Question 1 :
Direction cosines $l, m, n$ of two lines are connected by the equation $l-5m+3n=0$ and $7l^{2}+5m^{2}-3n^{2}=0$. The direction cosines of one of the lines are

Question 2 :
If $P(x, y, z)$ moves such that $x=0, z=0$, then the&#160;locus of $P$ is the line whose d.cs are<br/>

Question 3 :
The projections of a directed line segment on the coordinate axes are $12, 4, 3$ respectively.What are the direction cosines of the line segment?

Question 4 :
The projection of the join of the two points $(1,4,5), (6,7,2)$ on the line whose d.s's are $(4,5,6)$ is

Question 5 :
If the dr's the line are $(1+\lambda, 1-\lambda, 2)$ and it makes an angle ${60}^{o}$ with the Y-axis then $\lambda$ is

Question 6 :
Cosine of the angle between two diagonals of acube is equal to

Question 7 :
The direction cosines of the vectors $2\vec {i} + \vec {j} - 2\vec {k}$ is equal to

Question 8 :
What are the DR's of vector parallel to $\left( 2,-1,1 \right) $ and $\left( 3,4,-1 \right) $?

Question 10 :
<table class="table table-bordered"><tbody><tr><td>&#160;List I</td><td>List II&#160;</td></tr><tr><td>1) d.c's of $x -$ axis</td><td>a) $(1,1,1)$&#160;</td></tr><tr><td>2) d.c's of $y -$ axis</td><td>b)$\left(\displaystyle \frac{]}{\sqrt{3}}\frac{]}{\sqrt{3}},\frac{]}{\sqrt{3}}\right)$</td></tr><tr><td>3) d.c's of $z -$ axis</td><td>c) $(1,0,0)$<br/></td></tr><tr><td>4) d.c's of a line makes&#160;equal angles with axes</td><td>d) $(0,1,0)$</td></tr><tr><td>&#160;</td><td>e) $(0,0,1)$</td></tr></tbody></table>The correct order for 1, 2, 3, 4 is

Question 11 :
If a line has the direction ratio $18, 12, 4 $, then its direction cosines are:<br/>

Question 12 :
If a line makes the angles $ \alpha&#160;, \beta$ and $\gamma$ with the axes, then what is the value of $1+\cos 2\alpha +\cos 2\beta+\cos 2\gamma$&#160;equal to ?

Question 14 :
The straight line $\displaystyle \frac{x - 3}{3} = \frac{y - 2}{1} = \frac{z - 1}{0}$ is

Question 15 :
The direction angles of the line $x = 4z + 3, y = 2 - 3z$ are $\alpha, \beta$ and $\gamma$, then $\cos \alpha + \cos \beta + \cos \gamma =$ ________.

Question 16 :
If the $d.c's$ of two lines are connected by the equations $l + m + n = 0, l^2 + m^2 - n^2 = 0$, then angle between the lines is

Question 17 :
Direction cosines of ray from $P(1, -2, 4)$ to $Q(-1, 1, -2)$ are

Question 18 :
Direction cosines of the line $\cfrac { x+2 }{ 2 } =\cfrac { 2y-5 }{ 3 } ,z=-1$ are ____

Question 19 :
If the lines $x=1+a,y=-3-\lambda a,z=1+\lambda a$ and $x=\cfrac { b }{ 2 } ,y=1+b,z=2-b$ are coplanar, then $\lambda$ is equal to

Question 20 :
The direction cosines of a line which is equally inclined to axes, is given by

Question 21 :
A vector is equally inclined to the $x$-axis, $y$-axis and $z$-axis respectively, its direction cosines are

Question 22 :
The following lines are $\hat { r } =\left( \hat { i } +\hat { j } \right) +\lambda \left( \hat { i } +2\hat { j } -\hat { k } \right) +\mu \left( -\hat { i } +\hat { j } -\hat { 2k } \right) $

Question 23 :
The points with position vectors $60\hat{i}+3\hat{j}$, $40\hat{i}-8\hat{j}$, $a\hat{i}-52\hat{j}$ are collinear if

Question 24 :
From the point $P(3, -1, 11)$, a perpendicular is drawn on the line $L$ given by the equation $\dfrac {x}{2} = \dfrac {y - 2}{3} = \dfrac {z - 3}{4}$. Let $Q$ be the foot of the perpendicular.What are the direction ratios of the line segment $PQ$?

Question 25 :
A line makes an angle $\alpha,\beta,\gamma$ with the $X,Y,Z$ axes. Then $\sin^2\alpha+\sin^2\beta+\sin^2\gamma=$<br/>

Question 26 :
If $\bar {a}, \bar {b}$ and $\bar {c}$ are non-zero non collinear vectors and $\theta(\neq 0 , \pi)$ is the angle between $\bar {b}$ and $\bar {c}$ if $(\bar {a}\times \bar {b}) \times \bar {c}=\dfrac {1}{2} |\bar {b}|\bar {c}|\bar {a}$. then $\sin \theta =$

Question 27 :
The three points $ABC$ have position vectors $(1,x,3),(3,4,7)$ and $(y,-2,-5)$ are collinear then $(x,y)=$<br/>

Question 28 :
Can $\dfrac{1}{\sqrt{3}}, \dfrac{2}{\sqrt{3}}, \dfrac{-2}{\sqrt{3}}$ be the direction cosines of any directed line?

Question 29 :
A line passes through the points $(6, -7, -1)$ and $(2, -3, 1)$. The direction cosines of the line so directed that the angle made by it with the positive direction of x-axis is acute, is?