NCERT Solutions class 12 Maths Miscellaneous Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths chapter wise NCERT solution for Maths part 1 and Maths part 2 for all the chapters can be downloaded from our website and myCBSEguide mobile app for free.

**Download NCERT solutions for Three Dimensional Geometry as PDF.**

## NCERT Solutions class 12 Maths Three Dimensional Geometry** **

**1.Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points **** **

**Ans. **We know that direction ratios of the line joining the origin (0, 0, 0) to the point are

= 2 – 0, 1 – 0, 1 – 0 = 2, 1, 1 =

Similarly, direction ratios of the line joining the points and are

= = =

For these two lines,

= = 2 – 2 + 0 = 0

Therefore, the two given lines are perpendicular to each other.

#### NCERT Solutions class 12 Maths Miscellaneous

**2.If **** and **** are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are **** **

**Ans. ** and are direction cosines of two mutually perpendicular of two given lines L_{1} and L_{2}. (say)

Let and be the unit vectors along these lines L_{1} and L_{2}.

and

Let L be the line perpendicular to both the lines L_{1} and L_{2} and let be a unit vector along line L perpendicular both lines L_{1} and L_{2}.

Cross-product of two vectors =

[ L1 L2 (given, angle between them is ]

Since, is a unit vector, therefore its components are its direction cosines.

Thus, direction cosines of are

direction cosines of line L are

#### NCERT Solutions class 12 Maths Miscellaneous

**3.Find the angle between the lines whose direction ratios are **** and **** **

**Ans. **Direction ratios of one line are

A vector along this line is

Direction ratios of second line are

A vector along second line is

Let be the angle between the two lines, then

=

= = 0 =

#### NCERT Solutions class 12 Maths Miscellaneous

**4.Find the equation of the line parallel to ****axis and passing through the origin.**

**Ans. **We know that a unit vector along axis is

Direction cosines of axis are coefficients of in the unit vector

i.e., 1, 0, 0 =

Equation of the required line passing through the origin (0, 0, 0) and parallel to axis is

Vector equation of the required line is

[ and ]

#### NCERT Solutions class 12 Maths Miscellaneous

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

**Ans. **Given: Points A B C and D

Direction ratios of line AB are

A vector along the line AB is

Similarly, direction ratios of line CD are

A vector along the line AB is

Let be the angle between the two lines, then

= = = = = 1

=

Therefore, lines AB and CD are parallel.

#### NCERT Solutions class 12 Maths Miscellaneous

**6.If the lines **** and **** are perpendicular, find the value of **** **

**Ans. **Given: Equation of one line is

Direction ratios of this line are its denominators, i.e., =

A vector along this line is

Again, equation of second line is

Direction ratios of this line are its denominators, i.e., =

A vector along this line is

Since these given lines are perpendicular.

#### NCERT Solutions class 12 Maths Miscellaneous

**7.Find the vector equation of the line passing through (1, 2, 3) and perpendicular to the plane **** **

**Ans. **The required line passes through the point P (1, 2, 3).

Position vector (say) of point P is (1, 2, 3)

Equation of the given plane is

Comparing with

Since, the required line is perpendicular to the given plane, therefore, vector along the required line is

Equation of the required line is

#### NCERT Solutions class 12 Maths Miscellaneous

**8.Find the equation of the plane passing through **** and parallel to the plane **** **

**Ans. **Equation of any plane parallel to the plane is …..(i)

Plane (i) passes through

Putting in eq. (i), we get

Putting the value of in eq. (i), to get the required plane is

#### NCERT Solutions class 12 Maths Miscellaneous

**9.Find the shortest distance between lines **** and **** **

**Ans. **Given: Vector equation of one line is

Comparing with we get

and

Again given: Vector equation of another line is

Comparing with we get

and

We know that length of shortest distance between two (skew) lines is ..(i)

Now = =

Again

Expanding along first row,

=

=

And =

Putting these values in eq. (i), length of shortest distance =

#### NCERT Solutions class 12 Maths Miscellaneous

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

**Ans. **Given: A line through the points A (5, 1, 6) and B (3, 4, 1)

Direction ratios of this line AB are

3 – 5, 4 – 1, 1 – 6

Equation of the line AB is

……….(i)

Now we have to find the coordinates of the point where this line AB crosses the YZ-plane

i.e., ……….(ii)

Putting in eq. (i), we get

and

and and

and

Thus, required point is P

#### NCERT Solutions class 12 Maths Miscellaneous

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

**Ans. **Given: A line through the points A (5, 1, 6) and B (3, 4, 1)

Direction ratios of this line AB are

3 – 5, 4 – 1, 1 – 6

Equation of the line AB is

……….(i)

Now we have to find the coordinates of the point where this line AB crosses the ZX-plane

i.e., ……….(ii)

Putting in eq. (i), we get

and

and and

and

Thus, required point is P

#### NCERT Solutions class 12 Maths Miscellaneous

**12.Find the coordinates of the point where the line through **** and **** crosses the plane **** **

**Ans. **Direction ratios of the line joining the points A and B are

Equation of the line AB are ………(i)

Equation of the plane is ………(ii)

Now to find the point where line (i) crosses plane (ii),

From eq. (i) (say)

……….(iii)

