Angle between a Plane and a Line

Q.

If the acute angle between the two regression lines is $\theta$, then

1. ${\theta }_1={\theta }_2$

2. ${\theta }_1=2{\theta }_2$

3. $2{\theta }_1={\theta }_2$

4. none of these

Q.

The point of intersection $\frac{x-5}{3}=\frac{y-7}{-1}=\frac{z+2}{1}$ and $\frac{x+3}{\left(-36\right)}=\frac{y-3}{2}=\frac{z-6}{4}$ is

1. $(5,7,-2)$

2. $(-3,3,6)$

3. $(2,10,4)$

4. $(21,\frac{5}{3},\frac{10}{3})$

Q.

If a curve $y=f\left(x\right)$, passing through the point $(1,2)$ is the solution of the differential equation, $2x^{2}dy=(2xy+y^2)dx$, then $f\left(\frac{1}{2}\right)$ is equal to

1. $\frac{-1}{(1+{\log }_{e}2)}$

2. $1+{\log }_{e}2$

3. $\frac{1}{(1+{\log }_{e}2)}$

4. $\frac{1}{(1-{\log }_{e}2)}$

Q.

Find the equation of a straight line whose inclination is $60°$ and $y-intercept$ is $-3$.

Q. The image of the foot of the perpendicular drawn from the origin to the line joining the points $(-9,4,5)$ and $(10,0,-1)$ w.r.t. $XY-$ plane will be :

1. $(\frac{8}{9},\frac{2}{9},\frac{10}{9})$
2. $(\frac{58}{59},\frac{112}{59},\frac{-109}{59})$
3. $(\frac{58}{59},\frac{112}{59},\frac{109}{59})$
4. $(\frac{-8}{9},\frac{2}{9},\frac{-10}{9})$

Q. If an angle between the line, $\frac{x+1}{2}=\frac{y-2}{1}=\frac{z-3}{-2}$ and the plane, $x-2y-kz=3$ is ${\cos }^{-1}\left(\frac{2\sqrt{2}}{3}\right)$, then a value of $k$ is :

1. $\sqrt{\frac{5}{3}}$
2. $\sqrt{\frac{3}{5}}$
3. $-\frac{3}{5}$
4. $-\frac{5}{3}$

Q. On which of the following lines lies the point of intersection of the line, $\frac{x-4}{2}=\frac{y-5}{2}=\frac{z-3}{1}$ and the plane, $x+y+z=2$?

1. $\frac{x-1}{1}=\frac{y-3}{2}=\frac{z+4}{-5}$
2. $\frac{x-4}{1}=\frac{y-5}{1}=\frac{z-5}{-1}$
3. $\frac{x-2}{2}=\frac{y-3}{2}=\frac{z+3}{3}$
4. $\frac{x+3}{3}=\frac{4-y}{3}=\frac{z+1}{-2}$

Q. Angle between line
$\frac{x-5}{1}=\frac{y-2}{2}=\frac{z-8}{2}$ and plane $2x+y+2z+5=0$ is

1. ${\cos }^{-1}\left(\frac{8}{9}\right)$
2. ${\sin }^{-1}\left(\frac{8}{9}\right)$
3. ${\cos }^{-1}\left(\frac{1}{9}\right)$
4. ${\sin }^{-1}\left(\frac{1}{9}\right)$

Q. If the line
$\frac{x-x_1}{a_1}=\frac{y-y_1}{b_1}=\frac{z-z_1}{c_1}$
makes an angle $\theta$ with the plane $a_{2}x+b_{2}y+c_{2}z=d$, then which of the following is correct -

1. $\cos \left(\theta \right)=\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{(a_1^2+b_1^2+c_1^2)(a_2^2+b_2^2+c_2^2)}}$
2. $\sin \left(\theta \right)=\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{(a_1^2+b_1^2+c_1^2)(a_2^2+b_2^2+c_2^2)}}$
3. $\tan \left(\theta \right)=\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{(a_1^2+b_1^2+c_1^2)(a_2^2+b_2^2+c_2^2)}}$
4. $\cot \left(\theta \right)=\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{(a_1^2+b_1^2+c_1^2)(a_2^2+b_2^2+c_2^2)}}$

Q. The plane which bisects the line segment joining the points $(-3,-3,4)$ and $(3,7,6)$ at right angles, passes through which one of the following points?

1. $(4,-1,7)$
2. $(4,1,-2)$
3. $(2,1,3)$
4. $(-2,3,5)$

Q. Angle between line
$\frac{x-5}{1}=\frac{y-2}{2}=\frac{z-8}{2}$ and plane $2x+y+2z+5=0$ is

1. ${\cos }^{-1}\left(\frac{8}{9}\right)$
2. ${\sin }^{-1}\left(\frac{8}{9}\right)$
3. ${\cos }^{-1}\left(\frac{1}{9}\right)$
4. ${\sin }^{-1}\left(\frac{1}{9}\right)$