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NCERT solutions Mathematics Class 12 Part 2 chapter 11 Three Dimensional Geometry

Chapters

NCERT Mathematics Class 12 Part 2

Mathematics Textbook for Class 12 Part 2

Chapter 11 - Three Dimensional Geometry

Page 467

Find the direction cosines of a line which makes equal angles with the coordinate axes.

Q 2 | Page 467

Pages 467 - 478

If a line makes angles 90°, 135°, 45° with xy and z-axes respectively, find its direction cosines.

Q 1 | Page 467

If a line has the direction ratios −18, 12, −4, then what are its direction cosines?

Q 3 | Page 467

Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.

Q 4 | Page 467

Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector.`3hati+2hatj-2hatk`

Q 4 | Page 477

Find the equation of the line in vector and in Cartesian form that passes through the point with position vector `2hati -hatj+4hatk`  and is in the direction `hati + 2hatj - hatk`.

Q 5 | Page 477

Find the Direction Cosines of the Sides of the Triangle Whose Vertices Are (3, 5, -4), (-1, 1, 2) and (-5, -5, -2)

Q 5 | Page 467

Find the Cartesian equation of the line which passes through the point (−2, 4, −5) and parallel to the line given by `(x+3)/3 = (y-4)/5 = ("z"+8)/6`

Q 6 | Page 477

The Cartesian equation of a line is `(x-5)/3 = (y+4)/7 = ("z"-6)/2` Write its vector form.

The given line passes through the point (5, −4, 6). The position vector of this point is `veca = 5hati - 4hatj + 6hatk`

Also, the direction ratios of the given line are 3, 7, and 2.

This means that the line is in the direction of vector, `vecb =3hati +7hatj + 2hatk`

It is known that the line through position vector `veca` and in the direction of the vector `vecb`is given by the equation, `vecr = veca+lambdavecb, lambda in R`

This is the required equation of the given line in vector form.

Q 7 | Page 477

Find the vector and the Cartesian equations of the lines that pass through the origin and (5, −2, 3).

Q 8 | Page 477

Find the vector and the Cartesian equations of the line that passes through the points (3, −2, −5), (3, −2, 6).

 

Q 9 | Page 478

Pages 477 - 478

Show that the three lines with direction cosines `12/13,(-3)/13,(-4)/13; 4/13,12/13,3/13;3/13,(-4)/13,12/13 ` are mutually perpendicular.

Q 1 | Page 477

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).

Q 2 | Page 477

Show that the line through the points (4, 7, 8) (2, 3, 4) is parallel to the line through the points (−1, −2, 1), (1, 2, 5).

Q 3 | Page 477

Find the angle between the following pairs of lines:

`vecr = 2hati - 5hatj + hatk + lambda(3hati - 2hatj + 6hatk) and vecr = 7hati - 6hatk + mu(hati + 2hatj + 2hatk)`

Q 10.1 | Page 478

Find the angle between the following pairs of lines:

`vecr = 3hati + hatj - 2hatk + lambda(hati - hatj - 2hatk) and vecr = 2hati - hatj -56hatk + mu(3hati - 5hatj - 4hatk)`

Q 10.2 | Page 478

Find the angle between the following pairs of lines: 

`(x-2)/2 = (y-1)/5 = (z+3)/(-3)` and `(x+2)/(-1) = (y-4)/8 = (z -5)/4`

Q 11.1 | Page 478

Find the angle between the following pairs of lines:

`x/y = y/2 = z/1` and `(x-5)/4 = (y-2)/1 = (z - 3)/8`

Q 11.2 | Page 478

Find the values of p so the line `(1-x)/3 = (7y-14)/2p = (z-3)/2` and `(7-7x)/(3p) = (y -5)/1 = (6-z)/5` are at right angles.

Q 12 | Page 478

Show that the lines `(x-5)/7 = (y + 2)/(-5) = z/1` and `x/1 = y/2 = z/3` are perpendicular to each other.

Q 13 | Page 478

Find the shortest distance between the lines 

`vecr = (hati+2hatj+hatk) + lambda(hati-hatj+hatk)` and `vecr = 2hati - hatj - hatk + mu(2hati + hatj + 2hatk)`

Q 14 | Page 478

Find the shortest distance between the lines `(x+1)/7 = `(y+1)/(-6) = (z+1)/1` and (x-3)/1 = (y-5)/(-2) = (z-7)/1`

Q 15 | Page 478

Find the shortest distance between the lines whose vector equations are `vecr = (hati + 2hatj + 3hatk) + lambda(hati - 3hatj + 2hatk)` and `vecr = 4hati + 5hatj + 6hatk + mu(2hati + 3hatj + hatk)`

Q 16 | Page 478

Find the shortest distance between the lines whose vector equations are

`vecr = (1-t)hati + (t - 2)hatj + (3 -2t)hatk` and `vecr = (s+1)hati + (2s + 1)hatk`

Q 17 | Page 478

Pages 493 - 494

In following cases, determine the direction cosines of the normal to the plane and the distance from the origin.

z = 2

Q 1 | Page 493

In following cases, determine the direction cosines of the normal to the plane and the distance from the origin.

x + y + z = 1

Q 1.2 | Page 493

In following cases, determine the direction cosines of the normal to the plane and the distance from the origin.

2x + 3y – z = 5

Q 1.3 | Page 493

In following cases, determine the direction cosines of the normal to the plane and the distance from the origin.

5y + 8 = 0

Q 1.4 | Page 493

Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector.`3hati + 5hatj - 6hatk`

Q 2 | Page 493

In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.

x + y + z = 1

Q 3 | Page 493

Find the Cartesian equation of the following planes:

`vecr.(hati + hatj-hatk) = 2`

Q 3.1 | Page 493

Find the Cartesian equation of the following planes:

`vecr.(2hati + 3hatj-4hatk) = 1`

Q 3.2 | Page 493

Find the Cartesian equation of the following planes:

`vecr.[(s-2t)hati + (3 - t)hatj + (2s + t)hatk] = 15`

Q 3.3 | Page 493

In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.

2x + 3y + 4z – 12 = 0

Q 4.1 | Page 493

In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.

3y + 4z – 6 = 0

Q 4.2 | Page 493

In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.

5y + 8 = 0

Q 4.4 | Page 493

Find the vector and Cartesian equation of the planes that passes through the point (1, 0, −2) and the normal to the plane is `hati + hatj - hatk`

Q 5.1 | Page 493

Find the vector and Cartesian equation of the planes that passes through the point (1, 4, 6) and the normal vector to the plane is `hati -2hatj +  hatk`

Q 5.2 | Page 493

Find the equations of the planes that passes through three points.

(1, 1, −1), (6, 4, −5), (−4, −2, 3)

Q 6.1 | Page 493

Find the equations of the planes that passes through three points.

(1, 1, 0), (1, 2, 1), (−2, 2, −1)

Q 6.2 | Page 493

Find the intercepts cut off by the plane 2x + y – z = 5.

Q 7 | Page 493

Find the equation of the plane with intercept 3 on the y-axis and parallel to ZOX plane.

Q 8 | Page 493

Find the equation of the plane through the intersection of the planes 3x – y + 2z – 4 = 0 and x + y + z – 2 = 0 and the point (2, 2, 1).

Q 9 | Page 493

Find the vector equation of the plane passing through the intersection of the planes `vecr.(2hati + 2hatj - 3hatk) = 7, vecr.(2hati + 5hatj + 3hatk) = 9` and through the point (2, 1, 3)

Q 10 | Page 493

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 − y + z = 0. Also find the distance of the plane, obtained above, from the origin.

Q 11 | Page 493

Find the angle between the planes whose vector equations are `vecr.(2hati + 2hatj - 3hatk) = 5 and hatr.(3hati - 3hatj  + 5hatk) = 3`

Q 12 | Page 494

In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.

7x + 5y + 6z + 30 = 0 and 3x – y – 10z + 4 = 0

Q 13.1 | Page 494

In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.

2x + y + 3z – 2 = 0 and x – 2y + 5 = 0

 

Q 13.2 | Page 494

In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.

2x – 2y + 4z + 5 = 0 and 3x – 3y + 6z – 1 = 0

Q 13.3 | Page 494

In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.

2x – y + 3z – 1 = 0 and 2x – y + 3z + 3 = 0

Q 13.4 | Page 494

In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.

4x + 8y + z – 8 = 0 and y + z – 4 = 0

Q 13.5 | Page 494

In the given cases, find the distance of each of the given points from the corresponding given plane.

Point                    Plane
(0, 0, 0)           3x – 4y + 12 z = 3

Q 14.1 | Page 494

In the given cases, find the distance of each of the given points from the corresponding given plane

Point                   Plane

(3, – 2, 1)             2x – y + 2z + 3 = 0

Q 14.2 | Page 494

In the given cases, find the distance of each of the given points from the corresponding given plane.

