NCERT Solutions for Exercise 5.6 Class 12 Maths Chapter 5 - Continuity and Differentiability

NCERT Solutions for Exercise 5.6 Class 12 Maths Chapter 5 - Continuity and Differentiability

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Komal MiglaniUpdated on 22 Apr 2025, 11:35 PM IST

Continuity is like a peaceful river flowing without any gaps, and Differentiability is when each ripple of the river follows a certain pattern smoothly and precisely. Oftentimes, we come across some functions which can not simply be represented as $y=f(x)$, as both the variables are expressed in the form of a third variable. These types of form of functions are generally known as functions in parametric form. In exercise 5.6 of the chapter Continuity and Differentiability, we will learn about the derivatives of functions in parametric form. These concepts will help the students differentiate expressions involving variables $x$ and $y$ given in the form of a third variable or parameter. This article on the NCERT Solutions for Exercise 5.5 Class 12 Maths Chapter 5 - Continuity and Differentiability offers clear and step-by-step solutions for the problems given in the exercise, so that the students can develop their problem-solving ability and understand the logic behind these solutions. For syllabus, notes, and PDF, refer to this link: NCERT.

Class 12 Maths Chapter 5 Exercise 5.6 Solutions: Download PDF

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Continuity and Differentiability Exercise: 5.6

Question:1 If x and y are connected parametrically by the equations given in Exercises 1 to 10, without eliminating the parameter, Find dy/dx .

$x = 2at^2, y = at^4$

Answer:

Given equations are
$x = 2at^2, y = at^4$
Now, differentiate both w.r.t t
We get,
$\frac{dx}{dt}=\frac{d(2at^2)}{dt}= 4at$
Similarly,
$\frac{dy}{dt}=\frac{d(at^4)}{dt}= 4at^3$
Now, $\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}= \frac{4at^3}{4at} = t^2$
Therefore, the answer is $\frac{dy}{dx}= t^2$

Question:2 If x and y are connected parametrically by the equations given in Exercises 1 to 10, without eliminating the parameter, Find dy/dx .

$x= a \cos \theta , y = b \cos \theta$

Answer:

Given equations are
$x= a \cos \theta , y = b \cos \theta$
Now, differentiate both w.r.t $\theta$
We get,
$\frac{dx}{d\theta}=\frac{d(a\cos \theta)}{d\theta}= -a\sin \theta$
Similarly,
$\frac{dy}{d\theta}=\frac{d(b\cos \theta)}{d\theta}= -b\sin \theta$
Now, $\frac{dy}{dx}=\frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}= \frac{-b\sin \theta}{-a\sin \theta} = \frac{b}{a}$
Therefore, answer is $\frac{dy}{dx}= \frac{b}{a}$

Question:3 If x and y are connected parametrically by the equations given in Exercises 1 to 10, without eliminating the parameter, Find dy/dx . $x = \sin t , y = \cos 2 t$

Answer:

Given equations are
$x = \sin t , y = \cos 2 t$
Now, differentiate both w.r.t t
We get,
$\frac{dx}{dt}=\frac{d(\sin t)}{dt}= \cos t$
Similarly,
$\frac{dy}{dt}=\frac{d(\cos 2t)}{dt}= -2\sin 2t = -4\sin t \cos t \ \ \ \ \ (\because \sin 2x = \sin x\cos x)$
Now, $\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}= \frac{-4\sin t \cos t }{\cos t} = -4\sin t$
Therefore, the answer is $\frac{dy}{dx} = -4\sin t$

Question:4 If x and y are connected parametrically by the equations given in Exercises 1 to 10, without eliminating the parameter, Find dy/dx

$x = 4t , y = 4/t$

Answer:

Given equations are
$x = 4t , y = 4/t$
Now, differentiate both w.r.t t
We get,
$\frac{dx}{dt}=\frac{d(4 t)}{dt}= 4$
Similarly,
$\frac{dy}{dt}=\frac{d(\frac{4}{t})}{dt}= \frac{-4}{t^2}$
Now, $\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}= \frac{ \frac{-4}{t^2} }{4} = \frac{-1}{t^2}$
Therefore, the answer is $\frac{dy}{dx} = \frac{-1}{t^2}$

