3D Geometry · JEE Main previous year questions

If the lines $\vec{r}=(\hat{i}-\hat{j}+\hat{k})+\lambda (3\hat{j}-\hat{k})$ and $\vec{r}=(\alpha \hat{i}-\hat{j})+\mu (2\hat{i}-3\hat{k})$ are co-planar, then the distance of the plane containing these two lines from the point $(\alpha ,0,0)$ is :

1. A.$\frac{2}{9}$
2. B.$\frac{2}{11}$
3. C.$\frac{4}{11}$
4. D.2

Let $\mathrm{Q}$ be the mirror image of the point $\mathrm{P}(1,0,1)$ with respect to the plane $\mathrm{S}:x+y+z=5$. If a line $\mathrm{L}$ passing through $(1,-1,-1)$, parallel to the line $\mathrm{PQ}$ meets the plane $\mathrm{S}$ at $\mathrm{R}$, then ${\mathrm{QR}}^2$ is equal to :

1. A.$2$
2. B.$5$
3. C.$7$
4. D.$11$

Let the lines $L_1:\vec{r}=\lambda \left(\hat{i}+2\hat{j}+3\hat{k}\right)$, $\lambda \in \mathbb{R}$ $L_2:\vec{r}=\left(\hat{i}+3\hat{j}+\hat{k}\right)+\mu \left(\hat{i}+\hat{j}+5\hat{k}\right)$; $\mu \in \mathbb{R}$, intersect at the point $\mathrm{S}$. If a plane $ax+by-z+d=0$ passes through $\mathrm{S}$ and is parallel to both the lines $L_1$ and $L_2$, then the value of $a+b+d$ is equal to $\underline{}$.

Let $\mathrm{p}$ be the plane passing through the intersection of the planes $\vec{r}\cdot (\hat{i}+3\hat{j}-\hat{k})=5$ and $\vec{r}\cdot (2\hat{i}-\hat{j}+\hat{k})=3$, and the point $(2,1,-2)$. Let the position vectors of the points $\mathrm{X}$ and $\mathrm{Y}$ be $\hat{i}-2\hat{j}+4\hat{k}$ and $5\hat{i}-\hat{j}+2\hat{k}$ respectively. Then the points :

1. A.X and X + Y are on the same side of P
2. B.Y and Y - X are on the opposite sides of P
3. C.X and Y are on the opposite sides of P
4. D.X + Y and X - Y are on the same side of P

Let ${\mathrm{l}}_1$ be the line in xy-plane with x and y intercepts $\frac{1}{8}$ and $\frac{1}{4\sqrt{2}}$ respectively, and ${\mathrm{l}}_2$ be the line in zx-plane with x and z intercepts $-\frac{1}{8}$ and $-\frac{1}{6\sqrt{3}}$ respectively. If d is the shortest distance between the line ${\mathrm{l}}_1$ and ${\mathrm{l}}_2$, then $d^{-2}$ is equal to $\underline{}$.

Let the points on the plane P be equidistant from the points $(-4,2,1)$ and $(2,-2,3)$. Then the acute angle between the plane P and the plane $2x+y+3z=1$ is :

1. A.$\frac{\pi }{6}$
2. B.$\frac{\pi }{4}$
3. C.$\frac{\pi }{3}$
4. D.$\frac{5\pi }{12}$

Let the acute angle bisector of the two planes $x-2y-2z+1=0$ and $2x-3y-6z+1=0$ be the plane P. Then which of the following points lies on P?

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

Let P be the plane passing through the point $(1,2,3)$ and the line of intersection of the planes $\vec{r}\cdot (\hat{i}+\hat{j}+4\hat{k})=16$ and $\vec{r}\cdot (-\hat{i}+\hat{j}+\hat{k})=6$. Then which of the following points does NOT lie on P?

1. A.$(3,3,2)$
2. B.$(6,-6,2)$
3. C.$(4,2,2)$
4. D.$(-8,8,6)$

Let Q be the foot of the perpendicular from the point $P(7,-2,13)$ on the plane containing the lines $\frac{x+1}{6}=\frac{y-1}{7}=\frac{z-3}{8}$ and $\frac{x-1}{3}=\frac{y-2}{5}=\frac{z-3}{7}$. Then $(\mathrm{PQ}{)}^2$, is equal to ________.

For real numbers $\alpha$ and $\beta \ne 0$, if the point of intersection of the straight lines $\frac{x-\alpha }{1}=\frac{y-1}{2}=\frac{z-1}{3}$ and $\frac{x-4}{\beta }=\frac{y-6}{3}=\frac{z-7}{3}$, lies on the plane $x+2y-z=8$, then $\alpha -\beta$ is equal to :

1. A.5
2. B.9
3. C.3
4. D.7

The distance of the point P(3, 4, 4) from the point of intersection of the line joining the points Q(3, $-4$, $-5$) and R(2, $-3$, 1) and the plane 2x + y + z = 7, is equal to ____.

