31.

A circle of radius 4, drawn on a chord of the parabola y² = 8x as diameter, touches the axis of the parabola. Then, the slope of the chord is

• 1

• 2

The mid-point of a chord of the ellipse x² + 4y² - 2x + 20y = 0 is (2, - 4). The equation of the chord is

• x - 6y = 26

• x + 6y = 26

• 6x - y = 26

• 6x + y = 26

If the focus of the ellipse $\frac{{\mathrm{x}}^2}{25} + \frac{{\mathrm{y}}^2}{16} = 1$ and the hyperbola $\frac{{\mathrm{x}}^2}{4} - \frac{{\mathrm{y}}^2}{{\mathrm{b}}^2} = 1$ coincide, then b² =?

• 4

• 5

• 8

• 9

If a = 9 is a chord of contact of the hyperbola x² - y² = 9, then the equation of the tangent at one of the points of contact is 

• $\mathrm{x} + \sqrt{3}\mathrm{y} + 2 = 0$

• $3\mathrm{x} + 2\sqrt{2}\mathrm{y} - 3 = 0$

• $3\mathrm{x} - \sqrt{2}\mathrm{y} + 6 = 0$

• $\mathrm{x} - \sqrt{3}\mathrm{y} + 2 = 0$

The perpendicular distance from the point $\left(1, \mathrm{\pi }\right)$ to the line joining (1, 0⁰) and $\left(1, \frac{\mathrm{\pi }}{2}\right)$ (in polar coordinates) is

• 2

• $\sqrt{3}$

• 1

• $\sqrt{2}$

If D(2, 1, 0), E(2, 0, 0) and F(0, 1, 0) are mid-points of the sides BC, CA and AB of $∆$ABC, respectively. Then, the centroid of$∆$ABC is

• $\left(\frac{1}{3}, \frac{{\displaystyle 1}}{{\displaystyle 3}}, \frac{{\displaystyle 1}}{{\displaystyle 3}}\right)$

• $\left(\frac{4}{3}, \frac{2}{3}, 0\right)$

• $\left(- \frac{1}{3}, \frac{1}{3}, \frac{1}{3}\right)$

• $\left(\frac{2}{3}, \frac{1}{3}, \frac{1}{3}\right)$

The relation between pressure p and volume V is given by pV^1/4 constant. If the percentage decrease in volume is 1/2, then the percentage increase in pressure is

• $- \frac{1}{8}$

• $\frac{1}{16}$

• $\frac{1}{8}$

• $\frac{1}{2}$

The area (in sq units) bounded by the curves x = - 2y² and x = 1 - 3y² is

• $\frac{2}{3}$

• 1

• $\frac{4}{3}$

• $\frac{5}{3}$

$$\begin{gathered} \mathrm{If} \mathrm{I} \mathrm{is} \mathrm{the} \mathrm{identity} \mathrm{matrix} \mathrm{of} \mathrm{order} 2 \mathrm{and} \\ \mathrm{A} =\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right], \mathrm{then} \mathrm{for} \mathrm{n} \ge 1, \mathrm{mathematical} \mathrm{induction} \mathrm{gives} \end{gathered}$$

• ${\mathrm{A}}^{\mathrm{n}} = \mathrm{nA} - \left(\mathrm{n} - 1\right)\mathrm{I}$

• ${\mathrm{A}}^{\mathrm{n}} = \mathrm{nA} + \left(\mathrm{n} - 1\right)\mathrm{I}$

• ${\mathrm{A}}^{\mathrm{n}} = 2^{\mathrm{n}}\mathrm{A} - \left(\mathrm{n} +\hspace{0.17em}1\right)\mathrm{I}$

• ${\mathrm{A}}^{\mathrm{n}} = 2^{\mathrm{n} - 1}\mathrm{A} - \left(\mathrm{n} - 1\right)\mathrm{I}$

$\mathrm{If} \mathrm{A} = \left[\begin{array}{cc}- 8& 5\\ 2& 4\end{array}\right] \mathrm{satisfies} \mathrm{the} \mathrm{equation} {\mathrm{x}}^2 + 4\mathrm{x} - \mathrm{p} = 0, \mathrm{then} \mathrm{p} \mathrm{is} \mathrm{equal} \mathrm{to}$

• 64

• 42

• 36

• 24