111.

Let f(x) = [x], where [.] is the greatest integer function and g(x) = sin(x) be two real valued functions over 

Which of the following statements is correct ?

• Both f(x) and g(x) are continuous at x = 0.

• f(x) is continuous at x = 0, but g(x) is not continuous at x = 0

• g(x) is continuous at x = 0, but f(x) is not continuous at x = 0

• Both  f(x) and g(x) are discontinuous at x = 0

Let f(x) = [x], where [.] is the greatest integer function and g(x) = sin(x) be two real valued functions over $\mathrm{ℝ}$

Which one of the following statements is correct ?

• $\underset{\mathrm{x}\to 0}{\lim }\left(\mathrm{fog}\right)\left(\mathrm{x}\right) \mathrm{exists}$

• $\underset{\mathrm{x}\to 0}{\lim }\left(\mathrm{gof}\right)\left(\mathrm{x}\right) \mathrm{exists}$

• $\underset{\mathrm{x}\to 0^{-}}{\lim }\left(\mathrm{fog}\right)\left(\mathrm{x}\right) = \underset{\mathrm{x}\to 0^{-}}{\lim }\left(\mathrm{gof}\right)\left(\mathrm{x}\right)$

• $\underset{\mathrm{x}\to 0^{+}}{\lim }\left(\mathrm{fog}\right)\left(\mathrm{x}\right) = \underset{\mathrm{x}\to 0^{+}}{\lim }\left(\mathrm{gof}\right)\left(\mathrm{x}\right)$

Letf(x) = [x], where [.] is the greatest integer function and g(X) = sin(x) be two real valued functions over R.

Which of the following statements are correct ?

1. (fof) (x) = f (x).

2. (gog)(x) = g(x) only when x = 0.

3.(go(fog))(x) can take only three values.

Select the correct answer using the code given below :

• 1 and 2 only

• 2 and 3 only

• 1 and 3 only

• 1, 2 and 3

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

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

Consider the parabola y = x² + 7x + 2 and the straight liney = 3x - 3.

What are the coordinates of the point on the parabola which is closest to the straight line ?

• (0, 2)

• ( - 2,  - 8)

• ( - 7, 2)

• (1, 10)

Consider the parabola y = x² + 7x + 2 and the straight liney = 3x - 3.

What is the shortest distance from the above point on the parabola to the line ?

• $\frac{\sqrt{10}}{2}$

• $\frac{\sqrt{10}}{5}$

• $\frac{1}{\sqrt{10}}$

• $\frac{\sqrt{5}}{4}$

$$\begin{gathered} \mathrm{Let} \mathrm{f}\left(\mathrm{x}\right) = \left\{- 2, - 3 \le \mathrm{x} \le 0 \\ \mathrm{x} - 2, 0 < \mathrm{x} \le 3\right\} \mathrm{and} \\ \mathrm{g}\left(\mathrm{x}\right) = \mathrm{f}\left(\left|\mathrm{x}\right|\right) + \left|\mathrm{f}\left(\mathrm{x}\right)\right| \end{gathered}$$

Which of the following statements is/are correct ?

1. g(x) is differentiable at x = 0
2. g(x) is differentiable at x = 2.the parabola to the line ?

Select the correct answer using the code given below :

• 1 only

• 2 Only

• Both 1 and 2

• Neither 1 nor 2

$$\begin{gathered} \mathrm{Let} \mathrm{f}\left(\mathrm{x}\right) = \left\{- 2, - 3 \le \mathrm{x} \le 0 \\ \mathrm{x} - 2, 0 < \mathrm{x} \le 3\right\} \mathrm{and} \\ \mathrm{g}\left(\mathrm{x}\right) = \mathrm{f}\left(\left|\mathrm{x}\right|\right) + \left|\mathrm{f}\left(\mathrm{x}\right)\right| \end{gathered}$$

Which of the following statements is/are correct ?

1. g(x) is differentiable at x = 0
2. g(x) is differentiable at x = 2.the parabola to the line ?

What is the value of the differential coefficient of g(x) at x = - 2 ?

•  - 1

• 0

• 1

• 2

$$\begin{gathered} \mathrm{Let} \mathrm{f}\left(\mathrm{x}\right) = \left\{- 2, - 3 \le \mathrm{x} \le 0 \\ \mathrm{x} - 2, 0 < \mathrm{x} \le 3\right\} \mathrm{and} \\ \mathrm{g}\left(\mathrm{x}\right) = \mathrm{f}\left(\left|\mathrm{x}\right|\right) + \left|\mathrm{f}\left(\mathrm{x}\right)\right| \end{gathered}$$

Which of the following statements is/are correct ?

1. g(x) is continuous at x = 0.
2. g(x) is continuous at x = 2.
3. g(x) is continuous at x = - 1.

Select the correct answer using the code given below

• 1 and 2 only

• 2 and 3 only

• 1 and 3 only

• 1, 2 and 3