CUET UG 2023 — Geometry

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:

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

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 :

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:

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 :

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 :

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 :

If $\sin^{-1} x + \sin^{-1} y = \frac{2\pi}{3}$, then the value of $\cos^{-1} x + \cos^{-1} y$ is :

Let $a \leq \tan^{-1} x + \cot^{-1} x + \sin^{-1} x \leq b$. If $\alpha$ and $\beta$ denote the minimum and maximum possible values of a and b respectively, then :

Let $\tan^{-1}y = \tan^{-1}x + \tan^{-1}\left(\frac{2x}{1-x^2}\right)$. Then $y$ is :

Match List - I with List - II. | List - I | List - II | |----------|-----------| | (A) Range of $y = \text{cosec}^{-1}x$ | (I) $R - (-1, 1)$ | | (B) Domain of $\sec^{-1}x$ | (II) $(0, \pi)$ | | (C) Domain of $\sin^{-1}x$ | (III) $[-1, 1]$ | | (D) Range of $y = \cot^{-1}x$ | (IV) $\left[\frac{-\pi}{2}, \frac{\pi}{2}\right] - \{0\}$ | Choose the correct answer from the options given below :

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:

Match List I with List II | LIST I | LIST II | |---|---| | A. $\sin^{-1} x + \cos^{-1} x, x \in [-1,1]$ | I. $-\frac{\pi}{2}$ | | B. $\tan^{-1} \sqrt{3} - \cot^{-1}(-\sqrt{3})$ | II. $-\frac{\pi}{6}$ | | C. $\cos^{-1}\left(\cos\frac{13\pi}{6}\right)$ | III. $\frac{\pi}{2}$ | | D. $\sin^{-1}\left(-\frac{1}{2}\right)$ | IV. $\frac{\pi}{6}$ | Choose the correct answer from the options given below:

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:

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

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

The maximum value of $\sin x + \cos x, x \in R$ is:

The maximum value of $(\sin x)(\cos x)$ is :

The principal value of $\cot^{-1}\left(-\frac{1}{\sqrt{3}}\right)$

The principal value of $\cot^{-1}\left(\frac{-1}{\sqrt{3}}\right)$ is :

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 :

The simplest form of $\tan^{-1} \frac{\sqrt{1+x^2} - 1}{x}$, $x \neq 0$ is:

The simplest form of $\tan^{-1}\left\{\frac{x}{\sqrt{a^2 - x^2}}\right\}$ is, where $-a < x < a$.

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:

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

Value of $\frac{e^{\sin(\tan^{-1} x + \cot^{-1} x)}}{e^{\sin(\sin^{-1} x + \cos^{-1} x)}}, x \in [-1, 1]$, is: