Integral Calculus -7


20 Questions MCQ Test Topic-wise Tests & Solved Examples for IIT JAM Mathematics | Integral Calculus -7


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This mock test of Integral Calculus -7 for Mathematics helps you for every Mathematics entrance exam. This contains 20 Multiple Choice Questions for Mathematics Integral Calculus -7 (mcq) to study with solutions a complete question bank. The solved questions answers in this Integral Calculus -7 quiz give you a good mix of easy questions and tough questions. Mathematics students definitely take this Integral Calculus -7 exercise for a better result in the exam. You can find other Integral Calculus -7 extra questions, long questions & short questions for Mathematics on EduRev as well by searching above.
QUESTION: 1

What is the limit of the product 

Solution:



QUESTION: 2

The length of the cycloid with parametric equation x(t) = (t + sin t), y(t) = (1 - cos t) between (0, 0) and (π, 2) is:

Solution:

 The given curve is
x(t) = (t + sin t), y(t) = (1 - cos t)

QUESTION: 3

Which of the following is/are true?
1. 
2. 
3. 
4.  If f(x) ≥ φ(x) and both functions are integrable in [a, b]

Solution:
QUESTION: 4

  is revolved about major and minor axis respectively. Then the ratio of these solids made by two revolutions is:

Solution:

Case (i): When ellipse is rotated about major axis:

Take a small disc at a length of x from the centre of thickness dx. Then the volume of solid obtained by rotation will be ∫(−a to a)​(Area)dx

Area of disc =πr2

r can be calculated from the equation of ellipse as

x2/a2​ + r2/b= 1

⇒ r= b2(1−x2/a2​) = b2/a2​(a2−x2)

∴Volume of major axis​ = ∫(−a to a)​πa2/b2​(a2−x2)dx

= [πb2x − πb2x3/3a2​](−a to a)

​= 4/3​πab2

Case (ii): When ellipse is rotated about minor axis:

Following similar procedure as case (i),

r= a2/b2​(b− y2)

In this case, the area will be integrated w.r.t dy as it is rotated about the Y-axis.

∴Volume of minor axis​ = ∫(−a to a)​πa2/b2​(b2−y2)dx

= [πa2y − πa2y3/3b2​](−a to a)

​= 4/3​πab2

 

∴ Volume about major axis/Volume about minor axis

 = b/a

QUESTION: 5

Solution:

  ...(i)

  ...(ii)

QUESTION: 6

The surface area of the segment of a sphere of radius a and height h is given by:

Solution:

Let the sphere be generated by the revolution about the x - axis of the circle
x2 + y2=a2 ...(i)
Let OA =a, OC = b and OB = b + h
Hence The required surface


QUESTION: 7

 then the value of a is:

Solution:




QUESTION: 8

The moment of inertia about the axis of y of the region in the xy-plane bounded by y = 4 - x2 and the x-axis, is, proportional to

Solution:

The given curve is y = 4x2
Hence Required moment of inertia

QUESTION: 9

The value of 

Solution:

   ...(i)

   ...(ii)

QUESTION: 10

Match the list-I with list-II:

Solution:







QUESTION: 11

The length of the arc of the curve y = loge

Solution:

Given that
y = loge



QUESTION: 12

The intrinsic equations of the cardioids r = a (1 - cos θ) and r = a(1 + cos θ) measured from the pole are:

Solution:

The equations of given cardioids are
r= a (l -cos θ)    ...(i)
r= a (1 + cos θ)   ...(ii) 
Intrinsic equation for cardioid (i):


     ...(iii)



QUESTION: 13

Consider the Assertion (A) and Reason (R) given below:

Assertion (A)

Reason (R) - sin x is continuous in any closed interval [ 0 ,t] .
The correct answer is:

Solution:
QUESTION: 14

The value of 

Solution:



QUESTION: 15

Solution:

Let



QUESTION: 16

The perimeters of the cardioids r = a (1 - cos θ) and r = a (1 + cos θ) differ by:

Solution:

The curves are
r = a(1 - cos θ)     ...(i)
r = a(1 + cos θ)    ...(ii)
The perimeter for curve (i) is





Hence s1 - s2 = 8a - 8a = 0. 

QUESTION: 17

The length of the arc of the curve 6xy = x4 + 3 from x = 1 to x = 2 is:

Solution:

The given curve is
6xy = x4 + 3

Thus Length of the arc between x
= 1 and x = 2 is:


QUESTION: 18

The area of the region in the first quadrant bounded by the y-axis and curves y = sin x and y = cos x is:

Solution:

The given curves are
y = sin x   ...(i)
y = cos x   ...(ii)
The curves intersect at a point B 

Hence Required area = Area OABO

QUESTION: 19

The line y = x + 1 is revolved about x-axis. The volume of solid of revolution formed by revolving the area covered by the given curve, x-axis and lines x = 0, x = 2 is:

Solution:

The given curve is y = x + 1. This represents a straight line
Hence The required volume

QUESTION: 20

The length of the complete cycloid x = a (θ + sin θ), y = a(1 - cos θ) is given by:

Solution:

The equations of the curve are
x = a(θ + sin θ), y = a(1 - cos θ)

Hence The entire length of the curve

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