This set of Class 12 Maths Chapter 5 Multiple Choice Questions & Answers (MCQs) focuses on “Derivatives of Functions in Parametric Forms”.
1. Find \(\frac{dy}{dx}\), if x=3a2 cos2θ and y=4a sin2θ.
a) \(\frac{3}{4a}\)
b) –\(\frac{4}{3a}\)
c) \(\frac{4}{3a}\)
d) –\(\frac{3}{4a}\)
View Answer
Explanation: Given that, x=3a2 cos2θ and y=4a sin2θ
Then, \(\frac{dx}{dθ}\)=3a2.(2 cosθ)(-sinθ)=-3a2 sin2θ
\(\frac{dy}{dθ}\)=4a(2 sinθ)(cosθ)=4a sin2θ
\(\frac{dy}{dx}\)=\(\frac{dy}{dθ}×\frac{dθ}{dx}=-\frac{4a \,sin2θ}{3a^2 \,sin2θ}=-\frac{4}{3a}\)
2. Find \(\frac{dy}{dx}\), if x=9t4 and y=t.
a) \(\frac{1}{36t^3}\)
b) \(\frac{1}{36t^2}\)
c) \(\frac{-1}{36t^3}\)
d) \(\frac{1}{32t^3}\)
View Answer
Explanation: Given that, x=9t4 and y=t
\(\frac{dx}{dt}\)=36t3
\(\frac{dy}{dt}\)=1
∴\(\frac{dy}{dx}\)=\(\frac{dy}{dt}.\frac{dt}{dx}=\frac{1}{36t^3}\)
3. Find \(\frac{dy}{dx}\), if x=sin3t and y=t2 tan2t.
a) \(\frac{3t(tan2t+tsec^2 2t)}{4 cos3t}\)
b) \(\frac{(tan2t+tsec^2 2t)}{3 cos3t}\)
c) \(\frac{-2t(tan2t+tsec^2 2t)}{3 cos3t}\)
d) \(\frac{2t(tan2t+tsec^2 2t)}{3 cos3t}\)
View Answer
Explanation: Given that, x=sin3t and y=t2 tan2t
\(\frac{dx}{dt}\)=3 cos3t
By using u.v rule, we get
\(\frac{dy}{dt}\)=\(\frac{d}{dx} \,(t^2) \,tan2t+\frac{d}{dx} \,(tan2t)\) t2
\(\frac{dy}{dt}\)=2t tan2t+2t2 sec22t
\(\frac{dy}{dx}\)=\(\frac{dy}{dt}.\frac{dt}{dx}=\frac{(2t \,tan2t+2t^2 \,sec^22t)}{3 \,cos3t}\)
∴\(\frac{dy}{dx}=\frac{2t(tan2t+tsec^2 \,2t)}{3 \,cos3t}\)
4. Find \(\frac{dy}{dx}\), if x=logt2 and y=\(\frac{1}{t}\).
a) \(\frac{1}{2t}\)
b) –\(\frac{t}{2}\)
c) –\(\frac{1}{2t}\)
d) \(\frac{t}{2}\)
View Answer
Explanation: Given that, x=logt2 and y=\(\frac{1}{t}\)
\(\frac{dx}{dt}\)=\(\frac{1}{t^2}.2t=\frac{2}{t}\)
\(\frac{dy}{dt}\)=-\(\frac{1}{t^2}\)
∴\(\frac{dy}{dx}\)=\(\frac{dy}{dt}.\frac{dt}{dx}=-\frac{1}{t^2}.\frac{t}{2}=-\frac{1}{2t}\)
5. Find \(\frac{dy}{dx}\), if x=6 sin-12t and y=\(\frac{1}{\sqrt{1-4t^2}}\).
a) \(\frac{t}{1-4t^2}\)
b) –\(\frac{1}{3(1-4t^2)}\)
c) –\(\frac{t}{3(1-4t^2)}\)
d) \(\frac{1}{3(1-4t^2)}\)
View Answer
Explanation: Given that, x=6 sin-12t and y=\(\frac{1}{\sqrt{1-4t^2}}\)
\(\frac{dx}{dt}\)=\(\frac{6}{\sqrt{1-4t^2}}.2=\frac{12}{\sqrt{1-4t^2}}\)
\(\frac{dy}{dt}\)=-\(\frac{1}{2(1-4t^2)^{3/2}}.(-8t)=\frac{4t}{(1-4t^2)^{3/2}}\)
\(\frac{dy}{dx}\)=\(\frac{dy}{dt}.\frac{dt}{dx}=\frac{4t}{(1-4t^2)^{3/2}}.\frac{\sqrt{1-4t^2}}{12}\)
\(\frac{dy}{dx}\)=\(\frac{t}{3(1-4t^2)}\)
6. Find \(\frac{dy}{dx}\), if x=2t2 and y=6t6.
a) -9t4
b) 9t4
c) t4
d) 9t3
View Answer
Explanation: Given that, x=2t2 and y=6t6
\(\frac{dx}{dt}\)=4t
\(\frac{dy}{dt}\)=36t5
\(\frac{dy}{dx}\)=\(\frac{dy}{dt}.\frac{dt}{dx}=\frac{36t^5}{4t}=9t^4\)
7. Find \(\frac{dy}{dx}\), if x=2et and y=logt
a) \(\frac{1}{2te^t}\)
b) –\(\frac{1}{2te^t}\)
c) \(\frac{1}{te^t}\)
d) \(\frac{1}{e^t}\)
View Answer
Explanation: Given that, x=2et and y=logt
\(\frac{dx}{dt}\)=2et
\(\frac{dy}{dt}\)=1/t
∴\(\frac{dy}{dx}\)=\(\frac{dy}{dt}.\frac{dt}{dx}=\frac{1}{t}.\frac{1}{2e^t}=\frac{1}{2te^t}\).
8. Find \(\frac{dy}{dx}\), if x=tan2θ and y=cos2θ+sin2θ.
a) –\(\frac{tan^22θ \,sin2θ}{2}\)
b) \(\frac{3 tan^22θ sin2θ}{2}\)
c) 0
d) \(\frac{tan^22θ sin2θ}{2}\)
View Answer
Explanation: Given that, x=tan2θ and y=cos2θ+sin2θ
\(\frac{dx}{dθ}\)=2 sec22θ
\(\frac{dy}{dθ}\)=-2 sin2θ+2 sinθ cosθ=-2 sin2θ+sin2θ=-sin2θ
∴\(\frac{dy}{dx}\)=\(\frac{dy}{dθ}.\frac{dθ}{dx}=-\frac{sin2θ}{2 sec^22θ}=-\frac{sin2θ}{2 cos^22θ}.sin^22θ=-\frac{tan^22θ sin2θ}{2}\)
9. Find \(\frac{dy}{dx}\), if x=log(tant) and y=log(sint).
a) 2 cos2t
b) cos22t
c) cos2t
d) -cos2t
View Answer
Explanation: Given that, x=log(tant) and y=log(sint)
\(\frac{dx}{dt}\)=\(\frac{1}{tant}.sec^2t=cott sec^2t\)
\(\frac{dy}{dt}=\frac{1}{sin \,t}.cos \,t=cot \,t\)
∴\(\frac{dy}{dx}\)=\(\frac{dy}{dt}.\frac{dt}{dx}=\frac{cot\,t}{cot\,t sec^2t}=\frac{1}{sec^2t}=cos^2t\).
10. Find \(\frac{dy}{dx}\), if x=a2 t2 cotθ and y=at sinθ.
a) \(\frac{tanθ \,sinθ}{at}\)
b) \(\frac{tanθ \,sinθ}{2at}\)
c) \(\frac{tanθ \,sinθ}{2t}\)
d) \(\frac{tanθ \,sinθ}{2a}\)
View Answer
Explanation: Given that, x=a2 t2 cotθ and y=at sinθ
\(\frac{dx}{dt}\)=2ta2 cotθ
\(\frac{dy}{dt}\)=asinθ
\(\frac{dy}{dx}\)=\(\frac{asinθ}{2ta^2 \,cotθ}=\frac{a sinθ}{2a^2 t cosθ}.sinθ=\frac{tanθ \,sinθ}{2at}\)
Sanfoundry Global Education & Learning Series – Mathematics – Class 12.
To practice all chapters and topics of class 12 Mathematics, here is complete set of 1000+ Multiple Choice Questions and Answers.
If you find a mistake in question / option / answer, kindly take a screenshot and email to [email protected]
- Practice Class 11 - Mathematics MCQs
- Check Class 12 - Mathematics Books
- Practice Class 12 - Biology MCQs
- Practice Class 12 - Chemistry MCQs
- Practice Class 12 - Physics MCQs