Question:5 If x and y are connected parametrically by the equations given in Exercises 1 to 10, without eliminating the parameter, Find dy/dx $x = \cos \theta - \cos 2\theta , y = \sin \theta - \sin 2 \theta$

Answer:

Given equations are
$x = \cos \theta - \cos 2\theta , y = \sin \theta - \sin 2 \theta$
Now, differentiate both w.r.t $\theta$
We get,
$\frac{dx}{d\theta}=\frac{d(\cos \theta-\cos 2\theta)}{d\theta}= -\sin \theta -(-2\sin 2\theta) = 2\sin 2\theta - \sin \theta$
Similarly,
$\frac{dy}{d\theta}=\frac{d(\sin \theta - \sin 2\theta)}{d\theta}= \cos \theta -2\cos2 \theta$
Now, $\frac{dy}{dx}=\frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}= \frac{\cos \theta -2\cos 2 \theta}{2\sin2\theta-\sin \theta}$
Therefore, answer is $\frac{dy}{dx}= \frac{\cos \theta -2\cos 2 \theta}{2\sin2\theta-\sin \theta}$

Question:6 If x and y are connected parametrically by the equations given in Exercises 1 to 10, without eliminating the parameter, Find dy/dx $x = a ( \theta - \sin \theta ) , y = a ( 1+ \cos \theta )$

Answer:

Given equations are
$x = a ( \theta - \sin \theta ) , y = a ( 1+ \cos \theta )$
Now, differentiate both w.r.t $\theta$
We get,
$\frac{dx}{d\theta}=\frac{d(a(\theta- \sin \theta))}{d\theta}= a(1-\cos \theta)$
Similarly,
$\frac{dy}{d\theta}=\frac{d(a(1+\cos \theta))}{d\theta}=-a\sin \theta$
Now, $\frac{dy}{dx}=\frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}= \frac{-a\sin \theta}{a(1-\cos \theta)} = \frac{-\sin }{1-\cos \theta} =- \cot \frac{\theta}{2} \ \ \ \ \ \ \ (\cot \frac{x}{2}=\frac{\sin x}{1-\cos x})$
Therefore, the answer is $\frac{dy}{dx}=-\cot \frac{\theta}{2}$

Question:7 If x and y are connected parametrically by the equations given in Exercises 1 to 10, without eliminating the parameter, Find dy/dx $x = \frac{\sin ^3 t }{\sqrt {\cos 2t }} , y = \frac{\cos ^3 t }{\sqrt {\cos 2t }}$

Answer:

Given equations are

$x = \frac{\sin^3 t}{\sqrt{\cos 2t}}, \quad y = \frac{\cos^3 t}{\sqrt{\cos 2t}}$

Now, differentiate both w.r.t $t$:

$\frac{dx}{dt} = \frac{d\left( \frac{\sin^3 t}{\sqrt{\cos 2t}} \right)}{dt} = \frac{\sqrt{\cos 2t} \cdot \frac{d(\sin^3 t)}{dt} - \sin^3 t \cdot \frac{d(\sqrt{\cos 2t})}{dt}}{(\sqrt{\cos 2t})^2}$

$= \frac{3\sin^2 t \cos t \cdot \sqrt{\cos 2t} - \sin^3 t \cdot \frac{1}{2\sqrt{\cos 2t}} \cdot (-2\sin 2t)}{\cos 2t}$

$= \frac{3\sin^2 t \cos t \cdot \cos 2t + \sin^3 t \sin 2t}{\cos 2t \sqrt{\cos 2t}}$

$= \frac{\sin^3 t \sin 2t \left( 3 \cot t \cot 2t + 1 \right)}{\cos 2t \sqrt{\cos 2t}} \quad (\because \frac{\cos x}{\sin x} = \cot x)$

Similarly,

$\frac{dy}{dt} = \frac{d\left( \frac{\cos^3 t}{\sqrt{\cos 2t}} \right)}{dt} = \frac{\sqrt{\cos 2t} \cdot \frac{d(\cos^3 t)}{dt} - \cos^3 t \cdot \frac{d(\sqrt{\cos 2t})}{dt}}{(\sqrt{\cos 2t})^2}$

$= \frac{3\cos^2 t (-\sin t) \cdot \sqrt{\cos 2t} - \cos^3 t \cdot \frac{1}{2\sqrt{\cos 2t}} \cdot (-2\sin 2t)}{(\sqrt{\cos 2t})^2}$

$= \frac{-3\cos^2 t \sin t \cos 2t + \cos^3 t \sin 2t}{\cos 2t \sqrt{\cos 2t}}$

$= \frac{\sin 2t \cos^3 t \left( 1 - 3 \tan t \cot 2t \right)}{\cos 2t \sqrt{\cos 2t}}$

Now, $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{\frac{\sin 2t \cos^3 t \left( 1 - 3 \tan t \cot 2t \right)}{\cos 2t \sqrt{\cos 2t}}}{\frac{\sin^3 t \sin 2t \left( 3 \cot t \cot 2t + 1 \right)}{\cos 2t \sqrt{\cos 2t}}}$

$= \frac{\cot^3 t \left( 1 - 3 \tan t \cot 2t \right)}{3 \cot t \cot 2t + 1}$

$= \frac{\cos^3 t \left( 1 - 3 \frac{\sin t}{\cos t} \frac{\cos 2t}{\sin 2t} \right)}{\sin^3 t \left( 3 \frac{\cos t}{\sin t} \frac{\cos 2t}{\sin 2t} + 1 \right)}$

$= \frac{\cos^2 t \left( \cos t \sin 2t - 3 \sin t \cos 2t \right)}{\sin^2 t \left( 3 \cos t \cos 2t + \sin t \sin 2t \right)}$

$= \frac{\cos^2 t \left( 2 \sin t \cos^2 t - 3 \sin t \cos^2 t + 3 \sin t \right)}{\sin^2 t \left( 3 \cos t - 6 \cos t \sin^2 t + 2 \sin^2 \cos t \right)}$

$= \frac{\sin t \cos t \left( -4 \cos^3 t + 3 \cos t \right)}{\sin t \cos t \left( 3 \sin t - 4 \sin^3 t \right)}$

$\frac{dy}{dx} = \frac{-4 \cos^3 t + 3 \cos t}{3 \sin t - 4 \sin^3 t} = \frac{-\cos 3t}{\sin 3t} = -\cot 3t \quad (\because \sin 3t = 3 \sin t - 4 \sin^3 t \text{ and } \cos 3t = 4 \cos^3 t - 3 \cos t)$
Therefore, the answer is $\frac{dy}{dx} = -\cot 3t$

Question:8 If x and y are connected parametrically by the equations given in Exercises 1 to 10, without eliminating the parameter, Find dy/dx $x = a ( \cos t + \log \tan t/2 ),y = a \sin t$

Answer:

Given equations are
$x = a ( \cos t + \log \tan \frac{t}{2} ),y = a \sin t$
Now, differentiate both w.r.t t
We get,
$\frac{dx}{dt}=\frac{d(a ( \cos t + \log \tan \frac{t}{2} ))}{dt}= a(-\sin t + \frac{1}{\tan\frac{t}{2}}.\sec^2\frac{t}{2}.\frac{1}{2})$
$= a(-\sin t+\frac{1}{2}.\frac{\cos \frac{t}{2}}{\sin\frac{t}{2}}.\frac{1}{\cos^2\frac{t}{2}}) = a(-\sin t+\frac{1}{2\sin \frac{t}{2}\cos \frac{t}{2}})$
$=a(-\sin t+\frac{1}{\sin 2.\frac{t}{2}} ) = a(\frac{-\sin^2t+1}{\sin t})= a(\frac{\cos^2t}{\sin t})$
Similarly,
$\frac{dy}{dt}=\frac{d(a\sin t)}{dt}= a\cos t$
Now, $\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}= \frac{a \cos t }{ a(\frac{\cos^2t}{\sin t})} = \frac{\sin t}{\cos t} = \tan t$
Therefore, the answer is $\frac{dy}{dx} = \tan t$

Question:9 If x and y are connected parametrically by the equations given in Exercises 1 to 10, without eliminating the parameter, Find dy/dx $x = a \sec \theta , y = b \ tan \theta$

Answer:

Given equations are
$x = a \sec \theta , y = b \ tan \theta$
Now, differentiate both w.r.t $\theta$
We get,
$\frac{dx}{d\theta}=\frac{d(a\sec \theta)}{d\theta}= a\sec \theta \tan \theta$
Similarly,
$\frac{dy}{d\theta}=\frac{d(b\tan \theta)}{d\theta}= b\sec^2 \theta$
Now, $\frac{dy}{dx}=\frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}= \frac{b\sec^2 \theta}{a\sec\theta\tan \theta} = \frac{b\sec\theta}{a\tan \theta}= \frac{b\frac{1}{\cos\theta}}{a\frac{\sin \theta}{\cos \theta}} = \frac{b }{a\sin \theta} = \frac{b cosec \theta}{a}$
Therefore, the answer is $\frac{dy}{dx} = \frac{b cosec \theta}{a}$

Question:10 If x and y are connected parametrically by the equations given in Exercises 1 to 10, without eliminating the parameter, Find dy/dx $x = a ( \cos \theta + \theta \sin \theta ) , y = a ( \sin \theta - \theta \cos \theta )$

Answer:

Given equations are
$x = a ( \cos \theta + \theta \sin \theta ) , y = a ( \sin \theta - \theta \cos \theta )$
Now, differentiate both w.r.t $\theta$
We get,
$\frac{dx}{d\theta}=\frac{d(a(\cos \theta+ \theta\sin \theta))}{d\theta}= a(-\sin \theta+\sin \theta+ \theta\cos \theta)= a \theta\cos \theta$
Similarly,
$\frac{dy}{d\theta}=\frac{d(a(\sin \theta- \theta\cos \theta))}{d\theta}= a(\cos \theta-\cos \theta+ \theta\sin \theta) = a \theta\sin \theta$
Now, $\frac{dy}{dx}=\frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}= \frac{a \theta\sin \theta}{a \theta\cos \theta} = \tan \theta$
Therefore, the answer is $\frac{dy}{dx}= \tan \theta$

Answer:

Given equations are
$x = \sqrt {a ^{\sin ^{-1}t}} , y = \sqrt { a ^{ \cos ^{-1}t}}$

$xy=\sqrt{a^{sin^{-1}t+cos^{-1}t}}\\since\ sin^{-1}x+cos^{-1}x=\frac{\pi}{2}\\xy=a^{\frac{\pi}{2}}=constant=c$

differentiating with respect to x

$x\frac{dy}{dx}+y=0\\\frac{dy}{dx}=\frac{-y}{x}$

Also Read,

Topics covered in Chapter 5, Continuity and Differentiability: Exercise 5.6

The main topics covered in Chapter 5 of continuity and differentiability, exercises 5.6 are:

  • Derivatives of functions in parametric form: Understanding how to differentiate expressions when two variables are expressed as a third variable known as a parameter.
  • Application of the chain rule: Functions in parametric form can easily be differentiated with the help of the chain rule. For example, when $x$ and $y$ are expressed in terms of $t$, we can find the derivative as $\frac{dy}{dx}=\frac{dy}{dt}÷\frac{dx}{dt}$.

Also, read,

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NCERT Exemplar Solutions Subject Wise

Here are some links to subject-wise solutions for the NCERT exemplar class 12.

Frequently Asked Questions (FAQs)

Q: If x = sin (t) then find differentiation of x w.r.t t ?
A:

Given x = sin (t)

dx/dt = cos(t)

Q: If y = sin (t) then find the differentiation of y w.r.t x. ?
A:

y=sin(t)

As y is not dependent on the x.

dy/dx = 0

Q: Which book is best book for NCERT Class 12 Maths ?
A:

NCERT book is best for CBSE Class 12 Maths. You don't need other books for the CBSE board exams.

Q: Which book should i refer for Maths JEE main ?
A:

Mathematics book by M.L. khana is considered to be good book for the Maths JEE main.

Q: Do I need to but CBSE chapter wise solution book Class 12 Maths ?
A:

You don't need to buy any solution book for CBSE Class 12 Maths. Can follow NCERT book, solutions, NCERT exemplar and previous year solutions.

Q: Can i get chapter-wise solutions for Class 12 Maths ?
Q: Can I get marks distribution for Class 12 Maths ?
Q: Can I get NCERT solutions for Class 11 Maths ?
A:

Here you will get NCERT Solutions for Class 11 Maths. Solutions to each chapter are available with all the necessary steps.

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For the 12th CBSE Hindi Medium board exam, important questions usually come from core chapters like “Madhushala”, “Jhansi ki Rani”, and “Bharat ki Khoj”.
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