Direction Ratios

Q.

If $P(3,4,5),Q(4,6,3),R(-1,2,4),S(1,0,5)$then the projection of $RS$ on $PQ$ is

1. $-\frac{2}{3}$

2. $-\frac{4}{3}$

3. $\frac{1}{2}$

4. $2$

Q. If the direction cosines of a line are l, m and n and the direction ratios of the same line are a, b & c then which of the following is always correct ?

1. $\frac{l}{a}=\frac{m}{b}=\frac{n}{c}=1$
2. $\frac{l}{a}=\frac{m}{b}=\frac{n}{c}=2$
3. $\frac{l}{a}=\frac{m}{b}=\frac{n}{c}=K$, where $‘K^{\prime }$ is any constant real number.
4. None of these

Q. The plane denoted by $P_1:4x+7y+4z+81=0$ is rotated through a right angle about its line of intersection with the plane $P_2:5x+3y+10z=25$. If the plane in its new position is denoted by $P$, and the distance of this plane from the origin is $k$, then the value of $\left[\frac{k}{2}\right]$ is (where [.] represents greatest integer less than or equal to $k$).

Q. Direction ratios of the normal to the plane passing through the points (0, 1, 1), (1, 1, 2)and (-1, 2, - 2) are

1. (1, 1, 1)
2. (2, 1, -1)
3. (1, 2, -1)
4. (1, -2, -1)

Q. $ABC$ is a triangle in a plane with vertices $A(2,3,5),B(-1,3,2)$ and $C(\lambda ,5,\mu ).$ If the median through $A$ is equally inclined to the coordinate axes, then the value of $({\lambda }^3+{\mu }^3+5)$ is:

1. $1348$
2. $1130$
3. $1077$
4. $676$