{"id":538293,"date":"2021-09-20T08:08:12","date_gmt":"2021-09-20T08:08:12","guid":{"rendered":"https:\/\/www.indcareer.com\/schools\/?p=538293"},"modified":"2023-09-21T07:43:57","modified_gmt":"2023-09-21T07:43:57","slug":"ncert-solutions-for-12th-class-maths-chapter-11-three-dimensional-geometry","status":"publish","type":"post","link":"https:\/\/www.indcareer.com\/schools\/ncert-solutions-for-12th-class-maths-chapter-11-three-dimensional-geometry\/","title":{"rendered":"NCERT Solutions for 12th Class Maths: Chapter 11-Three Dimensional Geometry"},"content":{"rendered":"\n

Class 12: Maths Chapter 11 solutions. Complete Class 12 Maths Chapter 11 Notes.<\/p>\n\n\n\n

NCERT Solutions for 12th Class Maths: Chapter 11-Three Dimensional Geometry<\/h2>\n\n\n\n

Class 12: Maths Chapter 11 solutions. Complete Class 12 Maths Chapter 11 Notes.<\/p>\n\n\n\n

Ex 11.1 Class 12 Maths Question 1.<\/strong>
If a line makes angles 90\u00b0, 135\u00b0, 45\u00b0 with the and z axes respectively, find its direction cosines.
Solution:<\/strong>
Direction angles are 90\u00b0, 135\u00b0, 45\u00b0
Direction cosines are
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 11.1 Class 12 Maths Question 2.<\/strong>
Find the direction cosines of a line which makes equal angles with coordinate axes.
Solution:<\/strong>
Let direction angle be \u03b1 each
\u2234 Direction cosines are cos \u03b1, cos \u03b1, cos \u03b1
But l\u00b2 + m\u00b2 + n\u00b2 = 1
\u2234cos\u00b2 a + cos\u00b2 a + cos\u00b2 a = 1
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 11.1 Class 12 Maths Question 3.<\/strong>
If a line has the direction ratios \u2013 18,12, -4 then what are its direction cosines?
Solution:<\/strong>
Now given direction ratios of a line are -8,12,-4
\u2234 a = -18,b = 12,c = -4
Direction cosines are
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 11.1 Class 12 Maths Question 4.<\/strong>
Show that the points (2,3,4) (-1,-2,1), (5,8,7) are collinear.
Solution:<\/strong>
Let the points be A(2,3,4), B (-1, -2,1), C (5,8,7).
Let direction ratios of AB be
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 11.1 Class 12 Maths Question 5.<\/strong>
Find the direction cosines of the sides of the triangle whose vertices are (3,5, -4), (-1,1,2) and (-5,-5,-2).
Solution:<\/strong>
The vertices of triangle ABC are A (3, 5, -4), B (-1,1,2), C (-5, -5, -2)
(i) Direction ratios of AB are (-4,-4,6)
Direction cosines are
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 11.2 Class 12 Maths Question 1.<\/strong>
Show that the three lines with direction cosines:
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

are mutually perpendicular.
Solution:<\/strong>
Let the lines be L1<\/sub>,L2<\/sub> and L3<\/sub>.
\u2234 For lines L1 and L2
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 11.2 Class 12 Maths Question 2.<\/strong>
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:<\/strong>
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
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 11.2 Class 12 Maths Question 3.<\/strong>
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:<\/strong>
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
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 11.2 Class 12 Maths Question 4.<\/strong>
Find the equation of the line which passes through the point (1,2,3) and is parallel to the vector <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
Equation of the line passing through the point
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 11.2 Class 12 Maths Question 5.<\/strong>
Find the equation of the line in vector and in cartesian form that passes through the point with position vector <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

 and is in the direction <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

.
Solution:<\/strong>
The vector equation of a line passing through a point with position vector <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

 and parallel to the
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

Ex 11.2 Class 12 Maths Question 6.<\/strong>
Find the cartesian equation of the line which passes through the point (-2,4, -5) and parallel to the line is given by \"\\frac
Solution:<\/strong>
The cartesian equation of the line passing through the point (-2,4, -5) and parallel to the
\"NCERT<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 11.2 Class 12 Maths Question 7.<\/strong>
The cartesian equation of a line is
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


