Geometry PYQ — Page 2

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)

The value of $\cos(2\cos^{-1}x + \sin^{-1}x)$ at $x = \frac{1}{5}$ is

Consider $f(x) = \sin(3x) + 4, \forall x \in \mathbb{R}$. Then (A) Maximum value of $f(x)$ is 5 (B) Minimum value of $f(x)$ is 3 (C) Maximum value of $f(x)$ is attained at $x = \frac{\pi}{6}$ (D) Minimum value of $f(x)$ is attained at $x = 0$ Choose the correct answer from the options given below:

$\cos^{-1}\left(\cos\frac{7\pi}{6}\right)$ equals:

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

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

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:

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

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 value of $\tan^{-1}(1) + \cos^{-1}\left(-\frac{\sqrt{3}}{2}\right) + \sin^{-1}\left(\frac{1}{2}\right)$ is

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 simplified form of $\tan^{-1}\left(\frac{\cos x}{1+\sin x}\right)$, $-\frac{\pi}{2} < x < \frac{\pi}{2}$ is

The value of $\tan^{-1}(2) + \tan^{-1}(3)$ is equal to