Angle between Two Lines

Q. Observe the given figure and find the value of unknown angles x and y.

1. x = 72°, y = 22°
2. x = 72°, y = 40°
3. x = 112°, y = 40°
4. x = 112°, y = 22°

Q.

In the below Fig. $OA\text{and}OB$ are opposite rays:
$\left(ii\right)$ If $y={35}^{\circ },$ what is the value of $x?$

Q.

Two equal sides of an isosceles triangle have the equations $7x-y+3=0$and $x+y-3=0$ and its third side passes through the point $\left(1,-10\right)$. Find the equation of third side.

Q. One of the diagonals of a rhombus is equal to a side of the rhombus. The angles of the rhombus are

1. ${60}^o,{80}^o$
2. ${60}^o,{120}^o$
3. ${120}^o,{240}^o$
4. ${100}^o,{120}^o$

Q. Graphically, find the number of solution for the following pair of linear equations in two variables:
6x – 3y + 10 = 0
2x – y + 9 = 0

Q. Let the vectors a, b, c and d be such that $(a\times b)\times (c\times d)=0$. Let $P_1$and $P_2$be planes determined by pair of vectors a, b and c, d respectively. Then the angle between $P_1$and $P_2$ is

1. $0^{\circ }$
2. $\frac{\pi }{4}$
3. $\frac{\pi }{3}$
4. $\frac{\pi }{2}$

Q.

Solve for d: $3d^2+17d-20=0$

Q. Solve -
$35x^2-12ax+(a^2-b^2)=0$

Q. In quadrilateral ABCD, diagonal BD is bisected by the diagonal AC.
then $\Delta ABC$ and $\Delta ADC$ are equal in area.
State whether the above statement is true or false.

1. True
2. False

Q. A straight line $L$ through the point $(3,-2)$ is inclined at an angle ${60}^o$ to the line $\sqrt{3}x+y=1$. lf $L$ also intersects the $x-$axis, then the equation of $L$ is

1. $y+\sqrt{3}x+2-3\sqrt{3}=0$
2. $y-\sqrt{3}x+2+3\sqrt{3}=0$
3. $\sqrt{3}y-x+3+2\sqrt{3}=0$
4. $\sqrt{3}y+x-3+2\sqrt{3}=0$

Q. $A$ triangle has two of its sides along the axes, its third side touches the circle $x^2+y^2-2ax-2ay+a^2=0$, where $a>0$. If the locus of the circumcentre of the triangle passes through the point $(38,-37)$, then $a^2-2a$ is equal to

Q. The angle between the lines whose direction cosines are connected by the relations $l+m+n=0$ and $2lm+2nl-mn=0$ is

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

Q. $A\equiv (1,2)$ and $B\equiv (7,10)$ are two points. If $P(x,y)$ is a point such that $\angle APB={60}^{\circ }$ and area of $\Delta APB$ is maximum, then which of the following is (are) TRUE ?

1. $P$ lies on any line perpendicular to $AB$
2. $P$ lies on the perpendicular bisector of $AB$
3. $P$ lies on the line $3x+4y=36$
4. Radius of circumcircle of $\Delta PAB$ is $10$ units