250+ TOP MCQs on Derivatives of Functions in Parametric Forms | Class 12 Maths

Mathematics Problems for Class 12 on “Derivatives of Functions in Parametric Forms”.

1. Find (frac{dy}{dx}), if x=3a² cos²⁡θ and y=4a sin²⁡θ.
a) (frac{3}{4a})
b) –(frac{4}{3a})
c) (frac{4}{3a})
d) –(frac{3}{4a})
Answer: b
Clarification: Given that, x=3a² cos²⁡θ and y=4a sin²⁡θ
Then, (frac{dx}{dθ})=3a².(2 cos⁡θ)(-sin⁡θ)=-3a² sin⁡2θ
(frac{dy}{dθ})=4a(2 sin⁡θ)(cos⁡θ)=4a sin⁡2θ
(frac{dy}{dx})=(frac{dy}{dθ}×frac{dθ}{dx}=-frac{4a ,sin⁡2θ}{3a^2 ,sin⁡2θ}=-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})
Answer: a
Clarification: 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=sin⁡3t and y=t² tan⁡2t.
a) (frac{3t(tan⁡2t+tsec^2 2t)}{4 cos⁡3t})
b) (frac{(tan⁡2t+tsec^2 2t)}{3 cos⁡3t})
c) (frac{-2t(tan⁡2t+tsec^2 2t)}{3 cos⁡3t})
d) (frac{2t(tan⁡2t+tsec^2 2t)}{3 cos⁡3t})
Answer: d
Clarification: Given that, x=sin⁡3t and y=t² tan⁡2t
(frac{dx}{dt})=3 cos⁡3t
By using u.v rule, we get
(frac{dy}{dt})=(frac{d}{dx} ,(t^2) ,tan⁡2t+frac{d}{dx} ,(tan⁡2t)) t²
(frac{dy}{dt})=2t tan⁡2t+2t² sec²⁡2t
(frac{dy}{dx})=(frac{dy}{dt}.frac{dt}{dx}=frac{(2t ,tan⁡2t+2t^2 ,sec^2⁡2t)}{3 ,cos⁡3t})
∴(frac{dy}{dx}=frac{2t(tan⁡2t+tsec^2 ,2t)}{3 ,cos⁡3t})

4. Find (frac{dy}{dx}), if x=log⁡t² and y=(frac{1}{t}).
a) (frac{1}{2t})
b) –(frac{t}{2})
c) –(frac{1}{2t})
d) (frac{t}{2})
Answer: c
Clarification: Given that, x=log⁡t² 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^-1⁡2t 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)})
Answer: d
Clarification: Given that, x=6 sin^-1⁡2t 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
Answer: b
Clarification: 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=2e^t and y=log⁡t
a) (frac{1}{2te^t})
b) –(frac{1}{2te^t})
c) (frac{1}{te^t})
d) (frac{1}{e^t})