3D Geometry · JEE Main previous year questions

If the line, $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-2}{4}$ meets the plane, x + 2y + 3z = 15 at a point P, then the distance of P from the origin is :

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

Let P be the plane, which contains the line of intersection of the planes, $x+y+z-6=0$ and $2x+3y+z+5=0$ and it is perpendicular to the xy-plane. Then the distance of the point (0, 0, 256) from P is equal to :

1. A.$205\sqrt{5}$
2. B.$63\sqrt{5}$
3. C.$11/\sqrt{5}$
4. D.$17/\sqrt{5}$

Two vertical poles of heights, 20 m and 80 m stand a part on a horizontal plane. The height (in meters) of the point of intersection of the lines joining the top of each pole to the foot of the other, from this horizontal plane is :

1. A.12
2. B.16
3. C.15
4. D.18

The perpendicular distance from the origin to the plane containing the two lines, $$\frac{x+2}{3}=\frac{y-2}{5}=\frac{z+5}{7}\text{and}\frac{x-1}{1}=\frac{y-4}{4}=\frac{z+4}{7},\text{ is :}$$

1. A.$6\sqrt{11}$
2. B.$\frac{11}{\sqrt{6}}$
3. C.11
4. D.$11\sqrt{6}$

A tetrahedron has vertices $P(1,2,1)$, $Q(2,1,3)$, $R(-1,1,2)$ and $O(0,0,0)$. The angle between the faces OPQ and PQR is :

1. A.${\cos }^{-1}\left(\frac{17}{31}\right)$
2. B.${\cos }^{-1}\left(\frac{9}{35}\right)$
3. C.${\cos }^{-1}\left(\frac{19}{35}\right)$
4. D.${\cos }^{-1}\left(\frac{7}{31}\right)$

Let S be the set of all real values of $\lambda$ such that a plane passing through the points $(-{\lambda }^2,1,1)$, $(1,-{\lambda }^2,1)$ and $(1,1,-{\lambda }^2)$ also passes through the point $(-1,-1,1)$. Then S is equal to :

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

If an angle between the line, $\frac{x+1}{2}=\frac{y-2}{1}=\frac{z-3}{-2}$ and the plane, $x-2y-kz=3$ is ${\cos }^{-1}\left(\frac{2\sqrt{2}}{3}\right)$, then a value of k is :

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

Two lines $\frac{x-3}{1}=\frac{y+1}{3}=\frac{z-6}{-1}$ and $\frac{x+5}{7}=\frac{y-2}{-6}=\frac{z-3}{4}$ intersect at the point R. The reflection of R in the xy-plane has coordinates :

1. A.$(2,4,7)$
2. B.$(2,-4,-7)$
3. C.$(2,-4,7)$
4. D.$(-2,4,7)$

If the point $(2,\alpha ,\beta )$ lies on the plane which passes through the points $(3,4,2)$ and $(7,0,6)$ and is perpendicular to the plane $2x-5y=15$, then $2\alpha -3\beta$ is equal to

1. A.12
2. B.7
3. C.17
4. D.5

The equation of the line passing through $(-4,3,1)$, parallel to the plane $x+2y-z-5=0$ and intersecting the line $\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z-2}{-1}$ is :

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

The plane through the intersection of the planes x + y + z = 1 and 2x + 3y – z + 4 = 0 and parallel to y-axis also passes through the point :

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

The sum of the intercepts on the coordinate axes of the plane passing through the point $(-2,-2,2)$ and containing the line joining the points $(1,-1,2)$ and $(1,1,1)$ is :

1. A.4
2. B.$-4$
3. C.$-8$
4. D.12

A variable plane passes through a fixed point (3,2,1) and meets x, y and z axes at A, B and C respectively. A plane is drawn parallel to yz -plane through A, a second plane is drawn parallel zx-plane through B and a third plane is drawn parallel to xy-plane through C. Then the locus of the point of intersection of these three planes, is :

1. A.$\frac{x}{3}+\frac{y}{2}+\frac{z}{1}=1$
2. B.$x+y+z=6$
3. C.$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{11}{6}$
4. D.$\frac{3}{x}+\frac{2}{y}+\frac{1}{z}=1$

