Direction Cosines

Q.

Let $L_1$ and $L_2$ denotes the lines $\overrightarrow{r}=\hat{i}+\lambda \left(-\hat{i}+2\hat{j}+2\hat{k}\right),\lambda \in \mathrm{ℝ}$and $\overrightarrow{r}=\mu \left(2\hat{i}-\hat{j}+2\hat{k}\right),\mu \in \mathrm{ℝ}$ respectively. If $L_3$ is a line which is perpendicular to both $L_1$ and $L_2$ and cuts both of them, then which of the following options describe(s) $L_3$?

1. $\overrightarrow{r}=\frac{1}{3}(2\hat{i}+\hat{k})+t\left(2\hat{i}+2\hat{j}-\hat{k}\right),t\in \mathrm{ℝ}$

2. $\overrightarrow{r}=\frac{2}{9}(2\hat{i}-\hat{j}+2\hat{k})+t\left(2\hat{i}+2\hat{j}-\hat{k}\right),t\in \mathrm{ℝ}$

3. $\overrightarrow{r}=t\left(2\hat{i}+2\hat{j}-\hat{k}\right),t\in \mathrm{ℝ}$

4. $\overrightarrow{r}=\frac{2}{9}(4\hat{i}+\hat{j}+\hat{k})+t\left(2\hat{i}+2\hat{j}-\hat{k}\right),t\in \mathrm{ℝ}$

Q.

The equation of the plane passing through the point $(1,2,-3)$ and perpendicular to the planes $3x+y-2z=5$ and $2x-5y-z=7$, is:

1. $3x-10y+2z+11=0$

2. $6x-5y-2z-2=0$

3. $11x+y+17z+38=0$

4. $6x-5y+2z+10=0$

Q. The number of lines which are equally inclined to all the coordinate axes

Q.

The angle between the lines whose direction cosines satisfy the equations
$l+m+n=0$ and $l^2=m^2+n^2$, is

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

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

3. $\frac{\pi }{6}$

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

Q.

The summation of two unit vectors is a third unit vector, then the modulus of the difference of the unit vectors is?

1. $\sqrt{3}$

2. $1-\sqrt{3}$

3. $1+\sqrt{3}$

4. $-\sqrt{3}$

Q.

The angle between the lines represented by the equation $4x^2-24xy+11y^2=0$ are

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

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

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

4. ${\tan }^{-1}\left(\frac{1}{2}\right)$

Q.

The direction cosines of the line which is perpendicular to the lines whose direction cosines are proportional to $(1,-1,2)$ and $(2,1,-1)$ are

1. $\frac{-1}{\sqrt{35}},\frac{5}{\sqrt{35}},\frac{3}{\sqrt{35}}$

2. $\frac{13}{\sqrt{35}},\frac{-1}{\sqrt{35}},\frac{1}{\sqrt{35}}$

3. $\frac{2}{\sqrt{3}},\frac{5}{\sqrt{3}},\frac{7}{\sqrt{3}}$

4. $\frac{3}{\sqrt{35}},\frac{5}{\sqrt{35}},\frac{7}{\sqrt{35}}$

Q. If the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{2\pi }{3}$ and the projection of $\overrightarrow{a}$ in the direction of $\overrightarrow{b}$ is $-2$, then $|\overrightarrow{a}|=$

1. 2
2. 4
3. 1
4. 3

Q. The shortest distance between the line $x+y+2z-3=2x+3y+4z-4=0$ and $xz$-plane is units

Q.

Find the vector equation of the line passing through the point (1, 2, -4) and perpendicular to the two lines
$\frac{x-8}{3}=\frac{y+19}{-16}=\frac{z-10}{7}and\frac{x-15}{3}=\frac{y-29}{8}=\frac{z-5}{-5}$

Q.

The equation of the straight line which passes through the point $(1,-2)$ and cuts off equal intercepts from axes, is

1. $x+y=1$

2. $x-y=1$

3. $x+y+1=0$

4. $x–y–2=0$

Q. The locus of the centre of the circle which cuts off intercepts of length 2a and 2b on X-axis and Y-axis respectively, is

1. x + y = a + b
2. 
3. 
4. 

Q.

What is the Relation between the direction cosines of a line?

1. l² = m² = n²

2. l² + m² + n² = 1

3. l² = m² = n² = 0

4. there is no relation

Q.

The equation of the plane containing the line $\frac{x-x_1}{l}=\frac{y-y_1}{m}=\frac{z-z_1}{n}$ is (where $a,b,c$ are direction ratios normal to plane)

1. $a{x}_1+b{y}_1+c{z}_1=0$

2. $al+bm+cn=0$

3. $\frac{a}{l}=\frac{b}{m}=\frac{c}{n}$

4. $l{x}_1+m{y}_1+n_1=0$

Q.

Let $\overrightarrow{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ , $\overrightarrow{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$ and $\overrightarrow{c}=c_1\hat{i}+c_2\hat{j}+c_3\hat{k}$ be three non-zero vectors such that $c$ is a unit vector perpendicular to both $a$ and $b$. If the angle between $a$ and $b$ is $\frac{\mathrm{\pi }}{6}$, then ${\left|\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right|}^2$is equal to?

