Slope of a Line

Q. The slope of the line which is inclined at an angle of ${75}^{\circ }$ with the $x-$axis in anticlockwise direction is

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

Q. If ${\theta }_1$ and ${\theta }_2$ be the angles which the lines $(x^2+y^2)({\cos }^2\theta {\sin }^2\alpha +{\sin }^2\theta )=(x\tan \alpha -y\sin \theta {)}^2$ make with the axis of $x$, then if $\theta =\frac{\pi }{6},\tan {\theta }_1+\tan {\theta }_2$ is equal to

1. $(-\frac{8}{3})si{n}^2\alpha$
2. $(-\frac{8}{3})cosec2\alpha$
3. $-8\sqrt{3}cosec2\alpha$
4. $-4cosec2\alpha$

Q. If the line $y=x$ cuts the curve $y=2x^3+6x^2+x-4$ at three points $A,B$ and $C$. Then the value of $|OA.OB.OC|$ with $O$ being the origin is

1. $\sqrt{2}$
2. $2\sqrt{2}$
3. $4\sqrt{2}$
4. $8\sqrt{2}$

Q.

Two sides of a rhombus are along the lines x-y+1=0 and 7x-y-5=0. If its diagonals intersect at (-1, -2), then which one of the following is a vertex of this rhombus?

1. $(-3,-9)$

2. $(-3,-8)$

3. $(\frac{1}{3},-\frac{8}{3})$

4. $(-\frac{10}{3},-\frac{7}{3})$

Q. If ${\theta }_1$ and ${\theta }_2$ be the angles which the lines $(x^2+y^2)({\cos }^2\theta {\sin }^2\alpha +{\sin }^2\theta )=(x\tan \alpha -y\sin \theta {)}^2$ make with the axis of $x$, then if $\theta =\frac{\pi }{6},\tan {\theta }_1+\tan {\theta }_2$ is equal to

1. $(-\frac{8}{3})si{n}^2\alpha$
2. $(-\frac{8}{3})cosec2\alpha$
3. $-8\sqrt{3}cosec2\alpha$
4. $-4cosec2\alpha$

Q.

Which of the following statement is / are correct for equation 2x - 3y + 5 = 0

1. Slope of the straight line is $\frac{2}{3}$

2. x-intercept of the straight line is $\frac{-5}{2}$

3. y-intercept of the straight line is $\frac{5}{3}$

1. only 1 & 2

2. only 2 & 3

3. only 3 & 1

4. All 1, 2 & 3

Q. If the vertices of a quadrilateral $ABCD$ are $A(-4,2),B(2,6),C(8,5)$ and $D(9,-7)$, then the quadrilateral formed by joining the midpoints of the sides of the quadrilateral $ABCD$ will be a

1. Rhombus
2. Parallelogram
3. Rectangle
4. Square

Q. If the vertices of a quadrilateral $ABCD$ are $A(-4,2),B(2,6),C(8,5)$ and $D(9,-7)$, then the quadrilateral formed by joining the midpoints of the sides of the quadrilateral $ABCD$ will be a

1. Rhombus
2. Parallelogram
3. Rectangle
4. Square