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IIT JAM Mathematics Practice Test- 19 - Mathematics MCQ


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30 Questions MCQ Test IIT JAM Mathematics Mock Test Series - IIT JAM Mathematics Practice Test- 19

IIT JAM Mathematics Practice Test- 19 for Mathematics 2024 is part of IIT JAM Mathematics Mock Test Series preparation. The IIT JAM Mathematics Practice Test- 19 questions and answers have been prepared according to the Mathematics exam syllabus.The IIT JAM Mathematics Practice Test- 19 MCQs are made for Mathematics 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for IIT JAM Mathematics Practice Test- 19 below.
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IIT JAM Mathematics Practice Test- 19 - Question 1

The order of subgroup  <5>  <3> in Z30 Z12 is

Detailed Solution for IIT JAM Mathematics Practice Test- 19 - Question 1

Since 5 has order 6 in Z30 and 3 has order 4in Z12
Hence, <5>  <3> is a subgroup of order 24 in Z30 Z12.

IIT JAM Mathematics Practice Test- 19 - Question 2

For what value of n, double integral  where region R is bounded by x + y = 1 in the first quadrant?

Detailed Solution for IIT JAM Mathematics Practice Test- 19 - Question 2

Given that


⇒ n + 2 = 4
⇒ n = 2

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IIT JAM Mathematics Practice Test- 19 - Question 3

If  is harmonic in V then for any closed surface S consisting the volume V, the integral  is equal to 

Detailed Solution for IIT JAM Mathematics Practice Test- 19 - Question 3

By definition of gradient of 

Now,


Now by gauss - divergence theorem

Here 

IIT JAM Mathematics Practice Test- 19 - Question 4

The transformation T : R3 → R2 defined by, 
T(x,y,z) = (x +y, y+z) is,

Detailed Solution for IIT JAM Mathematics Practice Test- 19 - Question 4




⇒ T is linear 

IIT JAM Mathematics Practice Test- 19 - Question 5

The radius of convergence of power series  is,

Detailed Solution for IIT JAM Mathematics Practice Test- 19 - Question 5





⇒ R = 1/ea

IIT JAM Mathematics Practice Test- 19 - Question 6

A cauchy sequence in Q which does not have a limit in Q is

Detailed Solution for IIT JAM Mathematics Practice Test- 19 - Question 6

Here sequence in option (A) is not convergent ⇒ it is not cauchy. Sequence in option (C) is convergent which converge to 0.
⇒ It be a cauchy sequence which converge in Q.
Similarly sequence in Option (D) has two subsequences.
 and both are converging to 0. 
But sequence in option (B) will converge to e.
⇒ It be a cauchy sequence which is not converging in Q.

IIT JAM Mathematics Practice Test- 19 - Question 7

Suppose the matrix  has a certain complex no. λ ≠ 0 as an eigenvalue. Which of the following no. must also be an eigenvalue of A? 

Detailed Solution for IIT JAM Mathematics Practice Test- 19 - Question 7

We know that trace (A) 
i.e. 
also know that det (A) = 
Now 

IIT JAM Mathematics Practice Test- 19 - Question 8

If A and B are unbounded subsets of R, which of the following is necessarily unbounded?

Detailed Solution for IIT JAM Mathematics Practice Test- 19 - Question 8

Here A\B = A - B
So (A \ B) U (A ∩ B) = A, which is unbounded.

IIT JAM Mathematics Practice Test- 19 - Question 9

Let f: R3 → R3 be the matrix transformation defined by
 if we have,
 then for the solution, i.e. for the existence of values of x,y and z.

Detailed Solution for IIT JAM Mathematics Practice Test- 19 - Question 9

Given 



Thus for the solution, we must have
c + b - a = 0

IIT JAM Mathematics Practice Test- 19 - Question 10

Find the total no. of elements of order 12 in a cyclic group of order 60.

Detailed Solution for IIT JAM Mathematics Practice Test- 19 - Question 10

By a well-known theorem, we know that " If d is a positive divisor of n, the number of elements of order d in a cyclic group of order n is φ(d)".
Where φ (d) denotes Euler's φ - function
Here n = 60 and d = 12 (12 is divisor of 60)
then φ (12) = 1, 5, 7, 11
Hence total no. of elements of order 12 in cyclic group of order 60 is 4.

IIT JAM Mathematics Practice Test- 19 - Question 11

Which of the following pair of groups is isomorphic to each other?

Detailed Solution for IIT JAM Mathematics Practice Test- 19 - Question 11

Since U (10) and U (5) are cyclic group of order 4.
So U (10) ≌ U (5)
Now, Suppose that φ is an isomorphism from U (10) onto U (12). then 

and

IIT JAM Mathematics Practice Test- 19 - Question 12

If G is a group, then for every a ∈ G, what is (a−1)−1?

Detailed Solution for IIT JAM Mathematics Practice Test- 19 - Question 12

IIT JAM Mathematics Practice Test- 19 - Question 13

Let f' (sin x) < 0 and f' (cos x) > 0.  and g (x) = f (sin x) + f (cos x) then which of the following is true?

Detailed Solution for IIT JAM Mathematics Practice Test- 19 - Question 13



IIT JAM Mathematics Practice Test- 19 - Question 14

Let  be  a constant vector and V is the volume enclosed by the closed surface S. The integral  is equal to (where ñ is the outward going unit normal to the surface S)

Detailed Solution for IIT JAM Mathematics Practice Test- 19 - Question 14

If  be any vector function then by the result we know that


by comparison of the result it is clear that 
Hence curl 



[Where V is volume enclosed by closed surface S]

IIT JAM Mathematics Practice Test- 19 - Question 15

Find the value of the limit 

Detailed Solution for IIT JAM Mathematics Practice Test- 19 - Question 15

The rth term  where r varies form 1 to n.
Hence given limit is 
Hence, By the sum of the series by integration, this limit can be written as.

