A train moving with constant acceleration takes t seconds to pass a certain fixed point and the front and back end of the train pass the fixed point with velocities u and v respectively. Show that the length of the train is  $\frac{1}{2}\left(\mathrm{u} + \mathrm{v}\right)\mathrm{t}$.

22.

One possible condition for the three points (a, b), (b, a) and (a², - b²) to be collinear is

• a - b = 2

• a + b = 2

• a = 1 + b

• a = 1 - b

The coordinates of the foot of perpendicular from (a, 0) on the line y = mx + $\frac{\mathrm{a}}{\mathrm{m}}$ are

• $\left(0, \frac{\mathrm{a}}{\mathrm{m}}\right)$

• $\left(0, - \frac{\mathrm{a}}{\mathrm{m}}\right)$

• $\left(\frac{\mathrm{a}}{\mathrm{m}}, 0\right)$

• $\left(- \frac{\mathrm{a}}{\mathrm{m}}, 0\right)$

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

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)