Show that if x 2 + y 2 = 1 , then the point ( x , y , 1 − x 2 − y 2 ) is at a distance of 1 unit from the origin.

This is a short answer type question as classified in NCERT Exemplar Given is (x,y,1−x2−y2)∴ Distance between the origin and the point is=(x−0)2+(y−0)2+(1−x2−y2−0)2 =1=1Hence proved.

Name the octant in which each of the following points lies: i. (1, 2, 3), ii. (4, –2, 3), iii. (4, –2, –5), iv. (4, 2, –5), v. (–4, 2, 5), vi. (–3, –1, 6), vii. (2, –4, –7), viii. (–4, 2, –5).

This is a short answer type question as classified in NCERT Exemplar (i)Point(1,2,3) lies in IOctant (ii)Point(4,−2,3) lies in IVOctant (iii)Point(4,−2,−5) lies in VIII Octant (iv)Point(4,2,−5) lies in VOctant (v)Point(−4,2,5) lies in IIOctant (vi)Point(−3,−1,6) lies in IIIOctant (vii)Point (2,−4,−7) lies in VIIIOctant (viii)Point(−4,2,−5) lies in VIOctant

Let A , B , C be the feet of perpendiculars from a point P on the x -, y -, and z -axes, respectively. Find the coordinates of A , B , C in each of the following where the point P is: i. A = (3, 4, 2), ii. (–5, 3, 7), iii. (4, –3, –5).

This is a short answer type question as classified in NCERT Exemplar The coordinates of A, B and C are (i)A (3, 0, 0), B (0, 4, 0) and C (0, 0, 2) (ii)A (−5, 0, 0), B (0, 3, 0) and C (0, 0, 7) (iii)A (4, 0, 0), B (0, −3, 0) and C (0, 0, −5)

Let A , B , C be the feet of perpendiculars from a point P on the x y -, y z -, and z x -planes, respectively. Find the coordinates of A , B , C in each of the following where the point P is: i. (3, 4, 5), ii. (–5, 3, 7) iii. (4, –3, –5).

This is a short answer type question as classified in NCERT Exemplar T h e c o o r d i n a t e s o f A , B a n d C a r e ( i ) A ( 3 , 4 , 0 ) , B ( 0 , 4 , 5 ) a n d C ( 3 , 0 , 5 ) ( i i ) A ( − 5 , 3 , 0 ) , B ( 0 , 3 , 7 ) a n d C ( − 5 , 0 , 7 ) ( i i i ) A ( 4 , − 3 , 0 ) , B ( 0 , − 3 , − 5 ) a n d C ( 4 , 0 , − 5 )

How far apart are the points (2, 0, 0) and (–3, 0, 0)?

This is a short answer type question as classified in NCERT Exemplar Given points are (2, 0, 0), and (−3, 0, 0)∴ Distance between the given = (2+3)2+ (0−0)2+ (0−0)2=25=5Hence, the required =5.

Match each item given under Column C1 to its correct answer given under Column C2:

This is a matching answer type question as classified in NCERT Exemplar (a)In xy-plane, z-coordinate is zero. H e n c e , ( a ) ↔ ( i i i ) . (b)The point (2,3,4) lies in first octant. H e n c e , ( b ) ↔ ( i ) . (c)Locus of the points with x-coordinate is zero is yz-plane. H e n c e , ( c ) ↔ ( i i ) . (d)A line is parallel to x-axis if and only if all the points on the line have equal y and z-coordinates. H e n c e , ( d ) ↔ ( v i ) . (e)x=0, y=0 represents z-axis. H e n c e , ( e ) ↔ ( i v ) . (f)z=c represent a plane parallel to xy-plane. H e n c e , ( f ) ↔ ( v ) . (g)The planes x=a, y=b represent the line parallel to z-axis. H e n c e , ( g ) ↔ ( v i i i ) . (h)Coordinates of a point are the distances from the origin to the feet of perpendicular from the point on the respective axes, H e n c e , ( h ) ↔ ( v i i ) . ( i ) A b a l l i s s o l i d r e g i o n i n t h e s p a c e e n c l o s e d b y a s p h e r e . H e n c e , ( i ) ↔ ( x ) . ( j ) T h e r e g i o n i n t h e p l a n e e n c l o s e d b y a c i r c l e i s k n o w n a s a d i s c . H e n c e , ( i ) ↔ ( i x ) .

