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If the lines $x = ay+b$, $z = cy+d$ and $x = py+q$, $z = ry+s$ are perpendicular, then $ap + cr$ is equal to

A line is perpendicular to two lines having direction ratios $1,-2,-2$ and $0,2,1$ then the direction cosines of the line are:

The distance between the two lines $L_1$ and $L_2$ given by $L_1: \dfrac{x-1}{2} = \dfrac{y-2}{3} = \dfrac{-z-4}{-6}$ and $L_2: \dfrac{x-3}{2} = \dfrac{y-3}{3} = \dfrac{z+5}{6}$ is

The shortest distance between lines whose vector equations are: $\vec{r} = (1-t)\hat{i} + (t-2)\hat{j} + (3-2t)\hat{k}$ and $\vec{r} = (s+1)\hat{i} + (2s-1)\hat{j} - (2s+1)\hat{k}$, is:

The equation of line passing through the points (2, -1, 3) and (4, 2, 1) is (A) $\frac{x-2}{2} = \frac{y+1}{3} = \frac{z-3}{-2}$ (B) $\frac{x-4}{3} = \frac{y-2}{-2} = \frac{z-1}{2}$ (C) $\vec{r} = (2\hat{i}-\hat{j}+3\hat{k}) + \lambda(2\hat{i}+3\hat{j}-2\hat{k})$ (D) $\frac{x-4}{-2} = \frac{y-2}{-3} = \frac{z-1}{2}$ Choose the correct answer from the options given below:

Direction ratios of the line perpendicular to the lines $\frac{x-3}{2}=\frac{y+7}{-3}=\frac{z-2}{1}$ and $\frac{x+2}{1}=\frac{y+3}{2}=\frac{z-5}{-2}$ are

Given a line $L:\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$, then Match the LIST-I with LIST-II | LIST-I | LIST-II | |---|---| | A. The foot of perpendicular from point (1, 6, 3) on line L | I. $2\sqrt{13}$ | | B. Perpendicular distance from point (1, 6, 3) to line L | II. (1, 0, 7) | | C. Image of point (1, 6, 3) with respect to line L | III. (1, 3, 5) | | D. Distance between points (1, 6, 3) and (1, 0, 7) | IV. $\sqrt{13}$ | Choose the correct answer from the options given below:

The shortest distance between lines $L_1:\frac{x+3}{-4}=\frac{6-y}{-3}=\frac{z}{2}$ and $L_2:\frac{x+2}{-4}=\frac{y}{1}=\frac{z-7}{1}$ is:

The Cartesian equation of a line is $6x-2=3y+1=2z-2$. Then its vector from is:

Let $L_1$ and $L_2$ lines such that $L_1:\vec{r}=\hat{i}+\hat{j}+\lambda(2\hat{i}-\hat{j}+\hat{k})$, $L_2:\vec{r}=2\hat{i}+\hat{j}-\hat{k}+\mu(4\hat{i}-2\hat{j}+2\hat{k})$ and $\theta$ be the acute angle between $L_1$ and $L_2$, then Match the LIST-I with LIST-II | | LIST-I | | LIST-II | |---|---|---|---| | A. | $\cos\theta$ | I. | $\dfrac{\sqrt{66}}{6}$ | | B. | $\sin\theta$ | II. | $3$ | | C. | the shortest distance (in units) between $L_1$ and $L_2$ | III. | $1$ | | D. | (sum of squares of direction cosines of $L_1$) $+2$ | IV. | $0$ | Choose the correct answer from the options given below:

The perpendicular distance of point (1, 1, 3) on the line $\frac{x}{2}=\frac{y+1}{3}=\frac{-z-3}{-5}$ is

If $\lambda$ and $\mu$ are parameters, then Match the LIST-I with LIST-II | LIST-I | | LIST-II | |---|---|---| | A. | $\vec{r}=2\hat{i}+\hat{j}-3\hat{k}+\lambda(\hat{i}-\hat{j}+2\hat{k})$ parallel to the line | I. $\vec{r}=(\hat{j}+\hat{k})+\mu(-2\hat{i}+\hat{j}-2\hat{k})$ | | B. | $\vec{r}=\hat{i}+\hat{j}+\lambda(\hat{i}+\hat{j}-2\hat{k})$ perpendicular to the line | II. $\vec{r}=(2\hat{i}+\hat{k})+\mu(-2\hat{i}+\hat{j})$ | | C. | $\frac{x}{-2}=\frac{y-1}{1}=\frac{z-1}{-2}$ in vector form | III. $\vec{r}=(\hat{i}-\hat{j}+2\hat{k})+\mu(2\hat{i}-2\hat{j}+4\hat{k})$ | | D. | $\frac{2-x}{2}=y, z=1$ in vector form | IV. $\frac{x-1}{2}=\frac{y+1}{4}=\frac{z}{3}$ | Choose the correct answer from the options given below:

P is a point on the line segment joining the points (3, 2, -1) and (6, 2, -2) internally. If x coordinate of P is 5, then its z-coordinate is

If the shortest distance between the lines $\vec r=(\hat i+2\hat j+3\hat k)+\lambda(2\hat i+3\hat j+4\hat k)$ and $\vec r=(a\hat i+4\hat j+5\hat k)+\mu(4\hat i+6\hat j+8\hat k)$, $a \in N$, where $\lambda$ and $\mu$ are parameters, is $\sqrt{\dfrac{5}{29}}$, then $a$ is equal to

Which of the following statements are correct? (A) The angle between the vectors with direction ratios proportional to -3, 5, 4 and 4, 5, 3 is $\dfrac{\pi}{3}$ (B) The direction cosines of Y-axis is 0, 1, 0. (C) The cartesian equation of a line is $3x-6=2y+1=2z-1$, then its direction ratios are proportional to 2, 2, 3 (D) The vector equation of the line passing through the point $(2,-1,-1)$ and direction ratios 1, 2, 3 is $\vec r=(2\hat i-\hat j-\hat k)+\lambda(\hat i+2\hat j+3\hat k)$ Choose the correct answer from the options given below:

A line in space is uniquely determined if (A) It passes through a given point and has a given direction. (B) It passes through two given points. (C) It passes through a given point. (D) It has a given direction. Choose the correct answer from the options given below:

If the points $P(3,\mu,6), Q(-1,-1,2)$ and $R(2,8,5)$ are collinear, then the value of '$\mu$' is

Match the LIST-I with LIST-II | | LIST-I<br>Equation of line | | LIST-II<br>Point on the line, direction ratios of the line | |---|---|---|---| | A. | $\dfrac{x-5}{3} = \dfrac{y-2}{2} = \dfrac{z+4}{-8}$ | I. | $(-3, -2, 8), \langle 5, 2, -4 \rangle$ | | B. | $\dfrac{x+5}{3} = \dfrac{y+2}{2} = \dfrac{z-4}{-8}$ | II. | $(3, 2, -8), \langle 5, 2, -4 \rangle$ | | C. | $\dfrac{x-3}{5} = \dfrac{y-2}{2} = \dfrac{z+8}{-4}$ | III. | $(-5, -2, 4), \langle 3, 2, -8 \rangle$ | | D. | $\dfrac{x+3}{5} = \dfrac{y+2}{2} = \dfrac{z-8}{-4}$ | IV. | $(5, 2, -4), \langle 3, 2, -8 \rangle$ | Choose the correct answer from the options given below:

The value of $\lambda$ so that the lines $\dfrac{1-x}{3} = \dfrac{7y-14}{2\lambda} = \dfrac{z-3}{2}$ and $\dfrac{7-7x}{3\lambda} = \dfrac{y-5}{1} = \dfrac{6-z}{5}$ are at right angles is:

If shortest distance between the lines $\vec{r} = \hat{i} - \hat{j} + \lambda(2\hat{i}+\hat{k})$ and $\vec{r} = 2\hat{i} - \hat{j} + \mu(\hat{i}-\hat{j}-\hat{k})$ is $\dfrac{\sqrt{14}}{14}m$ units, the value of $m$ is

Two points are moving along the lines : $L_1: \vec{r} = (2\hat{i}+3\hat{j}+\hat{k}) + \lambda(\hat{i}+2\hat{j}+\hat{k})$ $L_2: \vec{r} = (\hat{i}+\hat{j}) + \mu(2\hat{i}+\hat{j}-\hat{k})$ where $\lambda$ and $\mu$ are real parameters , then which of the following statements are correct ? A. The direction ratios of the line perpendicular to both lines are $<-1,1,-1>$. B. The direction cosines of the second line $L_2$ are $<\dfrac{2}{\sqrt6},\dfrac{-1}{\sqrt6},\dfrac{-1}{\sqrt6}>$. C. Angle between two lines is $\pi/3$. D. The lines are skew lines. Choose the correct answer from the options given below:

The cartesian equation of the line passing through the point $(-4, 3, -2)$ and parallel to the line $\vec{r} = 5\hat{i} - \hat{k} + \lambda(\hat{i}-\hat{j}+2\hat{k})$ is:

The angle between the lines $\dfrac{3-x}{-7} = \dfrac{y+5}{-5} = \dfrac{z}{1}$ and $\dfrac{x}{1} = \dfrac{y}{2} = \dfrac{z}{3}$ is:

The points on the line $\frac{x+2}{3} = \frac{y+1}{2} = \frac{z-3}{2}$ at a distance of $\sqrt{17}$ from the point (-2, -1, 3) are:

Match List-I with List-II | List-I | List-II | |---|---| | (A) The distance of a point $P(a,b,c)$ from x-axis | (I) $\sqrt{a^2+b^2+c^2}$ | | (B) The distance of a point $P(a,b,c)$ from y-axis | (II) $\sqrt{b^2+c^2}$ | | (C) The distance of a point $P(a,b,c)$ from Z-axis | (III) $\sqrt{a^2+c^2}$ | | (D) The distance of a point $P(a,b,c)$ from origin | (IV) $\sqrt{a^2+b^2}$ | Choose the correct answer from the options given below:

The shortest distance (in units) between the lines $L_1:\frac{x-1}{2}=\frac{y-2}{3}=\frac{z+4}{6}$ and $L_2:\frac{x-3}{2}=\frac{3-y}{-3}=\frac{z+5}{6}$ is:

Match List-I with List-II | List-I (Lines) | List-II (Direction Ratios) | |---|---| | (A) $2x+3=y+1=z-1$ | (I) -1, 2, 0 | | (B) $\frac{1-2x}{2}=\frac{y}{2}, z=2$ | (II) 1, 1, 1 | | (C) $x=\frac{3y-1}{3}=z+1$ | (III) 2, 1, 1 | | (D) $x=2y+3, z=y+1$ | (IV) 1, 2, 2 | Choose the correct answer from the options given below:

Direction cosines of 2 lines are given by the equations $3l+m+5n=0, 6mn-2nl+5lm=0$ then which of the following statements are correct ? (A) Direction ratios of two lines are (1, 2, -1) and (-2, 1, 1). (B) Angle between lines is $\frac{\pi}{3}$. (C) The angle between the lines is $\cos^{-1}\left(-\frac{1}{6}\right)$. (D) Vectors parallel to these lines are $\hat i+2\hat j-\hat k$ and $-2\hat i+\hat j+\hat k$. Choose the correct answer from the options given below:

If the lines $\frac{1-x}{2}=\frac{2y}{p}=\frac{z-1}{1}$ and $\frac{3-2x}{p}=\frac{y-1}{2}=\frac{z}{1}$ are parallel then value of 'p' is

Let ABCD be a parallelogram with vertices $A(2,2,a), B(1,0,0), C(1,3,a)$ and D, where $a > 0$. If the area of parallelogram is 5 square units, then the value of $a$ is:

Match the LIST-I with LIST-II | LIST-I (Line) | | LIST-II (Direction cosiness) | |---|---|---| | A. $\vec{r} = (2\hat{i} - \hat{j} + \hat{k}) + \lambda(3\hat{i} + 2\hat{k})$ | | I. $\left\langle \dfrac{2}{\sqrt{5}}, \dfrac{1}{\sqrt{5}}, 0 \right\rangle$ | | B. $\dfrac{x-1}{2} = \dfrac{3-y}{1} = \dfrac{2z-1}{4}$ | | II. $\left\langle -\dfrac{1}{3}, \dfrac{2}{3}, -\dfrac{2}{3} \right\rangle$ | | C. $\dfrac{x+1}{1} = \dfrac{2y+1}{1}, z = -1$ | | III. $\left\langle \dfrac{3}{\sqrt{13}}, 0, \dfrac{2}{\sqrt{13}} \right\rangle$ | | D. $\dfrac{1-x}{1} = \dfrac{y}{2} = \dfrac{1-z}{2}$ | | IV. $\left\langle \dfrac{2}{3}, -\dfrac{1}{3}, \dfrac{2}{3} \right\rangle$ | Choose the correct answer from the options given below:

Let the square of the shortest distance between the lines $L_1: \vec{r} = (\hat{i}+2\hat{j}-4\hat{k}) + \lambda(2\hat{i}+3\hat{j}+6\hat{k})$ and $L_2: \vec{r} = (3\hat{i}+3\hat{j}-5\hat{k}) + \mu(2\hat{i}+3\hat{j}+6\hat{k})$ is $\dfrac{m}{n}$, where $\gcd(m,n)=1$. Then $m+n$ is equal to:

Let $L_1$ and $L_2$ be any two lines given by $L_1: 3\hat{i}+2\hat{j}-4\hat{k} + \lambda(\hat{i}+2\hat{j}+2\hat{k})$ $L_2: 5\hat{i}-2\hat{j} + \mu(3\hat{i}+2\hat{j}+6\hat{k})$ Where $\lambda$ and $\mu$ are parameters then which of the following statements are TRUE? A. Equation of a line passing through the point $(1,2,3)$ which is parallel to line $L_1$ is $\dfrac{x-1}{1} = \dfrac{y-2}{2} = \dfrac{z-3}{2}$ B. Direction cosines of line $L_2$ are $\dfrac{3}{7}, \dfrac{2}{7}, \dfrac{6}{7}$ C. Line $L_1$ is parallel to the line $L_2$ D. The angle between the two lines $L_1$ and $L_2$ is $\cos^{-1}\left(\dfrac{19}{21}\right)$ Choose the correct answer from the options given below:

If the lines $\frac{x-1}{2} = \frac{y+1}{\lambda-3} = \frac{z-2}{2}$ and $x = 0, \frac{2y-4}{-2} = \frac{z+1}{2-\lambda}$ are perpendicular, then value of $\lambda$ is

Match List-I with List-II | List-I (Straight Lines) | List-II (Direction Ratios) | |---|---| | (A) $\vec{r} = (\hat{i}+\hat{j}) + \lambda(2\hat{i}-\hat{j}+3\hat{k})$ | (I) 2,1,-3 | | (B) $\frac{x}{2} = \frac{1-y}{-1} = \frac{z+1}{3}$ | (II) -2,1,3 | | (C) $\frac{1-2x}{4} = \frac{y+1}{1} = \frac{2z+3}{6}$ | (III) 2,-1,3 | | (D) $\vec{r} = (2\hat{i}-\hat{k}) + \lambda(2\hat{i}+\hat{j}-3\hat{k})$ | (IV) 2,1,3 | Choose the correct answer from the options given below:

If $\vec{r} = \vec{a_1} + \lambda\vec{b_1}$ and $\vec{r} = \vec{a_2} + \mu\vec{b_2}$ are two skew lines such that $\vec{b_1} \times \vec{b_2} = 2\hat{i}+3\hat{j}+2\hat{k}$, $\vec{a_1} = \hat{i}+\hat{j}+\hat{k}$ and $\vec{a_2} = 2\hat{i}+3\hat{j}+\hat{k}$, then the shortest distance between the lines is

The angle between two lines whose direction ratio are proportional to $(-\theta, -2\theta, \theta)$ and $(-2\theta, \theta, \theta)$, where $\theta \in \mathbb{R} \sim \{0\}$, is

If the lines $\frac{x-1}{2} = \frac{y-m}{3} = \frac{z-3}{4}$ and $\frac{x-4}{5} = \frac{y-1}{2} = z$ are interesecting, then the value of $m$, is

If $\vec{a}$ and $\vec{b}$ are two perpendicular vector whose direction ratios are $(a_1,b_1,c_1)$ and $(a_2,b_2,c_2)$, then which of the following is a true statement?

Points on the line $\frac{x+2}{3} = \frac{y+1}{2} = \frac{z-3}{2}$ at a distance of 5 units from the point $(1, 3, 3)$, are

If $\theta$ is the angle between the lines $L_1$ and $L_2$, whose direction cosines are $l_1, m_1, n_1$ and $l_2, m_2, n_2$ respectively, then Match List-I with List-II | List-I | List-II | |---|---| | (A) $\cos\theta$ | (I) $l_1l_2+m_1m_2+n_1n_2$ | | (B) $\sin\theta$ | (II) $l_1l_2+m_1m_2+n_1n_2=0$ | | (C) $L_1$ and $L_2$ are perpendicular | (III) $\frac{l_1}{l_2}=\frac{m_1}{m_2}=\frac{n_1}{n_2}$ | | (D) $L_1$ and $L_2$ are parallel | (IV) $\sqrt{\sum(m_1n_2-m_2n_1)^2}$ | Choose the correct answer from the options given below:

The shortest distance between the lines whose vector equations are $\vec{r} = (6\hat{i}+2\hat{j}+2\hat{k}) + \lambda(\hat{i}-2\hat{j}+2\hat{k})$ and $\vec{r} = (-4\hat{i}-\hat{k}) + \mu(3\hat{i}-2\hat{j}-2\hat{k})$

Which of the following statements is/are TRUE? (A) The angle between the lines whose direction ratios are proportional to (4, -3, 5) and (3, 4, 5) is 60°. (B) If A(1,2,3) and B(2, 0, 5) are two points on a line, then its direction ratios are proportional to 1, -2, 2. (C) The direction cosines of z-axis are 0, 0, 1. (D) The vector equation of a line which passes through the point with position vector $3\hat{i}-\hat{j}+\hat{k}$ and is in the direction of $2\hat{i}+3\hat{j}+\hat{k}$ is $\vec{r} = (2\hat{i}+3\hat{j}+\hat{k}) + \lambda(3\hat{i}-\hat{j}+\hat{k})$, where $\lambda$ is a parameter. Choose the correct answer from the options given below:

If the two lines given by $x=py+q, z=ry+s$ and $x=p'y+q', z=r'y+s'$ are perpendicular, then which of the following is equal to -1 ?

The angle between the two lines whose direction cosines are given by the equations $4l+m+n=0$ and $2mn+24nl+4lm=0$, is

Let $(\alpha, 0, \beta)$ be the coordinates of the point, which divides the line segment joining the points (1,2,-3) and (1,-1,-4) internally, then $(\alpha^2+\beta^2)$ equals:

The cartesian equation of a line is $x = ay+b$, $z = cy+d$. It's direction ratios are

Equation of a line passing through (1, 2, -3) which is parallel to the line $\frac{x+2}{3} = \frac{y-3}{-4} = \frac{z+5}{5}$ is

The shortest distance between the lines $\vec{r} = 2\hat{i}+3\hat{j}+4\hat{k}+\lambda(\hat{i}-\hat{j}+\hat{k})$ and $\vec{r} = 5\hat{i}+5\hat{j}+5\hat{k}+\mu(-\hat{i}+\hat{j}-\hat{k})$ where $\lambda,\mu \in R$ (set of real numbers) is:

Match the LIST-I with LIST-II | | LIST-I (Equation of Line) | | LIST-II (Direction Ratio of line) | |---|---|---|---| | A. | $\dfrac{2-x}{1} = \dfrac{2y+1}{1} = \dfrac{z}{1}$ | I. | $2, 1, 1$ | | B. | $x = 2y+3,\ z = y+1$ | II. | $1, -2, 2$ | | C. | $\dfrac{x-1}{1} = \dfrac{y+1}{2},\ z = 1$ | III. | $-2, 1, 2$ | | D. | $\vec{r} = \hat{i} + \hat{j} + \lambda(\hat{i} - 2\hat{j} + 2\hat{k})$ | IV. | $1, 2, 0$ | Choose the correct answer from the options given below:

The shortest distance between the lines $\dfrac{x+1}{7}=\dfrac{-y-1}{6}=\dfrac{z+1}{1}$ and $\dfrac{x-3}{1}=\dfrac{-y+5}{2}=\dfrac{z-7}{1}$ is

If the lines $\dfrac{x+1}{3}=\dfrac{y+3}{5}=\dfrac{z+\lambda}{7}$ and $\dfrac{x-2}{1}=\dfrac{y-4}{3}=\dfrac{z-6}{5}$ are intersecting , then the value of '$\lambda$' is

The shortest distance between the lines whose vector equations are $\vec r=(\hat i+2\hat j+3\hat k)+\lambda(2\hat i+3\hat j+4\hat k)$ and $\vec r=(2\hat i+4\hat j+5\hat k)+\mu(4\hat i+6\hat j+8\hat k)$ is:

If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), (−4, 3, −6) and (2, 9, 2) respectively, then the angle between the lines AB and CD is

The image of the point (1, 6, 3) with respect to the line $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$ is

If the points $A(-1,3,2)$, $B(-4,2,-2)$ and $C(5,5,\lambda)$ are collinear, then value of $\lambda$ is :

If angle between the lines $\dfrac{x-1}{\propto} = \dfrac{y+2}{-5} = \dfrac{z+\frac{7}{3}}{\beta}$ and $x=z, y=0$ is $\dfrac{\pi}{4}$, then

For a given line L: $\dfrac{x}{2}=\dfrac{y+1}{3}=\dfrac{-z-3}{-5}$, which of the following is/are True ? (A) The direction ratio of line L is (2, 3, -5) (B) The perpendicular distance of the point (1, 1, 3) on the line L is $\sqrt{3}$ (C) The foot of perpendicular from the point (1, 1, 3) on line L is (2, 2, 1) (D) The image of point (1, 1, 3) with respect to line L is (3, 3, 1) Choose the **correct** answer from the options given below:

The direction ratios of the line perpendicular to the lines $\dfrac{x-5}{2} = \dfrac{y+11}{-3} = \dfrac{z+3}{1}$and $\dfrac{x-7}{1} = \dfrac{y+2}{2} = \dfrac{z-4}{-2}$ are proportional to:

The vector equation of line passing through $(-1, 3, -2)$ and perpendicular to the lines $\frac{x+4}{1} = \frac{y}{2} = \frac{z-3}{3}$ and $\frac{x+2}{-3} = \frac{y+5}{2} = \frac{z-6}{5}$ is

The angle at which the line, $\frac{x-1}{0} = \frac{2-y}{-1} = \frac{2z-3}{-2}$ is inclined with the positive direction of z-axis is

The shortest distance between the lines $\vec{r} = \hat{i} + \hat{j} + \lambda(2\hat{i} - \hat{j} + \hat{k})$ and $\vec{r} = 2\hat{i} + \hat{j} - \hat{k} + \mu(4\hat{i} - 2\hat{j} + 2\hat{k})$ is

The Cartesian equation of the line passing through the point (1, 2, -1) and parallel to the line $5x - 25 = 14 - 7y = 35z$ is

If the lines $\frac{x - 5}{5\lambda + 2} = \frac{2 - y}{5} = \frac{1 - z}{-1}$ and $x = \frac{y + 1/2}{2\lambda} = \frac{z - 1}{3}$ are perpendicular, then the value of $\lambda$ is equal to

The value of $\lambda$ so that the lines $\frac{1-x}{3} = \frac{7y-14}{2\lambda} = \frac{z-3}{2}$ and $\frac{7-7x}{3\lambda} = \frac{y-5}{1} = \frac{6-z}{5}$ are at right angle, is:

The foot of the perpendicular drawn from the point $(1,6,3)$ to the line $\frac{x}{1} = \frac{y-1}{2} = \frac{z-2}{3}$ is

If the direction ratios of two lines are $a, b, c$ and $(b-c), (c-a), (a-b)$ respectively, then the angle between these lines is:

If a line makes angles $\alpha, \beta, \gamma$ with the positive directions of the coordinate axes, then the value of $\cos 2\alpha + \cos 2\beta + \cos 2\gamma$ is

If z-coordinate of a point P on the line joining the points A (2, 2, 1) and B (5, 1, -2) is -1, than x-coordinate of point P is

The coordinates of the image of the point P (5, 4, 2) in the line $\vec{r} = (-\hat{i} + 3\hat{j} + \hat{k}) + \mu(2\hat{i} + 3\hat{j} - \hat{k})$, where $\lambda$ is a parameter, is

If a line makes angles $\alpha$, $\beta$ and $\gamma$ with the positive directions of x-axis, y-axis and z-axis respectively, then $\sin^2\alpha + \sin^2\beta + \sin^2\gamma$ is equal to

Consider the line $\vec{r} = -2\hat{i} + 3\hat{j} + \hat{k} + \lambda(5\hat{i} - 3\hat{j} - \hat{k})$. Match List-I with List-II | List-I | List-II | |---|---| | (A) A point on the given line | (I) $\left(\frac{5}{\sqrt{35}}, \frac{-3}{\sqrt{35}}, \frac{-1}{\sqrt{35}}\right)$ | | (B) Direction ratios of the given line | (II) (2, 3, 1) | | (C) Direction cosines of the given line | (III) (5, -3, -1) | | (D) Direction ratios of a line perpendicular to given line | (IV) (-2, 3, 1) | Choose the correct answer from the options given below:

The shortest distance between the lines $\vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(2\hat{i} + 3\hat{j} + 4\hat{k})$ and $\vec{r} = (2\hat{i} + 4\hat{j} + 5\hat{k}) + \mu(4\hat{i} + 6\hat{j} + 8\hat{k})$ is equal to

The direction cosines of a line which makes equal angles with co-ordinate axes are:

Let the equation of lines be as $L_1: \vec{r_1} = \vec{a_1} + \lambda\vec{b_1}$ and $L_2: \vec{r_2} = \vec{a_2} + \lambda\vec{b_2}$ such that $\vec{a_1} - \vec{a_2} = 2\hat{i} + 4\hat{j} + 4\hat{k}$ and $\vec{b_1} \times \vec{b_2} = 8\hat{i} - 4\hat{k}$. Then the shortest distance between $L_1$ and $L_2$ is

The vector equation of line passing through (2, -1, 3) and perpendicular to the lines $\frac{x-2}{3} = \frac{y-1}{1} = \frac{z+2}{2}$ and $\frac{x+3}{-4} = \frac{y-5}{-3} = \frac{z+1}{2}$ is (Here $\lambda$ is a parameter)

Cosine of the acute angle between the lines $\frac{x-3}{2} = \frac{y-2}{1} = \frac{z-5}{2}$ and $\frac{x-1}{6} = \frac{y-3}{-3} = \frac{z+6}{2}$ is

Let $L_1$ and $L_2$ be two lines, represented as, $L_1: \vec{r} = \hat{i} + \hat{j} + \lambda(2\hat{i} - \hat{j} + \hat{k})$ and $L_2: \vec{r} = 2\hat{i} + \hat{j} - \hat{k} + \mu(4\hat{i} - 2\hat{j} + 2\hat{k})$, where $\lambda$ and $\mu$ are scalars. Then which of the following are true? (A) $L_1$ is perpendicular to $L_2$. (B) $L_1$ is parallel to $L_2$. (C) $L_1$ passes through the point (1, 1, 0) (D) $L_2$ passes through the point (2, 1, -1) Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | |---|---| | (A) Line : $x = 2y + 1 = z - 1$ | (I) Crosses $xz$ plane at (1, 0, 1) | | (B) Line : $x + 1 = 2y + 1 = z$ | (II) Crosses $xz$ plane at (0, 0, 1) | | (C) Line : $x - 1 = 2y = z + 1$ | (III) Crosses $xz$ plane at (1, 0, -1) | | (D) Line : $x - 1 = 2y = z - 1$ | (IV) Crosses $xz$ plane at (1, 0, 2) | Choose the correct answer from the options given below:

If the points (-1, -1, 2), (2, m, 5) and (3, 11, 6) are collinear, then m equals

Which of the following statements are true? (A) The vector equation of the line through the point (5, 2, -4) and parallel to the vector $3\hat{i} + 2\hat{j} - 8\hat{k}$ is $\vec{r} = (5\hat{i} + 2\hat{j} - 4\hat{k}) + \lambda(3\hat{i} + 2\hat{j} - 8\hat{k})$ (B) Vector form of the equation of line $\frac{x-5}{3} = \frac{y+4}{7} = \frac{z-6}{2}$ is $\vec{r} = (5\hat{i} - 4\hat{j} + 6\hat{k}) + \lambda(3\hat{i} + 7\hat{j} + 2\hat{k})$ (C) The direction cosines of z-axis are (1, 1,0). (D) If a line has direction ratios 2, -1, -2, then its direction cosines are -2/3, -1/3, -2/3. Choose the correct answer from the options given below:

The value of p so that the lines $\frac{x-1}{-3} = \frac{2y-2}{2p} = \frac{z-3}{2}$ and $\frac{x-1}{-3p} = \frac{y-1}{4} = \frac{6-z}{5}$ are at right angles is

A line passes through the point with position vector $2\hat{i} - \hat{j} + 4\hat{k}$ and is in the direction of the vector $\hat{i} + \hat{j} - 2\hat{k}$. The equation of the line in Cartesian form is:

Acute angle between the lines $\frac{x}{3} = \frac{y}{4} = \frac{z}{5}$ and $\frac{x-1}{4} = \frac{y+1}{-3} = \frac{z+10}{5}$ is:

The co-ordinates of the point where the line $\frac{x+3}{3} = \frac{y-1}{-1} = \frac{z-5}{-5}$ cuts $yz$-plane are:

Match List-I with List-II | List-I | List-II | |---|---| | **Equation of line** | **Information** | | (A) $\vec{r} = (3\hat{i} - 2\hat{j} + \hat{k}) + \lambda(\hat{j} - 2\hat{k})$ | (I) Direction ratios are 2, 4, -1 | | (B) $\frac{2-x}{1} = \frac{2y+1}{4}$, $z = 2$ | (II) Perpendicular to $2\hat{i} - \hat{j} + \hat{k}$ | | (C) $\frac{x}{1} = \frac{y-3}{2} = \frac{3-4z}{2}$ | (III) Passing through the point $(3, -2, 1)$ | | (D) $\vec{r} = (3\hat{i} + 2\hat{j} + \hat{k}) + \lambda(2\hat{i} + \hat{j} - 3\hat{k})$ | (IV) Direction ratios are -1, 2, 0 | Choose the correct answer from the options given below:

If a line makes angle $\pi/3$ and $\pi/4$ with the positive directions of x-axis and y-axis respectively, then the acute angle made by the line with positive direction of z-axis is

Match **List-I** with **List-II** | List-I | List-II | |---|---| | (A) Equations of line through $(5, -4, 6)$ with direction ratios $3, 7, 2$ | (I) $\frac{x+3}{5} = \frac{y+7}{-4} = \frac{z+2}{6}$ | | (B) Equations of line through $(3, 7, 2)$ with direction ratios $5, -4, 6$ | (II) $\frac{x-3}{5} = \frac{y-7}{-4} = \frac{z-2}{6}$ | | (C) Equations of line through $(-5, 4, -6)$ with direction ratios $3, 7, 2$ | (III) $\frac{x-5}{3} = \frac{y+4}{7} = \frac{z-6}{2}$ | | (D) Equations of line through $(-3, -7, -2)$ with direction ratios $5, -4, 6$ | (IV) $\frac{x+5}{3} = \frac{y-4}{7} = \frac{z+6}{2}$ | Choose the correct answer from the options given below:

If the lines $\frac{1-x}{3} = \frac{y-2}{2\lambda} = \frac{z-3}{2}$ and $\frac{x-1}{3\lambda} = \frac{y-1}{1} = \frac{6-z}{5}$ are perpendicular, then $\lambda$ is equal to

The shortest distance between the following lines: $\vec{r} = (\hat{i} + \hat{j} - \hat{k}) + s(2\hat{i} + \hat{j} + \hat{k})$ $\vec{r} = (\hat{i} + \hat{j} + 2\hat{k}) + t(4\hat{i} + 2\hat{j} + 2\hat{k})$, where s and t are scalars, is:

The angle between the line $2x = 3y = z$ and $x$- axis is:

If lines $\frac{x+5}{5\lambda+2} = \frac{4-2y}{10} = \frac{1-3z}{-3}$ and $\frac{x-2}{1} = \frac{1+2y}{4\lambda} = \frac{2+z}{3}$ are perpendicular, than value of '$\lambda$' is

The value of k for which the lines $\frac{2x - 3}{4} = \frac{3 - y}{k} = \frac{z - 2}{-2}$ and $\frac{x - 2}{1} = \frac{y}{4} = \frac{5 - z}{3}$ are perpendicular to each other is:

If a line makes angles $\alpha$, $\beta$ and $\gamma$ with the positive directions of coordinate axes respectively, then $\cos 2\alpha + \cos 2\beta + \cos 2\gamma$ is equal to

The acute angle between the lines $\vec{r} = (4\hat{i} - \hat{j}) + \lambda(2\hat{i} + \hat{j} - 3\hat{k})$ and $\frac{x-1}{1} = \frac{y+1}{-3} = \frac{z-2}{2}$ is

The shortest distance between lines $\frac{-x-3}{4} = \frac{y-6}{3} = \frac{z}{2}$ and $\frac{-x-2}{4} = \frac{y}{1} = \frac{z-7}{1}$ is:

Distance of the point $(2, 4, -1)$ from the line $\frac{10+2x}{2} = \frac{y+3}{4} = \frac{6-z}{9}$ is

If the line $\frac{-x+1}{3} = \frac{-y-2}{-2k} = \frac{z+3}{2}$ and $\frac{-1+x}{3k} = \frac{-1 +y}{1} = \frac{-z+6}{5}$ are perpendicular, then the value of k is:

If a line makes angles α, β and γ with positive x-axis, y-axis and z-axis respectively, then the value of $sin^2\frac{α}{2}cos^2\frac{α}{2} + sin^2\frac{β}{2}cos^2\frac{β}{2} + sin^2\frac{γ}{2}cos^2\frac{γ}{2}$ is

Consider the line $\frac{x-2}{2} = \frac{2y-5}{-3}, z = -1$. Then which of the following is/are true? (A) It has Direction ratios (2, -3, -1) (B) It has Direction cosines $\left(\frac{4}{5}, \frac{-3}{5}, \frac{-1}{5}\right)$ (C) It has Direction ratios $\left(2, \frac{-3}{2}, 0\right)$ (D) It has Direction cosines $\left(\frac{4}{5}, \frac{-3}{5}, 0\right)$ Choose the correct answer from the options given below:

The angle between the pair of lines given by $\vec{r} = \hat{i} + 2\hat{j} - 3\hat{k} + \lambda (\hat{i} - 2\hat{j} + 2\hat{k})$ and $\vec{r} = 5\hat{i} + \hat{j} + \hat{k} + \mu (3\hat{i} - 2\hat{j} + 6\hat{k})$ is

Consider the equation of the line $\vec{r} = -\hat{i} + 2\hat{k} + \mu(4\hat{i} - \hat{j} + 2\hat{k})$. Match List-I with List-II | List-I | List-II | |---|---| | (A) It passes through the point | (I) 4, -1, 2 | | (B) Its direction ratios are | (II) $\frac{4}{\sqrt{21}}, \frac{-1}{\sqrt{21}}, \frac{2}{\sqrt{21}}$ | | (C) Its Cartesian form is | (III) (-1, 0, 2) | | (D) Its direction cosines are | (IV) $\frac{x+1}{4} = \frac{y}{-1} = \frac{z-2}{2}$ | Choose the correct answer from the options given below:

The co-ordinates of the point at which the line $\frac{x-3}{3} = \frac{y+1}{2} = \frac{z-4}{-2}$ crosses x-y plane, are

If the lines $\frac{x-5}{7} = \frac{y+2}{-5} = \frac{z}{\lambda}$ and $\frac{x}{1} = \frac{y}{2\lambda} = \frac{z}{3}$ are perpendicular to each other, then $\lambda$ is equal to

Consider the lines $l_1: \frac{x-1}{0} = \frac{y-1}{1} = \frac{2-z}{1}$ and $l_2: \frac{x}{2} = \frac{y}{0} = \frac{2z-1}{4}$, then which of the following are correct? (A) Direction Ratio's of $l_1 = <0, 1, 1>$ (B) Direction Ratio's of $l_2 = <2, 0, 2>$ (C) Angle between $l_1$ and $l_2 =$ $\frac{\pi}{3}$ (D) Angle between $l_1$ and $l_2 = $ $\frac{2\pi}{3}$ Choose the correct answer from the options given below:

The straight line $\frac{x+3}{3} = \frac{y+2}{4} = \frac{z+1}{0}$ is

If $\alpha, \beta$ and $\gamma$ are angle of inclinations of a line with x, y and z axes respectively, then the value of $2(\cos 2\alpha + \cos 2\beta + \cos 2\gamma)$ is

The angle between the lines $l_1: \frac{x + 1}{1} = \frac{2 - y}{2} = \frac{z - 1}{1}$ and $l_2: \frac{x - 1}{4} = \frac{2y - 4}{6} = \frac{z - 1}{2}$ is

Match **List-I** with **List-II** | List-I | List-II | |---|---| | **Lines** | **Direction Ratios** | | (A) $\frac{x - 1}{2} = \frac{2 - y}{1} = z$ | (I) 1, 3, -1 | | (B) $\frac{2x - 1}{2} = \frac{y + 1}{3} = \frac{1 - z}{1}$ | (II) 2, -2, 0 | | (C) $\frac{x + 1}{2} = \frac{3 - y}{2}, z = 2$ | (III) 2, -1, 1 | | (D) $\frac{2x - 3}{4} = \frac{1 - 2y}{2} = \frac{z}{5}$ | (IV) 2, -1, 5 | Choose the **correct** answer from the options given below:

Consider a line $\vec{r} = (\hat{i} + 4\hat{j}) + \lambda(2\hat{i} - 2\hat{j} + 3\hat{k})$, then which of the following statements are correct? (A) it passes through point (9, -4, 12) (B) it passes through point (1, 4, -1) (C) its direction cosine's are $\frac{2}{\sqrt{17}}, \frac{-2}{\sqrt{17}}, \frac{3}{\sqrt{17}}$ (D) its Cartesian equation is $\frac{x - 1}{2} = \frac{y - 4}{-2} = \frac{z}{3}$ Choose the **correct** answer from the options given below:

Consider the line $\vec{r} = \hat{i} - 2\hat{j} + 4\hat{k} + \lambda(-\hat{i} + 2\hat{j} - 4\hat{k})$ Match List-I with List-II | List-I | List-II | |---|---| | (A) A point on the given line | (I) $\left(\frac{-1}{\sqrt{21}}, \frac{2}{\sqrt{21}}, \frac{-4}{\sqrt{21}}\right)$ | | (B) direction ratios of the line | (II) $(4, -2, -2)$ | | (C) direction cosines of the line | (III) $(1, -2, 4)$ | | (D) direction ratios of a line perpendicular to given line | (IV) $(-1, 2, -4)$ | Choose the correct answer from the options given below:

The shortest distance between the lines $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $\frac{x-2}{4} = \frac{y-4}{6} = \frac{z-5}{8}$ is equal to

If a line makes angles $\alpha, \beta, \gamma$ with the positive directions of $x$-axis, $y$- axis and $z$-axis respectively, then $sin^2 \alpha + sin^2 \beta + sin^2 \gamma$ is equal to

The shortest distance between the lines $\vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(2\hat{i} + 3\hat{j} + 4\hat{k})$ and $\vec{r} = (2\hat{i} + 4\hat{j} + 5\hat{k}) + \mu(3\hat{i} + 4\hat{j} + 5\hat{k})$ is equal to

If the lines $\frac{1-x}{3} = \frac{3y-6}{k} = \frac{3-z}{-2}$ and $\frac{1-x}{2k} = \frac{y-5}{3} = \frac{6-z}{5}$ are perpendicular to each other, then $k$ is equal to

The direction cosines of a line equally inclined with the co-ordinate axes are

Consider two lines $l_1$ and $l_2$ with cartesian equations $\frac{x}{2} = \frac{1-y}{-2} = \frac{z}{1}$ and $\frac{2x-5}{16} = \frac{y-2}{-1} = \frac{z-5}{4}$ respectively. Which of the following is/are true? (A) Direction ratio of $l_1$ are 2, 2, 1 (B) Direction cosines of $l_1$ are $\frac{2}{3}, \frac{-2}{3}, \frac{1}{3}$ (C) Direction ratio of $l_2$ are 16, -1, 4 (D) Angle between $l_1$ and $l_2$ is $\cos^{-1}\left(\frac{38}{3\sqrt{273}}\right)$ Choose the correct answer from the options given below:

If a line makes angles $\alpha, \beta, \gamma$ with the positive directions of $x$ - axis, y -axis, z -axis respectively, then the value of $cos2\alpha + cos2\beta + cos2\gamma$ is equal to

The direction cosines of the line which is perpendicular to the lines with direction ratios $1,-2,-2$ and $0,2,1$ are :

The distance between the lines $\vec{r}=\hat{i}-2 \hat{j}+3 \hat{k}+\lambda(2 \hat{i}+3 \hat{j}+6 \hat{k})$ and $\vec{r}=3 \hat{i}-2 \hat{j}+1 \hat{k}+\mu(4 \hat{i}+6 \hat{j}+12 \hat{k})$ is :

The angle between the lines $\vec{r} = 3\hat{i} + 2\hat{j} - 4\hat{k} + \lambda(\hat{i} + 2\hat{j} + 2\hat{k})$ and $\vec{r} = 5\hat{j} - 2\hat{k} + \mu(3\hat{i} + 2\hat{j} + 6\hat{k})$ is :

The value of $\lambda$, so that the lines $\frac{1-x}{3} = \frac{7y-14}{2\lambda} = \frac{z-3}{2}$ and $\frac{7-7x}{3\lambda} = \frac{y-5}{1} = \frac{6-z}{5}$ are perpendicular is:

Match List I with List II | LIST I | LIST II | |---|---| | A. $lx + my + nz = d$ is | I. Equation of plane passing through a given point and normal to given vector | | B. $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$ is | II. Equation of plane in normal form | | C. $(\vec{r} - \vec{a}) \cdot \vec{n} = 0$ | III. Plane passing through the intersection of two planes | | D. $(a_1x + b_1y + c_1z + d_1) + \lambda(a_2x + b_2y + c_2z + d_2) = 0$ | IV. Intercept from of plane | Choose the correct answer from the options given below:

Distance between the point (3, 4, 5) and the point where the line $\frac{x-3}{1} = \frac{y-4}{2} = \frac{z-5}{2}$ meets the plane $x + y + z = 17$ is :

Cartesian equation of plane passing through the points (2, -4, 5) and perpendicular to the line with direction ratios (3, -1, 2) is :

If the shortest distance between the lines $l_1$ and $l_2$ given by $\vec{r} = a\hat{i} + 2\hat{j} - \hat{k} + \lambda(2\hat{i} - \hat{j} + \hat{k})$ and $\vec{r} = \hat{i} - \hat{j} + \hat{k} + \mu(2\hat{i} - \hat{j} + \hat{k})$ is $\sqrt{\frac{35}{6}}$ units, the values of 'a' can be :

The vector equation of the line joining the points $(-2, -3, -4)$ and $(1, -2, 4)$ is :

The angle between the two planes $x + y - z = 3$ and $3x + 2y + z = 5$ is :

If the equation of a floor of a room is given by $x + y - z + 4 = 0$ and the equation of roof is given by $x + y - z + 5 = 0$. Then, the height of the room is :

The equation of plane which cuts equal intercepts of unit length on the coordinate axes is :

If the straight lines $x = 1 + s$, $y = -3 - \lambda s$, $z = 1 + \lambda s$ and $x = \frac{t}{2}$, $y = 1 + t$, $z = 2 - t$ with parameters $s$ and $t$ respectively, are coplanar, then $\lambda$ is equal to :

The shortest distance between the lines $\frac{x+3}{1} = \frac{y-2}{2} = \frac{z+4}{3}$ and $\frac{x+3}{-3} = \frac{y+7}{2} = \frac{z-6}{4}$ is :

Find the root of perpendicular from origin to the line $\frac{x - 1}{3} = \frac{y - 2}{-2} = \frac{z + 1}{3}$.

The distance of the point $(2, 3, -5)$ from the plane $x + 2y - 2z = 9$ is :

Equation of y-axis in space, in vector form is :

The angle between 2 planes $4x + 8y + z - 8 = 0$ and $y + z - 4 = 0$ is :

The equation of plane that contains line $\frac{x - 1}{-1} = \frac{y + 1}{2} = \frac{z - 1}{3}$ and also pass through point $(0, 1, 0)$ is :

The direction ratio's of line of intersection of two planes : $2x + y + z + 47 = 0$ and $3x - 2y - z + 41 = 0$ are :

A line makes angle $\theta$ with $x$-axes as well as $z$-axis. If the angle $\beta$, which it makes with $y$-axis is such that $\sin^2\beta = 3\sin^2\theta$, then $\cos^2\theta$ is :

The equation of the plane, parallel to the plane $3x + 4y - 12z = 3$ and passes through $(1, 1, -1)$, is :

The distance of plane $\vec{r}\cdot(6\hat{i} - 3\hat{j} - 2\hat{k}) + 1 = 0$ from origin is :

The foot of perpendicular from point $(2, 4, -1)$ on the line $\frac{x+5}{1} = \frac{y+3}{4} = \frac{z-6}{-9}$ is :

If two lines $\frac{x-3}{2} = \frac{y-4}{5} = \frac{z}{4}$ and $\frac{x-4}{3} = \frac{y-5}{6} = \frac{1-z}{k}$, are coplanar, then $k$ is equal to

The foot of perpendicular from the point P (1, 2, -3) to the line $\frac{x+1}{2} = \frac{y-3}{-2} = \frac{z}{-1}$ is

The equation of plane passing through the point of (3, 2, 0) and containing the line $\frac{x-2}{2} = \frac{y+3}{4} = \frac{z-1}{1}$ is

The sum of all the values of $\lambda$ for which the distance of the point P (2, 3, $\lambda$) from the plane $x + 2y - 2z = 9$ is 3 units, is

The points of trisection of the segment joining the points (1, 0, 2) and (1, 3, 2) are: A. $(1, \frac{3}{2}, \frac{4}{3})$ B. (1, 1, 2) C. (1, 2, 2) D. $(1, \frac{3}{2}, 2)$ Choose the correct answer from the options given below:

The equation of line of intersection of the planes $x + y + 3z = 7$ and $x - y + 2z = 3$ is:

The distance of the point (1, 2, 0) from the line $\frac{x-3}{2} = \frac{y+4}{3} = \frac{z+6}{5}$ measured parallel to the plane $x + y + z = 3$ is

The line $\frac{x+2}{3} = \frac{y+3}{5} = \frac{z-6}{4}$ passes through (a, 2, c). The value of a and c are:

The distance of the plane $\vec{r} \cdot \left(\frac{2}{7}\hat{i} + \frac{3}{7}\hat{j} - \frac{6}{7}\hat{k}\right) = 2$ from the origin is

The direction ratios of the line $\frac{1-x}{3} = \frac{7y - 14}{2} = \frac{z-3}{2}$ are

If the vertices of a triangle ABC are $A(1, 2, 1)$, $B(4, 2, 3)$ and $C(2, 3, 1)$, then the equation of the median passing through the vertex $A$, is

A line makes the angle $\theta$ with each of the $x$ and $z$ axes. If the angle $\beta$ which it makes with $y$-axis is such that $\sin^2\beta = 3\sin^2\theta$, then the value of $\cos^2\theta$ is

The shortest distances of the point $(1, 2, 3)$ from $x$, $y$, $z$ axes respectively are

Distance between two planes $x + 2y - z = 5$ and $2x + 4y - 2z + 2 = 0$ is

The two lines given by $\vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \mu(\hat{i} - \hat{j} + \hat{k})$ and $\vec{r} = (2\hat{i} - \hat{j} - \hat{k}) + \mu(-\hat{i} + \hat{j} - \hat{k})$ A. are perpendicular B. are parallel. C. have shortest distance 0. D. have shortest distance $\sqrt{26}$. E. have shortest distance $\sqrt{78}$. Which of the above statements are true? Choose the correct answer from the options given below:

The correct order of steps from A to E, for finding value of p, so that the lines $\frac{1-x}{3} = \frac{7y-14}{2p} = \frac{z-3}{2}$ and $\frac{7-7x}{3p} = \frac{y-5}{1} = \frac{6-z}{5}$ are at right angle is: A. $p = \frac{70}{11}$ B. $(-3) \times \left(\frac{-3p}{7}\right) + 1 \times \left(\frac{2p}{7}\right) + 2 \times (-5) = 0$ C. $\frac{x-1}{-3} = \frac{y-2}{2\frac{p}{7}} = \frac{z-3}{2}$, $\frac{x-1}{-\frac{3p}{7}} = \frac{y-5}{1} = \frac{z-6}{-5}$ D. $\frac{9p}{7} + \frac{2p}{7} - 10 = 0$ E. $\frac{11p}{7} = 10$ Choose the correct answer from the options given below:

Image of origin with respect to plane $x + y + z = 3$ is:

The reflection of the point $(\alpha, \beta, \gamma)$ in the xz plane is

The angle between the planes $2x + y + 3z - 2 = 0$ and $x - 2y + 5 = 0$ is :

The angle between the pairs of lines $\vec{r} = (3\hat{i} + 2\hat{j} - 4\hat{k}) + \lambda(\hat{i} + 2\hat{j} + 2\hat{k})$ $\vec{r} = (5\hat{i} - 2\hat{j}) + \mu(3\hat{i} + 2\hat{j} + 6\hat{k})$ is :

The angle between the line $\frac{x+2}{3} = \frac{y-3}{2} = \frac{z+5}{6}$ and the plane $2x + 10y - 11z = 5$ is:

A. Equation of the line passing through the point (1, 2, 3) and parallel to the vector $3\hat{i} + 2\hat{j} - 2\hat{k}$ is $\frac{x-1}{3} = \frac{y-2}{2} = \frac{y-3}{-2}$. B. Equation of line passing through (1, 2, 3) and parallel to the line given by $\frac{x+3}{3} = \frac{4-y}{5} = \frac{z+8}{6}$ is $\frac{x-1}{3} = \frac{y-2}{5} = \frac{z+3}{6}$. C. Equation of line passing through the origin and (5, -2, 3) is $\frac{x}{5} = \frac{y}{-2} = \frac{z}{3}$. D. Equation of plane passing through the point (1, 2, 3) and perpendicular to the line with direction ratio's 2, 3, -1 is $2(x-1)+3(y-2)-1(z-3) = 0$. E. Equation of plane with intercepts 2, 3 and 4 on x, y and z-axis respectively is $2x + 3y + 4z = 1$. Choose the correct answer from the options given below:

If a line makes angles 90 degree, 60 degree and $\theta$ with $x, y$ and $z$ axis respectively, where $\theta$ is acute, then value of $\theta$ is: