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IIT JAM Mathematics Mock Test- 1 - Mathematics MCQ


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30 Questions MCQ Test - IIT JAM Mathematics Mock Test- 1

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

Let f : (0, 2) → R be defind by

then

Detailed Solution for IIT JAM Mathematics Mock Test- 1 - Question 1

Given that f : (0, 2) → R then

⇒ f(x) is differentiable only when x  = 1

i. e. f(x) is differentiable, exactly at one point.

IIT JAM Mathematics Mock Test- 1 - Question 2

Consider the differential equestion and y = 0 and  x → ∞ then  is

Detailed Solution for IIT JAM Mathematics Mock Test- 1 - Question 2

Given differential equestion is 

multiply bothe of side of equation by ey we get

Take ey = t

Integrating factor = 

solutions is

where c is an arbitrary constant

where u = ex

so 

now 

when x = log2

= not defind

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IIT JAM Mathematics Mock Test- 1 - Question 3

Solution of the following differential equation .

Detailed Solution for IIT JAM Mathematics Mock Test- 1 - Question 3

Leibnitz’s linear equation:

Solution is,

IIT JAM Mathematics Mock Test- 1 - Question 4

If , then f(x) is given by-

Detailed Solution for IIT JAM Mathematics Mock Test- 1 - Question 4

Putting x = 1 - x

Multiplying equation(2) by 2 and subtracting from (1)

IIT JAM Mathematics Mock Test- 1 - Question 5

Let T : R3 R3 be the linear transformation such that Y (1, 0, 1) = (0, 1, -1) and T(2, 1, 1) = (3, 2, 1) Then T(-1, -2, 1)

Detailed Solution for IIT JAM Mathematics Mock Test- 1 - Question 5

Since T :  

Such that

But let 

on solving -1 = 2a + b
-2  = a + 0
a + b = 1
⇒ b = 3
So applying transformation T on (1) bothside

IIT JAM Mathematics Mock Test- 1 - Question 6

Find the area of the region bounded by the curve 

Detailed Solution for IIT JAM Mathematics Mock Test- 1 - Question 6

We have,

y2 = 2x and x2 + y2 = 4x

y2 = 2x is parabola opening to the right of positive direction x-axis 

x2 + y2 = 4x

⇒ (x - 2)2 + y2 = 4 which is circle having center at (2,0) and radius 2 units

Solving the curves we get, 

When x = 0, y = 0 and when x = 2, y = ±2
Thus points of intersection are (0,0), (2,2) and (2,-2)

The graph is as shown in the following figure:

IIT JAM Mathematics Mock Test- 1 - Question 7

The Solution of  is

Detailed Solution for IIT JAM Mathematics Mock Test- 1 - Question 7

IIT JAM Mathematics Mock Test- 1 - Question 8

Evaluate .

Detailed Solution for IIT JAM Mathematics Mock Test- 1 - Question 8

Let,

Applying the general Leibnity's rule we obtain

 

IIT JAM Mathematics Mock Test- 1 - Question 9

Let Y = {y1} be a sequence such that

Detailed Solution for IIT JAM Mathematics Mock Test- 1 - Question 9

Let Y = (yn) be the sequence of real numbers given by


Clearly, Y is not a monotone sequence. However, if m > n, then


Since , it follows that if m > n, then


Therefore, it follows that (yn) is a Cauchy sequence. Hence it converges to a
limit y. At the present moment we cannot evaluate y directly; however, passing
to the limit (with respect to m) in the above inequality, we obtain

Hence we can calculate y to any desired accuracy by calculating the terms yn
for sufficiently large n. The reader should to this and show that y is approxi-
mately equal to 0.632 120559.

IIT JAM Mathematics Mock Test- 1 - Question 10

The general solution of the d.e is given by;

Detailed Solution for IIT JAM Mathematics Mock Test- 1 - Question 10

IIT JAM Mathematics Mock Test- 1 - Question 11

Choose a number n uniformly at random from the set {1,2,…,100}. Choose one of the first seven days of the year 2014 at random and consider n consecutive days starting from the chosen day. What is the probability that among the chosen �� days, the number of Sundays is different from the number of Mondays?

Detailed Solution for IIT JAM Mathematics Mock Test- 1 - Question 11

2014 starts with a Wednesday

1234567

WTFSSM T

So if Wednesday is day n = 1 then Sundays will come on days n = 5,12,19,…96, total 15 Sundays.

Similarly if we start with-

Thursday - 14 sundays 

Friday - 14 sundays 

Saturday - 15 sundays 

Sunday - 15 sundays 

Monday - 14 sundays 

Tuesday - 14 sundays

So, probability 

⇒ P = 2/7

IIT JAM Mathematics Mock Test- 1 - Question 12

Let p(x) be a non-zero polynomial of degree N the radius of convergence of the power series 

Detailed Solution for IIT JAM Mathematics Mock Test- 1 - Question 12

Since here an = p(n)

If let p(x) = 

Radius of convergence of power series is given by

IIT JAM Mathematics Mock Test- 1 - Question 13

Let T: R2 → R3 be the Linear transformation whose matrixwith respect to standard basis of R3 and R2 is  The T

Detailed Solution for IIT JAM Mathematics Mock Test- 1 - Question 13

Let i of T: R2 → R3 be a Linear transformation such that matrix a with respect to standard basis of T is 


here clearly the columns of A are linearly independent b ⇒ T is one one map-ping since matrix is 3 x 2 the column of A span R3 if Ahas 3 pivot positions but it is contradiction as A has 2 columns only

⇒ Associated Linear transformation is not onto Rank of matrix = Rank of Linear transformation = 2

IIT JAM Mathematics Mock Test- 1 - Question 14

Let  be two solutions of  then the set of initial conditions for which the above differential equation has No solution is:

Detailed Solution for IIT JAM Mathematics Mock Test- 1 - Question 14

Since  be two solutions of,

Then will satisfy (1),

So if 

On solving them we get,

Differential equation becomes,

And solution is of the form,

Take options,

On solving equation set (1) we get no solution.

Thus for the conditions  differential equation has no solution.

IIT JAM Mathematics Mock Test- 1 - Question 15

The differential equation  is:

Detailed Solution for IIT JAM Mathematics Mock Test- 1 - Question 15

Condition 1: 

(It is Homogeneous)

Condition 2: 

Equation (1) can be written as 

It is not a linear form.

It is in linear form.

Condition 3: 

So, it is an exact equation.

IIT JAM Mathematics Mock Test- 1 - Question 16

In an elastic collision of two particles, the following is conserved:

Detailed Solution for IIT JAM Mathematics Mock Test- 1 - Question 16

An elastic collision is a collision in which there is no net loss in kinetic energy in the system as a result of the collision. Both momentum and kinetic energy are conserved quantities in elastic collisions. Suppose two similar trolleys are traveling toward each other with equal speed.

IIT JAM Mathematics Mock Test- 1 - Question 17

The solutions sin x and cos x of the differential equation  are

Detailed Solution for IIT JAM Mathematics Mock Test- 1 - Question 17

We find that

for all real x. Thus, since W(sin x , cos x) ≠ 0 for all real x, we conclude that sin x and cos x are indeed linearly independent solutions of the given differen-tial equation on every real interval.

IIT JAM Mathematics Mock Test- 1 - Question 18

A group of order 49 is always a

Detailed Solution for IIT JAM Mathematics Mock Test- 1 - Question 18

Since, we know that every group of order p2 is cyclic, where p is a prime integer.

