**NCERT Solutions for Maths Class 12 Exercise 11.2**

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Class 12th is a very crucial stage of your student’s life, since you take all important decisions about your career on this stage. Mathematics plays a vital role to take decision for your career because if you are good in mathematics, you can choose engineering and technology field as your career.

**NCERT Solutions for Maths Class 12 Exercise 11.2** helps you to solve each and every problem with step by step explanation which makes you strong in mathematics.

All the schools affiliated with CBSE, follow the NCERT books for all subjects. You can check your syllabus from **NCERT Syllabus for Mathematics Class 12**.

**NCERT Solutions for Maths Class 12 Exercise 11.2** are prepared by the experienced teachers of CBSE board. If you are preparing for JEE Mains and NEET level exams, then it will definitely make your foundation strong.

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**NCERT Solutions for Maths Class 12 Exercise 11.1**

**NCERT Solutions for Maths Class 12 Exercise 11.3**

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**NCERT Solutions for Maths Class 12 Exercise 11.2**

**Maths Class
12 Ex 11.2 Question 1.**

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.

**Solution:**

Let the lines be L_{1},
L_{2} and L_{3}.

∴ For line L_{1}, direction
cosines are: *l*_{1} = 12/13;
*m*_{1} = –3/13;
*n*_{1} = –4/13;

For line L_{2}, direction cosines are: *l*_{2} = 4/13; *m*_{2}
= 12/13; *n*_{2}
= 3/13;

For line L_{3}, direction cosines are: *l*_{3} = 3/13; *m*_{3}
= –4/13; *n*_{3}
= 12/13;

Let the angle between lines L_{1 }and L_{2 }is Î¸_{1}.

Then, cos Î¸_{1
}= *l*_{1}* l*_{2 }+ *m*_{1}* m*_{2 }+
*n*_{1}* n*_{2}

= (12/13 × 4/13) + (–3/13
× 12/13) + (–4/13 × 3/13)

= 48/169 – 36/169 –
12/169 = 0

∴
Î¸_{1} = 90˚

Thus,
the angle between lines L_{1
}and L_{2 }is 90˚.

Let the angle between lines L_{2 }and L_{3 }is Î¸_{2}.

Then, cos Î¸_{2
}= *l*_{2}* l*_{3 }+ *m*_{2}* m*_{3 }+
*n*_{2}* n*_{3}

= (4/13 × 3/13) + (12/13 × –4/13)
+ (3/13 × 12/13)

= 12/169 – 48/169 + 36/169
= 0

∴
Î¸_{2} = 90˚

Thus,
the angle between lines L_{2
}and L_{3 }is 90˚.

Let the angle between lines L_{1 }and L_{3 }is Î¸_{3}.

Then, cos Î¸_{3
}= *l*_{1}* l*_{3 }+ *m*_{1}* m*_{3 }+
*n*_{1}* n*_{3}

= (12/13 × 3/13) + (–3/13
× –4/13) + (–4/13 × 12/13)

= 36/169 + 12/169 – 48/169
= 0

∴
Î¸_{3} = 90˚

Thus,
the angle between lines L_{1
}and L_{3 }is 90˚.

Hence,
the given lines are mutually perpendicular.

**Maths Class
12 Ex 11.2 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 and B be
the points (1, –1, 2) and (3, 4, –2), respectively.

Direction ratios
of AB are: a_{1} = 3 – 1 = 2, b_{1} = 4 – (–1) = 5, c_{1}
= –2 – 2 = –4

Let C and D be
the points (0, 3, 2) and (3, 5, 6), respectively.

Direction ratios
of CD are: a_{2} = 3 – 0 = 3, b_{2} = 5 – 3 = 2, c_{2}
= 6 – 2 = 4

Now, AB is perpendicular
to CD if

a_{1}a_{2}
+ b_{1}b_{2} + c_{1}c_{2} = 0

2 × 3 + 5 × 2 +
(–4) × 4 = 6 + 10 – 16 = 0

Hence, the lines
AB and CD are perpendicular.

**Maths Class
12 Ex 11.2 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) and D(1, 2, 5).

Now, direction ratios of AB are:

**Maths Class
12 Ex 11.2 Question 4.**

Find the equation
of the line which passes through the point (1, 2, 3) and is parallel to the
vector 3** i** + 2

**– 2**

*j***.**

*k***Solution:**

Equation of the
line passing through the point

**Maths Class
12 Ex 11.2 Question 5.**

Find the equation
of the line in vector and in Cartesian form that passes through the point with
position vector 2** i** –

**+ 4**

*j***and is in the direction**

*k***+ 2**

*i***–**

*j***.**

*k***Solution:**

The vector
equation of a line passing through a point with position vector ** a** = 2

**–**

*i***+ 4**

*j***and parallel to the vector**

*k***=**

*b***+ 2**

*i***–**

*j***is**

*k***Maths Class
12 Ex 11.2 Question 6.**

Find the Cartesian equation of the line which passes through the point P(–2, 4, –5) and parallel to the line is given by.

**Solution:**

Given that the line passes through the point (–2, 4, –5) and parallel to the line given by .

Thus, the line
passing through P(x_{1}, y_{1}, z_{1}) where x_{1}
= –2, y_{1} = 4, z_{1} = –5 and the direction ratios a_{1}
= 3, b_{1} = 5, c_{1} = 6 is

**Maths Class
12 Ex 11.2 Question 7.**

The Cartesian equation of a line is. Write its vector form.

**Solution:**

The Cartesian equation of the line is … (i)

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

**Maths Class 12 Ex 11.2 Question 8.**

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

**Solution: **

Given that the
line passes through point A (0, 0, 0) and B(5, –2, 3).

Therefore, ** a** = 0

**+ 0**

*i***+ 0**

*j***and**

*k***= 5**

*b***– 2**

*i***+ 3**

*j*

*k*Direction ratios of the line passing through the

**Maths Class
12 Ex 11.2 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).

**Maths Class
12 Ex 11.2 Question 10.**

Find the angle
between the following pairs of lines

**Solution:**

**(i)** Let Î¸ be the angle between the given lines.

The given lines are parallel to the vectors

**Maths Class
12 Ex 11.2 Question 11.**

Find the angle
between the following pairs of lines

**Solution:**

**Maths Class
12 Ex 11.2 Question 12.**

Find the values
of *p* so that the lines

are at right angles.

**Solution:**

The given
equations are not in the standard form.

The equations of given lines in standard form are

**Maths Class
12 Ex 11.2 Question 13.**

Show that the linesare perpendicular to each other

**Solution:**

Then the direction ratios of the given lines are: *a*_{1} = 7, *b*_{1}
= –5, *c*_{1} = 1 and *a*_{2} = 1, *b*_{2} = 2, *c*_{2}
= 3

**Maths Class
12 Ex 11.2 Question 14.**

Find the shortest
distance between the lines

**Solution:**

**Maths Class
12 Ex 11.2 Question 15.**

Find the shortest
distance between the lines

**Solution:**

Shortest distance
between the lines

**Maths Class
12 Ex 11.2 Question 16.**

Find the shortest
distance between the lines whose vector equations are:

**Solution:**

**Maths Class
12 Ex 11.2 Question 17.**

Find the shortest
distance between the lines whose vector equations are

**Solution:**