If A = { x ∈ R ; | x − 2 | > 1 } , B = { x ∈ R : x 2 − 3 > 1 } , C = { x ∈ R ; | x − 4 | ≥ 2 } and Z is the set of all integers, then the number of subsets of the set ( A ∩ B ∩ C ) C ∩ Z i s ________

| x − 2 | > 1 g i v e s x − 2 1 i.e. x 3 . (i) represent set A x 2 − 3 > 1 g i v e s x 2 − 3 > 1 o r x 2 − 4 > 0 x 2 . (ii) represent set B | x − 4 | ≥ 2 g i v e s x − 4 ≤ − 2 o r x − 4 ≥ 2 x ≤ 2 o r x ≥ 6 . (iii)respect set C so number of subset = 28 = 256

Let the equation x 2 + y 2 + p x + ( 1 − p ) y + 5 = 0 represent circles of varying radius r ∈ (0, 5]. Then the number of elements in the set S = {q : q = p2 and q is an integer) is________

r = ( p 2 ) 2 + ( 1 − p 2 ) 2 − 5 = p 2 + 1 + p 2 − 2 p − 2 0 4 = 2 p 2 − 2 p − 1 9 2 Since r ∈ ( 0 , 5 ) s o 0 ⇒ 2 p 2 − 2 p − 1 9 > 0 & 2 p 2 − 2 p − 1 9 P ∈ [ 1 − 2 3 9 2 , 1 − 3 9 2 ) ∪ ( 1 + 3 9 2 , 1 + 2 3 9 2 ] So number of integral values of P2 is 61.

The number of distance real roots of the equation 3 x 4 + 4 x 3 − 1 2 x 2 + 4 = 0 is________

Let f (x) = 3x4 + 4x3 – 12x2 + 4 So, f’ (x) = 12x3 + 12x2 – 24x = 12x (x2 + x – 2) =12x (x + 2) (x – 1) So, number of distinct real roots = 4

Let a → = i ^ + 5 j ^ + α k ^ , b → = i ^ + 3 j ^ + β k ^ a n d c → = − i ^ + 2 j ^ − 3 k ^ be three vectors such that, | b → × c → | = 5 3 and a → is perpendicular to b → . Then the greatest among the value of | a → | 2 is

b → × c → = | i ^ j ^ k ^ 1 3 β − 1 2 − 3 | = i ^ ( − 9 − 2 β ) − j ^ ( − 3 + β ) + k ^ ( 5 ) | b → × c → | = 5 3 g i v e s ( 9 − 2 β ) 2 + ( β − 3 ) 2 + 2 5 = 5 3 8 1 + 4 β 2 + 3 6 β + β 2 + 9 − 6 β + 2 5 = 7 5 β = − 2 , − 4 also a → ⊥ b → s o a → . b → = 0 i . e . 1 + 1 5 + α β = 0 So, | a → | 2 = 1 + 2 5 + α 2 = 9 0

If the minimum area of the triangle formed by a tangent to the ellipse x 2 b 2 + y 2 4 a 2 = 1 and the co-ordinate axis is kab, then k is equal to________

Equation of tangent to given ellipse at P : x . c o s θ b + y . s i n θ 2 a = 1 A (b sec θ. 0) 7 B (0, 2a cosec θ) area of Δ O A B = 2 a b 2 s i n θ c o s θ = 2 a b s i n 2 θ For minimum area sin 2θ = 1 So minimum area = 2ab =>k = 2

Distance of the point ( 3 , 4 , 5 ) from the origin ( 0 , 0 , 0 ) is (a) 5 0 (b) 3 (c) 4 (d) 5

This is an Objective Type Questions as classified in NCERT Exemplar Given points A (3, 4, 5) and the given O (0, 0, 0)∴ OA= (3−0)2+ (4−0)2+ (5−0)2=9+16+25=50Hence, the correct option is (a).

If the distance between the points (a,0,1) and (0,1,2) is 27 , then the value of a is (a) 5 (b) ± 5 (c) –5 (d) None of these

This is an Objective Type Questions as classified in NCERT Exemplar Let the given points be A(a,0,1) and B(0,1,2)∴ AB=(a−0)2+(0−1)2+(1−2)2 27=a2+1+1Squaring both sides, we get 27=a2+2 ⇒a2=25 ∴a=±5Hence, the correct option is (b).

Let f(x) = { x 3 − x 2 + 1 0 x − 7 , x ≤ 1 − 2 x + l o g 2 ( b 2 − 4 ) , x > 1 . Then the set of all values of b, for which f(x) has maximum value at x = 1, is:

[ − 6 , − 2 ) ∪ ( 2 , 6 ]

If x 2 + 9 y 2 − 4 x + 3 = 0 , x , y ∈ R ,  then x and y respectively lie in the intervals:

[1, 3] and  [ − 1 3 , 1 3 ]

[ − 1 3 , 1 3 ] a n d [ − 1 3 , 1 3 ]

The statement ( ∼ ( p ⇔ ∼ q ) ) ∧ q i s :

equivalent to ( p ⇒ q ) ∧ p

equivalent to ( p ⇒ q ) ∧ q

Let O be the origin and A be the point z 1 = 1 + 2i. If B is the point z 2 , Re(z 2 ) < 0, such that OAB is a right angled isosceles triangle with OB as hypotenuses, then which of the following is NOT true?

arg(z 1 – 2z 2 ) = -tan -1 <!-- [if gte vml 1]> 4 3

Maths NCERT Exemplar Solutions Class 11th Chapter Twelve: Overview, Questions, Preparation

Updated on Aug 6, 2025 10:57 IST

By Raj Pandey

Table of contents

Introduction to Three Dimensional Geometry Short Answer Type Questions
Introduction to Three Dimensional Geometry Long Answer Type Questions
Introduction to Three Dimensional Geometry Objective Type Questions
Introduction to Three Dimensional Geometry Fill in the blank Type Questions
Introduction to Three Dimensional Geometry Matching Type Questions
JEE Mains 2021

Introduction to Three Dimensional Geometry Long Answer Type Questions

1. Show that the three points $A (2, 3, 4)$ , $B (1, 2, 3)$ , and $C (4, 1, 10)$ are collinear and find the ratio in which $C$ divides $A B$ .

Sol:

$\begin{array}{l} G i v e n points a r e A (2, 3, 4), B (- 1, 2, - 3) a n d C (- 4, 1, - 10) \\ A B = \sqrt[]{{(2 + 1)}^{2} + {(3 - 2)}^{2} + {(4 + 3)}^{2}} = \sqrt[]{9 + 1 + 49} = \sqrt[]{59} \\ B C = \sqrt[]{{(- 1 + 4)}^{2} + {(2 - 1)}^{2} + {(- 3 + 10)}^{2}} = \sqrt[]{9 + 1 + 49} = \sqrt[]{59} \\ A C = \sqrt[]{{(2 + 4)}^{2} + {(3 - 1)}^{2} + {(4 + 10)}^{2}} = \sqrt[]{36 + 4 + 196} = \sqrt[]{236} = 2 \sqrt[]{59} \\ ∴ A B + B C = A C \\ \sqrt[]{59} + \sqrt[]{59} = 2 \sqrt[]{59} \\ H e n c e, A, B a n d C a r e c o l l i n e a r a n d A C : B C = 2 \sqrt[]{59} : \sqrt[]{59} = 2 : 1 \\ H e n c e, C d i v i d e s A B i s 2 : 1 e x t e r n a l l y . \end{array}$

2. Prove that the points $(0, 1, 7)$ , $(2, 1, 9)$ , and $(6, 5, 13)$ are collinear. Find the ratio in which the first point divides the join of the other two.

\begin{array}{l} L e t t h e g i v e n points a r e A (0, - 1, - 7), B (2, 1, - 9), C (6, 5, - 13) \\ A B = \sqrt[]{{(2 - 0)}^{2} + {(1 + 1)}^{2} + {(- 9 + 7)}^{2}} = \sqrt[]{4 + 4 + 4} = \sqrt[]{12} = 2 \sqrt[]{3} \\ B C = \sqrt[]{{(6 - 2)}^{2} + {(5 - 1)}^{2} + {(- 13 + 9)}^{2}} = \sqrt[]{16 + 16 + 16} = \sqrt[]{48} = 4 \sqrt[]{3} \\ A C = \sqrt[]{{(6 - 0)}^{2} + {(5 + 1)}^{2} + {(- 13 + 7)}^{2}} = \sqrt[]{36 + 36 + 36} = \sqrt[]{108} = 6 \sqrt[]{3} \\ 2 \sqrt[]{3} + 4 \sqrt[]{3} = 6 \sqrt[]{3} \\ i . e ., A B + B C = A C \\ ∴ A B : A C = 2 \sqrt[]{3} : 6 \sqrt[]{3} = 1 : 3 \\ H e n c e, point A d i v i d e s B a n d C i n 1 : 3 e x t e r n a l l y . \end{array}

Commonly asked questions

Show that the three points $A (2, 3, 4)$ , $B (1, 2, 3)$ , and $C (4, 1, 10)$ are collinear and find the ratio in which $C$ divides $A B$ .

This is a short answer type question as classified in NCERT Exemplar

Prove that the points $(0, 1, 7)$ , $(2, 1, 9)$ , and $(6, 5, 13)$ are collinear. Find the ratio in which the first point divides the join of the other two.

This is a long answer type question as classified in NCERT Exemplar

$\begin{array}{l} L e t t h e g i v e n points a r e A (0, - 1, - 7), B (2, 1, - 9), C (6, 5, - 13) \\ A B = \sqrt[]{{(2 - 0)}^{2} + {(1 + 1)}^{2} + {(- 9 + 7)}^{2}} = \sqrt[]{4 + 4 + 4} = \sqrt[]{12} = 2 \sqrt[]{3} \\ B C = \sqrt[]{{(6 - 2)}^{2} + {(5 - 1)}^{2} + {(- 13 + 9)}^{2}} = \sqrt[]{16 + 16 + 16} = \sqrt[]{48} = 4 \sqrt[]{3} \\ A C = \sqrt[]{{(6 - 0)}^{2} + {(5 + 1)}^{2} + {(- 13 + 7)}^{2}} = \sqrt[]{36 + 36 + 36} = \sqrt[]{108} = 6 \sqrt[]{3} \\ 2 \sqrt[]{3} + 4 \sqrt[]{3} = 6 \sqrt[]{3} \\ i . e ., A B + B C = A C \\ ∴ A B : A C = 2 \sqrt[]{3} : 6 \sqrt[]{3} = 1 : 3 \\ H e n c e, point A d i v i d e s B a n d C i n 1 : 3 e x t e r n a l l y . \end{array}$

What are the coordinates of the vertices of a cube whose edge is 2 units, one of whose vertices coincides with the origin and the three edges passing through the origin, coincides with the positive direction of the axes through the origin?

This is a long answer type question as classified in NCERT Exemplar

$\begin{array}{l} G i v e n t h a t e a c h e d g e o f t h e c u b o i d i s 2 u n i t s . \\ ∴ C o o r d i n a t e s o f t h e v e r t i c e s a r e \\ A (2, 0, 0), B (2, 2, 0), C (0, 2, 0), D (0, 2, 2), E (0, 0, 2), F (2, 0, 2), G (2, 2, 2) a n d O (0, 0, 0) . \end{array}$

Introduction to Three Dimensional Geometry Objective Type Questions

1. The distance of point $P (3, 4, 5)$ from the $y z$ -plane is

(a) 3 units

(b) 4 units

(c) 5 units

(d) 550

Sol.

$\begin{array}{l} G i v e n point i s P (3, 4, 5) \\ ∴ Distance o f P f r o m y z p l a n e = \sqrt[]{{(0 - 3)}^{2} + {(4 - 4)}^{2} + {(5 - 5)}^{2}} = \sqrt[]{9} = 3 u n i t s \\ H e n c e, t h e c o r r e c t o p t i o n i s (a) . \end{array}$

2. What is the length of the foot of the perpendicular drawn from the point $P (3, 4, 5)$ on the $y$ -axis?

(a) $\sqrt[]{41}$

(b) $\sqrt[]{34}$

(c) 5

(d) None of these

Sol.

\begin{array}{l} O n y - a x i s, x = 0 a n d z = 0 g i v e n point i s P (3, 4, 5) \\ ∴ T h e point A i s (0, 4, 0) \\ ∴ P A = \sqrt[]{{(0 - 3)}^{2} + {(4 - 4)}^{2} + {(0 - 5)}^{2}} = \sqrt[]{9 + 0 + 25} = \sqrt[]{34} \\ H e n c e, t h e c o r r e c t o p t i o n i s (b) . \end{array}

Commonly asked questions

The distance of point $P (3, 4, 5)$ from the $y z$ -plane is

(a) 3 units

(b) 4 units

(c) 5 units

(d) 550

This is an Objective Type Questions as classified in NCERT Exemplar

What is the length of the foot of the perpendicular drawn from the point $P (3, 4, 5)$ on the $y$ -axis?

(a) $\sqrt[]{41}$

(b) $\sqrt[]{34}$

(c) 5

(d) None of these

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l} O n y - a x i s, x = 0 a n d z = 0 g i v e n point i s P (3, 4, 5) \\ ∴ T h e point A i s (0, 4, 0) \\ ∴ P A = \sqrt[]{{(0 - 3)}^{2} + {(4 - 4)}^{2} + {(0 - 5)}^{2}} = \sqrt[]{9 + 0 + 25} = \sqrt[]{34} \\ H e n c e, t h e c o r r e c t o p t i o n i s (b) . \end{array}$

Distance of the point $(3, 4, 5)$ from the origin $(0, 0, 0)$ is

(a) $\sqrt[]{50}$

(b) 3

(c) 4

(d) 5

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l} G i v e n points A (3, 4, 5) a n d t h e g i v e n O (0, 0, 0) \\ ∴ O A = \sqrt[]{{(3 - 0)}^{2} + {(4 - 0)}^{2} + {(5 - 0)}^{2}} = \sqrt[]{9 + 16 + 25} = \sqrt[]{50} \\ H e n c e, t h e c o r r e c t o p t i o n i s (a) . \end{array}$

If the distance between the points $(a, 0, 1)$ and $(0, 1, 2)$ is $\sqrt[]{27}$ , then the value of $a$ is

(a) 5

(b) $\pm 5$

(c) –5

(d) None of these

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l} L e t t h e g i v e n points b e A (a, 0, 1) a n d B (0, 1, 2) \\ ∴ A B = \sqrt[]{{(a - 0)}^{2} + {(0 - 1)}^{2} + {(1 - 2)}^{2}} \\ \sqrt[]{27} = \sqrt[]{a^{2} + 1 + 1} \\ S q u a r i n g b o t h s i d e s, w e g e t \\ 27 = a^{2} + 2 \Rightarrow a^{2} = 25 ∴ a = \pm 5 \\ H e n c e, t h e c o r r e c t o p t i o n i s (b) . \end{array}$

$x$ -axis is the intersection of two planes

(a) $x y$ and $x z$

(b) $y z$ and $z x$

(c) $x y$ and $y z$

(d) None of these

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l} W e k n o w t h a t o n t h e x y a n d x z p l a n e s, t h e l i n e o f intersection i s x a x i s . \\ H e n c e, t h e c o r r e c t o p t i o n i s (a) . \end{array}$

Equation of $y$ -axis is considered as

(a) $x = 0, y = 0$

(b) $y = 0, z = 0$

(c)

(d) None of these

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l} O n y - a x i s, x = 0 a n d z = 0 \\ H e n c e, t h e c o r r e c t o p t i o n i s (c) . \end{array}$

The point $(2, 3, 4)$ lies in the

(a) First octant

(b) Seventh octant

(c) Second octant

(d) Eighth octant

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l} T h e point (- 2, - 3, - 4) l i e s i n s e v e n t h octant . \\ H e n c e, t h e c o r r e c t o p t i o n i s (b) . \end{array}$

A plane is parallel to yz-plane so it is perpendicular to:

(a) x -axis

(b) y-axis

(c) z-axis

(d) None of these

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l} A n y p l a n e p a r a l l e l t o y z - p l a n e i s p e r p e n d i c u l a r t o x - a x i s . \\ H e n c e, t h e c o r r e c t o p t i o n i s (a) . \end{array}$

The locus of a point for which $y = 0, z = 0$ is

(a) Equation of x-axis

(b) Equation of y-axis

(c) Equation of z-axis

(d) None of these

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l} W e k n o w t h a t o n e e q u a t i o n o f x a x i s, y = 0, z = 0 \\ H e n c e, t h e l o c u s o f t h e point i s e q u a t i o n o f x a x i s . \\ H e n c e, t h e c o r r e c t o p t i o n i s (a) . \end{array}$

The locus of a point for which $x = 0$ is

(a) $x y$ -plane

(b) $y z$ -plane

(c) $z x$ -plane

(d) None of these

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l} O n t h e y z p l a n e, x = 0 \\ H e n c e, t h e l o c u s o f t h e point i s y z p l a n e . \\ H e n c e, t h e c o r r e c t o p t i o n i s (b) . \end{array}$

If a parallelepiped is formed by planes drawn through the points $(5, 8, 10)$ and $(3, 6, 8)$ parallel to the coordinate planes, then the length of the diagonal of the parallelepiped is

(a) $2 \sqrt[]{3}$

(b) $3 \sqrt[]{2}$

(c) $\sqrt[]{2}$

(d) $\sqrt[]{3}$

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l} G i v e n points a r e A (5, 8, 10) a n d B (3, 6, 8) \\ ∴ A B = \sqrt[]{{(5 - 3)}^{2} + {(8 - 6)}^{2} + {(10 - 8)}^{2}} = \sqrt[]{4 + 4 + 4} = \sqrt[]{12} = 2 \sqrt[]{3} \\ H e n c e, t h e c o r r e c t o p t i o n i s (a) . \end{array}$

$L$ is the foot of the perpendicular drawn from a point $P (3, 4, 5)$ on the $x y$ -plane. The coordinates of point $L$ are

(a) $(3, 0, 0)$

(b) $(0, 4, 5)$

(c) $(3, 0, 5)$

(d) None of these

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l} W e k n o w t h a t o n x y p l a n e, z = 0 \\ S o, t h e c o o r d i n a t e s o f t h e point L a r e (3, 4, 0) . \\ H e n c e, t h e c o r r e c t o p t i o n i s (d) . \end{array}$

$L$ is the foot of the perpendicular drawn from a point $(3, 4, 5)$ on the $x$ -axis. The coordinates of $L$ are

(a) $(3, 0, 0)$

(b) $(0, 4, 0)$

(c) $(0, 0, 5)$

(d) None of these

This is an Objective Type Questions as classified in NCERT Exemplar

$\begin{array}{l} W e k n o w t h a t o n x - a x i s, y = 0 a n d z = 0 . \\ S o, t h e r e q u i r e d c o o r d i n a t e s a r e (3, 0, 0) . \\ H e n c e, t h e c o r r e c t o p t i o n i s (a) . \end{array}$

Introduction to Three Dimensional Geometry Fill in the blank Type Questions

1. The three axes $O X, O Y, O Z$ determine ____.

Sol:

$\begin{array}{l} T h e t h r e e a x e s O X, O Y a n d O Z determine t h r e e c o o r d i n a t e p l a n e s . \\ H e n c e, t h e f i l l e r v a l u e i s t h r e e c o o r d i n a t e p l a n e s . \end{array}$

2. The three planes determine a rectangular parallelepiped which has ____ rectangular faces.

Sol:

$\begin{array}{l} T h r e e p a i r s . \\ H e n c e, t h e v a l u e o f t h e f i l l e r i s t h r e e p a i r s . \end{array}$