write its vector form.
Solution:<\/strong>
The cartesian equation of the line is
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Clearly (i) passes through the point (5, \u2013 4, 6) and has 3,7,2 as its direction ratios.
=> Line (i) passes through the point A with
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 11.2 Class 12 Maths Question 8.<\/strong>
Find the vector and the cartesian equations of the lines that passes through the origin and (5,-2,3).
Solution:<\/strong>
The line passes through point
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Direction ratios of the line passing through the
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 11.2 Class 12 Maths Question 9.<\/strong>
Find the vector and cartesian equations of the line that passes through the points (3, -2, -5), (3,-2,6).
Solution:<\/strong>
The PQ passes through the point P(3, -2, -5)
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 11.2 Class 12 Maths Question 10.<\/strong>
Find the angle between the following pair of lines
(i) <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

(ii) <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Solution:<\/strong>
(i) Let \u03b8 be the angle between the given lines.The given lines are parallel to the vectors
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 11.2 Class 12 Maths Question 11.<\/strong>
Find the angle between the following pair of lines
(i) <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

(ii) <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Solution:<\/strong>
Given
(i) <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

(ii) <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 11.2 Class 12 Maths Question 12.<\/strong>
Find the values of p so that the lines
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


are at right angles
Solution:<\/strong>
The given equation are not in the standard form
The equation of given lines is
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 11.2 Class 12 Maths Question 13.<\/strong>
Show that the lines <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

 are perpendicular to each other
Solution:<\/strong>
Given lines
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

 \u2026(i)
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

 \u2026(ii)
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 11.2 Class 12 Maths Question 14.<\/strong>
Find the shortest distance between the lines
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

 and
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
The shortest distance between the lines
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 11.2 Class 12 Maths Question 15.<\/strong>
Find the shortest distance between the lines
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
Shortest distance between the lines
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 11.2 Class 12 Maths Question 16.<\/strong>
Find the distance between die lines whose vector equations are:
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

 and
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
Comparing the given equations with
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 11.2 Class 12 Maths Question 17.<\/strong>
Find the shortest distance between the lines whose vector equations are
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

 and
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
Comparing these equation with
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 11.3 Class 12 Maths Question 1.<\/strong>
In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin.
(a) z = 2
(b) x+y+z = 1
(c) 2x + 3y \u2013 z = 5
(d) 5y+8 = 0
Solution:<\/strong>
(a) Direction ratios of the normal to the plane are 0,0,1
=> a = 0, b = 0, c = 1
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 11.3 Class 12 Maths Question 2.<\/strong>
Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 11.3 Class 12 Maths Question 3.<\/strong>
Find the Cartesian equation of the following planes.
(a) <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


(b) <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


(c) <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
(a) \"\\overrightarrow is the position vector of any arbitrary point P (x, y, z) on the plane.
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 11.3 Class 12 Maths Question 4.<\/strong>
In the following cases find the coordinates of the foot of perpendicular drawn from the origin
(a) 2x + 3y + 4z \u2013 12 = 0
(b) 3y + 4z \u2013 6 = 0
(c) x + y + z = 1
(d) 5y + 8 = 0
Solution:<\/strong>
(a) Let N (x1, y1, z1) be the foot of the perpendicular from the origin to the plane 2x+3y+4z-12 = 0
\u2234 Direction ratios of the normal are 2, 3, 4.
Also the direction ratios of ON are (x1,y1,z1)
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Ex 11.3 Class 12 Maths Question 5.<\/strong>
Find the vector and cartesian equation of the planes
(a) that passes through the point (1,0, -2) and the normal to the plane is <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

(b) that passes through the point (1,4,6) and the normal vector to the plane is <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Solution:<\/strong>
(a) Normal to the plane is i + j \u2013 k and passes through (1,0,-2)
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Ex 11.3 Class 12 Maths Question 6.<\/strong>
Find the equations of the planes that passes through three points
(a) (1,1,-1) (6,4,-5), (-4, -2,3)
(b) (1,1,0), (1,2,1), (-2,2,-1)
Solution:<\/strong>
(a) The plane passes through the points (1,1,-1) (6,4,-5), (-4,-2,3)
Let the equation of the plane passing through(1,1,-1)be
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 11.3 Class 12 Maths Question 7.<\/strong>
Find the intercepts cut off by the plane 2x+y-z = 5.
Solution:<\/strong>
Equation of the plane is 2x + y- z = 5 x y z
Dividing by 5: <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

