Equation of a Plane : Intercept Form

Q.

If $\text{A}$ and $\text{B}$ are two events such that $\text{P(AUB)}$$=$$\text{P(A∩B)}$, then the true relation is

1. $\text{P(A)+P(B)=0}$

2. $\text{P(A) + P(B) = P(A)P(B/A)}$

3. $\text{P(A)+P(B) = 2P(A)P(B/A)}$

4. None of these

Q. Find the equation of the plane with intercepts 2, 3 and 4 on the x, y and z-axes respectively

1. 6x+4y+3z = 12
2. 4x+3y-6z = 2
3. x+y+z = 1
4. 7x+2y+9z = 10

Q. Find the number of solutions of |x-1|+|x-2|+|x-3|=10

Q.

The position vectors of the $A$ and $B$ to $O$ are $2i+2j+k$ and $2i+4j+4k$.

Length of internal bisector of $\angle BOA$ of $△AOB$ is

1. $\sqrt{\frac{136}{9}}$

2. $\sqrt{\frac{136}{3}}$

3. $\frac{20}{3}$

4. $\frac{25}{3}$

Q.

If , $\begin{array}{ll}\frac{(1+i)x-2i}{3+i}+\frac{(2-3i)y+i}{3-i}& =i\end{array}$then $(x,y)$ equal

1. $(3,1)$

2. $(3,-1)$

3. $(-3,1)$

4. $(-3,-1)$

Q. Let $S$ be the mirror image of the point $Q(1,3,4)$ with respect to the plane $2x–y+z+3=0$ and let $R(3,5,\gamma )$ be a point of this plane. Then the square of the length of the line segment $SR$ is

Q.

A variable plane $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$ at a unit distance from origin cuts the coordinate axes at A, B and C. Centroid (x, y, z) satisfies the equation $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=K$ is

1. 9

2. 3

3. $\frac{1}{9}$

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

Q.

If the coordinates of the points $A$ and $B$ be $(3,3)$ and $(7,6)$, then the length of the portion of the line $AB$ intercepted between the axes is

1. $\frac{5}{4}$

2. $\frac{\sqrt{10}}{4}$

3. $\frac{\sqrt{13}}{3}$

4. None of these

Q. Let $\pi$ be the plane parallel to $y-$axis and containing the points $(1,0,1)$ and $(3,2,-1).$ Also, $A\equiv (4,0,0)$ and $B\equiv (6,0,-2)$ are two points and $P\equiv (x_0,y_0,z_0)$ is a variable point on the plane $\pi .$ Then which of the following is/are CORRECT?

1. The equation of the plane $\pi$ is $x+z=2$
2. If $(PA+PB)$ is minimum, then $|4x_0+y_0+2z_0|$ is $12$
3. If $|PA-PB|\in [0,\sqrt{N}],$ then $N$ is $8$
4. If the reflection of the line $AB$ in the plane $\pi$ is $\frac{x-2}{1}=\frac{y-\alpha }{0}=\frac{z+\beta }{-1},$ then $({\alpha }^4+{\beta }^4)$ is $16$

Q. Find the equation of the plane with intercept 3 on the y -axis and parallel to ZOX plane.

Q. The coordinates of the circumcentre of the triangle formed by the points (3, 2, -5), (-3, 8, -5) (-3, 2, 1) are

1. (-1, 4, -3)
2. (1, 4, -3)
3. (-1, 4, 3)
4. (-1, -4, -3)

Q. A variable plane at a distance of $1$ unit from the origin cuts the coordinates axes at $A,B$ and $C.$ If the centroid $D(x,y,z)$ of triangle $ABC,$ satisfies the relation $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=k,$ then the value of $k$ is

1. $3$
2. $1$
3. $\frac{1}{3}$
4. $9$

Q.

A variable plane moves so that the sum of reciprocals of its intercepts on the three coordinate axes is constant $\lambda .$ It passes through a fixed point, which has coordinates

1. $(\lambda ,\lambda ,\lambda )$

2. $(\frac{1}{\lambda },\frac{1}{\lambda }.\frac{1}{\lambda })$
   

3. $(-\lambda ,-\lambda ,-\lambda )$

4. $(-\frac{1}{\lambda },-\frac{1}{\lambda },-\frac{1}{\lambda })$

Q. Let f(x)=4x(1-x), 0<=x<=1. The number of solutions of f(f(f(f(x))=x/5 is

Q. Let $A(0,6,8),B(15,20,0)$ are two given points and $P(\lambda ,0,0)$ is a point on $x$ - axis such that $PA+PB$ is minimum. Then which of the following is/are correct?

