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IIT JAM Mathematics Practice Test- 21 - Mathematics MCQ


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30 Questions MCQ Test - IIT JAM Mathematics Practice Test- 21

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IIT JAM Mathematics Practice Test- 21 - Question 1

If f(x) = xα In x and f(0) = 0, then the value of α for which Rolle’s theorem can be applied in [0,1] is,

Detailed Solution for IIT JAM Mathematics Practice Test- 21 - Question 1


The function has to be continuous in [0,1]

IIT JAM Mathematics Practice Test- 21 - Question 2

The value of  is _____.

Detailed Solution for IIT JAM Mathematics Practice Test- 21 - Question 2

Let 
The limit of integration are 


on reversing the order of integration 
The limit of x and y becomes

The integral is then 

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IIT JAM Mathematics Practice Test- 21 - Question 3

The solution of differential equation, 

Detailed Solution for IIT JAM Mathematics Practice Test- 21 - Question 3

Given equation can be written as,


the given equation becomes,

IIT JAM Mathematics Practice Test- 21 - Question 4

What is the laplce tranform of the first derivative of a function y(t) with respect to t : y’(t)?

Detailed Solution for IIT JAM Mathematics Practice Test- 21 - Question 4

Let  f(t) = y(t)

= −f(0) + sF(s)
= sY(0) − y(0)

IIT JAM Mathematics Practice Test- 21 - Question 5

Let V(F) be a vector space of dimension 10 and let U and W are two distinct 7 -dimensional vector subspaces of V. Then dimensions of the subspace (U∩W) cannot be?

Detailed Solution for IIT JAM Mathematics Practice Test- 21 - Question 5

We have dim U = dim W = 7 and U and W are distinct 
⇒ U + W subspace properly contains U and W.
⇒ dim(U + W) > 7
But also ( U + W ) be a subspace of V
⇒ dim(U + W) ≤ dim V = 10
⇒ (U + W) can have dimensions 8. 9 or 10.
Now dim (U ∩ W) = dim U + dim W - dim (U + W)
= 14 - dim(U + W)

IIT JAM Mathematics Practice Test- 21 - Question 6

let S be the boundary of the region consisting of the parabolic cylinder z = 1 -xand the planes y = 0 ,y = 2 and z = 0. Evaluate the integer  where  is the outward drawn unit normal to s.

Detailed Solution for IIT JAM Mathematics Practice Test- 21 - Question 6


By Gauss- divergence theorem

IIT JAM Mathematics Practice Test- 21 - Question 7

Solution of the differential equation  by taking y = z2 is,

Detailed Solution for IIT JAM Mathematics Practice Test- 21 - Question 7

y = z2

So the given equation can be written as,


Thus equation (1) can be written as

where D = d/dt

IIT JAM Mathematics Practice Test- 21 - Question 8

Consider three vector in the space P:(-2,1,5) A:(1,2,-5) and B:(7,5,-9) then find the perpendicular distance of the point p to the line joining the paints AB.

Detailed Solution for IIT JAM Mathematics Practice Test- 21 - Question 8




So required perpendixular length is

= 7

IIT JAM Mathematics Practice Test- 21 - Question 9

The general solution of yy" - ( y')2 = 0 is,

Detailed Solution for IIT JAM Mathematics Practice Test- 21 - Question 9

We have 

Integrate with respect to x,

IIT JAM Mathematics Practice Test- 21 - Question 10

The value of the integral  is

Detailed Solution for IIT JAM Mathematics Practice Test- 21 - Question 10

Region of integration is bounded by curves 

after changing the order of the integration we get
after changing the order of the integration we get

IIT JAM Mathematics Practice Test- 21 - Question 11

If f(x) = x2 + 2bx + 2c2 and g(x) = -x2 - 2cx + b2
such that minf(x) > maxg(x), then the relation between b and c is

Detailed Solution for IIT JAM Mathematics Practice Test- 21 - Question 11

When we calculate minimum of f(x) and maximum of g(x) then we find that f(x) has minimum at x = - b and g(x) has maximum at x = - c 
then minf(x) > maxg(x)

IIT JAM Mathematics Practice Test- 21 - Question 12

If  then value of a for which A2= B, is _____

Detailed Solution for IIT JAM Mathematics Practice Test- 21 - Question 12


⇒α2 = 1 and α + 1 = 5
which is not possible at the same time, 
∴ no real values of α exists.

IIT JAM Mathematics Practice Test- 21 - Question 13

Given a real valued function ‘f such that

hen the value of  is
(where {.} and [.] denotes fractional part and greatest integer functions)

Detailed Solution for IIT JAM Mathematics Practice Test- 21 - Question 13


IIT JAM Mathematics Practice Test- 21 - Question 14

Let X be a connected subset of real number’s. If every element of X is rational, then the cardinality of X is -

Detailed Solution for IIT JAM Mathematics Practice Test- 21 - Question 14

We know if X ⊂ R then X is connected 
⇔ X be an interval
So if X be a non-empty and non-singleton connected set then X always has rational and irrational elements.
So there is only possibility for X is X be singleton 
Set i.e. X = {a} where a ∈ Q 
⇒ cardinality of X is 1.

