Continuity and Differentiability Class 12 Mcq Test (120501)

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

What is the value of 'k' for which the function \(f(x) = \begin{cases} \frac{\sin(5x)}{3x} & \text{if } x \neq 0 \\ k & \text{if } x = 0 \end{cases}\) is continuous at \(x=0\)?

a) \(\frac{3}{5}\)

b) \(\frac{5}{3}\)

c) \(5\)

d) \(3\)

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) \( e^x \)

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) \( \cos(t) \)

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

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

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

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

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

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

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

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

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

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

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

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) \( e^x \)

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

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

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

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

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

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

C) \( e^x \)

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