IIT JAM Mathematics MCQ Test 1

IIT JAM Mathematics MCQ Test 1

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

IIT JAM Mathematics MCQ Test 1 for Mathematics 2023 is part of Mock Test Series for IIT JAM Mathematics preparation. The IIT JAM Mathematics MCQ Test 1 questions and answers have been prepared according to the Mathematics exam syllabus.The IIT JAM Mathematics MCQ Test 1 MCQs are made for Mathematics 2023 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for IIT JAM Mathematics MCQ Test 1 below.
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IIT JAM Mathematics MCQ Test 1 - Question 1

Let ( y - C)2 = Cx be the primitive of   The no. of integral curves which will pass through (1, 2) is,

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

IIT JAM Mathematics MCQ Test 1 - Question 2

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

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

Given differential equation is

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

reduce the given differential equation into the form

IIT JAM Mathematics MCQ Test 1 - Question 3

has a solution ___________, when t= 0, then

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

then its solution is

Now,

IIT JAM Mathematics MCQ Test 1 - Question 4

The solution of the differential equation

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

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

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

IIT JAM Mathematics MCQ Test 1 - Question 5

An integrating factor of the differential equation

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

Explanation :

IIT JAM Mathematics MCQ Test 1 - Question 6

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

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

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

x2 + (y-a)2 = a2

IIT JAM Mathematics MCQ Test 1 - Question 7

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

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

differential w.r. to x

IIT JAM Mathematics MCQ Test 1 - Question 8

The differential equation

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

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

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

IIT JAM Mathematics MCQ Test 1 - Question 9

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

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

IIT JAM Mathematics MCQ Test 1 - Question 10

Solution of the differential equation

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

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 =

IIT JAM Mathematics MCQ Test 1 - Question 11

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

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

x=0, y=1/2   (1)  ⇒ -2 = -3 +c ⇒ c=1

IIT JAM Mathematics MCQ Test 1 - Question 12

The slope of a curve at any point is the reciprocal of twice the ordinate at the point and it passes through the point (4, 3). The equation of the curve is

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

We have,
Slope = dy/dx ⇒ dy/dx = 1/2y ⇒ 2 y dy = dx
Integrating both sides, we get y2 = x + C
This passes through (4, 3)
∴ 9 = 4 + C ⇒ C = 5
So, the equation of the curve is y2 = x + 5

IIT JAM Mathematics MCQ Test 1 - Question 13

the solution of differential equation

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

Hence the given differential equation

IIT JAM Mathematics MCQ Test 1 - Question 14

The differential equation

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

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

IIT JAM Mathematics MCQ Test 1 - Question 15

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

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

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
IIT JAM Mathematics MCQ Test 1 - 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 ?

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

*Multiple options can be correct
IIT JAM Mathematics MCQ Test 1 - 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

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

Given differential equation is

*Multiple options can be correct
IIT JAM Mathematics MCQ Test 1 - 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.

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

Option (a), (b) are clearly correct

Now we have

Now from (4) and (5)

*Multiple options can be correct
IIT JAM Mathematics MCQ Test 1 - Question 19

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

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

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
IIT JAM Mathematics MCQ Test 1 - 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.

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

we have x2 y" - 2y = 0

*Answer can only contain numeric values
IIT JAM Mathematics MCQ Test 1 - 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 __________.

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

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
IIT JAM Mathematics MCQ Test 1 - 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 _______ .

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

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
IIT JAM Mathematics MCQ Test 1 - Question 23

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

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

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
IIT JAM Mathematics MCQ Test 1 - Question 24

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

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*Answer can only contain numeric values
IIT JAM Mathematics MCQ Test 1 - Question 25

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

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

where

v(x)= -1/6x2

∴ v(x)= -1/6(-1)2 = 0.1666666

*Answer can only contain numeric values
IIT JAM Mathematics MCQ Test 1 - 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 __________.

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

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
IIT JAM Mathematics MCQ Test 1 - Question 27

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

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

*Answer can only contain numeric values
IIT JAM Mathematics MCQ Test 1 - Question 28

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

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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
IIT JAM Mathematics MCQ Test 1 - Question 29

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

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*Answer can only contain numeric values
IIT JAM Mathematics MCQ Test 1 - 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

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

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

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