The normals at three points· P,Q and R of the parabola y² = 4ax meet at (h, k). The centroid of the $∆$PQR lies on

• x = 0

• y = 0

• x = - a

• y = a

A tower AB leans towards West making an angle $\mathrm{\alpha }$ with the vertical. The angular elevation of B, the top most point of the tower is $\mathrm{\beta }$ as observed from a point C due East of A at a distance 'd' from A. If the angular elevation of B from a point D due East of C at a distance 2d from C is r, then 2$\tan \left(\mathrm{\alpha }\right)$ can be given as

• $3\cot \left(\mathrm{\beta }\right) - 2\cot \left(\mathrm{\gamma }\right)$

• $3\cot \left(\mathrm{\gamma }\right) - 2\cot \left(\mathrm{\beta }\right)$

• $3\cot \left(\mathrm{\beta }\right) - \cot \left(\mathrm{\gamma }\right)$

• $\cot \left(\mathrm{\beta }\right) - 3\cot \left(\mathrm{\gamma }\right)$

P is a fixed point (a, a, a) on a line through the origin is equally inclined to the axes, then any plane through P perpendicular to Op, makes intercepts on the axes, the sum of whose reciprocal is equal to

• a

• $\frac{3}{2\mathrm{a}}$

• $\frac{3\mathrm{a}}{2}$

• None of these

The tangent at (1, 7) to the curve x² = y - 6 touches the circle x² + y2 + 16x + 12y + c = 0 at

• (6, 7)

• (- 6, 7)

• (6, - 7)

• (- 6, - 7)

In an equilateral triangle, the inradius, circumradius and one of the exradii are in the ratio

• 2 : 3 : 5

• 1 : 2 : 3

• 1 : 3 : 7

• 3 : 7 : 9

The perimeter of the triangle with vertices at (1, 0, 0), (0, 1, 0) and (0, 0, 1) is

• 3

• 2

• 2$\sqrt{2}$

• 3$\sqrt{2}$

If a line in the space makes angle $\mathrm{\alpha }, \mathrm{\beta } \mathrm{and} \mathrm{\gamma }$ with the coordinate axes, then

$\cos \left(2\mathrm{\alpha }\right) + \cos \left(2\mathrm{\beta }\right) + \cos \left(2\mathrm{\gamma }\right) + {\sin }^2\left(\mathrm{\alpha }\right) + {\sin }^2\left(\mathrm{\beta }\right) + {\sin }^2\left(\mathrm{\gamma }\right)$ equals

• - 1

• 0

• 1

• 2

Y-axis cuts the line joining the points (- 3, - 4) and (1, - 2) in the ratio

• 1 : 3

• 2 : 3

• 3 : 1

• 3 : 2

If the points (1, 1), (- 1, - 1), $\left(- \sqrt{3}, \sqrt{3}\right)$ are the vertices of a triangle, then this triangle is

• a right-angled triangle

• an isosceles triangle

• an equilateral triangle

• None of the above

40.

The point of intersection of the lines $\frac{\mathrm{x} - 5}{3} = \frac{\mathrm{y} - 7}{- 1} = \frac{\mathrm{z} + 2}{1}$ and $\frac{\mathrm{x} + 3}{- 36} = \frac{\mathrm{y} - 3}{2} = \frac{\mathrm{z} - 6}{4}$ is

• (2, 10, 4)

• (21, 5/3, 10/3)

• (5, 7, - 2)

• (- 3, 3, 6)