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 ) .

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.

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

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 0 < x < 1, then 3 2 x 2 + 5 3 x 3 + 7 4 x 4 + . . . . . . ,  is equal to

x ( 1 + x 1 − x ) + l o g e ( 1 − x )

1 − x 1 + x + l o g e ( 1 − x )

x ( 1 − x 1 + x ) + l o g e ( 1 − x )

1 + x 1 − x + l o g e ( 1 − x )

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

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 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 ]

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

Q: 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 Let the given points are A(0,−1,−7), B(2,1,−9), C(6,5,−13) AB=(2−0)2+(1+1)2+(−9+7)2=4+4+4=12=23 BC=(6−2)2+(5−1)2+(−13+9)2=16+16+16=48=43 AC=(6−0)2+(5+1)2+(−13+7)2=36+36+36=108=63 23+43=63i.e., AB+BC=AC∴ AB:AC=23:63=1:3Hence, point A divides B and C in 1:3 externally.

Q: 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 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 ) .

Q: The three axes OX,OY,OZ determine ____.

This is a Fill in the blanks Type Questions as classified in NCERT Exemplar The three axes OX, OY and OZ determine three coordinate planes.Hence, the filler value is three coordinate planes.

Introduction to Three Dimensional Geometry Short Answer Type Questions

Locate the following points:

i. (1, –1, 3),

ii. (–1, 2, 4)

iii. (–2, –4, –7),

iv. (–4, 2, –5).

Sol.

$\begin{array}{l} (i) L o c a t i o n o f P (1, - 1, 3) = (x, - y, z) = I Voctant \\ (i i) L o c a t i o n o f Q (- 1, 2, 4) = (- x, y, z) = I Ioctant \\ (i i i) L o c a t i o n o f R (- 2, - 4, - 7) = (- x, - y, - z) = V I Ioctant \\ (i v) L o c a t i o n o f S (- 4, 2, - 5) = (- x, y, - z) = V I octant \end{array}$

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).

Sol:

$\begin{array}{l} (i)Point (1, 2, 3) l i e s i n IOctant \\ (i i)Point (4, - 2, 3) l i e s i n I VOctant \\ (i i i)Point (4, - 2, - 5) l i e s i n V I I I Octant \\ (i v)Point (4, 2, - 5) l i e s i n VOctant \\ (v)Point (- 4, 2, 5) l i e s i n I IOctant \\ (v i)Point (- 3, - 1, 6) l i e s i n I I IOctant \\ (v i i)Point (2, - 4, - 7) l i e s i n V I I IOctant \\ (v i i i)Point (- 4, 2, - 5) l i e s i n V IOctant \end{array}$

Commonly asked questions

Q:

Show that if $x^{2} + y^{2} = 1$ , then the point $(x, y, \sqrt[]{1 - x^{2} - y^{2}})$ is at a distance of 1 unit from the origin.

Q:

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).

Q:

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).

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

Q:

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 $A B$ .

Q:

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.

Q:

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?

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

Q:

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

Q:

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

Q:

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

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

(b) 3

(c) 4

(d) 5

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:

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

Commonly asked questions

Q:

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

Q:

A line is parallel to $x y$ -plane if all the points on the line have equal ____.

Q:

If the midpoints of the sides of a triangle $A B, B C, C A$ are $D (1, 2, 3), E (3, 0, 1), F (1, 1, 4)$ , then the centroid of the triangle $A B C$ is ____.

Introduction to Three Dimensional Geometry Matching Type Questions

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

Sol.

\begin{array}{l} (a) I n x y - p l a n e, z - c o o r d i n a t e i s z e r o . \\ H e n c e, (a) \leftrightarrow (i i i) . \\ (b) T h e point (2, 3, 4) l i e s i n f i r s t octant . \\ H e n c e, (b) \leftrightarrow (i) . \\ (c) L o c u s o f t h e points w i t h x - c o o r d i n a t e i s z e r o i s y z - p l a n e . \\ H e n c e, (c) \leftrightarrow (i i) . \\ (d) A l i n e i s p a r a l l e l t o x - a x i s i f a n d o n l y i f a l l t h e points o n t h e l i n e \\ h a v e e q u a l y a n d z - c o o r d i n a t e s . \\ H e n c e, (d) \leftrightarrow (v i) . \\ (e) x = 0, y = 0 r e p r e s e n t s z - a x i s . \\ H e n c e, (e) \leftrightarrow (i v) . \\ (f) z = c r e p r e s e n t a p l a n e p a r a l l e l t o x y - p l a n e . \\ H e n c e, (f) \leftrightarrow (v) . \\ (g) T h e p l a n e s x = a, y = b r e p r e s e n t t h e l i n e p a r a l l e l t o z - a x i s . \\ H e n c e, (g) \leftrightarrow (v i i i) . \\ (h) C o o r d i n a t e s o f a point a r e t h e distances f r o m t h e o r i g i n t o t h e f e e t o f p e r p e n d i c u l a r f r o m \\ t h e point o n t h e r e s p e c t i v e a x e s, \\ H e n c e, (h) \leftrightarrow (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) \leftrightarrow (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) \leftrightarrow (i x) . \end{array}

Commonly asked questions

Q:

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

JEE Mains 2021

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

Introduction to Three Dimensional Geometry Short Answer Type Questions

Commonly asked questions

Try these practice questions

Introduction to Three Dimensional Geometry Long Answer Type Questions

Commonly asked questions

Introduction to Three Dimensional Geometry Objective Type Questions

Commonly asked questions

Introduction to Three Dimensional Geometry Fill in the blank Type Questions

Commonly asked questions

Introduction to Three Dimensional Geometry Matching Type Questions

Commonly asked questions

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