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# IIT JAM Mathematics MCQ Test 1

## 30 Questions MCQ Test Mock Test Series for IIT JAM Mathematics | IIT JAM Mathematics MCQ Test 1

Description
This mock test of IIT JAM Mathematics MCQ Test 1 for Mathematics helps you for every Mathematics entrance exam. This contains 30 Multiple Choice Questions for Mathematics IIT JAM Mathematics MCQ Test 1 (mcq) to study with solutions a complete question bank. The solved questions answers in this IIT JAM Mathematics MCQ Test 1 quiz give you a good mix of easy questions and tough questions. Mathematics students definitely take this IIT JAM Mathematics MCQ Test 1 exercise for a better result in the exam. You can find other IIT JAM Mathematics MCQ Test 1 extra questions, long questions & short questions for Mathematics on EduRev as well by searching above.
QUESTION: 1

Solution:

QUESTION: 2

### Which of the following transformations reduce the differential equation  into the form

Solution:

Given differential equation is

Here  which shows that the transformation  t= 1/log z

reduce the given differential equation into the form

QUESTION: 3

### has a solution ___________, when t= 0, then

Solution:

then its solution is

Now,

QUESTION: 4

The solution of the differential equation

where y(0) = 0 and y'(0) = -2 is

Solution:

Auxiliary equation is m2 - m-2 = 0

m=-1,2

∴ C.F = c1e-x +c2e2x

Solving these, we get c1= 1, c2= -1

∴   the required solution is y = e-x - e2x+ xe2x

QUESTION: 5

An integrating factor of the differential equation

Solution:

Explanation :

QUESTION: 6

The differential equation representing the family of circles touching x-axis at the origin is

Solution:

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

x2 + (y-a)2 = a2

QUESTION: 7

A curve passing through (2,3) and satisfying the differential equation

Solution:

differential w.r. to x

QUESTION: 8

The differential equation

(3a2x2 + by cos x)dx + (2 sin x - 4ay3)dy = 0 is exact for

Solution:

Given differential equation is

(3a2x2 + by cos x)dx + (2 sin x - 4ay3)dy = 0

since it si exact if

Here M = 3a2x2 + bycosx
N = 2sinx - 4ay3
then bcosx = 2cosx
=> cosx(b - 2) = 0

=> b= 2[∵ cos x ≠ 0]

which shows that given differential equation is exact for any value of a but b = 2

QUESTION: 9

The integrating factor of the equation (1 + y2)dx = (tan-1y - x)dy is -

Solution:

QUESTION: 10

Solution of the differential equation

Solution:

The given equation can be re-written as

Integrating

Putting y2 = v so that 2y (dy/dx) = (dv/dx). We then get

which is linear. Its I.F =

QUESTION: 11

Solution of equation , given y(0) = 1/2 is

Solution:

x=0, y=1/2   (1)  ⇒ -2 = -3 +c ⇒ c=1

QUESTION: 12

Integrating factor of differential equation

Solution:

Explanation : xcosx dy/dx + y(xsinx + cosx) = 1......(1)

=> dy/dx + y(tanx + 1/x) = secx/x............(2)

I.F of (2) e∫(tanx + 1/x) dx

= eln secx + lnx

= eln secx

= x secx

I.F of (1) is xsecx/xcosx

= sec2 x

QUESTION: 13

the solution of differential equation

Solution:

Hence the given differential equation

QUESTION: 14

The differential equation

Solution:

It is clear by the definition of linear differential equation and order of differential equation, that is linear and of second order.

QUESTION: 15

If (c1 logx + c2)x2 is the general solution of the differential equation  then k equals

Solution:

Given differential equation is

--(1)

and it’s general solution is

(c1 log x + c2) x2     ..(2)

Since (1) is homogeneous differential  equation so put then (1) reduce to

It’s A.E is m2 + m(k - 1) + 4 = 0 • • (3)
Now from (2) we have

By observation we can say that (3) has solution m = 2,

... (4)

On comparing (3) and (4). we have

k-1 = -4

k=-3

*Multiple options can be correct
QUESTION: 16

Consider a function g which has derivative g’(x) for every real x and which satisfies g’(0) = 2 and g(x + y) = ey g(x) + ex g(y) for all x and y. Then which of the followings is/are correct ?

Solution:

*Multiple options can be correct
QUESTION: 17

Consider the differential equation (3x2y4 + 2xy)dx+(2x3y3 - x2)dy=0 then which of the follwing(s) is/are not an I.F

Solution:

Given differential equation is

*Multiple options can be correct
QUESTION: 18

If y1 and y2 are two solution of the differential equation . Then which of the followings can be given as general soln of this differential eqn.

Solution:

Option (a), (b) are clearly correct

Now we have

Now from (4) and (5)

*Multiple options can be correct
QUESTION: 19

For the given differential equation, which of the following(s) statement are true,

Solution:

Given equation ca be written as,

not homogenous differential equation

compare with

Here at x = 0, P and Q both are not defined.

⇒ x=0 is a singular point of given differential equation.

Now consider x P(x) = -1/2

and both are defined and differential at x = 0

⇒ x= 0 is a regular singular point of given equation.

*Multiple options can be correct
QUESTION: 20

Given that φ(x) = x2 is a solution of the differential equation,   then which of the following(s) is/are not its 2nd  L.l. solution.

Solution:

we have x2 y" - 2y = 0

*Answer can only contain numeric values
QUESTION: 21

If the integrating factor of the differential equation, (x7y2 + 3y)dx + 3x8y - x) dy = 0 is of the form xmyn , then the sum of value of m and n is __________.

Solution:

Given differential equation is

(x7y2 + 3y)dx + 3x8y - x) dy = 0   ..(1)

If Integrating factor is xmyn then multiplying the given differential equation by I.F. , equation (1) becomes exact
i.e

is exact differential equation.

and for exact

on solving (1) and (2) we get

m=-7, n=1

m+n = -7+1 = -6

*Answer can only contain numeric values
QUESTION: 22

The sum of order and degree of the differential equation representing the family o f parabolas whose center is (-a.O) i s _______ .

Solution:

The equation of the families of parabolas whose centre (-a,0) is

y2 = 4a(x+a)    ..(1)

Differentiating both sides with respect to x, we get

Dividing (i) by (ii) we get

Substituting this value of a in (2), a is eliminated and we get

which shows that order of this differential equation is one and degree is 2. The sum of order of this differential equation and degree is 3.

*Answer can only contain numeric values
QUESTION: 23

The value o f y as t -> ∞ , for an initial value o f y(1) = 0, for the differential equation

Solution:

Hence solution of equation (i) is,

But, y(1)=0, therefore

From equation (ii), we have

From equation (ii), we have

*Answer can only contain numeric values
QUESTION: 24

If y = f(x) be a particular solution of differential equation y" + 4y = 4 cosec2 2x , then f(x) at is _________.

Solution:

*Answer can only contain numeric values
QUESTION: 25

Consider the differential equation   be its general solution and u(x) = x3 then the value of v(x) at x=(-1) is, _________

Solution:

where

v(x)= -1/6x2

∴ v(x)= -1/6(-1)2 = 0.1666666

*Answer can only contain numeric values
QUESTION: 26

No. of regular Singular points of the differential equation,  x2( x - 2 )2 y" + 2x (x - 2) y’+ (x + 1)y = 0. is __________.

Solution:

In the normalized form (6,3), this is

Clearly the singular points of the differential equation are x = 0 and x = 2. We investigate them one at a time.
Consider x = 0 first. and form the functions defined by the products

of the form. The product function defined by x2P2(x)f is analytical at x = 0, but that defined by xP1(x) is not. Thus x = 0 is an irregular singular point of Now consider x = 2. Forming the products for this point, we have

Both of the product functions thus defined are analytic at x = 2. and hence x = 2 is a regular singular point.

*Answer can only contain numeric values
QUESTION: 27

The value of I.F of the differential equation,   at x = 2 is, _______.

Solution:

*Answer can only contain numeric values
QUESTION: 28

The directional derivative of f(x,y) = x2 + xy at Po(1,2) in the direction of the unit vector

Solution:

The derivative of f at P0(x0,yo) in the direction of the unit vector u = u1i + u2j is the number

*Answer can only contain numeric values
QUESTION: 29

Find the directional Derivative of f(x,y) = x2 sin 2y at  the point

Solution:

*Answer can only contain numeric values
QUESTION: 30

Find the derivation of f(x,y,z) = x3 - xy2 -z at Po (1,1,0) in the direction of v= 2i - 3j+ 6k

Solution:

the partial derivatives of f at P0 are

The gradient of f ata p0 is

The derivative of f at Pin the direction of v is therefore