Determinant Form of a Line

Q. If the direction ratios of two lines are given by $3lm-4ln+mn=0$ and $l+2m+3n=0,$ then the angle between the lines is

1. $\frac{\pi }{2}$
2. $\frac{\pi }{3}$
3. $\frac{\pi }{4}$
4. $\frac{\pi }{6}$

Q. The distance between two parallel lines is unity. A point P lies between the lines at a distance a from one of them. The length of a side of an equilateral $\Delta$PQR, vertex Q of which lies on one of the parallel lines and vertex R lies on the other line, is

1. 
2. 
3. 
4. 

Q. A line $AB$ in three dimensional space makes angles ${45}^{\circ }$ and ${120}^{\circ }$ with the positive $x$ and $y-$axis respectively. If $AB$ makes an acute angle $\theta$ with the positive $z-$axis. Then $\theta$ equals:

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

Q. In $△ABC,\angle B=90°$ and perpendicular from $B$ on $AC$ intersects it at $D.$ If $AC=4BD,$ then the smallest angle of $△ABC$ is

1. $\frac{\pi }{8}$
2. $\frac{\pi }{6}$
3. $\frac{\pi }{5}$
4. $\frac{\pi }{12}$

Q. The angle between the pair of lines given by
${\overrightarrow{r}}_1=\hat{i}+6\hat{j}+4\hat{k}+\lambda (2\hat{i}-\hat{j}+3\hat{k})$ ${\overrightarrow{r}}_2=4\hat{i}+2\hat{j}-\hat{k}+\lambda (\hat{i}+2\hat{j}-\hat{k})$ is

$\left(a\right){\sin }^{-1}\left(\frac{3}{2\sqrt{21}}\right)\left(b\right){\cos }^{-1}\left(\frac{3}{2\sqrt{21}}\right)$
$\left(c\right)-{\sin }^{-1}\left(\frac{3}{2\sqrt{21}}\right)\left(d\right)-{\cos }^{-1}\left(\frac{3}{2\sqrt{21}}\right)$

Q. Acute angle bwtween the lines $ax+by+c=0$ and $x\cos \theta +y\sin \theta =c(c\ne 0)$ is ${45}^{\circ }.$ If both the lines meet with the line $y\cos \theta =x\sin \theta$ at same point, then the value of $a^2+b^2$ is

1. $2$
2. $1$
3. $4$
4. $3$

Q. If $\theta$ is the acute angle between the two lines whose direction cosines are given by $3l+m+5n=0$ and $6mn-2ln+5lm=0,$ then the value of $\sec \theta$ is equal to

Q. Acute angle bwtween the lines $ax+by+c=0$ and $x\cos \theta +y\sin \theta =c(c\ne 0)$ is ${45}^{\circ }.$ If both the lines meet with the line $y\cos \theta =x\sin \theta$ at same point, then the value of $a^2+b^2$ is

1. $2$
2. $1$
3. $4$
4. $3$

Q. In a circle with centre $O$, suppose $A,P,B$ are three points on its circumference such that $P$ is the mid-point of minor arc $AB$. Suppose when $\angle AOB=\theta$,
$\frac{\text{area}\left(△AOB\right)}{\text{area}\left(△APB\right)}=\sqrt{5}+2,$
If $\angle AOB$ is doubled to $2\theta$, then the ratio is $\frac{\text{area}\left(△AOB\right)}{\text{area}\left(△APB\right)}$ is

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

Q. Acute angle bwtween the lines $ax+by+c=0$ and $x\cos \theta +y\sin \theta =c(c\ne 0)$ is ${45}^{\circ }.$ If both the lines meet with the line $y\cos \theta =x\sin \theta$ at same point, then the value of $a^2+b^2$ is

1. $2$
2. $1$
3. $4$
4. $3$

Q. If $(\alpha ,\beta )$ is a point of intersection of the lines $x\cos \theta +y\sin \theta =3\text{and}x\sin \theta -y\cos \theta =4$ where $\theta$ is parameter, then maximum value of $2^{\frac{\alpha +\beta }{\sqrt{2}}}$ is

1. $16$
2. $64$
3. $128$
4. $256$
5. $32$

Q. A line makes the same angle $\theta$ with each of the $x$ and $z-$axes. If the angle $\beta ,$ which it makes with the $y-$axis, is such that ${\sin }^2\beta =3{\sin }^2\theta ,$ then ${\cos }^2\theta =$

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