3D Geometry · JEE Main previous year questions

The distance of the point $(1,0,2)$ from the point of intersection of the line $\frac{x-2}{3}=\frac{y+1}{4}=\frac{z-2}{12}$ and the plane $x-y+z=16$, is :

1. A.$3\sqrt{21}$
2. B.$13$
3. C.$2\sqrt{14}$
4. D.$8$

The image of the line $\frac{x-1}{3}=\frac{y-3}{1}=\frac{z-4}{-5}$ in the plane $2x-y+z+3=0$ is the line :

1. A.$\frac{x-3}{3}=\frac{y+5}{1}=\frac{z-2}{-5}$
2. B.$\frac{x-3}{-3}=\frac{y+5}{-1}=\frac{z-2}{5}$
3. C.$\frac{x+3}{3}=\frac{y-5}{1}=\frac{z-2}{-5}$
4. D.$\frac{x+3}{-3}=\frac{y-5}{-1}=\frac{z+2}{5}$

Consider the following planes $P:x+y-2z+7=0$ $Q:x+y+2z+2=0$ $R:3x+3y-6z-11=0$

1. A.$P$ and $R$ are perpendicular
2. B.$Q$ and $R$ are perpendicular
3. C.$P$ and $Q$ are parallel
4. D.$P$ and $R$ are parallel

The equation of a plane containing the line $\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z+2}{1}$ and the point $(0,7,-7)$ is

1. A.$x+y+z=0$
2. B.$x+2y+z=21$
3. C.$3x-2y+5z+35=0$
4. D.$3x+2y+5z+21=0$

If two vertical poles $20\mathrm{m}$ and $80\mathrm{m}$ high stand apart on a horizontal plane, then the height (in $\mathrm{m}$) of the point of intersection of the lines joining the top of each pole to the foot of other is

1. A.16
2. B.18
3. C.50
4. D.15

The values of $a$ for which the two points $(1,a,1)$ and $(-3,0,a)$ lie on the opposite sides of the plane $3x+4y-12z+13=0$, satisfy

1. A.$0<a<\frac{1}{3}$
2. B.$-1<a<0$
3. C.$a<-1\text{ or }a>\frac{1}{3}$
4. D.$a=0$

A line with positive direction cosines passes through the point $P(2,-1,2)$ and makes equal angles with the coordinate axes. If the line meets the plane $2x+y+z=9$ at point $Q$, then the length $PQ$ equals

1. A.$\sqrt{2}$
2. B.2
3. C.$\sqrt{3}$
4. D.1

A line $AB$ in three-dimensional space makes angles $45^{\circ }$ and $120^{\circ }$ with the positive $x$-axis and the positive $y$-axis respectively. If $AB$ makes an acute angle $\theta$ with the positive $z$-axis, then $\theta$ equals :

1. A.$45^{\circ }$
2. B.$60^{\circ }$
3. C.$75^{\circ }$
4. D.$30^{\circ }$

Statement-1 : The point $A(3,1,6)$ is the mirror image of the point $B(1,3,4)$ in the plane $x-y+z=5$. Statement-2 : The plane $x-y+z=5$ bisects the line segment joining $A(3,1,6)$ and $B(1,3,4)$.

1. A.Statement - 1 is true, Statement - 2 is true; Statement - 2 is not a correct explanation for Statement - 1.
2. B.Statement - 1 is true, Statement - 2 is false.
3. C.Statement - 1 is false , Statement - 2 is true.
4. D.Statement - 1 is true, Statement - 2 is true; Statement - 2 is a correct explanation for Statement - 1.

The shortest distance between the line $y-x=1$ and the curve $x=y^2$ is :

1. A.$\frac{2\sqrt{3}}{8}$
2. B.$\frac{3\sqrt{2}}{5}$
3. C.$\frac{\sqrt{3}}{4}$
4. D.$\frac{3\sqrt{2}}{8}$

Three distinct points A, B and C are given in the 2 -dimensional coordinates plane such that the ratio of the distance of any one of them from the point $(1,0)$ to the distance from the point $(-1,0)$ is equal to $\frac{1}{3}$. Then the circumcentre of the triangle ABC is at the point :

1. A.$(\frac{5}{4},0)$
2. B.$(\frac{5}{2},0)$
3. C.$(\frac{5}{3},0)$
4. D.$(0,0)$

Let the line $\frac{x-2}{3}=\frac{y-1}{-5}=\frac{z+2}{2}$ lie in the plane $x+3y-\alpha z+\beta =0$. Then $(\alpha ,\beta )$ equals

