii) Any n-dimensional vector space over the field F is isomorphic to

Then

A) Only i and ii are true.

B) i is true, but ii and iii are not true.

C) None of them is true

D) All of them are true.

**50) **Following are two statements:

(i) Two finite-dimensional vector spaces over the same field are isomorphic.

(ii) Two finite-dimensional vector spaces over the same field and of the same dimension are isomorphic.

Then

A) i is true but ii is not true.

B) ii is true, but i is not true.

C) None of them is true

D) All of them are true.

_______________________________________________________________________

**Answers:**

**1)** B , **2)** D , **3)** B , **4)** __ **5)** B , **6)** B , **7)** __ **8)** C , **9)** A , **10)** __

**11)** __ **12)** B , **13)** __ **14)** D , **15)** __ **16)** __ **17)** B , **18)** C , **19)** __ **20)** __

**21)** __ **22)** __ **23)** B , **24)** C , **25)** B , **26)** C , **27)** A , **28)** __ **29)** __ **30)** A ,

**31)** D , **32)** B , **33)** C , **34)** A , **35)** D , **36)** C , **37)** A , **38)** C , **39)** D , **40)** __

**41)** __ **42)** B , **43)** B , **44)** __ **45)** C , **46)** C , **47)** __ **48)** B , **49)** D , **50)** B.

#### Pankaj Jyoti Mahanta

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## Yogesh

Posted at 13:56h, 31 MayCan you tell me the answer of question no. 13?

## Gonit Sora

Posted at 20:12h, 01 JuneThe answers are mentioned at the end of the post.

## Sachin

Posted at 14:08h, 01 NovemberQ.13 answer is D