Slope of a Line

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 line
$(3x+14y+7)+k(5x+7y+6)=0$
is parallel to the y-axis, then the value of $k$ is

1. 1/3

2. −3/5

3. −2

4. 2

Q. The line $(3x-y+5)+\lambda (2x-3y-4)=0$ will be parallel to y-axis, if $\lambda$ =

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

Q. A square with each side equal to $a$ lies above the $x$-axis and has one vertex at the origin. One of the sides passing through the origin makes an angle $\alpha (0<\alpha <\frac{\pi }{4})$ with the positive direction of the $x$-axis. Equation of a diagonal of the square is

1. $y(\cos \alpha -\sin \alpha )=x(\sin \alpha +\cos \alpha )$

2. $y(\sin \alpha +\cos \alpha )+x(\cos \alpha -\sin \alpha )=a$

3. $x(\cos \alpha -\sin \alpha )=y(\cos \alpha +\sin \alpha )$

4. $x(\cos \alpha -\sin \alpha )-y(\cos \alpha +\sin \alpha )=a$

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. The slope of the line formed by joining the points A(1, 3) and B(4, 7) is .

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

Q. The real value of $\lambda$ for which the image of the point $(\lambda ,\lambda -1)$ with respect to the line $3x+y=6\lambda$ is the point $({\lambda }^2+1,\lambda )$ is

1. $-1$
2. $1$
3. $0$
4. $2$

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. A rectangle $ABCD$ has its side $AB$ parallel to line $y=x$ and vertices $A,B$ and $D$ lie on $y=1,x=2$ and $x=-2$ respectively. Locus of vertex ${}^{\prime }{C}^{\prime }$ is

1. $y=5$
2. $x-y=5$
3. $x=5$
4. $x+y=5$