3D Geometry · JEE Main previous year questions

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

1. A.0
2. B.$\frac{1}{2\sqrt{2}}$
3. C.$\frac{1}{\sqrt{2}}$
4. D.$\frac{1}{2}$

A plane passes through the points A(1, 2, 3), B(2, 3, 1) and C(2, 4, 2). If O is the origin and P is (2, $-1$, 1), then the projection of $\overrightarrow{OP}$ on this plane is of length :

1. A.$\sqrt{\frac{2}{7}}$
2. B.$\sqrt{\frac{2}{5}}$
3. C.$\sqrt{\frac{2}{3}}$
4. D.$\sqrt{\frac{2}{11}}$

The shortest distance between the lines $$\frac{x-1}{0}=\frac{y+1}{-1}=\frac{z}{1}$$ and $x+y+z+1=0$, $2x-y+z+3=0$ is :

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

The angle of elevation of the top of a hill from a point on the horizontal plane passing through the foot of the hill is found to be $45^{\circ }$. After walking a distance of 80 meters towards the top, up a slope inclined at an angle of $30^{\circ }$ to the horizontal plane, the angle of elevation of the top of the hill becomes $75^{\circ }$. Then the height of the hill (in meters) is ______ .

Let L denote the line in the xy-plane with x and y intercepts as 3 and 1 respectively. Then the image of the point $(-1,-4)$ in this line is :

1. A.$\left(\frac{11}{5},\frac{28}{5}\right)$
2. B.$\left(\frac{29}{5},\frac{11}{5}\right)$
3. C.$\left(\frac{29}{5},\frac{8}{5}\right)$
4. D.$\left(\frac{8}{5},\frac{29}{5}\right)$

A plane P meets the coordinate axes at A, B and C respectively. The centroid of $\Delta \mathrm{ABC}$ is given to be $(1,1,2)$. Then the equation of the line through this centroid and perpendicular to the plane P is :

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

The foot of the perpendicular drawn from the point (4, 2, 3) to the line joining the points (1, –2, 3) and (1, 1, 0) lies on the plane :

1. A.x – 2y + z = 1
2. B.x + 2y – z = 1
3. C.x – y – 2z = 1
4. D.2x + y – z = 1

The plane which bisects the line joining, the points (4, –2, 3) and (2, 4, –1) at right angles also passes through the point :

1. A.(4, 0, 1)
2. B.(0, –1, 1)
3. C.(0, 1, –1)
4. D.(4, 0, –1)

Let a plane P contain two lines $\vec{r}=\hat{i}+\lambda (\hat{i}+\hat{j}),\lambda \in \mathbb{R}$ and $\vec{r}=-\hat{j}+\mu (\hat{j}-\hat{k}),\mu \in \mathbb{R}$ If $\mathrm{Q}(\alpha ,\beta ,\gamma )$ is the foot of the perpendicular drawn from the point $\mathrm{M}(1,0,1)$ to P, then $3(\alpha +\beta +\gamma )$ equals _______.

The plane passing through the points (1, 2, 1), (2, 1, 2) and parallel to the line, 2x = 3y, z = 1 also passes through the point :

1. A.(0, 6, –2)
2. B.(–2, 0, 1)
3. C.(0, –6, 2)
4. D.(2, 0 –1)

A plane passing through the point $(3,1,1)$ contains two lines whose direction ratios are $1,-2,2$ and $2,3,-1$ respectively. If this plane also passes through the point $(\alpha ,-3,5)$, then $\alpha$ is equal to:

1. A.-10
2. B.10
3. C.5
4. D.-5

If for some $\alpha$ and $\beta$ in $\mathbb{R}$, the intersection of the following three planes $x+4y-2z=1$ $x+7y-5z=\beta$ $x+5y+\alpha z=5$ is a line in ${\mathbb{R}}^3$, then $\alpha +\beta$ is equal to :

1. A.-10
2. B.0
3. C.10
4. D.2

If the distance between the plane, $23x-10y-2z+48=0$ and the plane containing the lines $\frac{x+1}{2}=\frac{y-3}{4}=\frac{z+1}{3}$ and $\frac{x+3}{2}=\frac{y+2}{6}=\frac{z-1}{\lambda }(\lambda \in \mathbb{R})$ is equal to $\frac{k}{\sqrt{633}}$, then k is equal to _______.