IIT JAM Mathematics Practice Test- 19 - Question 16

If  then find the length of the curve y = f(x) from the vertex (0, 0) to the point (a, a).

Detailed Solution for IIT JAM Mathematics Practice Test- 19 - Question 16

Given that 
then required length of the curve is 
  

IIT JAM Mathematics Practice Test- 19 - Question 17

Change the order of integration in 

Detailed Solution for IIT JAM Mathematics Practice Test- 19 - Question 17


which is a parabola with vertex (1, 1).

Here limit is f(X), so we have to consider a horizontal strip

 the bounded region takes values less than 1.}

IIT JAM Mathematics Practice Test- 19 - Question 18

Given 
and G(x) = (x - 1/2)2, x ∈ R
what is the area of the portion bounded between G(x) and f(x) inn the interval 

Detailed Solution for IIT JAM Mathematics Practice Test- 19 - Question 18


Hence Area

IIT JAM Mathematics Practice Test- 19 - Question 19

 are three vectors with magnitude |a| = 4, |b| = 4, |c| = 2 and such that  is perpendicular to  is perpendicular to  is perpendicular to ,  then  is equal to 

Detailed Solution for IIT JAM Mathematics Practice Test- 19 - Question 19

Given |a| = 4, |b| = 4, |c| = 2
and  


adding (i) (ii) and (iii) we get



= 6

IIT JAM Mathematics Practice Test- 19 - Question 20

If  and  =  (1, a, a2)
 are non-coplanar vectors, then abc is equal to 

Detailed Solution for IIT JAM Mathematics Practice Test- 19 - Question 20

Since (1, a, a2), (1, b, b2) and (1, c, c2) are non-coplanar therefore

and 

IIT JAM Mathematics Practice Test- 19 - Question 21

The differential equation  determines a family of circles with,

Detailed Solution for IIT JAM Mathematics Practice Test- 19 - Question 21


IIT JAM Mathematics Practice Test- 19 - Question 22

The orthogonal trajectories of the hyperbolas xy = c is

Detailed Solution for IIT JAM Mathematics Practice Test- 19 - Question 22

xy = c

 : DE of required O.T

IIT JAM Mathematics Practice Test- 19 - Question 23

Let P : R → R be a polynomial of the form P(x)  with  and 


If |x| is the absolute value of x ∈ R, then

Detailed Solution for IIT JAM Mathematics Practice Test- 19 - Question 23


Now take


Now take

IIT JAM Mathematics Practice Test- 19 - Question 24

A: R6 → R6 be a linear transformation such that A2 = 0, then the rank of A is 

Detailed Solution for IIT JAM Mathematics Practice Test- 19 - Question 24

We know if A and B are n x n matrices, then Rank (A) + Rank (B) - n ≤ Rank (AB)
where n be the order of matrices,
Take A =  B then we have
Rank (A) + Rank (A) - n ≤ Rank (A2)

IIT JAM Mathematics Practice Test- 19 - Question 25

Let V be the space of twice differentiable functions on R satisfying f'' - 2f' + f = 0. Define T : V → R2 by T(f) = (f'(0), f(0)), Then T is

Detailed Solution for IIT JAM Mathematics Practice Test- 19 - Question 25




⇒ T cannot be onto
⇒ T not one-one

IIT JAM Mathematics Practice Test- 19 - Question 26

Which one of the following is true?

Detailed Solution for IIT JAM Mathematics Practice Test- 19 - Question 26

If T : R4 → R3, be a L.T. then this map cannot be one-one, since dim (R4) = 4.
So ker (T) ≠ {0}.
(B) We know if V be a V.S. of all (n x n) skew-symmetric matrices then

In this case we cannot find out matrix representation of T w.r. to basis  of Mn(Q) but if F =R1, Then always char. eqn of T is same as A.
⇒ only (D) is correct.

IIT JAM Mathematics Practice Test- 19 - Question 27

Let f :  denote the function defined by f(x) =   Which of the following statement is correct? 

Detailed Solution for IIT JAM Mathematics Practice Test- 19 - Question 27


Clearly f is continuous at  and hence on R
Now 


IIT JAM Mathematics Practice Test- 19 - Question 28

Let {an} be a sequence of real no. such that  Then

Detailed Solution for IIT JAM Mathematics Practice Test- 19 - Question 28

We know if {an} be a sequence s.t.

Then if ∑un is convergent then sequence {an} is Cauchy and hence convergent
Now here we have 

IIT JAM Mathematics Practice Test- 19 - Question 29

Consider the surface corresponding to the equation 4x2 + y2 + z = 0 A possible unit tangent to this surface at the point (1, 2, -8) is

Detailed Solution for IIT JAM Mathematics Practice Test- 19 - Question 29


at the point (1,2,-8) the normal vector the surface is given by

IIT JAM Mathematics Practice Test- 19 - Question 30

If the surface integral of the field  over the closed surface of an arbitrary unit sphere is to be zero, then the relationship between 

Detailed Solution for IIT JAM Mathematics Practice Test- 19 - Question 30

Here given surface be closed , so we can apply Gauss - divergence theorem, we have



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