1. Perpendicular distance of the origin from the plane passing through $P,A,B$ is $\frac{10}{3}$
2. Perpendicular distance of the origin from the plane passing through $P,A,B$ is $\frac{10}{9}$
3. Value of $\lambda$ is $5$
4. Value of $\lambda$ is $10$

Q. If from the point P(a, b, c) perpendiculars PL, PM be drawn to YOZ and ZOX planes, then the equation of the plane OLM is

1. $\frac{x}{a}+\frac{y}{b}-\frac{z}{c}=0$
2. $\frac{x}{a}-\frac{y}{b}+\frac{z}{c}=0$
3. $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=0$
4. $\frac{x}{a}-\frac{y}{b}-\frac{z}{c}=0$

Q. The plane $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$ meets the coordinate axes at A, B, C respectively. D and E are the mid-points of AB and AC respectively. Coordinates of the mid-point of DE are

1. $(a,\frac{b}{4},\frac{c}{4})$
2. $(\frac{a}{4},b,\frac{c}{4})$
3. $(\frac{a}{4},\frac{b}{4},c)$
4. $(\frac{a}{2},\frac{b}{4},\frac{c}{4})$.

Q. If a plane meets the co-ordinate axes in A, B, C such that the centroid of the triangle ABC is the point $(1,r,r^2),$ then equation of the plane is

1. $x+ry+r^{2}z=3r^2$
2. $r^{2}x+ry+z=3r^2$
3. $x+ry+r^{2}z=3$
4. $r^{2}x+ry+z=3$

Q. Algebraic sum of the intercepts made by the plane x + 3y - 4z + 6 = 0 on the axes is

1. $\frac{19}{2}$
2. $-\frac{13}{2}$
3. $-\frac{22}{3}$
4. $-\frac{26}{3}$

Q.

Find the vector equation of the plane passing through the intersection of the planes and through the point (2, 1, 3)

Q. Find the real number $x$ and $y$ if$(x–iy)(3+5i)$is the conjugate of $–6–24i.$

Q. The areas of triangles formed by a plane with the positive $X,Y;Y,Z;Z,X$ axes respectively are $12,9,6$ square units, then the equation of the plane is

1. $\frac{x}{6}+\frac{y}{4}+\frac{z}{3}=1$
2. $\frac{x}{4}+\frac{y}{6}+\frac{z}{3}=1$
3. $\frac{x}{3}+\frac{y}{6}+\frac{z}{4}=1$
4. $\frac{x}{4}+\frac{y}{3}+\frac{z}{6}=1$

Q.

Eliminate the parameter $t$. Find a rectangular equation for the plane curve defined by the parametric equations

$x=3t,y=t+3;-2<t<3$

1. $y=x^2+1;-2<x<2$

2. $y=-3x+3;-\infty <x<\infty$

3. $y=\frac{1}{3}x+3;-6<x<9$

4. $y=\frac{1}{3}x-3;-\infty <x<\infty$

Q. Plane $ax+by+cz=1$ intersects axes in $A,B,C$ respectively. If $G(\frac{1}{6},-\frac{1}{3},1)$ is a centroid of $\Delta ABC$, then $a+b+3c=$_____

1. $\frac{4}{3}$
2. $4$
3. $2$
4. $\frac{5}{6}$

Q. Verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:
$y=x^2+2x+C:y^{\prime }-2x-2=0$.

Q. If $X$ and $Y$ are $2\times 2$ matrices then solve the following matrix equations for $X$ and $Y$
$2X+3Y=\left[\begin{array}{cc}2& 3\\ 4& 0\end{array}\right],$

$3X+2Y=\left[\begin{array}{cc}-2& 2\\ 1& -5\end{array}\right]$

Q.

Find the equation of the plane with intercept 3 on the y-axis and parallel to ZOX plane.

Q. Let $f=\left\{\right(1,1),(2,3),(0,-1),(-1,3\left)\right\}$ be a function from $ztoz$ defined by : $f\left(x\right)=ax+b$ for some integers $a,b$. Determine $a,b$.

Q. 25 if the points (0, 1, -2), (3, L, -1) and (u, -3, -4) are collinear, then the value of L and uare given by

Q. The sum of lengths of intercepts of the plane passing through the points (-2, -2, 2 ), (1, 1, 1) and (1, -1, 2) on the coordinates axes is

1. 3
2. 4
3. 6
4. 12