The function $\mathrm{f}\left(\mathrm{x}\right) = \frac{\tan \left\{\mathrm{\pi }\left[\mathrm{x} - {\displaystyle \frac{\mathrm{\pi }}{2}}\right]\right\}}{2 + {\left[\mathrm{x}\right]}^2}$, where $\left[\mathrm{x}\right]$ denotes the greatest integer $\le$ x, is

• continuous for all values of x

• discontinuous at x = $\frac{\mathrm{\pi }}{2}$

• not differentiable for some values of x

• discontinuous at x = - 2

The function f (x) = $\mathrm{f}\left(\mathrm{x}\right) = \mathrm{asin}\left|\mathrm{x}\right| + {\mathrm{be}}^{\left|\mathrm{x}\right|}$ is differentiable at x = 0 when

• 3a + b = 0

• 3a - b = 0

• a + b = 0

• a - b = 0

Suppose that the equation f (x) = x² + bx + c = 0 has two distinct real roots $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$. The angle between the tangent to the curve y = f (x) at the point $\left(\frac{\mathrm{\alpha } + \mathrm{\beta }}{2}, \mathrm{f}\left(\frac{{\displaystyle \mathrm{\alpha } + \mathrm{\beta }}}{{\displaystyle 2}}\right)\right)$ and the positive direction of the x-axis is

• 0°

• 30°

• 60°

• 90°

The function f(x) = x² + bx + c, where b and c real constants, describes

• one - to - one mapping

• onto mapping

• not one-to-one but onto mapping

• neither one-to-one nor onto mapping

Let $\mathrm{n} \ge 2$ be an integer,

$\mathrm{A} = \left[\begin{array}{ccc}\cos \left(2\mathrm{\pi }/3\right)& \sin \left(2\mathrm{\pi }/\mathrm{n}\right)& 0\\ - \sin \left(2\mathrm{\pi }/\mathrm{n}\right)& \cos \left(2\mathrm{\pi }/\mathrm{n}\right)& 0\\ 0& 0& 1\end{array}\right]$ and I is the identity matrix of order 3. Then,

• Aⁿ = I and A^{n - 1} $\ne$ I

• A^m $\ne$ I for any positive integer m

• A is not invertible 

• A^m = 0 for a positive integer m 

Let R be the set of all real numbers and f : [- 1, 1] $\to$ R be defined by

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{\mathrm{xsin}\left(\frac{1}{\mathrm{x}}\right), \mathrm{x} \ne 0 \\ 0, \mathrm{x} = 0\right\} \end{gathered}$$ Then,

• f satisfies tile conditions of Rolle's theorem on [-1, 1]

• f satisfies the conditions of Lagrange's mean value theorem on [-1, 1]

• f satisfies the conditions of Rolle's theorem on [0, 1]

• f satisfies the conditions of Lagrange's mean value theorem on [0, 1]

Let I denote the 3 x 3 identity matrix and P be a matrix obtained by rearranging the columns of I. Then,

• there are six distinct choices for P and det (P) = 1

• there are six distinct choices for P and det (P) =  $\pm 1$

• there are more than one choices for P and some of them are not invertible

• there are more than one choices for P and P^{- 1} = I in each choice

58.

Suppose that f(x) is a differentiable function such that f'(x) is continuous, f'(0) = 1 and f''(0) does not exist. Let g(x) = xf'(x), Then,

• g'(0) does not exist

• g'(0) = 0

• g'(0) = 1

• g'(0) = 2

Applying Lagrange's Mean Value Theorem for a suitable function f(x) in [0, h], we have f(h) = f(0) + hf'($\mathrm{\theta h}$), $0 < \mathrm{\theta } < 1$. Then, for f(x) = cos(x), the value of $\underset{\mathrm{h}\to 0^{+}}{\lim }\mathrm{\theta }$ is

• 1

• 0

• 1/2

• 1/3

For any two real numbers $\mathrm{\theta } \mathrm{and} \mathrm{\varphi }$ we define $\mathrm{\theta R\varphi }$, if and only if ${\sec }^2\left(\mathrm{\theta }\right) - {\tan }^2\left(\mathrm{\varphi }\right)$ = 1. The relation R is

• reflexive but not transitive

• symmetric but not reflexive

• both reflexive and symmetric but not transitive

• an equivalence relation