Direction Ratios

Q. If 1, 2, 1 are the direction ratios of a line then which of the following could be direction cosines of this line?

1. 
2. 
3. 
4. None of these

Q. Coordinates of a point $P$ lying on the line $\frac{x-2}{4}=\frac{y-1}{3}=\frac{z+1}{2}$ such that the line joining the points $P$ and $A(3,1,5)$ is parallel to the plane $4x-2y+z=0$, is

1. $(2,1,-1)$
2. $(\frac{16}{3},\frac{7}{2},\frac{2}{3})$
3. $(4,\frac{5}{2},0)$
4. $(3,\frac{7}{4},\frac{-1}{2})$

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. If the direction ratios of a line are (1, 1, 2), then find the direction cosines of the line.

1. (1, 1, 2)
2. $\left(0,\frac{1}{\sqrt{6}},\frac{1}{\sqrt{6}}\right)$
3. $\left(\frac{1}{\sqrt{6}},\frac{1}{\sqrt{6}},\frac{1}{\sqrt{6}}\right)$
4. $\left(\frac{1}{\sqrt{6}},\frac{1}{\sqrt{6}},\frac{2}{\sqrt{6}}\right)$

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

Q. In $R^3$, Let $L$ be a straight line passing through the origin. Suppose that all the points on $L$ are at a constant distance from the two planes $P_1:x+2y-z+1=0$ and $P_2:2x-y+z-1=0.$ Let $M$ be the locus of the feet of the perpendiculars drawn from the points on $L$ on the plane $P_1.$ Which of the following points lie(s) on $M?$

1. $(0,-\frac{5}{6},-\frac{2}{3})$
2. $(-\frac{1}{6},-\frac{1}{3},\frac{1}{6})$
3. $(-\frac{5}{6},0,\frac{1}{6})$
4. $(-\frac{1}{3},0,\frac{2}{3})$

Q. Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.

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