Geometry PYQ

$\sin^{-1}(\cos\frac{3\pi}{5})$ equals

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

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

The minimum value of the function $f(x) = 3\sin x - 4\cos x, x \in [-4\pi, 4\pi]$ is equal to

The value of $\cot\left(\cos^{-1}\frac{7}{25}\right)$ 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 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:

for $|x| < 1$, $sin (tan^{-1}x)$ equal to

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

Match List-I with List-II | List-I | List-II | |---|---| | (A) $\tan^{-1}\frac{2}{11} + \tan^{-1}\frac{7}{24}$ | (I) $\frac{3\pi}{4}$ | | (B) $\tan^{-1}2 + \tan^{-1}3$ | (II) $\pi$ | | (C) $\tan^{-1}1 + \tan^{-1}2 + \tan^{-1}3$ | (III) $\tan^{-1}\frac{1}{2}$ | | (D) $\tan^{-1}\frac{1}{7} + \tan^{-1}\frac{1}{13}$ | (IV) $\tan^{-1}\frac{2}{9}$ | Choose the correct answer from the options given below:

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: