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IIT JAM Mathematics Practice Test- 13 - Mathematics MCQ


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30 Questions MCQ Test IIT JAM Mathematics Mock Test Series - IIT JAM Mathematics Practice Test- 13

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IIT JAM Mathematics Practice Test- 13 - Question 1

What is the order of the differential equation given by dy / dx + 4y = sinx?

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

Since the order of a differential equation is defined as the order of the highest derivative occurring in the differential equation, i.e for nth derivative dny / dxn if n = 1.
It has order 1→ differential equation contains only dy / dx derivative with variables and constants.

IIT JAM Mathematics Practice Test- 13 - Question 2

If y = x is a solution of x2y" + x y' - y = 0,then the second linearly independent solution of the above equation is.

Detailed Solution for IIT JAM Mathematics Practice Test- 13 - Question 2

We have x2y”+xy’-y = 0, ...(1)

Compare this with

and given y = x is a solution of (1)

Then it’s 2nd L.I. solution = 

2nd L.l. solution = 1/x

(we can remove constant in (*))

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IIT JAM Mathematics Practice Test- 13 - Question 3

Consider the differential equation y” + 9y = 0 with the boundary conditions, y(0) = 0, y(2π) = 1, then the differential equation has

Detailed Solution for IIT JAM Mathematics Practice Test- 13 - Question 3

We have (D2 + 9)y = 0

⇒ y(x) = c1cos 3x + c2sin3x

y(0) = 0 ⇒c1 = 0

y(2π) = 1 ⇒ c1 = 1

⇒ DE has no solution.

IIT JAM Mathematics Practice Test- 13 - Question 4

The integrating factor of the differential equation,

(xy2 sin xy + y cosxy )dx + (x2y sin x y - x cos xy )dy - 0 is

Detailed Solution for IIT JAM Mathematics Practice Test- 13 - Question 4

Here

M  = xy2sinxy + y cosxy

So thus the integrating factor will be given by

IIT JAM Mathematics Practice Test- 13 - Question 5

Consider the following limit, 

Detailed Solution for IIT JAM Mathematics Practice Test- 13 - Question 5

IIT JAM Mathematics Practice Test- 13 - Question 6

The difference between the greatest and the least values of the function, 

Detailed Solution for IIT JAM Mathematics Practice Test- 13 - Question 6

⇒ f(x) is an increasing function .

⇒ max f(x) =f(3) and min f(x)= f(2)

⇒ Difference = f(3) -f(2)

IIT JAM Mathematics Practice Test- 13 - Question 7

Let 

then number of points (where f(x) attains its minimum value )is ,

Detailed Solution for IIT JAM Mathematics Practice Test- 13 - Question 7

hence ,minimum value of f(x) is 0 at x = 0. 

hence, no of points =1.

IIT JAM Mathematics Practice Test- 13 - Question 8

In [0,1] .Lagrange’s mean value theorem is NOT .applicable to

Detailed Solution for IIT JAM Mathematics Practice Test- 13 - Question 8

There is only one function in option(A), where critical point  but in other part critical point . then we can say that function in option (B),(C) and (D) are continuous on [0,1] and differentiable in (0,1).

now for 

here

⇒ f is not differentiable at 

f is not differentiable at  

⇒ LMVT is not applicable to f(x) in [0,11.

IIT JAM Mathematics Practice Test- 13 - Question 9

The orthogonal trajectories for the family of circles touching the y-axis at the origin is ,

Detailed Solution for IIT JAM Mathematics Practice Test- 13 - Question 9

The equation of family of circles touching y-axis at the origin is

IIT JAM Mathematics Practice Test- 13 - Question 10

The global maxima of f(x) = [2{-x2 + x + 1}] is , where {x} denotes fractional part of x and [-] denotes greatest integer function

Detailed Solution for IIT JAM Mathematics Practice Test- 13 - Question 10

Here we know

global maxima of f(x) is 1.

IIT JAM Mathematics Practice Test- 13 - Question 11

Consider the differential equation  where a,b > 0 and y(0)=y0 when x → ∞ then solution y(x) tends to

Detailed Solution for IIT JAM Mathematics Practice Test- 13 - Question 11

then sol.

IIT JAM Mathematics Practice Test- 13 - Question 12

Detailed Solution for IIT JAM Mathematics Practice Test- 13 - Question 12

 the sign of f has no changes

 be a point of infection.

 the sign of f has changes from +ve to -ve

 be the point of local maxima and clearly it be the only point at which f has local maximum value.

IIT JAM Mathematics Practice Test- 13 - Question 13

A function f is such that   and f has local  maximum of -20 at x = a , then f(x) may be ,

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

here f(x)= - 20 +(x-a)n can not be for n ≥4 and n ≥ 5

Since if n = 4 then   

f(x) = -2 0 + ( x - a )4

similarly if n = 5 then

IIT JAM Mathematics Practice Test- 13 - Question 14

Consider the differential equation   where

 and y(0) = 0, if y(x) be the continuous solution on [ 0, ∞) then

Detailed Solution for IIT JAM Mathematics Practice Test- 13 - Question 14

when x > 1 then f(x) = 0 , so by (1) 

be the continuous solution

IIT JAM Mathematics Practice Test- 13 - Question 15

The initial value problem  has a unique solution if ( x0,y0 ) equals,

Detailed Solution for IIT JAM Mathematics Practice Test- 13 - Question 15

Given problem is,

C le a rly the v alue o f (x0,y0) can not be (1,1), (0,1),(0,1) 

T h u s the p ro b le m has unique solution if (x0, y0) =(2 -1)

IIT JAM Mathematics Practice Test- 13 - Question 16

A particular integral of    is

Detailed Solution for IIT JAM Mathematics Practice Test- 13 - Question 16

Given equation is,

IIT JAM Mathematics Practice Test- 13 - Question 17

Solution of  given that y =1 when x =1 is

Detailed Solution for IIT JAM Mathematics Practice Test- 13 - Question 17

Given y (1) = 1

IIT JAM Mathematics Practice Test- 13 - Question 18

The polynomial function f(x) of degree 6 ,which satisfies,

and has local maxima at x =1 and local minimum at x = 0 and x =2, is

Detailed Solution for IIT JAM Mathematics Practice Test- 13 - Question 18

The limit can only exist if f(x) is having least degree as that of 4.

Given that f(x) has local extremum at x = 0,1,2

IIT JAM Mathematics Practice Test- 13 - Question 19

The set of all the values of K for which the point of minimum of the function f(x) = 1 + K2x - x3, satisfy the inequality

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

Now f"(x) = -6x Now for minimum of f, f" should be greater than 0. 

IIT JAM Mathematics Practice Test- 13 - Question 20

If   are three solutions of a non - homogeneous linear differential equation  where P(x), Q(x) and R(x) are continuous function on [a, b] with a > 0, then its particular solution w.r. to the conditions y(0) = 0 y.O) = 1

Detailed Solution for IIT JAM Mathematics Practice Test- 13 - Question 20

and P.l. = 2ex

= general soln : y(x) = c1xex + c2x2 + 2ex

Now y(0) = 0 =>0 = 2 Not possible

=> No particular solution exist w.r. to given conditions

IIT JAM Mathematics Practice Test- 13 - Question 21

Let y(x) be the solution of the different equation  such that y(0) = 2 and y'(0) = 2α. Then the values of  such that the in infimum of the set   is greater than or equal to1 , are

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

Given equation is,

solving above equation we have, 

So the minimum value of y(x) is given by,

The value of α for which infimum is greater than equal to 1 is given by

IIT JAM Mathematics Practice Test- 13 - Question 22

The differential equation satisfied by the system of parabolas y2 = 4a(x+ a) is

Detailed Solution for IIT JAM Mathematics Practice Test- 13 - Question 22

We have y2 = 4a(x + a) ...(1) diff.(1)

w.r. to x

IIT JAM Mathematics Practice Test- 13 - Question 23

if  is defined by

Consider the differential equation   , where a,b > 0 and y(0) = y0 when

, then solution y(x) tends to

Detailed Solution for IIT JAM Mathematics Practice Test- 13 - Question 23

IIT JAM Mathematics Practice Test- 13 - Question 24

Consider the function,

Detailed Solution for IIT JAM Mathematics Practice Test- 13 - Question 24

Now for differentiable at (0,0)

  and 

Clearly when (h,k) - > (0,0) then 

f is differentiable at (0,0)

IIT JAM Mathematics Practice Test- 13 - Question 25

Let y(x) be the solution of the differential equation,

satisfying the condition y(0) = 2. then which of the following is not true ?

Detailed Solution for IIT JAM Mathematics Practice Test- 13 - Question 25

we have

now take

IIT JAM Mathematics Practice Test- 13 - Question 26

consider the following's two inqualities as:

then which one is true?

Detailed Solution for IIT JAM Mathematics Practice Test- 13 - Question 26

IIT JAM Mathematics Practice Test- 13 - Question 27

Consider the follwing two statements 

(I) for all x in [0,1] , let the second derivative f"(x) of a function f(x) exist and satisfy  if f(0) = f(1) then 

then which one is true? 

Detailed Solution for IIT JAM Mathematics Practice Test- 13 - Question 27

=> f adnd f' both are continous and differentiable on [0,1]

also given f(0) = f(1)

=> function f satisfy all the conditions of Rolle's theorem in [0,1] so 

Case-II: Now f' is continous in [0,1] and f' is diff. in (0,1) so we can apply firstly rake x> C, then apply LMVT on f' in [c,x] then we have

Now if take x<c then by LMVT on [x,c]

IIT JAM Mathematics Practice Test- 13 - Question 28

Let y(x) be a continous solution of the intial value problem

then which of the follwoing is true ?

Detailed Solution for IIT JAM Mathematics Practice Test- 13 - Question 28

so by given y(0) = 0

⇒ C = o

 ⇒ y is continous at x = 0

⇒ y is continous everywhere on R

Now differentiable at x = 0

f is not differentiable at x = 0

IIT JAM Mathematics Practice Test- 13 - Question 29

A differentiable function f(x) has relative minimum at x = 0, then the function y = f(x) +ax+ b, has a relative minimum at x = 0 for

Detailed Solution for IIT JAM Mathematics Practice Test- 13 - Question 29

now the relative minima of  y at x = 0

we have 

for maxima and minima  a = 0

y is minimum for all b and a = 0

IIT JAM Mathematics Practice Test- 13 - Question 30

At  t = 0 the function  has

Detailed Solution for IIT JAM Mathematics Practice Test- 13 - Question 30

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