IIT JAM Mathematics Practice Test- 21 - Question 15

Let 
be an identity in λ (i.e a polynomial in λ) where p, q, r, s and t are constants, then the value of t is

Detailed Solution for IIT JAM Mathematics Practice Test- 21 - Question 15

We put λ = 0 in the given equation, so that we can get

t = 0 ( 0 + 16) + 1( -1 2 ) + 3 ( 4 - 0 ) = -12 + 12 = 0
⇔ t = 0

IIT JAM Mathematics Practice Test- 21 - Question 16

Find the work done in moving a particle once around a circle C in the xy plane, if the circle has a centre (0, 0) and radius 1 and if the force field  is given by

Detailed Solution for IIT JAM Mathematics Practice Test- 21 - Question 16

Equation of the circle is x2 + y2=1
Let x = cosθ, y = sinθ
then dx = -sinθdθ, dy = cosθdθ

and z = 0 ⇒ dz = 0
So total work done is 

IIT JAM Mathematics Practice Test- 21 - Question 17

If T : R→ Ris given by T(x , y ,z) =( x - y, y + 3z ,x + 2y), then T-1 is

Detailed Solution for IIT JAM Mathematics Practice Test- 21 - Question 17

Here T is one-one and onto so T-1 is exist 
Let T ( x , y , z) = ( p , q , r)
⇒ T-1(p, q r) = (x, y, z)
So Now T(x , y , z) = (p ,q ,r )
⇒ (x - y, y + 3z. x + 2y) = (p, q. r)
x - y = p ...(i)
y + 3z = q ...(ii)
x + 2y = r ...(iii)
equation (1) and equation (3) ⇒ - 3y = p - r

then by equation(1), 

and by equation (2),

IIT JAM Mathematics Practice Test- 21 - Question 18

Consider the power series  Where a1 = 1/2 and  Then which of the following is the region of convergence of power series?

Detailed Solution for IIT JAM Mathematics Practice Test- 21 - Question 18


IIT JAM Mathematics Practice Test- 21 - Question 19

Which of the following statement is true?

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

(A) We know if T: U → V be a L.T and dim U =m,dim V=n .Then if m>n then then T can not be one-one.
here U = Rso its dim is 4 
and V = Rso its dim is 3 
Thus T can not be one-one
(B) We know if V be a V .S of all n x n, skew symmetric matrices, then

So here n = 4, so dim V = 4.3/2 = 6
(C) Clearly characteristic polynomial of A will be of degree n. while characteristic polynomial of T will be of degree n2
So they can not be equal.
Thus option A.B,C can not be true, 
hence option (D) is correct.
That vector space can have basis as {(1,0),(0,1)} or {(1,1),(0,1)} or {(1,1),(1,0)}

IIT JAM Mathematics Practice Test- 21 - Question 20

The value of
 is ____.

Detailed Solution for IIT JAM Mathematics Practice Test- 21 - Question 20

Applying L’ Hospital ( from), and using leibnitz's rule
we get

IIT JAM Mathematics Practice Test- 21 - Question 21

Consider the quotient group Q/Z of the additive group of rational numbers. The order of the element  is

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

IIT JAM Mathematics Practice Test- 21 - Question 22

The value of  is

Detailed Solution for IIT JAM Mathematics Practice Test- 21 - Question 22

IIT JAM Mathematics Practice Test- 21 - Question 23

Let A is the subset of R given as , Then which one is incorrect for the set A?

Detailed Solution for IIT JAM Mathematics Practice Test- 21 - Question 23




here A is the union of three open intervals. Hence A is open, since a not an interval ⇒ A not connected.
Clearly set of boundary points of is  and also A has no isolated points.

IIT JAM Mathematics Practice Test- 21 - Question 24

Which of the following is incorrect?

Detailed Solution for IIT JAM Mathematics Practice Test- 21 - Question 24

By Property of direct product
 iff ( m,n) = 1 i.e.g.c.d of m & n is 1.

IIT JAM Mathematics Practice Test- 21 - Question 25

Let G denotes the group of all 2 x 2 invertible matrices with entries from R. Let H1 ={ A ∈ G: det (A )= 1} and H2= { A ∈ G A is upper triangular} consider the following statement:
P = H1ΔG
Q = H2ΔG, Then,

Detailed Solution for IIT JAM Mathematics Practice Test- 21 - Question 25


 

Similarly, we can show that 

IIT JAM Mathematics Practice Test- 21 - Question 26

Which one is a group under multiplication modulo 11?

Detailed Solution for IIT JAM Mathematics Practice Test- 21 - Question 26

By the definition of operation table of a group.

IIT JAM Mathematics Practice Test- 21 - Question 27

let 2 be the centre of the groups G. if G/Z is cyclic, then

Detailed Solution for IIT JAM Mathematics Practice Test- 21 - Question 27

By D well known theorem of group theory we know that
“If Z be the centre of the group G and G/Z is cyclic then G is abelian”.

IIT JAM Mathematics Practice Test- 21 - Question 28

The integral  is equal to

Detailed Solution for IIT JAM Mathematics Practice Test- 21 - Question 28

Limits are x = 0 , x = 1 
y = x2, y = 2x
After changing the order of the integration, we get

IIT JAM Mathematics Practice Test- 21 - Question 29

Consider the double integral  After reversing the order of the integration, the integral becomes,

Detailed Solution for IIT JAM Mathematics Practice Test- 21 - Question 29

Region of integration is bounded by curves, 
y = x ; y = 2 + x ; x = 0 ; x = 1

IIT JAM Mathematics Practice Test- 21 - Question 30

Let f: R - {n} → R be a function defined by  where m ≠ n then 

Detailed Solution for IIT JAM Mathematics Practice Test- 21 - Question 30

One-One: Let x,y ∈ R such that.


So, f is one-one.
Onto: Let α ∈ R (co - domain)such that f(x) = α

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