Geometry PYQ — Page 6

A line makes angle $\theta$ with $x$-axes as well as $z$-axis. If the angle $\beta$, which it makes with $y$-axis is such that $\sin^2\beta = 3\sin^2\theta$, then $\cos^2\theta$ is :

The shortest distances of the point $(1, 2, 3)$ from $x$, $y$, $z$ axes respectively are

The Principal value of $\cos^{-1} \left( -\frac{1}{2} \right)$ is:

The distance of the point $(2, 3, -5)$ from the plane $x + 2y - 2z = 9$ is :

Equation of y-axis in space, in vector form is :

The equation of plane that contains line $\frac{x - 1}{-1} = \frac{y + 1}{2} = \frac{z - 1}{3}$ and also pass through point $(0, 1, 0)$ is :

Match List I with List II | List I (Functions) | List II (Principal value branches) | |---|---| | A. $f(x) = \cos^{-1} x$ | I. $[0, \pi]$ | | B. $f(x) = \tan^{-1} x$ | II. $[0, \pi] - \left\{\frac{\pi}{2}\right\}$ | | C. $f(x) = \text{cosec}^{-1} x$ | III. $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] - \{0\}$ | | D. $f(x) = \sec^{-1} x$ | IV. $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ | Choose the correct answer from the options given below:

The correct order of steps from A to E, for finding value of p, so that the lines $\frac{1-x}{3} = \frac{7y-14}{2p} = \frac{z-3}{2}$ and $\frac{7-7x}{3p} = \frac{y-5}{1} = \frac{6-z}{5}$ are at right angle is: A. $p = \frac{70}{11}$ B. $(-3) \times \left(\frac{-3p}{7}\right) + 1 \times \left(\frac{2p}{7}\right) + 2 \times (-5) = 0$ C. $\frac{x-1}{-3} = \frac{y-2}{2\frac{p}{7}} = \frac{z-3}{2}$, $\frac{x-1}{-\frac{3p}{7}} = \frac{y-5}{1} = \frac{z-6}{-5}$ D. $\frac{9p}{7} + \frac{2p}{7} - 10 = 0$ E. $\frac{11p}{7} = 10$ Choose the correct answer from the options given below:

The reflection of the point $(\alpha, \beta, \gamma)$ in the xz plane is

The equation of plane passing through the point of (3, 2, 0) and containing the line $\frac{x-2}{2} = \frac{y+3}{4} = \frac{z-1}{1}$ is

The line $\frac{x+2}{3} = \frac{y+3}{5} = \frac{z-6}{4}$ passes through (a, 2, c). The value of a and c are:

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

The angle between the planes $2x + y + 3z - 2 = 0$ and $x - 2y + 5 = 0$ is :

The points of trisection of the segment joining the points (1, 0, 2) and (1, 3, 2) are: A. $(1, \frac{3}{2}, \frac{4}{3})$ B. (1, 1, 2) C. (1, 2, 2) D. $(1, \frac{3}{2}, 2)$ Choose the correct answer from the options given below:

The angle between the pairs of lines $\vec{r} = (3\hat{i} + 2\hat{j} - 4\hat{k}) + \lambda(\hat{i} + 2\hat{j} + 2\hat{k})$ $\vec{r} = (5\hat{i} - 2\hat{j}) + \mu(3\hat{i} + 2\hat{j} + 6\hat{k})$ is :

Find the root of perpendicular from origin to the line $\frac{x - 1}{3} = \frac{y - 2}{-2} = \frac{z + 1}{3}$.

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

The distance of the point (1, 2, 0) from the line $\frac{x-3}{2} = \frac{y+4}{3} = \frac{z+6}{5}$ measured parallel to the plane $x + y + z = 3$ is

If two lines $\frac{x-3}{2} = \frac{y-4}{5} = \frac{z}{4}$ and $\frac{x-4}{3} = \frac{y-5}{6} = \frac{1-z}{k}$, are coplanar, then $k$ is equal to

The foot of perpendicular from point $(2, 4, -1)$ on the line $\frac{x+5}{1} = \frac{y+3}{4} = \frac{z-6}{-9}$ is :

The value of $\sin\left[2\cot^{-1}\left(\frac{-5}{12}\right)\right]$ is :

$\sin^{-1}(\cos x) = \frac{\pi}{2} - x$ is valid for :

The value of $\frac{1}{AB^2} + \frac{1}{BC^2}$ is:

If the vertices of a triangle ABC are $A(1, 2, 1)$, $B(4, 2, 3)$ and $C(2, 3, 1)$, then the equation of the median passing through the vertex $A$, is