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IIT JAM Mathematics Practice Test- 14 - Mathematics MCQ


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30 Questions MCQ Test - IIT JAM Mathematics Practice Test- 14

IIT JAM Mathematics Practice Test- 14 for Mathematics 2024 is part of Mathematics preparation. The IIT JAM Mathematics Practice Test- 14 questions and answers have been prepared according to the Mathematics exam syllabus.The IIT JAM Mathematics Practice Test- 14 MCQs are made for Mathematics 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for IIT JAM Mathematics Practice Test- 14 below.
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IIT JAM Mathematics Practice Test- 14 - Question 1

If  and   then the value of 

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if -1 to 2 integration of f(x) is 5 then 2 to -1 it will be -5.

IIT JAM Mathematics Practice Test- 14 - Question 2

 is equal to 

Detailed Solution for IIT JAM Mathematics Practice Test- 14 - Question 2

Let

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IIT JAM Mathematics Practice Test- 14 - Question 3

The value of integral  over the region in the positive quadrant for which x + y ≤ 1, is

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IIT JAM Mathematics Practice Test- 14 - Question 4

Evaluate  

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Put x = rcos θ, y = rsinθ

and dxdy = rdθdr

IIT JAM Mathematics Practice Test- 14 - Question 5

The area of smaller part between the circle x2 + y2 = 4 and line x = 1 is

Detailed Solution for IIT JAM Mathematics Practice Test- 14 - Question 5

 

The smaller area enclosed by the circle, x2+y2=4 and the line, x = 1 is represented by the shaded area ACBA as shown in the diagram.

It can be observed that,

2∫1 to 2 (4-x2)1/2dx

= 2 [x/2(4-x2) + 2sin^-1(x/2)]1 to 2

= 2[2*pi/2 - (3)1/2 /2 +2pi/6]

= 2[pi - (3)1/2 /2 +pi/3]

8pi/3 - (3)1/2

IIT JAM Mathematics Practice Test- 14 - Question 6

The area bounded by curve, y = x2 and y = 2 - x2 is

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Given y = x2 and y = 2 - x2 from these eq. we get

Required area

 

IIT JAM Mathematics Practice Test- 14 - Question 7

The volume of the region bounded by the surfaces y = x2, x = y2 and the planes z = 0, z = 3 is,

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= 2-1 = 1

IIT JAM Mathematics Practice Test- 14 - Question 8

The value of the integral  where  is;

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Region D is 2x ≤ x2 + y2 ≤ 4x

i.e. x2 + y2 = 2x and x2 + y2 = 4x (x - 1)2 + y2 = 1 and (x - 2)2 + y2 = 4 polar form

x → rcos θ

y → rsin θ

dxdy → rdθdr

then

IIT JAM Mathematics Practice Test- 14 - Question 9

If  then  f'(4) is equal to 

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IIT JAM Mathematics Practice Test- 14 - Question 10

if  then the value of 

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from equation (1) and equation (2)

IIT JAM Mathematics Practice Test- 14 - Question 11

The area of largest part between the circle x2 + y2 = 8 and the line is x+y = 2√2 is,

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Area of the smallest part is 

so, area of the largest part is area of the circle -area of the smallest part

IIT JAM Mathematics Practice Test- 14 - Question 12

Area bounded by the curve xy - 3x - 2y -10 = 0, x-axis and the lines x = 3,x = 4 is

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Given curve is xy - 3x - 2y -10 = 0 

=>(x - 2)y = 3x +10

= 3 (1 ) + 16 [log(2) - log (1 )] = 3 + 16 log 2

IIT JAM Mathematics Practice Test- 14 - Question 13

The value of the integral  is

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Limits are y = x, y = ∞

X = 0, X = ∞

Changing the order of the integration

IIT JAM Mathematics Practice Test- 14 - Question 14

Evaluate 

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IIT JAM Mathematics Practice Test- 14 - Question 15

The value of the limit 

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By the sum of the series

IIT JAM Mathematics Practice Test- 14 - Question 16

The value of the double integral  where R = [0 ,1 ; 0, 3] and  [ x + y ] denotes the greatest integer less than or equal to (x + y) is,

Detailed Solution for IIT JAM Mathematics Practice Test- 14 - Question 16

R = [0, 1 : 0, 3] i.e. 0 < x < 1 , 0 < y < 3 , 0 < x + y < 4

IIT JAM Mathematics Practice Test- 14 - Question 17

Find the value 

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IIT JAM Mathematics Practice Test- 14 - Question 18

Find the value of 

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IIT JAM Mathematics Practice Test- 14 - Question 19

The area of the loop of the curve ay2 = x2(a - x) is 

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Required Area

IIT JAM Mathematics Practice Test- 14 - Question 20

The length of the arc of the curve

x sinθ + y cosθ = f'(θ)

x cosθ - y sinθ = f"(θ) is given by

Detailed Solution for IIT JAM Mathematics Practice Test- 14 - Question 20

The equations of the given curve are 

x sinθ  + y cosθ  = f'(θ )

x cosθ  - y sinθ  = f"(θ  ) 

Solving these equations, we get 

x = sinθ  f'(θ ) + cosθ  f"(θ) 

y = cosθ  f'(θ ) - sinθ  f"(θ ) 

Now differentiate these equation w.r. to θ we get

Hence required length of arc of the curve is

IIT JAM Mathematics Practice Test- 14 - Question 21

Evaluate  were R is bounded by y = x2 and y = x3

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IIT JAM Mathematics Practice Test- 14 - Question 22

The area bounded by the curves  and x = 4 is given by

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The given equation of the cuives are

Hence the required area is

IIT JAM Mathematics Practice Test- 14 - Question 23

Which of the following is the volume generated by rotating the area bounded by  and y = 2 about the x - axis ?

Detailed Solution for IIT JAM Mathematics Practice Test- 14 - Question 23

The intersecting points of the graphs are

 and r(x) = 2

therefore, the volume is

IIT JAM Mathematics Practice Test- 14 - Question 24

Let F : R → R be a continuous function and a > 0. Then tine integral is equal to 

Detailed Solution for IIT JAM Mathematics Practice Test- 14 - Question 24

Given double integral is

 this integral is not solvable.

Limits are y = 0, y = x

and x = 0, x = a

then by changing the order of integration we get

IIT JAM Mathematics Practice Test- 14 - Question 25

Find the length of the arc of the semi cubical parabola ay2 = x3 from its vertex to the point (a, a).

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Differentiating the equation of the curve


Therefore the required length

IIT JAM Mathematics Practice Test- 14 - Question 26

Let  then the total no. of real distinct roots of the equation x2 - f'(x)= 0 is

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Now equation becomes

⇒ equation (1) has 3 real distinct roots.

IIT JAM Mathematics Practice Test- 14 - Question 27

Compute the integral  along the arc of the parabola x = y2 fro m (1 , - 1 )to(1, 1)

Detailed Solution for IIT JAM Mathematics Practice Test- 14 - Question 27

When the equation of the curve of integration is taken by expressing x as a single valued function of y, 

we put x = y2

so that

IIT JAM Mathematics Practice Test- 14 - Question 28

By changing the order of integration, the integral  can be represented as   determine the value of A.

Detailed Solution for IIT JAM Mathematics Practice Test- 14 - Question 28

Here the limits of y are given by the circle and the straight line y = x +2a. The limits of x are given by the straight lines x = 0 and x = a

Therefore, the integral extends to all points in the space bounded by the axis of y, the circle AB, the straight line AL and the straight line LM. Draw BH and M perpendiculars to AL. Now when the order of integration is changed, we take strips paralles to axis of x instead of that is y, thus the integral breaks up into three parts: First corresponding to the area BAH, second to the rectangle BHKM and the third to the triangle MKL, hence

so

IIT JAM Mathematics Practice Test- 14 - Question 29

The value of the limit 

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IIT JAM Mathematics Practice Test- 14 - Question 30

 is equal to 

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