$$\begin{gathered} \mathrm{Let} \mathrm{f} : \mathrm{ℝ} \to \mathrm{ℝ} \mathrm{be} \mathrm{a} \mathrm{function} \mathrm{such} \mathrm{that} \\ \mathrm{f}\left(\mathrm{x}\right) = {\mathrm{x}}^3 + {\mathrm{x}}^2\mathrm{f}\text{'}\left(1\right) + \mathrm{xf}\text{'}\text{'}\left(2\right) + \mathrm{f}\text{'}\text{'}\text{'}\left(3\right) \\ \mathrm{for} \mathrm{x} \in \mathrm{ℝ} \end{gathered}$$

Consider the following :

1. f(2) = f(1) - f(0)
2. f"(2) - 2f'(1) = 12

Which of the above is/are correct ?

• 1 only

• 2 only

• Both 1 and 2

• Neither 1 nor 2

Consider the following in respect of the function

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{2 +\mathrm{x}, \mathrm{x} \ge 0 \\ 2 - \mathrm{x}, \mathrm{x}< 0\right\} \\ 1. \underset{\mathrm{x}\to 1}{\lim }\mathrm{f}\left(\mathrm{x}\right) \mathrm{does} \mathrm{not} \mathrm{exist} \\ 2. \mathrm{f}\left(\mathrm{x}\right) \mathrm{is} \mathrm{differentiable} \mathrm{at} \mathrm{x} = 0 \\ 3. \mathrm{f}\left(\mathrm{x}\right) \mathrm{is} \mathrm{continuous} \mathrm{at} \mathrm{x} = 0 \end{gathered}$$

Which of the above statements is/are correct ?

• 1 only

• 3 only

• 2 and 3 only

• 1 and 3 only

Let  $\mathrm{f} :\hspace{0.17em}\mathrm{A} \to \mathrm{ℝ}, \mathrm{where} \mathrm{A} = \mathrm{ℝ}╲\left(0\right) \mathrm{is} \mathrm{such} \mathrm{that} \mathrm{f}\left(\mathrm{x}\right) = \frac{\mathrm{x} + \left|\mathrm{x}\right|}{\mathrm{x}}.$

On which one of the following sets is f (x) continuous ?

• A

• $\mathrm{B} = \left\{\mathrm{x} \in \mathrm{ℝ} : \mathrm{x} \ge 0\right\}$

• $\mathrm{C} = \left\{\mathrm{x} \in \mathrm{ℝ} : \mathrm{x} \le 0\right\}$

• $\mathrm{D} = \mathrm{ℝ}$

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{3{\mathrm{x}}^2 + 12\mathrm{x} - 1, - 1 \le \mathrm{x} \le 2 \\ 37 - \mathrm{x}, 2 < \mathrm{x} \le 3 \right\} \end{gathered}$$

Which of the following statements is/are correct ? 

1. f(x) is continuous at x = 2.
2. f(x) attains greatest value at x = 2.
3. f(x) is differentiable at x = 2

Select the correct answer using the code given below :

• 1 and 3 only

• 2 and 3 only

• Both 1 and 2

• 1, 2 and 3

35.

Which of the following equations is/are correct ?

1. f( - 2) = f(5)

2. f"( - 2) + f"(0.5) + f"(3) = 4

Select the correct answer using the code given below :

• 1 Only

• 2 Only

• Both 1 and 2 

• Neither 1 nor 2

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

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

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

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

If ${\mathrm{x}}^{{\log }_7\left(\mathrm{x}\right)} > 7 \mathrm{where} \mathrm{x} > 0,$ then which one of the following is correct ?

• $\mathrm{x} \in \left(0, \infty \right)$

• $\mathrm{x} \in \left(\frac{1}{7}, 7\right)$

• $\mathrm{x} \in \left(0, \frac{1}{7}\right) \cup \left(7, \infty \right)$

• $\mathrm{x} \in \left(\frac{1}{7}, \infty \right)$