NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry Ex 11.2

Get Free NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry Ex 11.2 PDF in Hindi and English Medium. Sets Class 12 Maths NCERT Solutions are extremely helpful while doing your homework. Three Dimensional Geometry Exercise 11.2 Class 12 Maths NCERT Solutions were prepared by Experienced LearnCBSE.in Teachers. Detailed answers of all the questions in Chapter 11 Class 12 Three Dimensional Geometry Ex 11.2 provided in NCERT Textbook.

Free download NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry Ex 11.2 PDF in Hindi Medium as well as in English Medium for CBSE, Uttarakhand, Bihar, MP Board, Gujarat Board, BIE, Intermediate and UP Board students, who are using NCERT Books based on updated CBSE Syllabus for the session 2019-20.

NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry Ex 11.2

Ex 11.2 Class 12 Maths Question 1.
Show that the three lines with direction cosines:
Image may be NSFW.
Clik here to view. $\frac { 12 }{ 13 } ,\frac { -3 }{ 13 } ,\frac { -4 }{ 13 } ,\frac { 4 }{ 13 } ,\frac { 12 }{ 13 } ,\frac { 3 }{ 13 } ,\frac { 3 }{ 13 } ,\frac { -4 }{ 13 } ,\frac { 12 }{ 13 }$
are mutually perpendicular.
Solution:
Let the lines be L1,L2 and L3.
∴ For lines L1 and L2
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Ex 11.2 Class 12 Maths Question 2.
Show that the line through the points (1,-1,2) (3,4, -2) is perpendicular to the line through the points (0,3,2) and (3,5,6).
Solution:
Let A, B be the points (1, -1, 2), (3, 4, -2) respectively Direction ratios of AB are 2,5, -4
Let C, D be the points (0, 3, 2) and (3, 5, 6) respectively Direction ratios of CD are 3, 2,4 AB is Perpendicular to CD if
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Ex 11.2 Class 12 Maths Question 3.
Show that the line through the points (4,7,8) (2,3,4) is parallel to the line through the points (-1,-2,1) and (1,2,5).
Solution:
Let the points be A(4,7,8), B (2,3,4), C (-1,-2,1) andD(1,2,5).
Now direction ratios of AB are
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Ex 11.2 Class 12 Maths Question 4.
Find the equation of the line which passes through the point (1,2,3) and is parallel to the vector Image may be NSFW.
Clik here to view. $3\hat { i } +2\hat { j } -2\hat { k }$
Solution:
Equation of the line passing through the point
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Ex 11.2 Class 12 Maths Question 5.
Find the equation of the line in vector and in cartesian form that passes through the point with position vector Image may be NSFW.
Clik here to view. $2\hat { i } -\hat { j } +4\hat { k }$ and is in the direction Image may be NSFW.
Clik here to view. $\hat { i } +2\hat { j } -\hat { k }$ .
Solution:
The vector equation of a line passing through a point with position vector Image may be NSFW.
Clik here to view. $\overrightarrow { a }$ and parallel to the
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Ex 11.2 Class 12 Maths Question 6.
Find the cartesian equation of the line which passes through the point (-2,4, -5) and parallel to the line is given by Image may be NSFW.
Clik here to view. $\frac { x+3 }{ 3 } =\frac { y-4 }{ 5 } =\frac { z+8 }{ 6 }$
Solution:
The cartesian equation of the line passing through the point (-2,4, -5) and parallel to the
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Ex 11.2 Class 12 Maths Question 7.
The cartesian equation of a line is
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Clik here to view. $\frac { x-5 }{ 3 } =\frac { y+4 }{ 7 } =\frac { z-6 }{ 2 }$
write its vector form.
Solution:
The cartesian equation of the line is
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Clik here to view. $\frac { x-5 }{ 3 } =\frac { y+4 }{ 7 } =\frac { z-6 }{ 2 }$
Clearly (i) passes through the point (5, – 4, 6) and has 3,7,2 as its direction ratios.
=> Line (i) passes through the point A with
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Ex 11.2 Class 12 Maths Question 8.
Find the vector and the cartesian equations of the lines that passes through the origin and (5,-2,3).
Solution:
The line passes through point
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Clik here to view. $\therefore \overrightarrow { a } =\overrightarrow { 0 }$
Direction ratios of the line passing through the
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Ex 11.2 Class 12 Maths Question 9.
Find the vector and cartesian equations of the line that passes through the points (3, -2, -5), (3,-2,6).
Solution:
The PQ passes through the point P(3, -2, -5)
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Ex 11.2 Class 12 Maths Question 10.
Find the angle between the following pair of lines
(i) Image may be NSFW.
Clik here to view. $\overrightarrow { r } =2\hat { i } -5\hat { j } +\hat { k } +\lambda (3\hat { i } +2\hat { j } +6\hat { k } )$
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Clik here to view. $and\quad \overrightarrow { r } =7\hat { i } -6\hat { j } +\mu (\hat { i } +2\hat { j } +2\hat { k } )$
(ii) Image may be NSFW.
Clik here to view. $\overrightarrow { r } =3\hat { i } +\hat { j } -2\hat { k } +\lambda (\hat { i } -\hat { j } -2\hat { k } )$
Image may be NSFW.
Clik here to view. $\overrightarrow { r } =2\hat { i } -\hat { j } -56\hat { k } +\mu (3\hat { i } -5\hat { j } -4\hat { k } )$
Solution:
(i) Let θ be the angle between the given lines.
The given lines are parallel to the vectors
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Ex 11.2 Class 12 Maths Question 11.
Find the angle between the following pair of lines
(i) Image may be NSFW.
Clik here to view. $\frac { x-2 }{ 2 } =\frac { y-1 }{ 5 } =\frac { z+3 }{ -3 } and\frac { x+2 }{ -1 } =\frac { y-4 }{ 8 } =\frac { z-5 }{ 4 }$
(ii) Image may be NSFW.
Clik here to view. $\frac { x }{ 2 } =\frac { y }{ 2 } =\frac { z }{ 1 } and\frac { x-5 }{ 4 } =\frac { y-2 }{ 1 } =\frac { z-3 }{ 8 }$
Solution:
Given
(i) Image may be NSFW.
Clik here to view. $\frac { x-2 }{ 2 } =\frac { y-1 }{ 5 } =\frac { z+3 }{ -3 } and\frac { x+2 }{ -1 } =\frac { y-4 }{ 8 } =\frac { z-5 }{ 4 }$
(ii) Image may be NSFW.
Clik here to view. $\frac { x }{ 2 } =\frac { y }{ 2 } =\frac { z }{ 1 } and\frac { x-5 }{ 4 } =\frac { y-2 }{ 1 } =\frac { z-3 }{ 8 }$
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Clik here to view. NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry 11

