Direction Cosines

Q.

Find the angle between the lines vector $\overrightarrow{r}=\left(2i-3j+k\right)+l\left(i+j+3k\right)$ and vector $=\left(i-j-k\right)+\mu \left(2i-3j+k\right)$

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

2. ${\cos }^{-1}\left(\frac{9}{\sqrt{91}}\right)$

3. ${\cos }^{-1}\left(\frac{2}{\sqrt{154}}\right)$

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

Q. 41. The equation of the curve in which the portion of the tangent included between the coordinate axes is bisected at the point of contact is

Q.

The angle between the lines represented by the equation $a{x}^2+2hxy+b{y}^2=0$ is given by

1. 

2. 

3. 

4. 

Q. The lenght of the minor axis (along y-axis) of an ellipse in the standard form is 4/3^(1/2). If this ellipse touches the line x + 6y = 8, then its eccentricity is :

Q. The angle of intersection of the parabolas y^2=4ax and x^2=4ay at the origin

Q. Direction cosines of Z - axis are -

1. 0, 0, 1
2. 0, 1, 0
3. 1, 0, 0
4. 0, 1, 1

Q. Let $PM$ be the perpendicular from the point $P(1,2,3)$ to the $xy$ plane. If $\overrightarrow{OP}$ makes an angle $\theta$ with the positive direction of the $z$-axis and $\overrightarrow{OM}$ makes an angle $\varphi$ with the positive direction of $x$-axis, where $O$ is the origin and $\theta$ and $\varphi$ are acute angles, then

1. $\cos \theta \cos \varphi =\frac{1}{\sqrt{14}}$
   
2. $\sin \theta \sin \varphi =\frac{2}{\sqrt{14}}$
   
3. $\tan \varphi =2$
   
4. $\tan \theta =\frac{\sqrt{5}}{3}$

Q. A straight line, makes an angle of ${60}^{\circ }$ with each of y and z-axis, then the inclination of the line with x-axis is

1. ${60}^{\circ }$
2. ${30}^{\circ }$
3. ${45}^{\circ }$
4. ${75}^{\circ }$

Q. find the equation of hyperbola if its foci are(6, 4), (-4, 4) and eccentricity is 2

Q. If set A and set B are two subsets of universal set of set U and n(A) = 40 , n(B) = 30 , n(A intersection B) = 10 then what is the value of n(A intersection B) ?

Q. The projection of any line on co-ordinate axes be respectively 3, 4, 5 then its length is

1. 12
2. 50
3. $5\sqrt{2}$
4. 25