A rod of length l slides with its ends on two perpendicular lines. Then, the locus of its mid point is

• ${\mathrm{x}}^2 - {\mathrm{y}}^2 = \frac{{\mathrm{l}}^2}{4}$

• None of these

The equation of straight line through the intersection of line 2x +y = 1 and 3x + 2y = 5 and passing through the origin is

• 7x + 3y = 0

• 7x - y = 0

• 3x + 2y = 0

• x + y = 0

The line joining (5, 0) to ($10\cos \left(\mathrm{\theta }\right), 10\sin \left(\mathrm{\theta }\right)$) is divided internally in the ratio 2 : 3 at P. If 0 varies, then the locus of P is

• a straight line

• a pair of straight lines

• a circle

• None of the above

If 2x + y + k = 0 is a normal to the parabola y² = - 8x, then the value of k, is

• 8

• 16

• 24

• 32

$\underset{\mathrm{n}\to \infty }{\lim }\left[\frac{1}{1 . 2} + \frac{1}{2 . 3} + \frac{1}{3 . 4} + ... + \frac{1}{\mathrm{n}\left(\mathrm{n} + 1\right)}\right]$ is equal to

• 1

• - 1

• 0

• None of these

The condition that the line lx + my = 1 may be normal to the curve y² = 4ax, is

• al³ - 2alm² = m²

• al² + 2alm³ = m²

• al³ + 2alm² = m³

• al³ + 2alm² = m²

If the equation of an ellipse is 3x² + 2y² + 6x - 8y + 5 = 0, then which of the following are true?

• e = $\frac{1}{\sqrt{3}}$

• centre is (- 1, 2)

• foci are (- 1, 1) are (- 1, 3)

• All of the above

The equation of the common tangents to the two hyperbolas $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2} - \frac{{\mathrm{y}}^2}{{\mathrm{b}}^2} = 1 \mathrm{and} \frac{{\mathrm{y}}^2}{{\mathrm{a}}^2} - \frac{{\mathrm{x}}^2}{{\mathrm{b}}^2} = 1, \mathrm{are}$

• $\mathrm{y} = \pm \mathrm{x} \pm \sqrt{{\mathrm{b}}^2 - {\mathrm{a}}^2}$

• $\mathrm{y} = \pm \mathrm{x} \pm \sqrt{{\mathrm{a}}^2 - {\mathrm{b}}^2}$

• $\mathrm{y} = \pm \mathrm{x} \pm \sqrt{{\mathrm{a}}^2 + {\mathrm{b}}^2}$

• $\mathrm{y} = \pm \mathrm{x} \pm \left({\mathrm{a}}^2 - {\mathrm{b}}^2\right)$

Domain of the function f(x) = log_x(cos(x)), is

• $\left(- \frac{\mathrm{\pi }}{2}, \frac{\mathrm{\pi }}{2}\right) - \left\{1\right\}$

• $\left[- \frac{\mathrm{\pi }}{2}, \frac{\mathrm{\pi }}{2}\right] - \left\{1\right\}$

• $\left(- \frac{\mathrm{\pi }}{2}, \frac{\mathrm{\pi }}{2}\right)$

• None of these

20.

If x = $\sec \left(\mathrm{\theta }\right) - \cos \left(\mathrm{\theta }\right), \mathrm{y} = {\sec }^{\mathrm{n}}\left(\mathrm{\theta }\right) - {\cos }^{\mathrm{n}}\left(\mathrm{\theta }\right)$, then $\left({\mathrm{x}}^2 + 4\right){\left(\frac{d\mathrm{y}}{d\mathrm{x}}\right)}^2$ is equal to

• n²(y² - 4)

• n²(4 - y²)

• n²(y² + 4)

• None of these