An angle between the plane, $x+y+z=5$ and the line of intersection of the planes, $3x+4y+z-1=0$ and $5x+8y+2z+14=0$, is :

1. A.${\sin }^{-1}\left(\sqrt{\frac{3}{17}}\right)$
2. B.${\cos }^{-1}\left(\sqrt{\frac{3}{17}}\right)$
3. C.${\cos }^{-1}(3/17)$
4. D.${\sin }^{-1}(3/17)$

An aeroplane flying at a constant speed, parallel to the horizontal ground, $\sqrt{3}\text{km}$ above it, is observed at an elevation of $60^{\circ }$ from a point on the ground. If, after five seconds, its elevation from the same point, is $30^{\circ }$, then the speed (in km / hr) of the aeroplane, is :

1. A.1500
2. B.1440
3. C.750
4. D.720

If ${\mathrm{L}}_1$ is the line of intersection of the planes $2x-2y+3z-2=0$, $x-y+z+1=0$ and ${\mathrm{L}}_2$ is the line of intersection of the planes $x+2y-z-3=0$, $3x-y+2z-1=0$, then the distance of the origin from the plane, containing the lines ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$, is :

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

The length of the projection of the line segment joining the points (5, -1, 4) and (4, -1, 3) on the plane, $x+y+z=7$ is :

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

If $x=a,y=b,z=c$ is a solution of the system of linear equations $x+8y+7z=0$ $9x+2y+3z=0$ $x+y+z=0$ such that the point (a, b, c) lies on the plane $x+2y+z=6$, then $2a+b+c$ equals :

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

If a variable plane, at a distance of 3 units from the origin, intersects the coordinate axes at A, B and C, then the locus of the centroid of $\Delta \mathrm{ABC}$ is :

1. A.$\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=1$
2. B.$\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=3$
3. C.$\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=\frac{1}{9}$
4. D.$\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=9$

If the line, $\frac{x-3}{1}=\frac{y+2}{-1}=\frac{z+\lambda }{-2}$ lies in the plane, $2x-4y+3z=2$, then the shortest distance between this line and the line, $\frac{x-1}{12}=\frac{y}{9}=\frac{z}{4}$ is :

1. A.$2$
2. B.$1$
3. C.$0$
4. D.$3$

The coordinates of the foot of the perpendicular from the point $(1,-2,1)$ on the plane containing the lines, $\frac{x+1}{6}=\frac{y-1}{7}=\frac{z-3}{8}$ and $\frac{x-1}{3}=\frac{y-2}{5}=\frac{z-3}{7}$, is :

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

The line of intersection of the planes $\vec{r}\cdot (3\hat{i}-\hat{j}+\hat{k})=1$ and $\vec{r}\cdot (\hat{i}+4\hat{j}-2\hat{k})=2$, is :

1. A.$\frac{x-\frac{4}{7}}{-2}=\frac{y}{7}=\frac{z-\frac{5}{7}}{13}$
2. B.$\frac{x-\frac{4}{7}}{2}=\frac{y}{-7}=\frac{z+\frac{5}{7}}{13}$
3. C.$\frac{x-\frac{6}{13}}{2}=\frac{y-\frac{5}{13}}{-7}=\frac{z}{-13}$
4. D.$\frac{x-\frac{6}{13}}{2}=\frac{y-\frac{5}{13}}{7}=\frac{z}{-13}$

If the line, $\frac{x-3}{2}=\frac{y+2}{-1}=\frac{z+4}{3}$ lies in the planes, $lx+my-z=9$, then $l^2+m^2$ is equal to :

1. A.$5$
2. B.$2$
3. C.$26$
4. D.$18$

The distance of the point $(1,-5,9)$ from the plane $x-y+z=5$ measured along the line $x=y=z$ is :

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

The equation of the plane containing the line $2x-5y+z=3$; $x+y+4z=5$, and parallel to the plane, $x+3y+6z=1$, is :

1. A.$x+3y+6z=7$
2. B.$2x+6y+12z=-13$
3. C.$2x+6y+12z=13$
4. D.$x+3y+6z=-7$