Slope of a Line Connecting Two Points

Q.

Find the value of $k$ for which the equation $9x^2+3kx+4=0$ have equal roots.

Q. Find the equation of the line, which makes intercepts $–3\text{and}2$ on the $x$ and $y$ axes respectively.

Q.

If the curve $x^2+2y^2=2$ intersects the line $x+y=1$ at two points $P$ and $Q$, then the angle subtended by the line segment $PQ$ at the origin is :

1. $\frac{\pi }{2}+{\tan }^{-1}\left(\frac{1}{4}\right)$

2. $\frac{\pi }{2}-{\tan }^{-1}\left(\frac{1}{4}\right)$

3. $\frac{\pi }{2}+{\tan }^{-1}\left(\frac{1}{3}\right)$

4. $\frac{\pi }{2}-{\tan }^{-1}\left(\frac{1}{3}\right)$

Q.

If one root of the quadratic equation $3x^2-kx-2=0$ is $2$, find the value of ‘$k$’.

Q.

Write the discriminant of the following:

$3\sqrt{3}{x}^2+10x+\sqrt{3}=0$

Q. If the real part of $\frac{\overline{z}+2}{\overline{z}-1}$ is $4$, then show that the locus of the point representing z in the complex plane is a circle.

Q.

The condition that the two lines represented by the equation $a{x}^2+2hxy+b{y}^2=0$ to be perpendicular are

1. $ab=-1$

2. $a=-b$

3. $a=b$

4. $ab=1$

Q.

If one root is $3+\sqrt{5}$, then the quadratic equation will be:

1. $x^2+6x-4=0$

2. $x^2+6x+4=0$

3. $x^2-6x+4=0$

4. $x^2-6x-4=0$

Q.

Mark the correct alternative in each of the following :

The acute angle between the medians drawn from the acute angles of a right angled isosceles triangle is

1. $co{s}^{-1}\left(\frac{2}{3}\right)$

2. $co{s}^{-1}\left(\frac{3}{4}\right)$

3. $co{s}^{-1}\left(\frac{4}{5}\right)$

4. $co{s}^{-1}\left(\frac{5}{6}\right)$

Q.

Find the area of the smaller part of the circle $x^2+y^2=a^2$ cut-off by the line $x=\frac{a}{\sqrt{2}}.$

Q.

Find the slope of a line passing through the following points:

$\left(i\right)(-3,2)and(1,4)\left(ii\right)(a{t}_1^2,2a{t}_1)and(a{t}_2^2,2a{t}_2)\left(iii\right)(3,-5),and(1,2)$

Q. Find the distance of the plane $2x-3y+4z-6=0$ from the origin.

Q. Find the area between the curves $y=x\text{and}y=x^2$

Q. The angle between the $x-$axis and the line joining the points $(3,-1)$ and $(4,-2)$ is

1. ${45}^{\circ }$
2. ${75}^{\circ }$
3. ${120}^{\circ }$
4. ${150}^{\circ }$

Q.

Find the value of $c$ such that equation $4x^2-2\left(c+1\right)x+\left(c+4\right)=0$ has real and equal roots.

Q. The distance of the plane $\overrightarrow{r}\cdot (\frac{2}{7}\hat{i}+\frac{3}{7}\hat{ȷ}-\frac{6}{7}\hat{k})=1$ from the origin is

1. $7$
2. $\frac{1}{7}$
3. None of these
4. $1$

Q.

Solve system of linear equations, using matrix method.
$2x+y+z=1$
$x-2y-z=\frac{3}{2}$
$3y-5z=9$

Q. Show that the differential equation $xdy-ydx=\sqrt{x^2+y^2}dx$ is homogeneous and solve it.

Q.

Show that the points $(1,4),(3,-2)$ and $(-3,16)$ are collinear. Find equation of the line through them.

Q.

Find the angle between the X-axis and the line joining the points (3, -1) and (4, -2).

Q.

The $x$ coordinate of the incentre of the triangle that has the coordinates of mid - points of its sides as $(0,1),(1,1)$ and $(1,0)$ is

1. $2+\sqrt{2}$

2. $2-\sqrt{2}$

3. $1+\sqrt{2}$

4. $1-\sqrt{2}$

Q. Find local maximum and local minimum values of the function $f$ given by $f\left(x\right)=3x^4+4x^3-12x^2+12$.

Q.

Find the value of $m$ so that the roots of the equation:

$\left(4-m\right)x^2+\left(2m+4\right)x+\left(8m+1\right)=0$ are equal.

Q. If $x_1,x_2,x_3,x_4,x_5,x_6$ are non-zero distinct real roots of the equation $x^6+a{x}^4+bx+\lambda =0$ for some $a,b,\lambda \in R,$ then the value of ${\prod }_{i=1}^6(1-x_i)-{\prod }_{i=1}^6(1+x_i)$ is equal to

1. $2\lambda$
2. Zero
3. $-2b$
4. $2b$

Q. The unit vector normal to the plane $x+2y+3z-6=0$ is $\frac{1}{\sqrt{14}}\hat{i}+\frac{2}{\sqrt{14}}\hat{j}+\frac{3}{\sqrt{14}}\hat{i}$
State true or false

1. True
2. False

Q. Find all the points of local maxima and local minima of the function $f$ given by $f\left(x\right)=2x^3-6x^2+6x+5$.

Q.

Consider the following population and year graph:

Find the slope of the line AB and using it, find what will be the population in the year 2010.

Q. Find the complex number satisfying the equation $z+\sqrt{2}|(z+1)|+i=0$

Q.

Angle between$x=2andx-3y=6is$

1. $\infty$

2. ${\tan }^{-1}\left(3\right)$

3. ${\tan }^{-1}\left(\frac{1}{3}\right)$

4. $noneofthese$

Q. The foot of the perpendicular drawn from the point $(1,8,4)$ on the line joining the points $(0,–11,4)$ and $(2,–3,1)$ is

1. $(4,5,2)$
2. $(–4,5,2)$
3. $(4,–5,2)$
4. $(4,5,–2)$