Angle between a Plane and a Line

Q.

Let $L$denote the line in the $xy$-plane with $x$ and$y$ intercepts as $3$ and $1$ respectively. Then the image of the point $(-1,-4)$in this line is

1. $\left(\frac{11}{5},\frac{28}{5}\right)$

2. $\left(\frac{8}{5},\frac{29}{5}\right)$

3. $\left(\frac{29}{5},\frac{11}{5}\right)$

4. $\left(\frac{29}{5},\frac{8}{5}\right)$

Q. Foot of the perpendicular drawn from the point $(1,3,4)$ to the plane $2x–y+z+3=0$ is

1. $(1,2,-3)$
2. $(-1,4,3)$
3. $(0,-4,-7)$
4. $(-3,5,2)$

Q.

Angle between two line of regression is given by.

1. ${\tan }^{-1}[b_{xy}–(1/b_{yx}\left)\right]/[1–(b_{xy}/b_{yx}\left)\right]$

2. ${\tan }^{-1}[b_{yx}.b_{xy}–1]/\left[\right(b_{yx}+b_{xy}\left)\right]$

3. ${\tan }^{-1}[b_{xy}–(1/b_{yx}\left)\right]/[1+(b_{xy}/b_{yx}\left)\right]$

4. ${\tan }^{-1}[b_{yx}–b_{xy}]/[1+(b_{yx}.b_{xy}\left)\right]$

Q. The plane which bisects the line segment joining the points $(-3,-3,4)$ and $(3,7,6)$ at right angles, passes through which one of the following points?

1. $(4,-1,7)$
2. $(4,1,-2)$
3. $(2,1,3)$
4. $(-2,3,5)$

Q.

If $l_1,m_1,n_1,l_2,m_2,n_{2}and{l}_3,m_3,n_3$ are the direction cosines of three mutually perpendicular lines, then prove that the line whose direction cosines are proportional to $l_1+l_2+l_3,m_1+m_2+m_{3}and{n}_1+n_2+n_3$ makes equal angles with them.

Q. On which of the following lines lies the point of intersection of the line, $\frac{x-4}{2}=\frac{y-5}{2}=\frac{z-3}{1}$ and the plane, $x+y+z=2$?

1. $\frac{x-1}{1}=\frac{y-3}{2}=\frac{z+4}{-5}$
2. $\frac{x-4}{1}=\frac{y-5}{1}=\frac{z-5}{-1}$
3. $\frac{x-2}{2}=\frac{y-3}{2}=\frac{z+3}{3}$
4. $\frac{x+3}{3}=\frac{4-y}{3}=\frac{z+1}{-2}$

Q.

Let $f\left(x\right)=\left\{\begin{array}{r}\sin \left[\frac{|x^2-5x+6|}{x^2-5x+6}\right]x\ne 2,3\\ 1x=2\mathrm{or}3\end{array}\right\}$ , the set of all points where $f$ is differentiable is

1. $\left(-\infty ,\infty \right)$

2. $\left(-\infty ,\infty \right)-\left\{2\right\}$

3. $\left(-\infty ,\infty \right)-\left\{3\right\}$

4. $\left(-\infty ,\infty \right)-\left\{2,3\right\}$

Q.

If $\sin \left(\theta +\alpha \right)=a$ and $\sin \left(\theta +\beta \right)=b$, then $\cos 2\left(\alpha -\beta \right)-\left(4ab\right)\cos \left(\alpha -\beta \right)$ is equal to

1. $1–a^2–b^2$

2. $1–2a^2–2b^2$

3. $2+a^2+b^2$

4. $2-a^2-b^2$

Q.

The locus of point $P$ which divides the line joining $\left(1,0\right)$ and $\left(2\cos \theta ,2\sin \theta \right)$ internally in the ratio $2:3$ for all $\theta$, is a

1. Straight line

2. Circle

3. Pair of straight line

4. Parabola

Q.

The angle between the line $\frac{x-1}{2}=\frac{y-2}{1}=\frac{z+3}{-2}$ and the plane $x+y+4=0$ is

1. $0°$

2. $30°$

3. $45°$

4. $90°$

Q.

In a general quadratic equation, if $f(x,y)=0,∆=0\text{and}{h}^2=ab,$then the equation represents

1. Two parallel straight lines

2. Two perpendicular straight lines

3. Two intersecting lines

4. None of these

Q. If the length of the projection of the line segment with points $(1,0,-1)$ and $(-1,2,2)$ to the plane $x+3y-5z=6$ is $d$, then the value of $\left[d/2\right],$ is
where [.] represent greatest interger function

Q. If an angle between the line, $\frac{x+1}{2}=\frac{y-2}{1}=\frac{z-3}{-2}$ and the plane, $x-2y-kz=3$ is ${\cos }^{-1}\left(\frac{2\sqrt{2}}{3}\right)$, then a value of $k$ is :

1. $\sqrt{\frac{5}{3}}$
2. $\sqrt{\frac{3}{5}}$
3. $-\frac{3}{5}$
4. $-\frac{5}{3}$

Q. The distance (in units) between the parallel planes $x+2y-3z=2$ and $2x+4y-6z+7=0$ is:

1. $\frac{2}{\sqrt{14}}$
2. $\frac{11}{\sqrt{56}}$
3. $\frac{7}{\sqrt{56}}$
4. $\frac{9}{\sqrt{14}}$

