The normal at (a, 2a) on y² = 4ax, meets the curve again at (at, 2at ), then the value of t is

• - 1

• 1

• - 3

• 3

The value of ${\left[\sqrt{2}\left\{\cos \left(56°15\text{'}\right) +\hspace{0.17em}\mathrm{isin}\left(56°15\text{'}\right)\right\}\right]}^8$

• - 16i

• 16i

• 8i

• 4i

3.

If $\mathrm{\alpha }$ is an nth root of unity, then $1 + 2\mathrm{\alpha } + 3{\mathrm{\alpha }}^2 + ... + {\mathrm{n\alpha }}^{\mathrm{n} - 1}$equals

• $- \frac{\mathrm{n}}{\left(1 - \mathrm{\alpha }\right)}$

• $- \frac{\mathrm{n}}{{\left(1 + \mathrm{\alpha }\right)}^2}$

• $\frac{\mathrm{n}}{\left(1 - \mathrm{\alpha }\right)}$

• None of these

Given, $\mathrm{f}\left(\mathrm{x}\right) = \log \left(\frac{1 + \mathrm{x}}{1 - \mathrm{x}}\right)$ and $\mathrm{g}\left(\mathrm{x}\right) = \frac{3\mathrm{x} + {\mathrm{x}}^3}{1 +\hspace{0.17em}3{\mathrm{x}}^2}$, then fog(x) equals

• [f(x)]³

• - f(x)

• 3f(x)

• None of these

If a set A contains n elements, then which of the following. cannot be the number of reflexive relations on the set A?

• 2^{n + 1}

• 2^{n - 1}

• 2ⁿ

• $2^{{\mathrm{n}}^2} - 1$

The relation on the set A = {x : $\left|\mathrm{x}\right| < 3, \mathrm{x} \in \mathrm{Z}$} is defined by R = {(x, y) : y = $\left|\mathrm{x}\right|, \mathrm{x} \ne - 1$} Then, the number of elements in the power set of R is

• 8

• 16

• 32

• 64

Which of the following proposition is a tautology?

• $\left(~ \mathrm{P} \vee ~ \mathrm{q}\right) \wedge \left(\mathrm{p} \vee ~ \mathrm{q}\right)$

• $~ \mathrm{q} \wedge \left(\mathrm{p} \vee ~ \mathrm{q}\right)$

• $~ \mathrm{P} \wedge \left(\mathrm{p} \vee ~ \mathrm{q}\right)$

• $\left(~ \mathrm{P} \vee ~ \mathrm{q}\right) \vee \left(\mathrm{p} \vee ~ \mathrm{q}\right)$

The combined equation of the asymptotes of the hyperbola 2x² + 5xy + 2y² + 4x + 5y = 0 is

• 2x² + 5xy + 2y² + 4x + 5y - 2 = 0

• 2x² + 5xy + 2y² = 0

• 2x² + 5xy + 2y² + 4x + 5y + 2 = 0

• None of the above

A point on the ellipse : $\frac{{\mathrm{x}}^2}{16} + \frac{{\mathrm{y}}^2}{9} = 1$ at a distance equal to the mean of length of the semi-major and semi-minor axes from the centre, is

• $\left(\frac{2\sqrt{91}}{7}, \frac{- 3\sqrt{91}}{14}\right)$

• $\left(\frac{- 2\sqrt{105}}{7}, \frac{\sqrt{91}}{14}\right)$

• $\left(\frac{- 2\sqrt{105}}{7}, \frac{- 3\sqrt{91}}{14}\right)$

• $\left(\frac{2\sqrt{91}}{7}, \frac{3\sqrt{105}}{14}\right)$

The parametric coordinates of any point on the parabola whose focus is (0, 1) and the directrix is x + 2 = 0, are

• (t² - 1, 2t + 1)

• (t² + 1, 2t + 1)

• (t², 2t)

• (t² + 1, 2t - 1)