Let the foot of perpendicular from a point P(1, 2, $-1$) to the straight line $L:\frac{x}{1}=\frac{y}{0}=\frac{z}{-1}$ be N. Let a line be drawn from P parallel to the plane x + y + 2z = 0 which meets L at point Q. If $\alpha$ is the acute angle between the lines PN and PQ, then $\cos \alpha$ is equal to ____.

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

Let L be the line of intersection of planes $\vec{r}\cdot (\hat{i}-\hat{j}+2\hat{k})=2$ and $\vec{r}\cdot (2\hat{i}+\hat{j}-\hat{k})=2$. If $P(\alpha ,\beta ,\gamma )$ is the foot of perpendicular on L from the point $(1,2,0)$, then the value of $35(\alpha +\beta +\gamma )$ is equal to :

1. A.$101$
2. B.$119$
3. C.$143$
4. D.$134$

The area (in sq. units) of the region bounded by the curves $x^2+2y-1=0$, $y^2+4x-4=0$ and $y^2-4x-4=0$, in the upper half plane is ________.

If the shortest distance between the lines ${\vec{r}}_1=\alpha \hat{i}+2\hat{j}+2\hat{k}+\lambda (\hat{i}-2\hat{j}+2\hat{k})$, $\lambda \in \mathbb{R}$, $\alpha >0$ and ${\vec{r}}_2=-4\hat{i}-\hat{k}+\mu (3\hat{i}-2\hat{j}-2\hat{k})$, $\mu \in \mathbb{R}$ is 9, then $\alpha$ is equal to ________.

Let the plane $ax+by+cz+d=0$ bisect the line joining the points $(4,-3,1)$ and $(2,3,-5)$ at the right angles. If a, b, c, d are integers, then the minimum value of $(a^2+b^2+c^2+d^2)$ is _______.

The equation of the planes parallel to the plane $x-2y+2z-3=0$ which are at unit distance from the point $(1,2,3)$ is $ax+by+cz+d=0$. If $(b-d)=k(c-a)$, then the positive value of k is :

Let the mirror image of the point (1, 3, a) with respect to the plane $\vec{r}\cdot \left(2\hat{i}-\hat{j}+\hat{k}\right)-b=0$ be ($-3$, 5, 2). Then, the value of | a + b | is equal to ____.

Let P be a plane containing the line $\frac{x-1}{3}=\frac{y+6}{4}=\frac{z+5}{2}$ and parallel to the line $\frac{x-1}{4}=\frac{y-2}{-3}=\frac{z+5}{7}$. If the point $(1,-1,\alpha )$ lies on the plane P, then the value of $|5\alpha |$ is equal to ____.

Let P be a plane $lx+my+nz=0$ containing the line, $\frac{1-x}{1}=\frac{y+4}{2}=\frac{z+2}{3}$. If plane P divides the line segment AB joining points $A(-3,-6,1)$ and $B(2,4,-3)$ in ratio $k:1$ then the value of k is equal to :

1. A.2
2. B.3
3. C.1.5
4. D.4

If for $a>0$, the feet of perpendiculars from the points $A(a,-2a,3)$ and $B(0,4,5)$ on the plane $lx+my+nz=0$ are points $C(0,-a,-1)$ and D respectively, then the length of line segment CD is equal to :

1. A.$\sqrt{41}$
2. B.$\sqrt{55}$
3. C.$\sqrt{31}$
4. D.$\sqrt{66}$

Consider the three planes $P_1:3x+15y+21z=9$, $P_2:x-3y-z=5$, and $P_3:2x+10y+14z=5$ Then, which one of the following is true?

1. A.$P_1$ and $P_2$ are parallel.
2. B.$P_1,P_2$ and $P_3$ all are parallel.
3. C.$P_1$ and $P_3$ are parallel.
4. D.$P_2$ and $P_3$ are parallel.

Let $(\lambda ,2,1)$ be a point on the plane which passes through the point $(4,-2,2)$. If the plane is perpendicular to the line joining the points $(-2,-21,29)$ and $(-1,-16,23)$, then ${\left(\frac{\lambda }{11}\right)}^2-\frac{4\lambda }{11}-4$ is equal to _________.

If the mirror image of the point $(1,3,5)$ with respect to the plane $4x-5y+2z=8$ is $(\alpha ,\beta ,\gamma )$, then $5(\alpha +\beta +\gamma )$ equals :

1. A.39
2. B.41
3. C.47
4. D.43

Let L be a line obtained from the intersection of two planes $x+2y+z=6$ and $y+2z=4$. If point $P(\alpha ,\beta ,\gamma )$ is the foot of perpendicular from $(3,2,1)$ on L, then the value of $21(\alpha +\beta +\gamma )$ equals :

1. A.102
2. B.142
3. C.136
4. D.68