If $\underset{\mathrm{x}\to 0}{\lim }\mathrm{\varphi }\left(\mathrm{x}\right) = {\mathrm{a}}^2, \mathrm{where} \mathrm{a} \ne 0, \mathrm{then} \mathrm{what} \mathrm{is} \underset{\mathrm{x}\to 0}{\lim }\mathrm{\varphi }\left(\frac{\mathrm{x}}{\mathrm{a}}\right) = ?$

• a²

• a ^{- 2}

•  - a²

•  - a

What is $\underset{\mathrm{x}\to 0}{\lim }{\mathrm{e}}^{ - \frac{1}{{\mathrm{x}}^2}} = ?$

• 0

• 1

•  - 1

• limit does not exist

What is f'(x) = ? when x > 1 

• 0

• 2x - 1

• 4x - 2

• 8x - 4

What is f'(x) equal to when 0 < x < 1 ?

• 0

• 2x - 1

• 4x - 2

• 8x - 4

$$\begin{gathered} \mathrm{Let} \mathrm{f}\left(\mathrm{x}\right) = \left\{\frac{{\mathrm{e}}^{\mathrm{x}} - 1}{\mathrm{x}}, \mathrm{x} > 0 \\ 0, \mathrm{x} = 0\right\} \end{gathered}$$

be a real valued function.

Which one of the following statements is correct ?

• f (x) is a strictly decreasing function in (0, x)

• f(x) is a strictly increasing function in (0, x)

• f(x) is neither increasing nor decreasing in (0, x).

• f(x) is not decreasing in (0, x)

$$\begin{gathered} \mathrm{Let} \mathrm{f}\left(\mathrm{x}\right) = \left\{\frac{{\mathrm{e}}^{\mathrm{x}} - 1}{\mathrm{x}}, \mathrm{x} > 0 \\ 0, \mathrm{x} = 0\right\} \end{gathered}$$

be a real valued function.

Which one of the following statements is correct ?

1. f(x) is right continuous at x = 0.
2. f(x) is discontinuous at x = 1.

Select the correct answer using the code given below

• 1 only

• 2 only

• Both 1 and 2

• Neither 1 nor 2

27.

Let f(x) = x² + 2x – 5 and g(x) = 5x + 30

If h(x) = 5 f(x) – xg(x), then what is the derivative of h(x)

•  - 40

•  - 20

• - 10

• 0

Let f(x) = x² + 2x – 5 and g(x) = 5x + 30

What are the roots of the equation g[f(x)] = 0 ?

• 1, - 1

•  - 1, - 1

• 1, 1

• 0, 1

Consider the equation x^y = e^x - y. What is $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}$ at  x = 1 equal to ?

• 0

• 1

• 2

• 4

Consider the equation x^y = e^x - y

What is $\frac{\mathrm{dy}}{\mathrm{dx}}$ at  x = 1 equal to ?

• 0

• 1

• 2

• 4