Geometry PYQ — Page 7

The foot of perpendicular from the point P (1, 2, -3) to the line $\frac{x+1}{2} = \frac{y-3}{-2} = \frac{z}{-1}$ is

The two lines given by $\vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \mu(\hat{i} - \hat{j} + \hat{k})$ and $\vec{r} = (2\hat{i} - \hat{j} - \hat{k}) + \mu(-\hat{i} + \hat{j} - \hat{k})$ A. are perpendicular B. are parallel. C. have shortest distance 0. D. have shortest distance $\sqrt{26}$. E. have shortest distance $\sqrt{78}$. Which of the above statements are true? Choose the correct answer from the options given below:

The value of $\text{cosec}^{-1}(-2) - 2\sec^{-1}(-2)$ is equal to:

The value of $\alpha$ is

$\tan(\alpha + \beta) =$

$\angle ABC$ is equal to:

Match List - I with List - II | List - I | List - II | |---|---| | A. $\tan^{-1}\left(\tan\frac{7\pi}{6}\right)$ | I. $\frac{5\pi}{6}$ | | B. $\tan^{-1}\left(\tan\frac{8\pi}{6}\right)$ | II. $\frac{\pi}{2}$ | | C. $\tan^{-1}\frac{1}{\sqrt{3}} + cosec^{-1}\frac{2}{\sqrt{3}}$ | III. $\frac{\pi}{6}$ | | D. $\cos^{-1}\left(\cos\frac{5\pi}{6}\right)$ | IV. $\frac{\pi}{3}$ | Choose the correct answer from the options given below:

The angle between 2 planes $4x + 8y + z - 8 = 0$ and $y + z - 4 = 0$ is :

The equation of line of intersection of the planes $x + y + 3z = 7$ and $x - y + 2z = 3$ is:

The distance of plane $\vec{r}\cdot(6\hat{i} - 3\hat{j} - 2\hat{k}) + 1 = 0$ from origin is :

The sum of all the values of $\lambda$ for which the distance of the point P (2, 3, $\lambda$) from the plane $x + 2y - 2z = 9$ is 3 units, is

The value of $\sin\left[\frac{\pi}{2} - \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right]$ is :

Area of $\triangle ABD$ is

$\sin^{-1}(1 - x) - 2\sin^{-1}x = \frac{\pi}{2}$, than $x$ is equal to (a) $0$ (b) $1$ (c) $\frac{1}{2}$ (d) $2$ Choose the most appropriate answer from the options given below: