Geometry PYQ — Page 3

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 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 $\frac{x - 5}{5\lambda + 2} = \frac{2 - y}{5} = \frac{1 - z}{-1}$ 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 $\frac{1-x}{3} = \frac{y-2}{2\lambda} = \frac{z-3}{2}$ and $\frac{x-1}{3\lambda} = \frac{y-1}{1} = \frac{6-z}{5}$ are perpendicular, then $\lambda$ is equal to

If the 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 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 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 $\alpha, \beta$ 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 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 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 the coordinate axes, then the value of $\cos 2\alpha + \cos 2\beta + \cos 2\gamma$ is

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 $\alpha$, $\beta$ 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 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 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

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

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

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

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

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

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

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

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:

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: