If $l_1, m_1, n_1$ and $l_2, m_2, n_2$ are direction cosines of lines $L_1$ and $L_2$ respectively and $\theta$ is the acute angle between them, then :
(A) $\cos \theta = l_1 l_2 + m_1 m_2 + n_1 n_2$
(B) $\sin \theta = l_1 l_2 + m_1 m_2 + n_1 n_2$
(C) $\tan \theta = \frac{l_1}{l_2} + \frac{m_1}{m_2}...
CBSE CLASS 12 • MATHEMATICS • CHAPTER 11
Three-Dimensional Geometry
65 distinct questions across 94 board paper set appearances (2022–2026).
Direction ratios of a vector parallel to line $\frac{x-1}{2} = -y = \frac{2z+1}{6}$ are : (A) 2, − 1, 6 (B) 2, 1, 6 (C) 2, 1, 3 (D) 2, − 1, 3
Assertion (A) : If a line makes angles $\alpha, \beta, \gamma$ with positive direction of the coordinate axes, then $\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 2$.
Reason (R) : The sum of squares of the direction cosines of a line is $1$.
The solution set of the inequation $3x + 5y < 7$ is :
(a) whole $xy$-plane except the points lying on the line $3x + 5y = 7$.
(b) whole $xy$-plane along with the points lying on the line $3x + 5y = 7$.
(c) open half plane containing the origin except the points of line $3x + 5y = 7$.
(d) open half p...
(a) Find the vector equation of the line passing through the point $(2, 1, 3)$ and perpendicular to both the lines
$$\frac{x - 1}{1} = \frac{y - 2}{2} = \frac{z - 3}{3}; \frac{x}{-3} = \frac{y}{2} = \frac{z}{5}.$$
OR
(b) The equations of a line are $5x - 3 = 15y + 7 = 3 - 10z$. Write the direction c...
Find the values of $\lambda$, for which the distance of point $(2, 1, \lambda)$ from plane $3x + 5y + 4z = 11$ is $2\sqrt{2}$ units.
Find the equation of the plane passing through points (2, 1, 0), (3, -2, -2) and (1, 1, -7). Also, obtain its distance from the origin.
The foot of a perpendicular drawn from the point $(-2, -1, -3)$ on a plane is $(1, -3, 3)$. Find the equation of the plane.
(a) If a line makes $60^\circ$ and $45^\circ$ angles with the positive directions of $x$-axis and $z$-axis respectively, then find the angle that it makes with the positive direction of $y$-axis. Hence, write the direction cosines of the line. \
OR \
(b) Check whether the lines $\frac{x - 1}{2} = \f...
Assertion (A) : A line can have direction cosines $<1, 1, 1>$
Reason (R) : $\cos heta = 1$ is possible for $ heta = 0$.
In a school, the swimming pool is shaped as a cuboid, and the breadth of the pool is represented by line $\frac{-x + 2}{-5} = y = z - 3$. Two students A and B are swimming along the length of the pool.
Based on the above information, answer the following questions :
(i) If the path of student A is r...
If the direction cosines of a line are $\sqrt{3} \text{ k}, \sqrt{3} \text{ k}, \sqrt{3} \text{ k}$, then the value of k is :
(A) $\pm 1$
(B) $\pm \sqrt{3}$
(C) $\pm 3$
(D) $\pm \frac{1}{3}$
An engineer is designing a new metro rail network in a city.
Initially, two metro lines, Line A and Line B, each consisting of multiple stations are designed. The track for Line A is represented by $l_1 : \frac{x - 2}{3} = \frac{y + 1}{-2} = \frac{z - 3}{4}$, while the track for Line B is represente...
(i) Find the probability that the airplane will not crash.
If a line makes angles of $90^\circ, 135^\circ$ and $45^\circ$ with the $x, y$ and $z$ axes respectively, then its direction cosines are
(a) $0, -\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}$
(b) $-\frac{1}{\sqrt{2}}, 0, \frac{1}{\sqrt{2}}$
(c) $\frac{1}{\sqrt{2}}, 0, -\frac{1}{\sqrt{2}}$
(d) $0, \frac{1}...
The point which lies in the half-plane $2x + y - 4 \le 0$ is :
(a) $(0, 8)$
(b) $(1, 1)$
(c) $(5, 5)$
(d) $(2, 2)$
(a) Show that the lines $\frac{x + 1}{3} = \frac{y + 3}{5} = \frac{z + 5}{7}$ and $\frac{x - 2}{1} = \frac{y - 4}{3} = \frac{z - 6}{5}$ intersect. Also find their point of intersection.
OR
(b) Find the shortest distance between the pair of lines $\frac{x - 1}{2} = \frac{y + 1}{3} = z$ and $\frac{x +...
Find the distance of the point $(-1, -5, -10)$ from the point of intersection of the line $\frac{x - 2}{3} = \frac{y + 1}{4} = \frac{z - 2}{12}$ and the plane $x - y + z = 5$.
If the distance of the point $(1, 1, 1)$ from the plane $x - y + z + \lambda = 0$ is $\frac{5}{\sqrt{3}}$, find the value(s) of $\lambda$.
(b) Find the coordinates of the point where the line through the points (- 1, 1, - 8) and (5, - 2, 10) crosses the ZX-plane.
Direction cosines of the line given by equations : $\frac{2x - 1}{4} = \frac{1 - y}{3} = \frac{-z}{6}$ are
(A) $2, -3, -6$
(B) $\frac{2}{7}, \frac{-3}{7}, \frac{-6}{7}$
(C) $\frac{2}{7}, \frac{-3}{7}, \frac{6}{7}$
(D) $\frac{4}{\sqrt{61}}, \frac{-3}{\sqrt{61}}, \frac{-6}{\sqrt{61}}$
Assertion (A) : A line can have direction cosines $<1, 1, 1>$
Reason (R) : $\cos \theta = 1$ is possible for $\theta = 0$.
Direction cosines of line $x = y = 1 - z$ are
(A) $1, 1, 1$
(B) $\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{-1}{\sqrt{3}}$
(C) $0, 0, 1$
(D) $\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}$
Direction cosines of line $\frac{1-x}{0} = y = z$ are
(A) $1, 1, 1$
(B) $0, \frac{-1}{\sqrt{2}}, \frac{-1}{\sqrt{2}}$
(C) $1, 0, 0$
(D) $0, -1, -1$
Direction ratios of lines $l_1$ and $l_2$ are $<12, -3, 9>$ and $<4, q, -p>$ respectively. The values of p and q for which $l_1$ and $l_2$ are parallel are respectively :
(A) $-1, 3$
(B) $3, 1$
(C) $-3, -1$
(D) $-1, -3$
Direction ratios of lines $l_1$ and $l_2$ respectively are $<1, -2, 3>$ and $<-2, p, -6>$. The value of p for which $l_1 \parallel l_2$, is :
(A) $-4$
(B) $4$
(C) $-10$
(D) $10$
Direction ratios of lines $l_1$ and $l_2$ respectively are $<1, 0, 0>$ and $<0, -1, 0>$. The direction ratios of the line perpendicular to both $l_1$ and $l_2$ are :
(A) $<1, 1, 0>$
(B) $<0, 0, -1>$
(C) $<1, 1, 1>$
(D) $<1, 0, -1>$
If the lines $\frac{x-3}{1} = \frac{1-y}{1} = \frac{z+2}{p}$ and $\frac{2-x}{3} = \frac{y+1}{5} = \frac{z+56}{2p}$ are perpendicular to each other, then find the value(s) of $p$.
OR
Find the angle between the lines whose direction ratios are $1, 1, 2$ and $\sqrt{3}-1, -\sqrt{3}-1, 4$.
Assertion ($A$) : If a line makes angles $45^\circ$, $90^\circ$ and $45^\circ$ with the positive $x$, $y$ and $z$-axis respectively, then direction cosines of the line are $<\frac{1}{\sqrt{2}}, 0, \frac{1}{\sqrt{2}}>$.
Reason ($R$) : If a line makes angles $\alpha$, $\beta$, $\gamma$ with positive a...
(iii) (b) If a bird flies across the top of the swimming pool with its path represented by $\frac{x + 4}{2} = \frac{y + 3}{8} = \frac{z - 5}{-18}$, then find the shortest distance between its path and the path of student A.
The graph of the inequality $3x + 2y > 6$ is :
(A) entire XOY plane
(B) whole XOY plane excluding the points on the line $3x + 2y = 6$
(C) half plane that contains the origin
(D) half plane that neither contains the origin nor the points on the line $3x + 2y = 6$
The solution set of the inequality $5x + 4y < 7$ is :
(A) Open half-plane containing the origin.
(B) Whole $xy$-plane except the points lying on the line $5x + 4y = 7$.
(C) Open half-plane not containing the origin.
(D) Closed half-plane not containing the origin.
Assertion (A) : $\langle \frac{1}{\sqrt{5}}, \frac{1}{\sqrt{5}}, \frac{1}{\sqrt{5}} \rangle$ cannot be the direction cosines of a line.
Reason (R) : If $l$, $m$, $n$ are the direction cosines of a line, then $l^2 + m^2 + n^2 = 1$.
If the equations of a line are $\frac{x - 2}{2} = \frac{2y - 5}{-3}$, $z = -1$, then find the direction ratios of the line and also a point on the line.
Case Study - 2
According to recent research, air turbulence has increased in various regions around the world due to climate change. Turbulence makes flights bumpy and often delays the flights.
Assume that, an airplane observes severe turbulence, moderate turbulence or light turbulence with equal ...
If the direction cosines of a line are $\lambda, \lambda, \lambda$, then $\lambda$ is equal to : (A) $-\frac{1}{\sqrt{3}}$ (B) 1 (C) $\frac{1}{\sqrt{3}}$ (D) $\pm\frac{1}{\sqrt{3}}$
Case Study - 2
An engineer is designing a new metro rail network in a city.
Initially, two metro lines, Line A and Line B, each consisting of multiple stations are designed. The track for Line A is represented by $l_1 : \frac{x-2}{3} = \frac{y+1}{-2} = \frac{z-3}{4}$, while the track for Line B is r...
According to recent research, air turbulence has increased in various regions around the world due to climate change. Turbulence makes flights bumpy and often delays the flights. Assume that, an airplane observes severe turbulence, moderate turbulence or light turbulence with equal probabilities. Fu...
A student observes an open-air Honeybee nest on the branch of a tree, whose plane figure is parabolic shape given by $x^2 = 4y$. Then the area (in sq units) of the region bounded by parabola $x^2 = 4y$ and the line $y = 4$ is
(A) $\frac{32}{3}$
(B) $\frac{64}{3}$
(C) $\frac{128}{3}$
(D) $\frac{256}{...
Case Study - 3
Airplanes are by far the safest mode of transportation when the number of transported passengers are measured against personal injuries and fatality totals.
Previous records state that the probability of an airplane crash is $0.00001%$. Further, there are $95%$ chances that there wil...
The Cartesian equations of a line are given as $6\text{x} - 2 = 3\text{y} + 1 = 2\text{z} - 2$
The direction ratios of the line are :
(A) $2, -1, 3$
(B) $1, -2, -3$
(C) $1, 2, 3$
(D) $3, 1, 2$
The solution set of the inequation $2\text{x} + 3\text{y} < 6$ is :
(A) open half-plane not containing origin
(B) whole xy-plane except the points lying on the line $2\text{x} + 3\text{y} = 6$
(C) open half-plane containing origin
(D) half-plane containing the origin and the points lying on the line...
