Angle between Two Planes

Q.

What is the condition for the vectors $2i+3j-4k$ and $3i-aj+bk$to be parallel?

Q. The direction ratios of normal to the plane through the points $(0,-1,0)$ and $(0,0,1)$ and making an angle $\frac{\pi }{4}$ with the plane $y-z+5=0$ are :

1. $\sqrt{2},1,-1$
2. $2\sqrt{3},1,-1$
3. $2,-1,1$
4. $2,\sqrt{2},-\sqrt{2}$

Q.

If $z+z^3=0$ then which of the following must be true on the complex plane?

1. Re(z) < 0

2. z⁴ = 1

3. Re(z) = 0

4. Im(z) = 0

Q.

The angle between the planes 3x - 4y + 5z = 0 and 2x -y - 2z = 5 is [MP PET 1988]

1. 
   

2. 
   

3. 
   

4. None of these

Q. Let the plane passing through the point $(-1,0,-2)$ and perpendicular to each of the planes $2x+y-z=2$ and $x-y-z=3$ be $ax+by+cz+8=0$. Then the value of $a+b+c$ is equal to

1. $3$
2. $5$
3. $8$
4. $4$

Q.

The equation of the plane passing through the origin and containing the line $\frac{x-1}{5}=\frac{y-2}{4}=\frac{z-3}{5}$ is

1. $x+5y-3z=0$

2. $x-5y+3z=0$

3. $3x–10y+5z=0$

4. $x+5y+3z=0$

Q. If the direction cosines of a line $L$ are $(ab,b,b)$ and the angle between $L$ and $X$ axis is $\frac{\pi }{3}$. Then the value of $\frac{1}{a^2}+\frac{1}{b^2}$ is

1. $\frac{25}{6}$
2. $\frac{6}{25}$
3. $\frac{25}{8}$
4. $\frac{35}{6}$

Q.

For the hyperbola , $\frac{x^2}{{\cos }^{2}a}-\frac{y^2}{{\sin }^{2}a}=1$which of the following remains constant when $ɑ$ varies?

1. Eccentricity

2. Directrix

3. Abscissae of vertices

4. Abscissae of foci

Q. Given that vectorP+vectorQ=vectorR and that VectorR is perpendicular to vector P.If magnitude of vector P=magnitude of vector R, then what is the angle between P & Q?

Q. The $\text{cosine}$ of the angle between any two diagonals of a cube is

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

Q.

Equation of the plane perpendicular to the line $\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$ and passing through the point $(2,3,4)$ is

1. $2x+3y+z=17$

2. $x+2y+3z=9$

3. $3x+2y+z=16$

4. $x+2y+3z=20$

Q. Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two non-parallel unit vectors in a plane. If the vectors $(\alpha \overrightarrow{a}+\overrightarrow{b})$ bisects the internal angle between $\overrightarrow{a}$ and $\overrightarrow{b}$, then $\alpha$ is

1. $\frac{\pi }{2}$
2. $1$
3. $2$
4. $4$

Q. The equation of the plane through the point (0, –4, –6) and (–2, 9, 3) and perpendicular to the plane
x – 4y – 2z = 8 is

1. 3x + 3y – 2z = 0
2. x – 2y + z = 2
3. 2x + y – z = 2
4. 5x – 3y + 2z = 0

Q. The equation of the plane through the point (2, 5, –3) perpendicular to the planes x + 2y + 2z = 1 and x – 2y + 3z = 4 is

1. 3x – 4y + 2z – 20 = 0
2. 7x – y + 5z = 0
3. x – 2y + = 11
4. 10x – y – 4z = 27

Q. Find the planes bisecting the angle between the planes x + 2y + 2z = 9
and 4x – 3y + 12z + 13 = 0.

1. 25x+17y+62z-78=0
2. 25x+17y+62z+78=0
3. x+35y-10z+156=0
4. x+35y-10z-156=0

Q. Area of the region bounded by rays $|x|+y=1$ and $X-$axis is

1. $\frac{1}{4}$
2. $2$
3. $\frac{1}{2}$
4. $1$

Q.

The angle between the lines $x\cos 30°+y\sin 30°=3$and $x\cos 60°+y\sin 60°=5$is

1. $90°$

2. $30°$

3. $60°$

4. None of these

Q.

Find the position vector of a point A in space such that $\overrightarrow{OA}$ is inclined at ${60}^{\circ }$ to OX and at ${45}^{\circ }$ to OY and $\overrightarrow{|OA|}$= 10 units.

Q. If $P$ be the point $(2,6,3),$ then the equation of the plane through $P$, at right angles to $OP$, where $O$ is the origin is:

1. $2x+6y+3z=7$
2. $2x-6y+3z=7$
3. $2x+6y-3z=49$
4. $2x+6y+3z=49$

Q. Let $P_1$ denote the equation of a plane to which the vector $(\hat{i}+\hat{j})$ is normal and which contains the line whose equation is $\overrightarrow{r}=\hat{i}+\hat{j}+\hat{k}+\lambda (\hat{i}-\hat{j}-\hat{k})$ and $P_2$ denote the equation of the plane containing the line $L$ and a point with position vector $\hat{j}$. Which of the following holds good?

1. The equation of $P_1$ is $x+y=2$
2. The equation of $P_2$ is $\overrightarrow{r}.(\hat{i}-2\hat{j}+\hat{k})=2$
3. The acute angle between $P_1$ and $P_2$ is ${\cot }^{-1}\left(\sqrt{3}\right)$
4. The angle between the plane $P_2$ and the line $L$ is ${\tan }^{-1}\sqrt{3}$

Q. If the distance between the plane $ax-4y+2z=k$ and the plane containing the lines $\frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5}\text{and}\frac{x-3}{4}=\frac{y-4}{5}=\frac{z-5}{6}\text{is,}2\sqrt{6}\text{then}\frac{|k|}{3}$ is equal to

Q. The angle between the planes $2x-y+3z=6$ and $x+y+2z=7$ is:

1. ${\cos }^{-1}\left(\sqrt{\frac{7}{3}}\right)$
2. ${\cos }^{-1}\left(\frac{1}{2}\sqrt{\frac{7}{3}}\right)$
3. ${45}^{\circ }$
4. ${60}^{\circ }$

Q. The equation of the plane, which is equidistant from lines $\frac{x-1}{1}=\frac{y-2}{2}=\frac{z-2}{3}$ and $\frac{x-2}{2}=\frac{y-2}{1}=\frac{z-1}{2}$ is

1. $x+19y-5z=11$
2. $x+4y-3z=5$
3. $x+7y-5z=9$
4. $3x+2y-3z=10$

Q. Angle between the planes $2x+2y-3z-5=0$ and $3x-3y+5z-3=0$ is ${\cos }^{-1}\frac{k}{\sqrt{17}\sqrt{43}}$ then $k$ is

Q.

$OA$and $OB$ are two roads enclosing an angle of ${120}^{°}$ $X$and $Y$ start from $O$ at the same time. $X$ travels along $OA$ with a speed $4\text{km/hour}$ and $Y$ travels along $OB$ with a speed $3\text{km/hour}$. The rate at which the shortest distance between $X$and $Y$ is increasing after $1\text{hour}$ is

1. $\sqrt{37}\text{km/h}$

2. $37\text{km/h}$

3. $13\text{km/h}$

4. $\sqrt{13}\text{km/h}$

Q. Find the slope of the line whose inclination is:​
(iii) ${72}^{\circ }{30}^{\prime }$

Q.

The angle at which the curve $y=a{e}^{ax}$ intersects the $y-\text{axis}$ is

1. ${\tan }^{-1}\left(a^2\right)$

2. ${\sin }^{-1}\left(\frac{1}{\sqrt{1+a^4}}\right)$

3. ${\text{cot}}^{-1}\left(a^2\right)$

4. ${\sin }^{-1}\left(\sqrt{1+a^4}\right)$

Q. the real value of n for which substitution y=u^n will tranform differential equation 2x^4ydy/dx+y^4=4x^6, into a homogeneous equatio

Q. The equation of the plane passing through the points (1, -3, -2) and perpendicular to planes x +2y+2z=5 and 3x+3y+2z=8, is

1. 2x -4y +3z-8 =0
2. 2x-4y-3z+8 =0
3. 2x+4y+3z+8 =0
4. x-3y+z-5=0

Q. 13. In the following cases, determine whether the given planes are parallel orperpendicular, and in case they are neither, find the angles between them.(a) 7x + 5y + 6z +30 0 and 3x - y -10z 4-0(b) 2r + y 3z-2-0 and x - 2y +5 0(c) 2r- 2y +4z +5-0 and 3x - 3y 6z-1 0(d) 2r - y 3z-10 and 2x -y +3z+3-0(e) 4x + 8y +z- 8-0 and y +z- 4 0