Geometry PYQ — Page 3

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:

Match List-I with List-II | List-I | List-II | |---|---| | (A) $\cos^{-1} x + \cos^{-1}(-x)$ | (I) $\frac{\pi}{3}$ | | (B) $\text{cosec}^{-1}(-x) + \sec^{-1}(-x)$ | (II) $-\frac{\pi}{3}$ | | (C) $\tan^{-1}\sqrt{3} - \sec^{-1}(-2)$ | (III) $\pi$ | | (D) $\tan^{-1}\left(\tan\frac{4\pi}{3}\right)$ | (IV) $\frac{\pi}{2}$ | Choose the correct answer from the options given below:

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 value of $-\cosec^2(\cot^{-1}y) + \sec^2( \tan^{-1}x)$ is equal to

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

Arrange the principal values of the following functions in ascending order (A) $\cosec^{-1}(2)$ (B) $\tan^{-1}(-\sqrt{3})$ (C) $\tan^{-1}(1)$ (D) $\tan^{-1}\left(\cos\frac{3\pi}{7}\right)$ Choose the correct answer from the options given below:

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

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

Match List-I with List-II | List-I | List-II | |---|---| | (A) $\sin^{-1}(-1)$ | (I) $\frac{5\pi}{6}$ | | (B) $\cot^{-1}(-1)$ | (II) $\frac{-\pi}{2}$ | | (C) $\sec^{-1}\left(\frac{-2}{\sqrt{3}}\right)$ | (III) $\frac{\pi}{4}$ | | (D) $\tan^{-1}(1)$ | (IV) $\frac{3\pi}{4}$ | Choose the correct answer from the options given below:

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

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 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:

For $x \in [-1,1]$, if $4\sin^{-1}x + \cos^{-1}x = \pi$ then $x$ is equal to

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

Match List-I with List-II | List-I | List-II | |---|---| | Inverse Trigonometric function | Principal values of arguments | | (A) $sin^{-1}\left(\frac{-1}{2}\right)$ | (I) $\frac{-\pi}{3}$ | | (B) $cos^{-1}\left(\frac{-1}{2}\right)$ | (II) $\frac{3\pi}{4}$ | | (C) $tan^{-1}(-\sqrt{3})$ | (III) $\frac{-\pi}{6}$ | | (D) $sec^{-1}(-\sqrt{2})$ | (IV) $\frac{2\pi}{3}$ | Choose the correct answer from the options given below:

The maximum value of $\sin x \cdot \cos x$ 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:

Match List-I with List-II | List-I | List-II | |---|---| | (Inverse Trigonometric Function) | (Principal Value) | | (A) $\sin^{-1}(-\frac{1}{2})$ | (I) ${\pi}/{6}$ | | (B) $\cos^{-1}(-\frac{1}{2})$ | (II) $-{\pi}/{6}$ | | (C) $\tan^{-1}(-\sqrt{3})$ | (III) ${2\pi}/{3}$ | | (D) $\cot^{-1}(\sqrt{3})$ | (IV) $-{\pi}/{3}$ | Choose the correct answer from the options given below: