If the function f : R $- \left\{1, - 1\right\} \to \mathrm{A}$ defined by $\mathrm{f}\left(\mathrm{x}\right) = \frac{{\mathrm{x}}^2}{1 - {\mathrm{x}}^2}$, is surjective, then A is equal to :

• R - (- 1, 0)

• R - [- 1, 0)

• - {- 1}

• $[0, \infty )$

If the function f defined on $\left(\frac{\mathrm{\pi }}{6}, \frac{\mathrm{\pi }}{3}\right)$ by f(x) = $$\begin{gathered} \left\{\frac{\sqrt{2}\cos \left(\mathrm{x}\right) - 1}{\cot \left(\mathrm{x}\right) - 1}, \mathrm{x} \ne \frac{\mathrm{\pi }}{4} \\ \mathrm{k}, \mathrm{x} = \frac{{\displaystyle \mathrm{\pi }}}{{\displaystyle 4}}\right\} \end{gathered}$$ is continuous, then k is equal to :

• 2

• $\frac{1}{2}$

• $\frac{1}{\sqrt{2}}$

• 1

3.

Let $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ be the roots of the equation x² + x + 1 = 0 . Then for y $\ne$ 0 in R, $\left|\begin{array}{ccc}\mathrm{y} + 1& \mathrm{\alpha }& \mathrm{\beta }\\ \mathrm{\alpha }& \mathrm{y} +\hspace{0.17em}\mathrm{\beta }& 1\\ \mathrm{\beta }& 1& \mathrm{y} + \mathrm{\alpha }\end{array}\right|$ is equal to :

• y(y² - 1)

• y³

• y(y² - 3)

• y³ - 1

If the line y = mx + 7$\sqrt{3}$ is normal to the hyperbola $\frac{{\mathrm{x}}^2}{24} - \frac{{\mathrm{y}}^2}{18} = 1$, then a value of m is :

• $\frac{\sqrt{5}}{2}$

• $\frac{\sqrt{15}}{2}$

• $\frac{{\displaystyle 2}}{{\displaystyle \sqrt{15}}}$

• $\frac{{\displaystyle 3}}{{\displaystyle \sqrt{15}}}$

Let $\sum _{\mathrm{k} = 1}^0\mathrm{f}\left(\mathrm{a} + \mathrm{k}\right) = 16\left(2^{10} - 1\right)$, where function satisfies f(x + y) = f(x)f(y) for all natural numbers x, y and f(1) = 2 . Then the natural number ‘a’ is

• 16

• 22

• 20

• 25

If the tangent to the curve, y = x³ + ax - b at the point (- 1, - 5) is perpendicular to the line, - x + y + 4 = 0, then which one of the following points lies on the curve ?

• (2, - 1)

• (- 2, 2)

• (2, - 2)

• (- 2, 1)

If $\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 3\\ 0& 1\end{array}\right] ... \left[\begin{array}{cc}1& \mathrm{n} - 1\\ 0& 1\end{array}\right] = \left[\begin{array}{cc}1& 78\\ 0& 1\end{array}\right]$ then the inverse of $\left[\begin{array}{cc}1& \mathrm{n}\\ 0& 1\end{array}\right]$ is

• $\left[\begin{array}{cc}1& 12\\ 0& 1\end{array}\right]$

• $\left[\begin{array}{cc}1& 0\\ 12& 1\end{array}\right]$

• $\left[\begin{array}{cc}1& 0\\ 13& 1\end{array}\right]$

• $\left[\begin{array}{cc}1& - 13\\ 0& 1\end{array}\right]$

Let f(x) = 15 - $\left|\mathrm{x} - 10\right|$ ; x $\in$ R. Then the set of all values of x, at which the function, g(x) = f(f(x) is not differentiable, is:

• {5, 10, 15}

• {10}

• {10, 15}

• {5, 0, 15, 20}

Let $\overline{\mathrm{\alpha }} = 3\hat{\mathrm{i}} + \hat{\mathrm{j}} \mathrm{and} \mathrm{\beta } = 2\hat{\mathrm{i}} - \hat{\mathrm{j}} + 3\hat{\mathrm{k}}$. If $\overline{\mathrm{\beta }} = \overline{{\mathrm{\beta }}_1} - \overline{{\mathrm{\beta }}_2}$, where $\overline{{\mathrm{\beta }}_1}$ is paralle to $\overline{\mathrm{\alpha }}$ and $\overline{{\mathrm{\beta }}_2}$ is perpendicular to $\overline{\mathrm{\alpha }}$, then $\overline{{\mathrm{\beta }}_1} \times \overline{{\mathrm{\beta }}_2}$

• $\frac{1}{2}\left(- 3\overline{\mathrm{i}} + 9\overline{\mathrm{j}} + 5\overline{\mathrm{k}}\right)$

• $\frac{1}{2}\left(3\overline{\mathrm{i}} - 9\overline{\mathrm{j}} + 5\overline{\mathrm{k}}\right)$

• $3\overline{\mathrm{i}} - 9\overline{\mathrm{j}} - 5\overline{\mathrm{k}}$

• $- 3\overline{\mathrm{i}} + 9\overline{\mathrm{j}} + 5\overline{\mathrm{k}}$

The value of ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{{\sin }^3\left(\mathrm{x}\right)}{\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)}\mathrm{dx}$ is :

• $\frac{\mathrm{\pi } - 2}{4}$

• $\frac{\mathrm{\pi } - 2}{8}$

• $\frac{\mathrm{\pi } - 1}{4}$

• $\frac{\mathrm{\pi } - 1}{2}$