Integral Calculus NAT Level - 2


10 Questions MCQ Test Topic wise Tests for IIT JAM Physics | Integral Calculus NAT Level - 2


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*Answer can only contain numeric values
QUESTION: 1

Show that the area common to the ellipses a2x2 + b2y2 = 1, b2x2 + a2y2 = 1. when 0 < a < b is   Find the value of λ.


Solution:


This is a horizontal ellipse 

This is a vertical ellipse
Total area = 4 x Area of OABC







Total area in first quadrant 
Total area 
Hence, value of λ = 4
The correct answer is: 4

*Answer can only contain numeric values
QUESTION: 2

If the area shaded in the given figure is of the form λa2. Find the value of λ given the cardioids are r = a(1 + cosθ) and r = a(1 – cosθ) and a given circle r = a.


Solution:

Let us first calculate the area included between two cardioids

Shaded area = 4 × Area OL in first quadrant



Now, this shaded portion lies entirely inside the circle r = a with area πa2

Hence, required area

The correct answer is: 2.428

*Answer can only contain numeric values
QUESTION: 3

Transform the integral  to polar coordinates and if   Find the value of λ.


Solution:

Slope of this curve is equation of the curve x = a cos3t, y = b sin3t.

Total Area = 4 × area OABO
This is of parametric form


The correct answer is: 2

*Answer can only contain numeric values
QUESTION: 4

Find the area of one loop of the curve  If the area is of the form λπ.  Find the value of λ.


Solution:

The equation of the curve is 

Turn the initial line through an angle π/18 and putting θ as   the above equation reduces to r = 2 cos 3θ and the tracing of this curve is as in figure.

Also for r = 2 cos 3θ when r = 0 we get cos 3θ = 0 or  and these is also symmetry about the initial line.

The correct answer is: 0.333

*Answer can only contain numeric values
QUESTION: 5

If the whole area of the curve given by the equation x = a cos3 t,  y = b sin3 t or   is of the form λπab.  Find the value of λ.


Solution:

Slope of this curve is equation of the curve 

Total Area = 4 × area OABO
This is of parametric form


The correct answer is: 0.375

*Answer can only contain numeric values
QUESTION: 6

If the area between the curve x(x2 + y2) = a(x2 – y2) and its asymptote is A1 and the area of the loop is A2 Then value of A1 + A2 = λ·a2. Find the value of λ.


Solution:

The curve is symmetrical about x-axis. The loop is situated between lines x = 0 and x = a. The line x = a is asymptote of the curve,

We have,

For any point on arc OLA

For any point on arc OMB

Area between curve and its asymptotes 

Area of the loop is given as 

Hence, λ = 4.
The correct answer is: 4

*Answer can only contain numeric values
QUESTION: 7

Find the area lying outside the circle r = 2acosθ and inside the cardioid r = a(1 + cosθ). This is of form λπa2. Find value of λ.


Solution:

Hence, the required area  

Taking Mod sign we get 

The correct answer is: 0.5

*Answer can only contain numeric values
QUESTION: 8

Evaluate  where R is the region in the first quadrant that is outside the circle r = 2 and inside the cardioid r = 2(1 + cosθ).


Solution:


The correct answer is: 2.666

*Answer can only contain numeric values
QUESTION: 9

The value of  sin x sin-1 (sin x sin y) dxdy is


Solution:

Let 
Then   keeping x constant when

Hence, θ varies from 0 to x.


Changing the order of integration with the help of figure

The correct answer is: 0.894

*Answer can only contain numeric values
QUESTION: 10

If the ratio of the two parts into which the parabola 2a = r(1 + cosθ) divides the area of the cardiod r = 2a(1 + cosθ)  is of the form  Find the value of β/α.


Solution:

Solving the given equation  


Therefore shaded area = area OLMN
= 2 × area OMNO
2(area OMN + area ONO)

By putting  in the second integral and by putting θ/2 =  u in the first integral

Now unshaded Area = Whole Cardioid - Shaded Area

The correct answer is: 1.777

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