Locus of centroid of triangle whose vertices are $\left(\mathrm{acos}\left(\mathrm{t}\right), \mathrm{asin}\left(\mathrm{t}\right)\right) \left(\mathrm{bsin}\left(\mathrm{t}\right), - \mathrm{bcos}\left(\mathrm{t}\right)\right)$ and (1, 0) where t is a parameter is

• (3x - 1)² + (3y)² = a² - b²

• (3x - 1)² + (3y)² = a² + b²

• (3x + 1)² + (3y)² = a² + b²

• (3x + 1)² + (3y)² = a² - b²

A house of height 100 m subtends a right angle at the window of an opposite house. If the height of the window be 64 m, then the distance between the two houses is

• 48 m

• 36 m

• 54 m

• 72 m

The distance travelled by a motor car in t seconds after the brakes are applied is s feet, wheres = 22t - 12t². The distance travelled by the car before it stops, is

• 10.08 ft

• 10ft

• 11ft

• 11.5ft

The sides of triangle are in the ratio 1 : √3 : 2, then the angles of the triangle are in ratio

• 1 : 3 : 5

• 2 : 3 : 1

• 3 : 2 : 1

• 1 : 2 : 3

If the distance between the plane Ax - 2 y + z = d and the plane containing the lines $\frac{\mathrm{x} - 1}{2} = \frac{\mathrm{y} - 2}{3} = \frac{\mathrm{z} - 3}{4}$ and $\frac{\mathrm{x} - 2}{3} = \frac{\mathrm{y} - 3}{4} = \frac{\mathrm{z} - 4}{5}$ is $\sqrt{6}$, then $\left|\mathrm{d}\right|$ is equal to

• 3

• 4

• 6

• 1

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

157.

The shortest distance from the plane 12x + y + 3z = 327 to the sphere x² + y² + z² + 4x - 2y - 6z = 155 is

• 13

• 26

• 39

• $41\frac{4}{13}$

If the vertices of a· triangle are A(0, 4, 1), B(2, 3, - 1) and C(4, 5, 0), then the orthocentre of $∆$ABC, is

• (4, 5, 0)

• (2, 3, - 1)

• (- 2, 3, - 1)

• (2, 0, 2)

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

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

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

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

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