  • Group Of order p2 is abelian also.
  • Group of order 72 i.e, 49 is abelian and cyclic.
IIT JAM Mathematics Mock Test- 1 - Question 19

If  and if  is

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

IIT JAM Mathematics Mock Test- 1 - Question 20

Evaluate 

Detailed Solution for IIT JAM Mathematics Mock Test- 1 - Question 20

Adding (i) and (ii), we get

IIT JAM Mathematics Mock Test- 1 - Question 21

The least number which when divided by 4,5,6 and 7 leaves 3 as remainder, but when divided by 9 leaves no remainder is:

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

The LCM of 4,5,6 and 7 = 420

so, the required number = 420k + 3

which is exactly divisible by 9 for some value of k

Now, 420K + 3 = 46 x 9k + 6k + 3

When k = 1, 6k + 3 = 9, which is divisible by 9

So, the required number = 420 x 1 + 3 = 423

IIT JAM Mathematics Mock Test- 1 - Question 22

Which one of the following is the differential equation that represents the family of curves   where c is an arbitrary constant?

Detailed Solution for IIT JAM Mathematics Mock Test- 1 - Question 22

We know that

Now by differentiating equation (1) with respect to x, we get:

IIT JAM Mathematics Mock Test- 1 - Question 23

Let V be the vector space of real polynomials of degree atmost 2.which defines a linear operator then the matrix of with respect to the basis (1 , x, x2) is

Detailed Solution for IIT JAM Mathematics Mock Test- 1 - Question 23

Hence 

IIT JAM Mathematics Mock Test- 1 - Question 24

Value of the is (where C are the Wvo circles of radius 2 and 1 centered at the origin with positive orientation.)

Detailed Solution for IIT JAM Mathematics Mock Test- 1 - Question 24

In this case the region D will be the region beWeen these tvvo circles and that will only change the limits in the double integral.
Here is the work for this integral.

IIT JAM Mathematics Mock Test- 1 - Question 25

The equation of the curve which passes through the point (2a, a) and for which the sum of the Cartesian sub tangent and the abscissa is equal to the constant a, is:

Detailed Solution for IIT JAM Mathematics Mock Test- 1 - Question 25

We have, Cartesian sub tangent + abscissa = constant

Integrating, we get 

As the curve passes through the point (2a, a), we have c = a2

The required curve is .

IIT JAM Mathematics Mock Test- 1 - Question 26

The function sinx(1 + cosx) have maximum value at:

Detailed Solution for IIT JAM Mathematics Mock Test- 1 - Question 26

We know that for maximum or minimum value of 

Therefore f(x) has maximum value at .

IIT JAM Mathematics Mock Test- 1 - Question 27

Which one of the following options contains two solutions of the differential equation 

Detailed Solution for IIT JAM Mathematics Mock Test- 1 - Question 27

Given,

The given equation can be solved using the variable separable method as,

Integrating both sides, we get,

Now, from equation (1), we get

∴ y = constant = 1  is also a solution to the given differential equation.

IIT JAM Mathematics Mock Test- 1 - Question 28

The mass of a solid right circular cylinder of height h and radius of base b, if density (mass per unit volume) is numerically equal to the square of the distance from the axis of the cylinder.is

Detailed Solution for IIT JAM Mathematics Mock Test- 1 - Question 28

IIT JAM Mathematics Mock Test- 1 - Question 29

Two different families A and B are blessed with an equal number of children. There are 3 tickets to be distributed amongst the children of these families so that no child gets more than one ticket. If the probability that all the tickets go to the children of the family B is 1 / 12, then the number of children in each family is :

Detailed Solution for IIT JAM Mathematics Mock Test- 1 - Question 29

Let n be the number of children in each family 1/12 = 

IIT JAM Mathematics Mock Test- 1 - Question 30

Using the method of Lagrange multipliers the greatest and smallest value that the function f (x, y) = xy takes on the ellipse  is

Detailed Solution for IIT JAM Mathematics Mock Test- 1 - Question 30

We want extreme values of f(x, y) = xy subject to the constraints

To do so, we first find the value of x,y and λ, for which

the gradient eq. in eq (1) gives

from which we find

and
so y = 0 or λ = ±2. we now consider these Wvo cases
case I if
y = O then x = y = 0 But (0, 0) is not on the ellipse therefore y + 0
case II If y ≠ 0 then λ = ± 2 and x = substituting this in the eq. g(x,y) = 0
given

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