Distinguish Acute Angle Bisectors and Obtuse Angle Bisectors

Q. The angle between the lines $2x=3y=-z\text{and}6x=-y=-4z$ is

1. $0^{\circ }$
2. ${90}^{\circ }$
3. ${45}^{\circ }$
4. ${30}^{\circ }$

Q. Let the acute angle bisector of the two planes $x-2y-2z+1=0$ and $2x-3y-6z+1=0$ be the plane $P.$ Then which of the following points lies on $P?$

1. $(-2,0,-\frac{1}{2})$
2. $(0,2,-4)$
3. $(4,0,-2)$
4. $(3,1,-\frac{1}{2})$

Q. The equation of the obtuse angle bisector of lines $12x-5y+7=0$ and $3y-4x-1=0$ is

1. $4x+7y+11=0$
2. $4x+7y-11=0$
3. $7x+4y+11=0$
4. $7x+4y-11=0$

Q. Let the sides of $△ABC$ be $3x+4y=0,4x+3y=0$ and $x=3$. If $(h,k)$ be the centre of the circle inscribed in $△ABC$, then the value of $(h+k)$ is

Q. if 3x^2+xy-y^2-3x+6y+k=0 represents a pair of lines then k= 0, 1, 9, -9

Q. If the two lines $l_1:\frac{x-2}{3}=\frac{y+1}{-2},z=2$ and $l_2:\frac{x-1}{1}=\frac{2y+3}{\alpha }=\frac{z+5}{2}$ are perpendicular, then an angle between the lines $l_2$ and $l_3:\frac{1-x}{3}=\frac{2y-1}{-4}=\frac{z}{4}$ is :

Q. If $m$ is the slope of obtuse angle bisector between the lines $3x-4y+7=0$ and $12x+5y-2=0,$ then the value of $|55m|$ is equal to

Q. If the dc's of two lines parallel lines are given by $2l+3m+kn=0$ and $l^2-m^2+5n^2=0$ then the values of k are:

1. $5$
2. $3$
3. $-5$
4. $-3$

Q. Centre of the ellipse $4(x-2y+1{)}^2+9(2x+y+2{)}^2=5$ is

1. $(-2,2)$
2. $(1,5)$
3. $(-5,2)$
4. $(-1,0)$

Q. The equation of the bisector of the lines 4x+3y-6=0 and 5x+12y+9=0 containing the origin is given by .

1. 7x-9y-3=0
2. 7x+9y+3=0
3. 7x+9y-3=0
4. 7x-9y+3=0

Q. Solve for a and b, $\frac{\sqrt{2}+\sqrt{3}}{3\sqrt{2}-2\sqrt{3}}=a-b\sqrt{6}$. Find $6(a+b)$

Q. Solve$\sqrt{27+\sqrt{70+\sqrt{121}}}=?$

1. $6$
2. $7$
3. $8$
4. $9$

Q. The value of $k$ for which the lines $\frac{x+1}{-3}=\frac{y+2}{2k}=\frac{z-3}{2}\text{and}\frac{x-1}{3k}=\frac{y+5}{1}=\frac{z+6}{7}$ may be perpendicular is:

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

Q. The obtuse angle between the lines y = -2 and y = x +2 is __ (degree)

Q. Let $(x_0,y_0)$ be thhe solution of the following equations $(2x{)}^{ln2}=(3y{)}^{ln3}$ and $3^{ln3}$ and $3^{lnx}=2^{lny}$. Then $xy=$

1. $1/2$
2. $0$
3. $1/3$
4. $1/6$

Q. Show that the lines $\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z+2}{1}\mathrm{and}\frac{x}{1}=\frac{y-7}{-3}=\frac{z+7}{2}$ are coplanar. Also, find the equation of the plane containing them.

Q. If $y={\cos }^{-1}\left(\frac{x^n-x^{-n}}{x^n+x^{-n}}\right)$ then $y^{\prime }\left(x\right)$ is equal to

1. $\frac{2n{x}^{n-1}}{x^{2n}+1},$ if $n$ is even
2. $\frac{2n{x}^{n-1}}{x^{2n}+1},$
3. $\frac{2n{x}^n}{\left|x\right|(x^{2n}+1)}$ if $n$ is odd
4. $-\frac{2n{x}^n}{\left|x\right|(x^{2n}+1)}$ if $n$ is odd

Q. Solve: $2\sqrt{2}+5\sqrt{3}+\sqrt{2}-3\sqrt{3}$

Q. Find the tangent of the angle between the lines $2x+3y-8=0$ and $5x-y+7=0$.

Q. Integrate:
$\int \frac{1}{{\cos }^{2}x(1-\tan x{)}^2}dx$

Q. If $x$ is a positive acute angle and $\tan x=p$, $\cos x$ is equal to

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

Q. The obtuse angle between the lines y = -2 and y = x +2 is __ (degree)

Q. Find the equations of tangents to the parabola $y^2=12x$ which make an angle of ${45}^{\circ }$ with the line $y=3x+7.$

1. $x-2y+12=0$
2. $x+2y+12=0$
3. $4x+2y+3=0$
4. $4x-2y-3=0$

Q. I: lf the equation $4x^2+mxy-3y^2=0$ represents a pair of real and distinct lines, then $m\in R.$
II: lf the difference of the slopes of the lines $x^2-12kxy+y^2=0$ is $2,$ then $k$ is $\frac{1}{30}$.

1. only I is true
2. Only II is true
3. both I and II are true
4. neither I nor II are true

Q. 5x-3=15y+7=3-10z equation of line then find direction cosine

Q. Determine the length of $y=log\left(secx\right)$ between $0\le x\le \frac{\pi }{4}$.

1. $log(\sqrt{2}+1)$
2. $log(\sqrt{2}-1)$
3. $log(\sqrt{3}-1)$
4. $log(\sqrt{3}+1)$

Q. Solve for $x:2\sqrt{x+5}=8$

1. $11$
2. $4$
3. $6$
4. $9$