Ex 11.2 Class 12 Maths Question 12.
Find the values of p so that the lines
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Clik here to view. $\frac { 1-x }{ 3 } =\frac { 7y-14 }{ 2p } =\frac { z-3 }{ 2 } and\frac { 7-7x }{ 3p } =\frac { y-5 }{ 1 } =\frac { 6-z }{ 5 }$
are at right angles
Solution:
The given equation are not in the standard form
The equation of given lines is
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Ex 11.2 Class 12 Maths Question 13.
Show that the lines Image may be NSFW.
Clik here to view. $\frac { x-5 }{ 7 } =\frac { y+2 }{ -5 } =\frac { z }{ 1 } and\frac { x }{ 1 } =\frac { y }{ 2 } =\frac { z }{ 3 }$ are perpendicular to each other
Solution:
Given lines
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Clik here to view. $\frac { x-5 }{ 7 } =\frac { y+2 }{ -5 } =\frac { z }{ 1 }$ …(i)
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Clik here to view. $\frac { x }{ 1 } =\frac { y }{ 2 } =\frac { z }{ 3 }$ …(ii)
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Ex 11.2 Class 12 Maths Question 14.
Find the shortest distance between the lines
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Clik here to view. $\overrightarrow { r } =(\hat { i } +2\hat { j } +\hat { k } )+\lambda (\hat { i } -\hat { j } +\hat { k } )$ and
Image may be NSFW.
Clik here to view. $\overrightarrow { r } =(2\hat { i } -\hat { j } -\hat { k } )+\mu (2\hat { i } +\hat { j } +2\hat { k } )$
Solution:
The shortest distance between the lines
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Ex 11.2 Class 12 Maths Question 15.
Find the shortest distance between the lines
Image may be NSFW.
Clik here to view. $\frac { x+1 }{ 7 } =\frac { y+1 }{ -6 } =\frac { z+1 }{ 1 } and\frac { x-3 }{ 1 } =\frac { y-5 }{ -2 } =\frac { z-7 }{ 1 }$
Solution:
Shortest distance between the lines
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Clik here to view. NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry 15

Ex 11.2 Class 12 Maths Question 16.
Find the distance between die lines whose vector equations are:
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Clik here to view. $\overrightarrow { r } =(\hat { i } +2\hat { j } +3\hat { k) } +\lambda (\hat { i } -3\hat { j } +2\hat { k } )$ and
Image may be NSFW.
Clik here to view. $\overrightarrow { r } =(4\hat { i } +5\hat { j } +6\hat { k) } +\mu (2\hat { i } +3\hat { j } +\hat { k } )$
Solution:
Comparing the given equations with
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Clik here to view. NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry 16
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Ex 11.2 Class 12 Maths Question 17.
Find the shortest distance between the lines whose vector equations are
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Clik here to view. $\overrightarrow { r } =(1-t)\hat { i } +(t-2)\hat { j } +(3-2t)\hat { k }$ and
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Clik here to view. $\overrightarrow { r } =(s+1)\hat { i } +(2s-1)\hat { j } -(2s+1)\hat { k }$
Solution:
Comparing these equation with
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NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry Hindi Medium Ex 11.2

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Clik here to view. NCERT Solutions for Class 12 Maths Exercise 11.2 of Three Dimensional Geometry
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Clik here to view. NCERT Solutions for Class 12 Maths Exercise 11.2 download
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Clik here to view. Exercise 11.2 class 12
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Clik here to view. 12 Maths 11.2
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Clik here to view. 12 Maths solutions 11.2
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Clik here to view. 12 Maths Exercise 11.2 answers
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Clik here to view. Exercise 11.2 class 12 Maths
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