The shortest distance between the lines $$\frac{x-3}{3}=\frac{y-8}{-1}=\frac{z-3}{1}\text{and}\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-6}{4}\text{is :}$$

1. A.3
2. B.$\frac{7}{2}\sqrt{30}$
3. C.$3\sqrt{30}$
4. D.$2\sqrt{30}$

The mirror image of the point $(1,2,3)$ in a plane is $\left(-\frac{7}{3},-\frac{4}{3},-\frac{1}{3}\right)$. Which of the following points lies on this plane ?

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

Let P be a plane passing through the points (2, 1, 0), (4, 1, 1) and (5, 0, 1) and R be any point (2, 1, 6). Then the image of R in the plane P is :

1. A.(4, 3, 2)
2. B.(6, 5, - 2)
3. C.(3, 4, -2)
4. D.(6, 5, 2)

If the foot of the perpendicular drawn from the point $(1,0,3)$ on a line passing through $(\alpha ,7,1)$ is $\left(\frac{5}{3},\frac{7}{3},\frac{17}{3}\right)$, then $\alpha$ is equal to _______.

If the line $\frac{x-2}{3}=\frac{y+1}{2}=\frac{z-1}{-1}$ intersects the plane $2x+3y-z+13=0$ at a point P and the plane $3x+y+4z=16$ at a point Q, then PQ is equal to :

1. A.$2\sqrt{7}$
2. B.14
3. C.$2\sqrt{14}$
4. D.$\sqrt{14}$

A plane which bisects the angle between the two given planes 2x – y + 2z – 4 = 0 and x + 2y + 2z – 2 = 0, passes through the point :

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

The length of the perpendicular drawn from the point $(2,1,4)$ to the plane containing the lines $\vec{r}=(\hat{i}+\hat{j})+\lambda (\hat{i}+2\hat{j}-\hat{k})$ and $\vec{r}=(\hat{i}+\hat{j})+\mu (-\hat{i}+\hat{j}-2\hat{k})$ is :

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

The angle of elevation of the top of a vertical tower standing on a horizontal plane is observed to be $45^{\circ }$ from a point A on the plane. Let B be the point 30 m vertically above the point A. If the angle of elevation of the top of the tower from B be $30^{\circ }$, then the distance (in m) of the foot of the tower from the point A is :

1. A.$15\left(1+\sqrt{3}\right)$
2. B.$15\left(3-\sqrt{3}\right)$
3. C.$15\left(3+\sqrt{3}\right)$
4. D.$15\left(5-\sqrt{3}\right)$

If $\mathrm{Q}(0,-1,-3)$ is the image of the point P in the plane $3x-y+4z=2$ and R is the point (3, -1, -2), then the area (in sq. units) of $\Delta \mathrm{PQR}$ is :

1. A.$\frac{\sqrt{65}}{2}$
2. B.$2\sqrt{13}$
3. C.$\frac{\sqrt{91}}{2}$
4. D.$\frac{\sqrt{91}}{4}$

If the plane $2x-y+2z+3=0$ has the distances $\frac{1}{3}$ and $\frac{2}{3}$ units from the planes $4x-2y+4z+\lambda =0$ and $2x-y+2z+\mu =0$, respectively, then the maximum value of $\lambda +\mu$ is equal to :

1. A.13
2. B.9
3. C.5
4. D.15

A perpendicular is drawn from a point on the line $\frac{x-1}{2}=\frac{y+1}{-1}=\frac{z}{1}$ to the plane $x+y+z=3$ such that the foot of the perpendicular Q also lies on the plane $x-y+z=3$. Then the co-ordinates of Q are :

1. A.(4, 0, - 1)
2. B.(2, 0, 1)
3. C.(1, 0, 2)
4. D.(- 1, 0, 4)

A plane passing through the points (0, −1, 0) and (0, 0, 1) and making an angle $\frac{\pi }{4}$ with the plane y − z + 5 = 0, also passes through the point

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