1. $0$

2. $1$

3. $\frac{1}{4}\left({a_1}^2+{a_2}^2+{a_3}^2\right)\left({b_1}^2+{b_2}^2+{b_3}^2\right)$

4. $\frac{3}{4}\left({a_1}^2+{a_2}^2+{a_3}^2\right)\left({b_1}^2+{b_2}^2+{b_3}^2\right)$

Q. If a line has the direction ratios −18, 12, −4, then what are its direction cosines?

Q.

If a line makes angles $\alpha ,\beta$, $\gamma$ with the coordinate axes, then

1. $\cos 2\alpha +\cos 2\beta +\cos 2\gamma –1=0$

2. $\cos 2\alpha +\cos 2\beta +\cos 2\gamma –2=0$

3. $\cos 2\alpha +\cos 2\beta +\cos 2\gamma +1=0$

4. $\cos 2\alpha +\cos 2\beta +\cos 2\gamma +2=0$

Q.

A line passes through the point of intersection of the lines $3x+y+1=0$ and $2x-y+3=0$ and makes equal intercepts with axes. Then, equation of the line is

1. $5x+5y–3=0$

2. $x+5y–3=0$

3. $5x–y–3=0$

4. $5x+5y+3=0$

Q. A mirror and a source of light are situated at the origin O and a point on OX(x-axis) respectively. A ray of light from the source strikes the mirror and is reflected. If the DRs of the normal to the plane of mirror are 1, – 1, 1, the DCs for the reflected ray are

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

Q.

The points $A(12),B(2,4)$ and $C(4,8)$ form a/an

1. isosceles triangle

2. equilateral triangle

3. straight line

4. right-angled triangle

Q.

The co-ordinates of the point $P$ are $(x,y,z)$ and the direction cosines of the line $OP$ are $l,m,n$ when $O$ is the origin, . If$OP=r$, then

1. $l=x,m=y,n=z$

2. $l=xr,m=yr,n=zr$

3. $x=lr,y=mr,z=nr$

4. None of these

Q.

A line makes the same angle $\theta$ with each of the $xandz$- axes. If it makes the angle $\beta$with $y$-axis such that

${\sin }^2\beta =3{\sin }^2\theta ,$ then ${\cos }^2\theta$ equals.

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

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

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

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

Q.

The coordinates of the foot of the perpendicular drawn from $(2,4)$ to the line $x+y=1$ is

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

2. $\left(-\frac{1}{2},\frac{3}{2}\right)$

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

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

Q.

If $O$ is the origin and $OP=3$ with direction ratios $-1,2,-2$, then co-ordinates of $P$ are

1. $\left(1,2,2\right)$

2. $\left(-1,2,-2\right)$

3. $\left(-3,6,9\right)$

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

Q. The angle between two adjacent sides $\overrightarrow{a}$ and $\overrightarrow{b}$ of a parallelogram is $\frac{\pi }{6}$. If $\overrightarrow{a}=(2,2,-1)$ and $|\overrightarrow{b}|=2|\overrightarrow{a}|$, then area of this parallelogram is

1. $9$
2. $18$
3. $\frac{9}{2}$
4. $\frac{3}{4}$

Q.

Equation of the hyperbola with centre $(0,0)$ distance between the foci is $18$ and distance between directrices $8$.

Q.

In a parallelogram OABC, vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are respectively the position vectors of vertices A, B, C with reference to O is origin. A point $\overrightarrow{E}$ is taken on the side BC which divides it in the ratio of 2 : 1. Also, the line segment AE intersects the line bisecting the angel O internally in point P. If CP, when extended meets AB in point F. Then

The position vector of point F is

1. $\overrightarrow{a}+\frac{1}{3}\frac{|\overrightarrow{a}|}{|\overrightarrow{c}|}\overrightarrow{c}$

2. $\overrightarrow{a}+\frac{2|\overrightarrow{a}|}{|\overrightarrow{c}|}\overrightarrow{c}$

3. $\overrightarrow{a}+\frac{1}{2}\frac{|\overrightarrow{[}a]|}{|\overrightarrow{c}|}\overrightarrow{c}$

4. $\overrightarrow{a}+\frac{|\overrightarrow{a}|}{|\overrightarrow{c}|}\overrightarrow{c}$

Q. A straight line, makes an angle of ${60}^{\circ }$ with each of y and z-axis, then the inclination of the line with x-axis is

1. ${60}^{\circ }$
2. ${30}^{\circ }$
3. ${45}^{\circ }$
4. ${75}^{\circ }$

Q.

Two points $(a,0)$ and $(0,b)$are joined by a straight line, another point on this line is

1. $(3a,–2b)$

2. $(a^2,ab)$

3. $(–3a,2b)$

4. $(a,b)$

Q.

If $x+y=8$, then the maximum value of $x^{2}y$ is:

1. $\frac{2048}{9}$

2. $\frac{2048}{81}$

3. $\frac{2048}{3}$

4. $\frac{2048}{27}$