Putting the values of in eq. (ii), we get

Putting in eq. (iii), point of intersection of line (i) and plane (ii) is

Thus, required point of intersection is

#### NCERT Solutions class 12 Maths Miscellaneous

**13.Find the equation of the plane passing through the point **** and perpendicular to each of the planes **** and **** **

**Ans. **Since equation of any plane through the point is

……….(i)

This required plane is perpendicular to the plane

Product of coefficients ……….(ii)

Again the required plane is perpendicular to the plane

Product of coefficients ……….(iii)

Solving eq. (ii) and (iii), we get

Putting these values of in eq. (i), we get

#### NCERT Solutions class 12 Maths Miscellaneous

**14. If the points **** and **** be equidistant from the plane **** then find the value of **** **

**Ans. **Equation of the given plane is

……….(i)

Also, the point and are equidistant from plane (i)

(Perpendicular) distance of point from plane (i)

= Distance of point from plane (i)

[ If then]

Taking positive sign,

Taking negative sign,

Hence, the values of are 1 or

#### NCERT Solutions class 12 Maths Miscellaneous

**15.Find the equation of the plane passing through the line of intersection of the planes **** and **** and parallel to ****axis.**

**Ans. **Equation of one plane is

……….(i)

Equation of the second plane is ……….(ii)

Since, equation of any plane passing through the line intersection of these two planes is

L.H.S. of I + (L.H.S. of II) = 0

……….(i)

Comparing we have

Now required plane (i) is parallel to axis ( a vector along axis is = )

Putting in eq. (i), the equation of required plane,

#### NCERT Solutions class 12 Maths Miscellaneous

**16.If O be the origin and the coordinates of P be **** then find the equation of the plane passing through P and perpendicular to OP.**

**Ans. **Given: Origin O (0, 0, 0) and point P

To find: Equation of the plane passing through P =

Direction ratios of normal OP to the plane are

Equation of the required plane is

#### NCERT Solutions class 12 Maths Miscellaneous

**17.Find the equation of the plane which contains the line of intersection of the planes **** **** and which is perpendicular to the plane **

**Ans. **Equation of any plane passing through (or containing) the line of intersection of the planes and is L.H.S. of I + (L.H.S. of II) = 0

……….(i)

Comparing with we have,

Now plane (i) is perpendicular to the given plane

Comparing with we have,

For perpendicular planes

Putting in eq. (i), equation of required plane is

#### NCERT Solutions class 12 Maths Miscellaneous

**18.Find the distance of the point **** from the point of intersection of the line **** and the plane **

**Ans. **Given: A point P (say)

and equation of the line ……….(i)

equation of the plane is

Putting the value of from eq. (i) in eq. (ii),

Putting in eq. (i),

Therefore, Point of intersection is =

Distance of the given point P from the point of intersection is

=

=

#### NCERT Solutions class 12 Maths Miscellaneous

**19.Find the vector equation of the line passing through **** and parallel to the plane **** and **

**Ans. **The required line passes through the point A (1, 2, 3) =

= Position vector of point A =

Let be any vector along the required line.

Vector equation of required line is

……….(i)

Since required line is parallel to the plane

and

Comparing with we have,

And Comparing with we have,

Since is perpendicular to both and

=

Expanding along first row,

=

Putting this value of in eq. (i), vector equation of required line,

#### NCERT Solutions class 12 Maths Miscellaneous

**20.Find the vector equation of the line passing through the point **** and perpendicular to the two lines: **** and **** **

**Ans. **Given: A point on the required line is A

Position vector of point A is

Also given equations of two lines

and

Direction ratios of given two lines are and

Now =

Expanding along first row,

= =

Equation of the required line is

Again replacing by

#### NCERT Solutions class 12 Maths Miscellaneous

**21.Prove that if a plane has the intercepts **** and is at a distance of **** units from the origin, then **** **

**Ans. **We know that equation of plane making intercepts (on the axes) is

Given: Perpendicular distance of the origin (0, 0, 0) from plane =

=

Squaring both sides,

#### NCERT Solutions class 12 Maths Miscellaneous

**Choose the correct answer in Exercise Q. 22 and 23.**

**22.Distance between the two planes: **** and **** is**

**(A) 2 units(B) 4 units(C) 8 units(D) units**

**Ans. **Equation of one plane is

Equation of second plane is

Here

Since, therefore, the given two lines are parallel.

We know that the distance of the parallel lines =

=

=

Therefore, option (D) is correct.

#### NCERT Solutions class 12 Maths Miscellaneous

**23. The planes: **** and **** are **

**(A) Perpendicular(B) Parallel**

**(C) intersect axis(D) passes through **

**Ans. **Equations of the given planes are

and

For perpendicular = =

Planes are not perpendicular.

For parallel

given planes are parallel.

Therefore, option (B) is correct.

## NCERT Solutions class 12 Maths Miscellaneous

NCERT Solutions Class 12 Maths PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 12 Maths includes text book solutions from both part 1 and part 2. NCERT Solutions for CBSE Class 12 Maths have total 13 chapters. 12 Maths NCERT Solutions in PDF for free Download on our website. Ncert Maths class 12 solutions PDF and Maths ncert class 12 PDF solutions with latest modifications and as per the latest CBSE syllabus are only available in myCBSEguide

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