Q. The measure of the angle between the line $\overrightarrow{r}=(2,-3,1)+k(2,2,1);k\in R$ and the plane $2x-2y+z+7=0$ is

1. ${\tan }^{-1}\frac{1}{4\sqrt{5}}$
2. ${\cos }^{-1}\frac{1}{9}$
3. ${\sin }^{-1}\frac{1}{3}$
4. $\frac{\pi }{2}$

Q. If $OABCD$ is a pyramid with square base $ABCD$ of unit side and vertex $O$ such that $OA=OB=OC=OD=1$ unit, then

1. angle between the line $AC$ and $OD$ is $\frac{\pi }{3}.$
2. angle between the line $AC$ and $OD$ is $\frac{\pi }{2}.$
3. shortest distance between $AC$ and $OD$ is $\frac{1}{\sqrt{2}}.$ unit
4. shortest distance between $AC$ and $OD$ is $\frac{1}{2}$ unit

Q. An angle between the plane, $x+y+z=5$ and the line of intersection of the planes, $3x+4y+z-1=0$ and $5x+8y+2z+14=0$, is :

1. ${\sin }^{-1}\left(\sqrt{\frac{3}{17}}\right)$
2. ${\cos }^{-1}\left(\sqrt{\frac{3}{17}}\right)$
3. ${\cos }^{-1}\left(\frac{3}{\sqrt{17}}\right)$
4. ${\sin }^{-1}\left(\frac{3}{\sqrt{17}}\right)$

Q.

If the median $AD$ of a triangle $ABC$ divides the angle $\angle BAC$ in the ratio $1:2$, then $\frac{\sin \left(B\right)}{\sin \left(C\right)}$ is equal to

1. $2\cos \left(\frac{A}{3}\right)$

2. $\frac{1}{2}\sec \left(\frac{A}{3}\right)$

3. $\frac{1}{2}\sin \left(\frac{A}{3}\right)$

4. $2\csc \left(\frac{A}{3}\right)$

Q. The intercepts made on the axes by the plane which bisects the line joining the points (1, 2, 3) and (–3, 4, 5, ) at right angles are

1. $(-\frac{9}{2},9,9)$
2. $(9,-\frac{9}{2},9)$
3. $(9,\frac{9}{2},9)$
4. $(\frac{9}{2},9,9)$

Q. The acute angle between two lines whose direction cosines are given by the relation between $l+m+n=0$ and $l^2+m^2-n^2=0$ is

1. $\frac{\pi }{3}$
2. $\frac{\pi }{4}$
3. $\frac{\pi }{6}$
4. $\frac{\pi }{2}$

Q. If angle between line and plane is $\theta$ then the angle between line and the normal to plane is $\frac{\pi }{2}-\theta$

1. True
2. False

Q. If the angle $\theta$ between the lines $\frac{x+1}{1}=\frac{y-1}{2}=\frac{z-2}{2}$ and the plane $2x-y+\sqrt{\lambda z+4}=0$ such that $sin\theta =\frac{1}{3}$, then value of $\lambda$ is

1. $\frac{3}{4}$
   
2. $\frac{-4}{3}$
   
3. $\frac{5}{3}$
   
4. $\frac{-3}{5}$

Q. If an angle between the line, $\frac{x+1}{2}=\frac{y-2}{1}=\frac{z-3}{-2}$ and the plane, $x-2y-kz=3$ is ${\cos }^{-1}\left(\frac{2\sqrt{2}}{3}\right)$, then a value of $k$ is :

1. $\sqrt{\frac{5}{3}}$
2. $\sqrt{\frac{3}{5}}$
3. $-\frac{3}{5}$
4. $-\frac{5}{3}$

Q.

The measure of an angle in standard position is given.

Find two positive angles and two negative angles that are co terminal with the given angle. (Enter your answers as a comma-separated list.)

$-\frac{\mathrm{\pi }}{4}$

Q.

If $2(\sin a+\sin b)x-2\sin (a-b)y=3$ and $2(\cos a+\cos b)x+2\cos (a-b)y=5$ are perpendicular to each other then, $\sin 2a\sin 2b$ equals to

Q.

Find the equation of the line passing through the point (3, 0, 1) and parallel ti the planes x+2y=0 and 3y-z=0.

Q. If the distance of the plane passing through the point $P(1,1,1)$ and perpendicular to the line $\frac{x-1}{3}=\frac{y-1}{0}=\frac{z-1}{4}$ from the origin is $\frac{k}{5}$ then the value of $k$ is

Q. The acute angle between the line $\frac{x+1}{2}=\frac{y}{3}=\frac{z-1}{6}$ and the plane $10x+2y-11z=3$ is

1. ${\cos }^{-1}\left(\frac{3}{2\sqrt{10}}\right)$
2. ${\cos }^{-1}\left(\frac{8}{21}\right)$
3. ${\sin }^{-1}\left(\frac{3}{2\sqrt{10}}\right)$
4. ${\sin }^{-1}\left(\frac{8}{21}\right)$

Q. Equation of the plane passing through the line $\frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-3}{4}$ and perpendicular to the plane $x+2y+z=12$ is given by $ax+by+cz+4=0$ then -

1. $a=8,b=-2,c=5$
2. $a=-8,b=2,c=-5$
3. $a=-9,b=-2,c=-5$
4. $a=9,b=-2,c=-5$

Q. If angle between line and plane is $\theta$ then the angle between line and the normal to plane is $\frac{\pi }{2}-\theta$

1. False
2. True