Point                 Plane

(2, 3, – 5)           x + 2y – 2z = 9

Q 14.3 | Page 494

In the given cases, find the distance of each of the given points from the corresponding given plane.

Point              Plane

(– 6, 0, 0)        2x – 3y + 6z – 2 = 0

Q 14.4 | Page 494

Pages 497 - 499

Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, – 1), (4, 3, – 1).

Q 1 | Page 497

If l1m1n1 and l2m2n2 are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are m1n2 − m2n1n1l2 − n2l1l1m2 ­− l2m1.

Q 2 | Page 497

Find the angle between the lines whose direction ratios are aband b − cc − aa − b.

Q 3 | Page 498

Find the equation of a line parallel to x-axis and passing through the origin.

Q 4 | Page 498

If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), (­−4, 3, −6) and (2, 9, 2) respectively, then find the angle between the lines AB and CD.

Q 5 | Page 498

If the lines `(x-1)/(-3) = (y -2)/(2k) = (z-3)/2 and (x-1)/(3k) = (y-1)/1 = (z -6)/(-5)` are perpendicular, find the value of k.

Q 6 | Page 498

Find the vector equation of the plane passing through (1, 2, 3) and perpendicular to the plane `vecr.(hati + 2hatj -5hatk) + 9 = 0`

Q 7 | Page 498

Find the equation of the plane passing through (abc) and parallel to the plane `vecr.(hati + hatj + hatk) = 2`

Q 8 | Page 498

Find the shortest distance between lines `vecr = 6hati + 2hatj + 2hatk + lambda(hati - 2hatj + 2hatk)` and `vecr =-4hati - hatk + mu(3hati - 2hatj - 2hatk)`

Q 9 | Page 498

Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the YZ-plane

Q 10 | Page 498

Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the ZX − plane.

Q 11 | Page 498

Find the coordinates of the point where the line through (3, ­−4, −5) and (2, − 3, 1) crosses the plane 2x + z = 7).

Q 12 | Page 498

Find the equation of the plane passing through the point (−1, 3, 2) and perpendicular to each of the planes x + 2+ 3z = 5 and 3x + 3z = 0.

Q 13 | Page 498

If the points (1, 1, p) and (−3, 0, 1) be equidistant from the plane `vecr.(3hati + 4hatj - 12hatk)+ 13 = 0`, then find the value of p.

Q 14 | Page 498

Find the equation of the plane passing through the line of intersection of the planes `vecr.(hati + hatj + hatk) = 1` and `vecr.(2hati + 3hatj -hatk) + 4 = 0` and parallel to x-axis.

Q 15 | Page 498

If O be the origin and the coordinates of P be (1, 2, −3), then find the equation of the plane passing through P and perpendicular to OP.

Q 16 | Page 498

Find the equation of the plane which contains the line of intersection of the planes `vecrr.(hati + 2hatj + 3hatk) - 4 = 0, vecr.(2hati + htj - hatk) + 5 = 0`,  and which is perpendicular to the plane `vecr.(5hati + 3hatj - 6hatk) + 8 = 0`.

Q 17 | Page 498

Find the distance of the point (−1, −5, −­10) from the point of intersection of the line `vecr = 2hati -hatj + 2hatk + lambda(3hati + 4hatj + 2hatk)` and the plane `vecr.(hati -hatj + hatk) = 5`.

Q 18 | Page 499

Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes `vecr = (hati - hatj + 2hatk)  = 5`and `vecr.(3hati + hatj + hatk) = 6`

Q 19 | Page 499

Find the vector equation of the line passing through the point (1, 2, − 4) and perpendicular to the two lines:  `(x -8)/3 = (y+19)/(-16) = (z - 10)/7 and (x - 15)/3 = (y - 29)/8 = (z- 5)/(-5)`

Q 20 | Page 499

Prove that if a plane has the intercepts abc and is at a distance of P units from the origin, then `1/a^2 + 1/b^2 + 1/c^2 = 1/p^2`

Q 21 | Page 499

Distance between the two planes: 2x + 3y + 4z = 4 and 4x + 6y + 8z = 12 is

(A) 2 units

(B) 4 units

(C) 8 units

(D)`2/sqrt29 "units"`

Q 22 | Page 499

The planes: 2− y + 4z = 5 and 5x − 2.5y + 10z = 6 are

(A) Perpendicular

(B) Parallel

(C) intersect y-axis

(C) passes through `(0,0,5/4)`

Q 23 | Page 499

NCERT Mathematics Class 12 Part 2

Mathematics Textbook for Class 12 Part 2
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