3D Geometry · JEE Main previous year questions

The shortest distance between the lines $\frac{x-4}{1}=\frac{y-3}{2}=\frac{z-2}{-3}$ and $\frac{x+2}{2}=\frac{y-6}{4}=\frac{z-5}{-5}$ is :

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

Let the image of the point $\mathrm{P}(0,-5,0)$ in the line $\frac{x-1}{2}=\frac{y}{1}=\frac{z+1}{-2}$ be the point $\mathrm{R}$ and the image of the point $\mathrm{Q}\left(0,\frac{-1}{2},0\right)$ in the line $\frac{x-1}{-1}=\frac{y+9}{4}=\frac{z+1}{1}$ be the point $\mathrm{S}$. Then the square of the area of the parallelogram $\mathrm{PQRS}$ is ____ .

If the distance of the point $(a,2,5)$ from the image of the point $(1,2,7)$ in the line $\frac{x}{1}=\frac{y-1}{1}=\frac{z-2}{2}$ is 4, then the sum of all possible values of $a$ is equal to :

1. A.$11$
2. B.$9$
3. C.$6$
4. D.$4$

A line with direction ratios $1,-1,2$ intersects the lines $\frac{x}{2}=\frac{y}{3}=\frac{z+1}{3}$ and $\frac{x+1}{-1}=\frac{y-2}{1}=\frac{z}{4}$ at the points P and Q, respectively. If the length of the line segment PQ is $\alpha$, then $225{\alpha }^2$ is equal to:

1. A.1024
2. B.1014
3. C.1104
4. D.1204

The shortest distance between the lines $$\vec{r}=\left(\frac{1}{3}\hat{\mathrm{i}}+2\hat{\mathrm{j}}+\frac{8}{3}\hat{\mathrm{k}}\right)+\lambda (2\hat{\mathrm{i}}-5\hat{\mathrm{j}}+6\hat{\mathrm{k}})$$ and $\vec{r}=\left(-\frac{2}{3}\hat{\mathrm{i}}-\frac{1}{3}\hat{\mathrm{k}}\right)+\mu (\hat{\mathrm{j}}-\hat{\mathrm{k}})$, $\lambda ,\mu \in \mathbb{R}$, is:

1. A.$\sqrt{5}$
2. B.3
3. C.$2\sqrt{3}$
4. D.$\sqrt{15}$

If the point of intersection of the lines $\frac{x+1}{3}=\frac{y+a}{5}=\frac{z+b+1}{7}$ and $\frac{x-2}{1}=\frac{y-b}{4}=\frac{z-2a}{7}$ lies on xy-plane, then the value of $a+b$ is:

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

Two adjacent sides of a parallelogram PQRS are given by $\overrightarrow{\mathrm{PQ}}=\hat{j}+\hat{k}$ and $\overrightarrow{\mathrm{PS}}=\hat{i}-\hat{j}$. If the side PS is rotated about the point P by an acute angle $\alpha$ in the plane of the parallelogram so that it becomes perpendicular to the side PQ, then ${\sin }^2\left(\frac{5\alpha }{2}\right)-{\sin }^2\left(\frac{\alpha }{2}\right)$ is equal to :

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

Let P be a point in the plane of the vectors $\overrightarrow{AB}=3\hat{i}+\hat{j}-\hat{k}$ and $\overrightarrow{AC}=\hat{i}-\hat{j}+3\hat{k}$ such that P is equidistant from the lines AB and AC. If $|\overrightarrow{AP}|=\frac{\sqrt{5}}{2}$, then the area of the triangle ABP is :

1. A.$\frac{3}{2}$
2. B.$\frac{\sqrt{30}}{4}$
3. C.$\frac{\sqrt{26}}{4}$
4. D.2

If the distance of the point $P(43,\alpha ,\beta ),\beta <0$, from the line $\vec{r}=4\hat{i}-\hat{k}+\mu (2\hat{i}+3\hat{k}),\mu \in \mathbb{R}$ along a line with direction ratios $3,-1,0$ is $13\sqrt{10}$, then ${\alpha }^2+{\beta }^2$ is equal to ________

Let a line L passing through the point $\mathrm{P}(1,1,1)$ be perpendicular to the lines $\frac{x-4}{4}=\frac{y-1}{1}=\frac{z-1}{1}$ and $\frac{x-17}{1}=\frac{y-71}{1}=\frac{z}{0}$. Let the line L intersect the $yz-$ plane at the point Q . Another line parallel to L and passing through the point $\mathrm{S}(1,0,-1)$ intersects the $yz$-plane at the point R . Then the square of the area of the parallelogram PQRS is equal to ____ .

Let the direction cosines of two lines satisfy the equations : $4l+m-n=0$ and $2mn+10nl+3lm=0$. Then the cosine of the acute angle between these lines is :

1. A.$\frac{10}{7\sqrt{38}}$
2. B.$\frac{10}{\sqrt{38}}$
3. C.$\frac{10}{3\sqrt{38}}$
4. D.$\frac{20}{3\sqrt{38}}$

If the image of the point $P(a,2,a)$ in the line $\frac{x}{2}=\frac{y+a}{1}=\frac{z}{1}$ is $Q$ and the image of $Q$ in the line $\frac{x-2b}{2}=\frac{y-a}{1}=\frac{z+2b}{-5}$ is $P$, then $a+b$ is equal to _____.

Let $\mathrm{P}(\alpha ,\beta ,\gamma )$ be the point on the line $\frac{x-1}{2}=\frac{y+1}{-3}=z$ at a distance $4\sqrt{14}$ from the point $(1,-1,0)$ and nearer to the origin. Then the shortest distance, between the lines $\frac{x-\alpha }{1}=\frac{y-\beta }{2}=\frac{z-\gamma }{3}$ and $\frac{x+5}{2}=\frac{y-10}{1}=\frac{z-3}{1}$, is equal to

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

There are 12 points in a plane, no three of which are in the same straight line, except 5 points which are collinear. Then the total number of triangles that can be formed with the vertices at any three of these 12 points is

1. A.230
2. B.210
3. C.200
4. D.220

Let the shortest distance between the lines $\frac{x-3}{3}=\frac{y-\alpha }{-1}=\frac{z-3}{1}$ and $\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-\beta }{4}$ be $3\sqrt{30}$. Then the positive value of $5\alpha +\beta$ is

1. A.$42$
2. B.$40$
3. C.$48$
4. D.$46$

Line $L_1$ passes through the point $(1,2,3)$ and is parallel to $z$-axis. Line $L_2$ passes through the point $(\lambda ,5,6)$ and is parallel to $y$-axis. Let for $\lambda ={\lambda }_1,{\lambda }_2$, ${\lambda }_2<{\lambda }_1$, the shortest distance between the two lines be 3 . Then the square of the distance of the point $({\lambda }_1,{\lambda }_2,7)$ from the line $L_1$ is

1. A.25
2. B.32
3. C.40
4. D.37

The shortest distance between the curves $y^2=8x$ and $x^2+y^2+12y+35=0$ is:

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

Let $ABCD$ be a tetrahedron such that the edges $AB,AC$ and $AD$ are mutually perpendicular. Let the areas of the triangles $ABC,ACD$ and $ADB$ be $5,6$ and $7$ square units respectively. Then the area (in square units) of the $△BCD$ is equal to :

1. A.$\sqrt{110}$
2. B.12
3. C.$\sqrt{340}$
4. D.$7\sqrt{3}$

If the image of the point $\mathrm{P}(1,0,3)$ in the line joining the points $\mathrm{A}(4,7,1)$ and $\mathrm{B}(3,5,3)$ is $Q(\alpha ,\beta ,\gamma )$, then $\alpha +\beta +\gamma$ is equal to :

1. A.$\frac{46}{3}$
2. B.18
3. C.13
4. D.$\frac{47}{3}$

The line $L_1$ is parallel to the vector $\vec{a}=-3\hat{i}+2\hat{j}+4\hat{k}$ and passes through the point $(7,6,2)$ and the line $L_2$ is parallel to the vector $\vec{b}=2\hat{i}+\hat{j}+3\hat{k}$ and passes through the point $(5,3,4)$. The shortest distance between the lines $L_1$ and $L_2$ is :

1. A.$\frac{23}{\sqrt{38}}$
2. B.$\frac{21}{\sqrt{38}}$
3. C.$\frac{23}{\sqrt{57}}$
4. D.$\frac{21}{\sqrt{57}}$

Let a straight line $L$ pass through the point $P(2,-1,3)$ and be perpendicular to the lines $\frac{x-1}{2}=\frac{y+1}{1}=\frac{z-3}{-2}$ and $\frac{x-3}{1}=\frac{y-2}{3}=\frac{z+2}{4}$ . If the line $L$ intersects the $yz$-plane at the point $Q$, then the distance between the points $P$ and $Q$ is:

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

Let P be the foot of the perpendicular from the point $(1,2,2)$ on the line $L:\frac{x-1}{1}=\frac{y+1}{-1}=\frac{z-2}{2}$. Let the line $\vec{r}=(-\hat{i}+\hat{j}-2\hat{k})+\lambda (\hat{i}-\hat{j}+\hat{k}),\lambda \in \mathbf{R}$, intersect the line L at Q . Then $2(\mathrm{PQ}{)}^2$ is equal to :

1. A.$25$
2. B.$27$
3. C.$19$
4. D.$29$

Let $\mathrm{A}(x,y,z)$ be a point in $xy$-plane, which is equidistant from three points $(0,3,2)$, $(2,0,3)$ and $(0,0,1)$. Let $\mathrm{B}=(1,4,-1)$ and $\mathrm{C}=(2,0,-2)$. Then among the statements (S1) : $△\mathrm{ABC}$ is an isosceles right angled triangle, and (S2) : the area of $△\mathrm{ABC}$ is $\frac{9\sqrt{2}}{2}$,

1. A.both are false
2. B.only (S2) is true
3. C.only (S1) is true
4. D.both are true

If the image of the point $(4,4,3)$ in the line $\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-1}{3}$ is $(\alpha ,\beta ,\gamma )$, then $\alpha +\beta +\gamma$ is equal to

1. A.12
2. B.9
3. C.7
4. D.8

Let P be the image of the point $\mathrm{Q}(7,-2,5)$ in the line $\mathrm{L}:\frac{x-1}{2}=\frac{y+1}{3}=\frac{z}{4}$ and $\mathrm{R}(5,\mathrm{p},\mathrm{q})$ be a point on $\mathrm{L}$. Then the square of the area of $△PQR$ is ________.