1. A.$(-6,7)$
2. B.$(5,-15)$
3. C.$(-5,5)$
4. D.$(6,-17)$

The image of the point $(-1,3,4)$ in the plane $x-2y=0$ is :

1. A.$(-\frac{17}{3},-\frac{19}{3},4)$
2. B.$(15,11,4)$
3. C.$(-\frac{17}{3},-\frac{19}{3},1)$
4. D.None of these

Let $a,b$ and $c$ be distinct non-negative numbers. If the vectors $a\hat{i}+a\hat{j}+c\hat{k}$, $\hat{i}+\hat{k}$ and $c\hat{i}+c\hat{j}+b\hat{k}$ lie in a plane, then $c$ is :

1. A.the Geometric Mean of $a$ and $b$
2. B.the Arithmetic Mean of $a$ and $b$
3. C.equal to zero
4. D.the Harmonic Mean of $a$ and $b$

$A$ and $B$ are two like parallel forces. A couple of moment $H$ lies in the plane of $A$ and $B$ and is combined with them. The resultant of $A$ and $B$ after combining is displaced through a distance

1. A.$\frac{2H}{A-B}$
2. B.$\frac{H}{A+B}$
3. C.$\frac{H}{2(A+B)}$
4. D.$\frac{H}{A-B}$

If the angle $\theta$ between the line $\frac{x+1}{1}=\frac{y-1}{2}=\frac{z-2}{2}$ and the plane $2x-y+\sqrt{\lambda }z+4=0$ is such that $\sin \theta =\frac{1}{3}$ then value of $\lambda$ is :

1. A.$\frac{5}{3}$
2. B.$\frac{-3}{5}$
3. C.$\frac{3}{4}$
4. D.$\frac{-4}{3}$

If the plane $2ax-3ay+4az+6=0$ passes through the midpoint of the line joining the centres of the spheres $$x^2+y^2+z^2+6x-8y-2z=13\text{ and}$$ $$x^2+y^2+z^2-10x+4y-2z=8\text{ then }a\text{ equals :}$$

1. A.$-1$
2. B.$1$
3. C.$-2$
4. D.$2$

The distance between the line $$\vec{r}=2\hat{i}-2\hat{j}+3\hat{k}+\lambda (i-j+4k),\text{ and the plane}$$ $$\vec{r}\cdot \left(\hat{i}+5\hat{j}+\hat{k}\right)=5\text{ is}$$

1. A.$\frac{10}{9}$
2. B.$\frac{10}{3\sqrt{3}}$
3. C.$\frac{3}{10}$
4. D.$\frac{10}{3}$

A line with direction cosines proportional to $2,1,2$ meets each of the lines $x=y+a=z$ and $x+a=2y=2z$ . The co-ordinates of each of the points of intersection are given by :

1. A.$(2a,3a,3a),(2a,a,a)$
2. B.$(3a,2a,3a),(a,a,a)$
3. C.$(3a,2a,3a),(a,a,2a)$
4. D.$(3a,3a,3a),(a,a,a)$

Distance between two parallel planes $2x+y+2z=8$ and $4x+2y+4z+5=0$ is :

1. A.$\frac{9}{2}$
2. B.$\frac{5}{2}$
3. C.$\frac{7}{2}$
4. D.$\frac{3}{2}$

The intersection of the spheres $x^2+y^2+z^2+7x-2y-z=13$ and $x^2+y^2+z^2-3x+3y+4z=8$ is the same as the intersection of one of the sphere and the plane

1. A.$2x-y-z=1$
2. B.$x-2y-z=1$
3. C.$x-y-2z=1$
4. D.$x-y-z=1$

A plane which passes through the point $(3,2,0)$ and the line $\frac{x-4}{1}=\frac{y-7}{5}=\frac{z-4}{4}$ is :

1. A.$x-y+z=1$
2. B.$x+y+z=5$
3. C.$x+2y-z=1$
4. D.$2x-y+z=5$

The $d.r.$ of normal to the plane through $(1,0,0),(0,1,0)$ which makes an angle $\pi /4$ with plane $x+y=3$ are :

1. A.$1,\sqrt{2},1$
2. B.$1,1,\sqrt{2}$
3. C.$1,1,2$
4. D.$\sqrt{2},1,1$