\u2234 The intercepts on the axes OX, OY, OZ are <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

, 5, -5 respectively<\/p>\n\n\n\n

Ex 11.3 Class 12 Maths Question 8.<\/strong>
Find the equation of the plane with intercept 3 on the y- axis and parallel to ZOX plane.
Solution:<\/strong>
Any plane parallel to ZOX plane is y=b where b is the intercept on y-axis.
\u2234 b = 3.
Hence equation of the required plane is y = 3.<\/p>\n\n\n\n

Ex 11.3 Class 12 Maths Question 9.<\/strong>
Find the equation of the plane through the intersection of the planes 3x \u2013 y + 2z \u2013 4 = 0 and x + y + z \u2013 2 = 0 and the point (2,2,1).
Solution:<\/strong>
Given planes are:
3x \u2013 y + 2z \u2013 4 = 0 and x + y + z \u2013 2 = 0
Any plane through their intersection is
3x \u2013 y + 2z \u2013 4 + \u03bb(x + y + z \u2013 2) = 0
point (2,2,1) lies on it,
\u22343 x 2 \u2013 2 + 2 x 1 \u2013 4 +\u03bb(2+2+1-2)=0
=>\u03bb = \"\\frac
Now required equation is 7x \u2013 5y + 4z \u2013 8 = 0<\/p>\n\n\n\n

Ex 11.3 Class 12 Maths Question 10.<\/strong>
Find the vector equation of the plane passing through the intersection of the planes <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

 and through the point (2,1,3).
Solution:<\/strong>
Equation of the plane passing through the line of intersection of the planes
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 11.3 Class 12 Maths Question 11.<\/strong>
Find the equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x \u2013 y + z = 0.
Solution:<\/strong>
Given planes are
x + y + z \u2013 1 = 0 \u2026(i)
2x + 3y + 4z \u2013 5 = 0 \u2026(ii)
x \u2013 y + z = 0 \u2026.(iii)
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 11.3 Class 12 Maths Question 12.<\/strong>
Find the angle between the planes whose vector equations are <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
The angle \u03b8 between the given planes is
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 11.3 Class 12 Maths Question 13.<\/strong>
In the following determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angle between them.
(a) 7x + 5y + 6z + 30 = 0 and 3x \u2013 y \u2013 10z + 4 = 0
(b) 2x + y + 3z \u2013 2 = 0 and x \u2013 2y + 5 = 0
(c) 2x \u2013 2y + 4z + 5 = 0 and 3x \u2013 3y + 6z \u2013 1 = 0
(d) 2x \u2013 y + 3z \u2013 1 = 0 and 2x \u2013 y + 3z + 3 = 0
(e) 4x + 8y + z \u2013 8 = 0 and y + z \u2013 4 = 0.
Solution:<\/strong>
(a) Direction ratios of the normal of the planes 7x + 5y + 6z + 30 = 0 are 7,5,6
Direction ratios of the normal of the plane 3x \u2013 y \u2013 10z + 4 = 0 are 3,-1,-10
The plane 7x + 5y + 6z + 30 = 0 \u2026(i)
3x \u2013 y \u2013 10z + y = 0 \u2026(ii)
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 11.3 Class 12 Maths Question 14.<\/strong>
In the following cases, find the distance of each of the given points from the corresponding given plane.
Point           Plane
(a) (0, 0,0) 3x \u2013 4y + 12z = 3
(b) (3,-2,1) 2x \u2013 y + 2z + 3 = 0.
(c) (2,3,-5) x + 2y \u2013 2z = 9
(d) (-6,0,0) 2x \u2013 3y + 6z \u2013 2 = 0
Solution:<\/strong>
(a) Given plane: 3x \u2013 4y + 12z \u2013 3 = 0
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

NCERT Solutions for 12th Class Maths: Chapter 11: Download PDF<\/strong><\/h2>\n\n\n\n

NCERT Solutions for 12th Class Maths: Chapter 11-Three Dimensional Geometry<\/p>\n\n\n\n

Download PDF<\/strong>: NCERT Solutions for 12th Class Maths: Chapter 11-Three Dimensional Geometry PDF<\/a><\/p>\n\n\n\n

Chapterwise NCERT Solutions for Class 12 Maths :<\/strong><\/h2>\n\n\n\n