Dcs and drs practice · Mixed difficulty

If $P$ is a point in space such that $O P$ is inclined to $O X$ at $45^{\circ}$ and $O Y$ to $60^{\circ}$ then $O P$ is inclined to $O Z$ at

Two lines $\frac{x-x_i}{l_1}=\frac{y-y_i}{m_1}=\frac{z-z_i}{n_1}(i=1,2)$ are perpendicular to each other if their direction ratios satisfy

If $\left(\frac{1}{2}, \frac{1}{3}, n\right)$ are the direction cosines of a line, then the value of $n$ is

The angle between the lines $A B$ and $C D$, where $A=(0,0,0), B=(1,1,1), C=(-1,-1,-1)$ and $D=(0,1,0)$ is given by

The angle between the lines whose direction cosines are given by the equations $l^2+m^2-n^2=0, l+m+n=0$ is

If the direction cosines of two lines are such that $l+m+n=0, l^2+m^2-n^2=0$, then the angle between them is

If a line makes an angle of $\frac{\pi}{4}$ with the positive directions of each of $X$-axis and $Y$-axis, then the angle that the line makes with the positive direction of the $Z$-axis is

Let $A_{x y}, A_{y z}, A_{z x}$ be the area of the projection of a plane area $A$ on the $x y, y z, z x$ plane respectively Then $A^2=$

The angle between a line with direction ratios proportional to $2,2,1$ and a line joining $(3,1,4)$ to $(7,2,12)$, is

The direction ratios of line $I_1$ passing through $P(1,3,4)$ and perpendicular to line $I_2 \frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ (where, $I_1$ and $I_2$ are coplanar) is

If two straight lines whose direction cosines are given by the relations $l + m - n = 0,\,3{l^2} + {m^2} + cnl = 0$ are parallel, then the positive value of $c$ is:

The distance between two points $P$ and $Q$ is $d$ and the length of their projections of $P Q$ on the co-ordinate planes are $d_1, d_2, d_3$. Then $d_1^2+d_2^2+d_3^2=k d^2$ where ' $k$ ' is

The angle between the lines whose direction cosines are given by $2 l-m+2 n=0, l m+m n+n l=0$ is

If a line makes angle $\alpha, \beta$ and $\gamma$ with the coordinates axes, then

Let ${P}({x}, {y}, {z})$ be a point in the first octant, whose projection in the $x y$-plane is the point ${Q}$. Let ${OP}=\gamma$; the angle between ${OQ}$ and the positive $x$-axis be $\theta$; and the angle between OP and the positive ${z}$-axis be $\phi$, where ${O}$ is the origin. Then the distance of ${P}$ from the ${x}$-axis is:

The angle between the two lines $\frac{x+1}{2}=\frac{y+3}{2}=\frac{z-4}{-1}$ & $\frac{{x}-4}{1}=\frac{{y}+4}{2}=\frac{{z}+1}{2}$ is

Consider the following 3 lines in space
${L_1}:r = 3i - j + 2k + \lambda (2i + 4j - k)$
${L_2}:r = i + j - 3k + \mu (4i + 2j + 4k)$
${L_3}:r = 3i + 2j - 2k + t(2i + j + 2k)$
Then, which one of the following pair(s) is/are in the same plane?

If direction cosines of two lines are proportional to $(2,3,-6)$ and $(3,-4,5)$, then the acute angle between them is

If the direction cosines of a line are $\left(\frac{1}{c}, \frac{1}{c}, \frac{1}{c}\right)$ then

If the co-ordinates of the points $P$ and $Q$ be (1, 2, 1) and $(2,3,4)$ and $O$ be the origin, then

The plane containing the two lines $\frac{x-3}{1}=\frac{y-2}{4}=\frac{z-1}{5}$ and $\frac{x-2}{1}=\frac{y+3}{-4}=\frac{z+1}{5}$ is $11 x+m y+n z=28$ where-

The vector equations of two lines $L_1$ and $L_2$ are respectively
$r = 17i - 9j + 9k + \lambda (3i + j + 5k)$
and $r = 15i - 8j - k + \mu (4i + 3j)$
I. $L_1$ and $L_2$ are skew lines.

II. $(11,-11,-1)$ is the point of intersection of $L_1$ and $L_2$.

III. $(-11,11,1)$ is the point of intersection of $L_1$ and $L_2$.

IV. $\cos ^{-1}\left(\frac{3}{\sqrt{35}}\right)$ is the acute angle between, $L_1$ and $L_2$.

Then, which of the following is true?

The equation of the straight line through the origin and parallel to the line $(b+c) x+(c+a) y$ $+(a+b) z=k=(b-c) x+(c-a) y+(a-b) z$ are

If a point $R (4, y, z)$ lies on the line segment joining the points $P (2, -3, 4)$ and $Q (8, 0, 10),$ then the distance of $R$ from the origin is

If the distance between line $3(x+3)=2(y-5)=z+1$ and a line passing through $(1,-2,0)$ and having direction ratios $(2,3,6)$ is $K$ units, then $K^2$ is equal to

Consider the lines $L_{1}$ and $L_{2}$ given by

$\begin{array}{l} L_{1}: \frac{x-1}{2}=\frac{y-3}{1}=\frac{z-2}{2} \\ L_{2}: \frac{x-2}{1}=\frac{y-2}{2}=\frac{z-3}{3} \end{array}$

A line $L_{3}$ having direction ratios, $1,-1,-2$, intersects $L_{1}$ and $L_{2}$ at the points $P$ and $Q$ respectively. Then the length of line segment $P Q$ is

The projection of a line $P Q$, where $P$ is $(0,-1,3)$ and $Q$ is $(4,5,6)$ on a line whose direction ratios are $1,2,3$ is

The projection of a line segment on the coordinate axes are 2, 3, 6. Then, the length of the line segment is

If the direction cosines of a line are $\left(\frac{1}{c}, \frac{1}{c}, \frac{1}{c}\right)$ then

The projections of a vector on the three coordinate axes are $6,-3,2$ respectively. The direction cosines of the vector are