Angle between Two Line Segments

Q.

Show that the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}and\frac{x-4}{5}=\frac{y-1}{2}=z$ intersect. Also, find their point of intersection.

Q. The distance of the point $(-1,2,-2)$ from the line of intersection of the planes $2x+3y+2z=0$ and $x-2y+z=0$ is

1. $\frac{5}{2}$
2. $\frac{\sqrt{34}}{2}$
3. $\frac{\sqrt{42}}{2}$
4. $\frac{1}{\sqrt{2}}$

Q. Consider the line $L$ given by the equation $\frac{x-3}{2}=\frac{y-1}{1}=\frac{z-2}{1}.$ Let $Q$ be the mirror image of the point $(2,3,–1)$ with respect to $L.$ Let a plane $P$ be such that it passes through $Q,$ and the line $L$ is perpendicular to $P.$ Then which of the following points is on the plane $P$:

1. $(–1,1,2)$
2. $(1,2,2)$
3. $(1,1,1)$
4. $(1,1,2)$

Q. For all values of $\theta$ the locus of the point of intersection of the lines x cos $\theta$+ y sin $\theta$ = a and x sin $\theta$ - y cos $\theta$ =b is

1. A parabola

2. An ellipse

3. A hyperbola

4. A circle

Q.

The distance of the lines $2\mathrm{x}-3\mathrm{y}=4$ from the point $\left(1,1\right)$ measured parallel to the line $\mathrm{x}+\mathrm{y}=1$ is

1. $\sqrt{2}$

2. $\frac{5}{\sqrt{2}}$

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

4. $6$

Q.

The straight lines $3x+4y-5=0$ and $4x-3y=15$ intersected at the point $P.$

On these lines, the points $Q$ and $R$ are chosen so that $PQ=PR.$

The slopes of the line $QR$ passing through $(1,2)$ are

1. $\left(-7,\frac{1}{7}\right)$

2. $\left(7,\frac{1}{7}\right)$

3. $\left(7,-\frac{1}{7}\right)$

4. $\left(3,-\frac{1}{3}\right)$

Q. The lines $\frac{x+1}{1}=\frac{y-1}{2}=\frac{z-2}{-1},\frac{x-1}{2}=\frac{y}{1}=\frac{z+1}{4}$ are

1. skew lines
2. parallel lines
3. perpendicular lines
4. intersecting lines

Q.

Show that the straight lines whose direction cosines are given by 2l+2m-n=0 and mn+nl+lm=0 are at right angles.

Q.

The length of the perpendicular from the point $\left(a\cos \alpha ,a\sin \alpha \right)$, upon the straight line $y=x\tan \alpha +c$ (where $c>0$ ) is

1. $c$

2. $c{\sin }^2\alpha$

3. $c\cos \alpha$

4. $cse{c}^2\alpha$

Q. The line $\frac{x+6}{5}=\frac{y+10}{3}=\frac{z+14}{8}$ is the hypotenuse of an isosceles right-angled triangle whose opposite vertex is $(7,2,4)$. Then which of the following is not the side of the triangle?

1. $\frac{x-7}{2}=\frac{y-2}{-3}=\frac{z-4}{6}$
2. $\frac{x-7}{3}=\frac{y-2}{6}=\frac{z-4}{2}$
3. $\frac{x-7}{3}=\frac{y-2}{5}=\frac{z-4}{-1}$
4. none of these

Q. If the vector $\overrightarrow{b}$ is collinear with $\overrightarrow{a}=(2\sqrt{2},-1,4)$ and $|\overrightarrow{b}|=10,$ then

1. $\overrightarrow{a}=\pm \overrightarrow{b}$
2. $\overrightarrow{a}=\pm 2\overrightarrow{b}$
3. $\overrightarrow{b}=\pm \overrightarrow{a}$
4. $\overrightarrow{b}=\pm 2\overrightarrow{a}$

Q.

Find the value of $k$ if the straight line $2x+3y+4+k(6x-y+12)=0$ is perpendicular to the line $7x+5y-4=0.$

Q.

The distance of the point on $y=x^4+3x^2+2x$ which is nearest to the line $y=2x-1$ is

1. $\frac{2}{\sqrt{5}}$

2. $\sqrt{5}$

3. $\frac{1}{\sqrt{5}}$

4. $5\sqrt{5}$

Q.

If $z$ is a complex number such that $\left|z\right|\ge 2$ then the minimum value of $\left|z+\frac{1}{2}\right|$

1. is equal to $\frac{5}{2}$

2. lies in the interval $\left(1,2\right)$

3. is slightly greater than $\frac{5}{2}$

4. is strictly greater than $\frac{3}{2}$ but less than $\frac{5}{2}$

Q.

Two lines are given by${(x-2y)}^2+k(x-2y)=0$. The value of $k$ so that the distance between them is $3$, is

1. $\frac{2}{5}$

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

3. $\pm 3\sqrt{5}$

4. None of these

Q.

If $\theta$ is the angle between two regression lines with correlation coefficient $\gamma$, then

1. $\sin \theta \ge 1-{\gamma }^2$

2. $\sin \theta \le 1-{\gamma }^2$

3. $\sin \theta \le {\gamma }^2+1$

4. $\sin \theta \le {\gamma }^2-1$

Q.

The terminal side of $\mathrm{\theta }$ in standard position contains $\left(-6,8\right)$, how do you find the exact values of the six trigonometric functions of $\mathrm{\theta }$?

Q.

If the angle between the lines whose direction ratios are 2, -1 , 2 and a, 3, 5 be ${45}^{\circ }$, then a =

1. 1

2. 4

3. 2

4. 3

Q. If the solution curve $y=y\left(x\right)$ of the differential equation $y^{2}dx+(x^2–xy+y^2)dy=0$, which passes through the point $(1,1)$ and intersects the line $y=\sqrt{3}x$ at the point $(\alpha ,\sqrt{3}\alpha )$ , then value of ${\log }_e\left(\sqrt{3}\alpha \right)$ is equal to:

Q. The distance of the point $(1,0,-3)$ from the plane $x-y-z=9$ measured parallel to the line $\frac{x-2}{2}=\frac{y+2}{3}=\frac{z-6}{-6}$ is

1. $4$
2. $8$
3. $6$
4. $7$

Q. If a line with direction ratios$(2,2,1)$ intersects the line $\frac{x-7}{3}=\frac{y-5}{2}=\frac{z+3}{1}$ and $\frac{x-1}{2}=\frac{y+1}{4}=\frac{z+1}{3}$ at A and B then AB=.

1. $2$
2. $\sqrt{2}$
3. $\sqrt{3}$
4. $3$

Q.

The angle between the lines represented by the equation $2x^2+3xy-5y^2=0$, is

1. $\frac{\mathrm{\pi }}{3}$

2. $\frac{\mathrm{\pi }}{2}$

3. ${\tan }^{-1}\left|\frac{12}{5}\right|$

4. ${\tan }^{-1}\left(\frac{7}{3}\right)$

Q. A line makes angles $a,b,c,d$ with the four diagonals of a cube, then ${\cos }^{2}a+{\cos }^{2}b+{\cos }^{2}c+{\cos }^{2}d=$

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

Q. Let $P,Q$ are the points on the line $\frac{x-1}{3}=\frac{y-2}{-1}=\frac{z-3}{4}$ which are at a distance of $4\sqrt{3}$ from the plane $x+y-z=1.$ Then the value of $|PQ|$ is

1. $4\sqrt{26}$
2. $20$
3. $10\sqrt{13}$
4. $12\sqrt{26}$

Q.

The angle between any two diagonals of a cube is:

1. $co{s}^{-1}\left(\frac{1}{2}\right)$

2. $co{s}^{-1}\left(\frac{1}{3}\right)$

3. $co{s}^{-1}\left(\frac{1}{4}\right)$

4. $\frac{\pi }{2}$

Q.

Find the angle between vector P and vector Q if $\left|\overrightarrow{p}\right|=\left|\overrightarrow{q}\right|\&\left|\overrightarrow{p}-\overrightarrow{q}\right|=\left|\overrightarrow{p}+\overrightarrow{q}\right|$

1. $45°$

2. $90°$

3. $135°$

4. $150°$

Q.

Find the angle of intersection of the curves $y=4-x^{2}andy=x^2$

Q. The angle between the lines $x-3y-4=0,$ $4y-z+5=0$ and $x+3y-11=0,$ $2y-z+6=0$ is

1. ${\cos }^{-1}\left(\sqrt{\frac{1}{14}}\right)$
2. ${\cos }^{-1}\left(\sqrt{\frac{2}{13}}\right)$
3. $\frac{\pi }{2}$
4. $0$

Q. The angle between the straight lines whose direction cosines are given by $2l+2m-n=0,mn+nl+lm=0$, is

1. $\frac{\pi }{2}$
2. None of these
3. $\frac{\pi }{3}$
4. $\frac{\pi }{4}$

Q. 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

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