(a) Write the vector equations of the following lines and hence find the shortest distance between them : $\frac{x + 1}{2} = \frac{y + 1}{- 6} = \frac{z + 1}{1}$ and $\frac{x - 3}{1} = \frac{y - 5}{- 2} = \frac{z - 7}{1}$
The direction ratios of the line $\frac{x-1}{3}=\frac{2-y}{1}=\frac{3z}{2}$ are:
Direction cosines of the line $\frac{x - 1}{2} = \frac{1 - y}{3} = \frac{2z - 1}{12}$ are :
Direction cosines of a line perpendicular to both x-axis and z-axis are :
(a) 1, 0, 1
(b) 1, 1, 1
(c) 0, 0, 1
(d) 0, 1, 0
The direction ratios of a line parallel to z-axis are :
(a) $< 1, 1, 0 >$
(b) $< 1, 1, 1 >$
(c) $< 0, 0, 0 >$
(d) $< 0, 0, 1 >$
If the direction cosines of a line are $\langle \frac{1}{c}, \frac{1}{c}, \frac{1}{c} \rangle$, then
(a) $0 < c < 1$
(b) $c > 2$
(c) $c = \pm\sqrt{2}$
(d) $c = \pm\sqrt{3}$
If the equation of a line is $x = ay + b$, $z = cy + d$, then find the direction ratios of the line and a point on the line.
Find the equation of the plane passing through the points $(2, 2, -1)$, $(3, 4, 2)$ and $(7, 0, 6)$. Also, find the distance of this plane from the point $(4, 3, 1)$.
The solution set of the inequation $2x + y \geq 5$ is :
(a) half plane that contains the origin
(b) open half plane not containing the origin and not containing the points on the line $2x + y = 5$.
(c) whole xy-plane except the points lying on the line $2x + y = 5$.
(d) open half plane not containin...
If the direction cosines of a line are $\left(\frac{1}{a}, \frac{1}{a}, \frac{1}{a}\right)$, then :
(a) $0 < a < 1$
(b) $a > 2$
(c) $a > 0$
(d) $a = \pm\sqrt{3}$
If the direction cosines of a line are $\left$ \frac{1}{a}, \frac{1}{a}, \frac{1}{a}
ight$$, then :
(a) $0 < a < 1$
(b) $a > 2$
(c) $a > 0$
(d) $a = \pm \sqrt{3}$
The point $(x, y, 0)$ on the $xy$-plane divides the line segment joining the points $(1, 2, 3)$ and $(3, 2, 1)$ in the ratio :
(a) $1 : 2$ internally
(b) $2 : 1$ internally
(c) $3 : 1$ internally
(d) $3 : 1$ externally
The point which lies in the half-plane 2x + y - 4 $\leq$ 0 is :
(a) (0, 8)
(b) (1, 1)
(c) (5, 5)
(d) (2, 2)
Find the shortest distance between the pair of lines $\frac{x - 1}{2} = \frac{y + 1}{3} = z$ and $\frac{x + 1}{5} = \frac{y - 2}{1} ; z = 2$.
The direction ratios of a line whose Cartesian equations are $3x - 3 = 2y + 1 = 5 - 6z$, are :
(a) $2, 3, -1$
(b) $3, -2, 1$
(c) $2, 1, -3$
(d) $3, 2, -1$
Find the co-ordinates of point where the line $\frac{x - 3}{-1} = \frac{y + 4}{1} = \frac{z + 5}{6}$ crosses the plane passing through points $(\frac{7}{2}, 0, 0)$, $(0, 7, 0)$ and $(0, 0, 7)$.
Find the direction ratio and direction cosines of a line parallel to the line whose equations are $6x - 12 = 3y + 9 = 2z - 2$
Case-Study 3: Read the following passage and answer the questions given below.
There are two antiaircraft guns, named as A and B. The probabilities that the shell fired from them hits an airplane are $0.3$ and $0.2$ respectively. Both of them fired one shell at an airplane at the same time.
(i) What...
Find the coordinates of the point where the line through $(5, 1, 6)$ and $(3, 4, 1)$ crosses the zx-plane.
If the distance of the point $(1, 1, 1)$ from the plane $x - y + z + \lambda = 0$ is $\frac{5}{\sqrt{3}}$, find the value(s) of $\lambda$.
Find the equation of the plane passing through three points whose position vectors are -ĵ, 3î + 3ĵ and î + ĵ + ĵ.
Find the distance between the point $(3, 4, 5)$ and the point where the line $\frac{x - 3}{1} = \frac{y - 4}{2} = \frac{z - 5}{2}$ meets the plane $x + y + z = 17$.
Find the equation of the plane passing through the points P(4, 3, 4), Q(5, 3, 1) and R(7, 6, 2). Hence, find the distance of this plane from origin.