For what value of k is the function $$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{2\mathrm{x} + \frac{1}{4} \mathrm{x} < 0 \\ \mathrm{k} \mathrm{x} = 0 \\ {\left(\mathrm{x} + \frac{1}{2}\right)}^2 \mathrm{x} > 0\right\} \end{gathered}$$ continuous?

• $\frac{1}{4}$

• $\frac{1}{2}$

• 1

• 2

What is the derivative of $2^{{\left(\sin \left(\mathrm{x}\right)\right)}^2}$ with respect to sin(x) ?

• $\sin \left(\mathrm{x}\right)2^{{\left(\sin \left(\mathrm{x}\right)\right)}^2}\ln \left(4\right)$

• $2\sin \left(\mathrm{x}\right)2^{{\left(\sin \left(\mathrm{x}\right)\right)}^2}\ln \left(4\right)$

• $\ln \left(\sin \left(\mathrm{x}\right)\right)2^{{\left(\sin \left(\mathrm{x}\right)\right)}^2}$

• $2\sin \left(\mathrm{x}\right)\cos \left(\mathrm{x}\right)2^{{\left(\sin \left(\mathrm{x}\right)\right)}^2}$

Consider the following statements in respect of the function $\mathrm{f}\left(\mathrm{x}\right) = \sin \left(\frac{1}{\mathrm{x}}\right)$ for x ≠ 0 and f(0) = 0 :

1) $\underset{\mathrm{x}\to 0}{\lim }\mathrm{f}\left(\mathrm{x}\right)$ exists

2) f(x) is continuous at x = 0

Which of the above statements is/are correct ?

• 1 only

• 2 only

• Both 1 and 2

• Neither 1 nor 2

What is the derivative of tan^-1(x) with respect to cot^-1(x) ?

•  - 1

• 1

• $\frac{1}{{\mathrm{x}}^2 + 1}$

• $\frac{\mathrm{x}}{{\mathrm{x}}^2 + 1}$

45.

Consider the following statements:

f(x) = e^-|x|:

1) The function is continuous at x = 0.

2) The function is differentiable at x = 0.

Which of the above statements is/are correct ?

• 1 only

• 2 only

• Both 1 and 2

• Neither 1 nor 2

If $\mathrm{f}\left(\mathrm{x}\right) = \frac{\sin \left(\mathrm{x}\right)}{\mathrm{x}}$ where x ∈ R, is to be continuous at x = 0, then the value of the function at x = 0

• should be 0

• should be 1

• should be 2

• cannot be determined

If e^θφ = c + 4θφ, where c is an arbitrary constant and φ is a function of θ, then what is φdθ equal to ?

• θdφ

• - θdφ

• 4θ dφ

• - 4θ dφ