CUET UG 2025 — Geometry

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:

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 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 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:

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:

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:

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:

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:

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:

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:

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

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

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

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

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

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

For the principal value branch, the value of $\sin\left(\frac{\pi}{2} - \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right)$ 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

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

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

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

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 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

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

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

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 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 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 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

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

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

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

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

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

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:

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

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:

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:

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:

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:

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:

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:

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:

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:

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:

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

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

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

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

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

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

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 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

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

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

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:

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 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 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 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

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

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

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

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

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

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

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

The value of $-\cosec^2(\cot^{-1}y) + \sec^2( \tan^{-1}x)$ 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 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:

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

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 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 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

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: