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Continuity and Differentiability Class 12 Multiple Choice Test

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Continuity and Differentiability Class 12 (120501)


General Instruction:

1. There are 10 MCQ’s in the Test.

2. Passing %age is 50.

3. After time gets over, test will be submitted itself.

1 / 10

If \( x(t) = e^t \) and \( y(t) = \ln(t) \), what is \( \frac{dy}{dx} \)?

A) \( \frac{1}{t} \)

B) \( \frac{1}{e^t} \)

C) \( t \)

D) \( \frac{t}{e^t} \)

2 / 10

What is the derivative of \( y = \log(log(x)) \)?

A) \( \frac{1}{x \log(x)} \)

B) \( x \log(x) \)

C) \( \frac{1}{x} \)

D) \( \frac{1}{y} \)

3 / 10

If \( f(x) = \sin^{-1}(e^x) \), what is \( f'(x) \)?

A) \( \frac{e^x}{\sqrt{1-e^{2x}}} \)

B) \( \frac{e^x}{1+e^{2x}} \)

C) \( \frac{1}{\sqrt{1-e^{2x}}} \)

D) \( e^x \)

4 / 10

If \( f(x) = e^{2x} \), what is \( f”(x) \)?

A) \( 2e^{2x} \)
B) \( 4e^{2x} \)
C) \( e^{2x} \)
D) \( 4x \cdot e^{2x} \)

5 / 10

Given \( x^2 + y^2 = 1 \), what is \( \frac{dy}{dx} \) at any point on the curve?

A) \( -\frac{x}{y} \)

B) \( \frac{x}{y} \)

C) \( -\frac{y}{x} \)

D) \( \frac{y}{x} \)

6 / 10

Given \( y = x^x \), what is \( \frac{dy}{dx} \)?

A) \( x^x(\ln(x) + 1) \)

B) \( x^{x-1} \)

C) \( e^x \)

D) \( x \ln(x) \)

7 / 10

What is the derivative of \( \cos^{-1}(x) \)?

A) \( \frac{1}{\sqrt{1-x^2}} \)

B) \( \frac{1}{1+x^2} \)

C) \( -\frac{1}{\sqrt{1-x^2}} \)

D) \( \frac{1}{|x|\sqrt{x^2-1}} \)

8 / 10

What is the derivative of \( \arcsec(x) \)?

A) \( \frac{1}{x\sqrt{x^2-1}} \)

B) \( \frac{1}{\sqrt{x^2-1}} \)

C) \( \frac{1}{|x|\sqrt{x^2-1}} \)

D) \( -\frac{1}{|x|\sqrt{x^2-1}} \)

9 / 10

Given \( y = x^2 \cdot e^x \), what is \( \frac{dy}{dx} \)?

A) \( 2x \cdot e^x + x^2 \cdot e^x \)

B) \( 2x \cdot e^x \)

C) \( x^2 \cdot e^x \)

D) \( e^x \)

10 / 10

Given \( x = a \cos(t) \) and \( y = a \sin(t) \), what is \( \frac{dy}{dx} \)?

A) \( \tan(t) \)

B) \( \cot(t) \)

C) \( \sin(t) \)

D) \( \cos(t) \)

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