Relative Position of a Point with Respect to a Line

Q. For a point P in the plane, let $d_1$(p) be the distance of the point $P$ from the lines $x-y=0$ and $x+y=0$ respectively. The area of the region $R$ consisting of all points $P$ lying in the first quadrant of the plane and satisfying
$2\le d_1\left(P\right)+d_2\left(P\right)\le 4,$ is ___

Q. Number of integral coordinates strictly lying inside the triangle formed by the line $x+y=21$ with coordinate axes are

Q. If the point $(\alpha ,{\alpha }^2)$ lies between $x+y-2=0$ and $4x+4y=3,$ then the range of $\alpha$ is

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

Q. The zeroes of the polynomial $f\left(x\right)=6x^2-x-2$ is/are

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

Q.

Find the values of $\alpha$ so that the point $P({\alpha }^2,\alpha )$ lies inside or on the triangle formed by the lines $x-5y+6=0$, $x-3y+2=0$ and $x-2y-3=0$

Q. Abscissae and ordinates of $n$ given points are in $A.P.,$ with first term $a$ and common difference $1$ and $2$ respectively. If algebraic sum of perpendiculars drawn from these given points on a variable line which always passes through the point $(\frac{13}{2},11)$ is $0,$ then the value of $a\cdot n$ is

Q. The combined equation of three sides of a triangle is $(x^2-y^2)(2x+3y-6)=0$ such that the points $(-2,a)$ and $(b,1)$ be the interior point and extrerior point of the triangle respectively. If $k=\left[\frac{a}{b}\right],$ then maximum value of $|k|$ is
(where [.] denotes the greatest integer function.)

Q. If $P$ is a point $(x,y)$ on the line $y=-3x$ such that $P$ and the point $(3,4)$ are on the opposite sides of the line $3x-4y=8$, then

1. $x>\frac{8}{15}$
2. $x>\frac{8}{5}$
3. $y<-\frac{8}{5}$
4. $y<-\frac{8}{15}$

Q. The range of $\alpha$ for which the points $(\alpha ,\alpha +2)$ and $(\frac{3\alpha }{2},{\alpha }^2)$ lie on opposite sides of the line $2x+3y-6=0$ is

1. $(-\infty ,-2)\cup (1,\infty )$
2. $(-1,0)\cup (1,2)$
3. $(-\infty ,-2)\cup (0,1)$
4. $(-\infty ,0)\cup (1,2)$

Q. For points $P=(x_1,y_1)\text{and}Q=(x_2,y_2)$ of the co-ordinate plane, a new distance $d(P,Q)$ is defined by $d(P,Q)=|x_1-x_2|+|y_1-y_2|$. Let $O=(0,0)$ and $A=(3,2)$. Prove that the set of points in the first quadrant which are equidistant (with respect to the new distance) from $O$ and $A$ consinsts of the union of a line segment of finite length and an infinite ray. Sketch this set in a labelled diagram.

Q. The range of values of $\theta$ in the interval $(0,\pi )$ such that the points $(3,5)$ and $(\sin \theta ,\cos \theta )$ lie on the same side of the line $x+y-1=0$, is

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

Q.

Determine whether the point (-3, 2) lies inside or outside the triangle whose sides are given by the equations $x+y-4=0$, $3x-7y+8=0$, $4x-y-31=0$.

Q. If the coordinates of the foot of the perpendicular drawn from the point $(1,-2)$ on the line $y=2x+1$ is $(\alpha ,\beta )$, then the value of $|\alpha +\beta |$ is

Q. Let the lines $y-k_{1}x-\beta =0$ and $y-k_{2}x-\beta =0,(k_1\ne k_2),k_1,k_2\in R$ intersect at $P$ and the lines $x-p_{1}y-\alpha =0$ and $x-p_{2}y-\alpha =0,(p_1\ne p_2),p_1,p_2\in R$ intersect at $Q$. If the points $P$ and $Q$ always lies on or inside the triangle formed by the lines $2x-3y-6=0,3x-y+3=0$ and $3x+4y-12=0$, then

1. $\alpha \in [-1,3]$
2. $\alpha \in [-2,4]$
3. $\beta \in [-3,-2)$
4. $\beta \in [-2,3]$

Q. The position of the point (8, -9) with respect to the lines 2x+3y-4=0 and 6x+9y+8=0 is

1. Point lies on the same side of the lines
2. Point lies on the different sides of the line
3. Point lies on one of the line
4. None of these

Q.

Number of real roots of the equation $(x^2+2{)}^2+8x^2=6x(x^2+2)$ is :

1. $2$

2. None

3. $4$

4. $0$

Q. The number of non-negative integral values of $b$ for which the origin and point $(1,1)$ lie on the same side of straight line $a^{2}x+aby+1=0,\forall a\in R-\left\{0\right\},$ is

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

Q. The position of the points $(2,3)$ and $(-4,5)$ with respect to the line $3x-4y=8$ is

1. on same side
2. on opposite side
3. lie on the line
4. one point on the line other outside of the line

Q.

Find the values of the parameter a so that the point (a, 2) is an interior point of the triangle formed by the lines $x+y-4=0$, $3x-7y-8=0$ and $4x-y-31=0$

Q. The number of distinct common root(s) of the equations $x^5-x^3+x^2-1=0$ and $x^4-1=0$ is

Q. A line makes angles $\alpha ,\beta ,\gamma$ with the coordinate axes. If $\alpha +\beta =\frac{\pi }{2},$ then $(\cos \alpha +\cos \beta +\cos \gamma {)}^2$ is equal to

1. $1+\sin 2\alpha$
2. $1+\cos 2\alpha$
3. $1-\sin 2\alpha$
4. $1$

Q. Let the lines $y-k_{1}x-\beta =0$ and $y-k_{2}x-\beta =0,(k_1\ne k_2),k_1,k_2\in R$ intersect at $P$ and the lines $x-p_{1}y-\alpha =0$ and $x-p_{2}y-\alpha =0,(p_1\ne p_2),p_1,p_2\in R$ intersect at $Q$. If the points $P$ and $Q$ always lies on or inside the triangle formed by the lines $2x-3y-6=0,3x-y+3=0$ and $3x+4y-12=0$, then

1. $\alpha \in [-1,3]$
2. $\alpha \in [-2,4]$
3. $\beta \in [-3,-2)$
4. $\beta \in [-2,3]$

Q. The roots of the equation $\frac{3}{x-5}+\frac{2x}{x-3}=5$ are (where $x\ne 3,5$)

1. $5$
2. $6$
3. $7$
4. $4$

Q. The equation of the plane passing through the points $P(1,1,1),Q(3,-1,2),R(-3,5,-4)$ is

1. $x+y=2$
2. $y+z=2$
3. $z+x=2$
4. $x+y+z=2$

Q. If the angles made by a straight line with the coordinate axes are $\alpha ,\frac{\pi }{2}-\alpha ,\beta ,\text{then}\beta =$

1. $0$
2. $\frac{\pi }{6}$
3. $\frac{\pi }{2}$
4. $\pi$

Q. The linear factor(s) of the equation $9x^2-24xy+16y^2-12x+16y-12=0$ is/are

1. $3x-2y+2=0$
2. $3x-2y+6=0$
3. $3x-4y-6=0$
4. $3x-4y+2=0$

Q. The number of distinct zeroes of the polynomial $f\left(x\right)=x^2-8x+16$ is

Q. If the points $(1,2)$ and $(3,4)$ lie on the same side of the straight line $3x-5y+a=0$ then a lies in the set

1. $[7,\infty )$
2. $(-\infty ,11]$
3. $[7,11]$
4. $R-[7,11]$

Q. The roots of the equation $x+\frac{1}{x}=3,x\ne 0$ are

1. $\frac{3+\sqrt{7}}{2},\frac{3-\sqrt{7}}{2}$
2. $\frac{-3+\sqrt{5}}{2},\frac{-3-\sqrt{5}}{2}$
3. $\frac{3+\sqrt{5}}{2},\frac{3-\sqrt{5}}{2}$
4. $\frac{-3+\sqrt{7}}{2},\frac{-3-\sqrt{7}}{2}$

Q. The linear factor(s) of the equation $x^2+4xy+4y^2+3x+6y-4=0$ is/are

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