NCERT Solutions for Class 12 Maths Chapter 10 Miscellaneous Exercise
NCERT Solutions for Class 12 Maths Chapter 10 Miscellaneous Exercise contains all the questions with detailed solutions. Students are advised to solve these questions for better understanding of the concepts in chapter 10.
Krati Saraswat12 Feb, 2024
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NCERT Solutions for Class 12 Maths Chapter 10 Miscellaneous Exercise (Vector Algebra)
NCERT Solutions for Class 12 Maths Chapter 10 Miscellaneous Exercise Vector Algebra is prepared by the academic team of Physics Wallah. We have prepared NCERT Solutions for all exercise of Chapter 10. Given below are step-by-step solutions to all questions given in the NCERT Solutions for Class 12 Maths Chapter 10 Miscellaneous Exercise.NCERT Solutions for Class 12 Maths Chapter 10 Exercise 10.1
NCERT Solutions for Class 12 Maths Chapter 10 Miscellaneous Exercise Overview
NCERT Solutions for Class 12 Maths Chapter 10 Miscellaneous Excercise addresses these significant subjects. In order to fully comprehend the concepts presented in the chapter and make effective use of the provided solutions, it is recommended that students go over each topic in great detail. The intention is for students to effortlessly achieve excellent exam scores after reviewing and practicing these responses.NCERT Solutions for Class 12 Maths Chapter 10 Miscellaneous Exercise
Solve The Following Questions of NCERT Solutions for Class 12 Maths Chapter 10 Miscellaneous Exercise:
Question 1. Write down a unit vector in XY-plane making an angle of 30° with the positive direction of x-axis. Solution : If
is a unit vector in the XY-plane, then
Here,
θ
is the angle made by the unit vector with the positive direction of the
x
-axis.
Therefore, for
θ
= 30°:
Solution :
Given points are
Hence, the scalar components and the magnitude of the vector joining the given points are respectively
NCERT Solutions for Class 12 Maths Chapter 10 Exercise 10.2
Question 3. A girl walks 4 km towards west, then she walks 3 km in a direction 30° east of north and stops. Determine the girl’s displacement from her initial point of departure. Solution : Let O and B be the initial and final positions of the girl respectively. Then, the girl’s position can be shown as:
Hence, the girl’s displacement from her initial point of departure is
Question 4. If
then is it true that
Justify your answer.
Solution :
NCERT Solutions for Class 12 Maths Chapter 10 Exercise 10.3
Question 5. Find the value of x for which x (.png)
) is a unit vector.
Solution :
x (
)
is a unit vector if |
x (
)| = 1
Hence, the required value of
x
is ± 1/√3.
Question 6. Find a vector of magnitude 5 units and parallel to the resultant of the vectors
Solution :
Given: Vectors
Let
.png)
be the resultant of
.png)
and
.png)
Hence, the vector of magnitude 5 units and parallel to the resultant of vectors
and
is
Question 7. If
find a unit vector parallel to the vector
Solution :
Given: Vectors
Question 8. Show that the points A (1, –2, –8), B (5, 0, –2) and C (11, 3, 7) are collinear and find the ratio in which B divides AC.
Solution :
The given points are A (1, –2, –8), B (5, 0, –2), and C (11, 3, 7).
Hence, point B divides AC in the ratio 2 : 3.
Question 9. Find the position vector of a point R which divides the line joining the two points P and Q whose position vectors are (2
+
) and (
- 3
) externally in the ratio 1 : 2. Also, show that P is the middle point of line segment RQ.
.
It is given that point R divides a line segment joining two points P and Q externally in the ratio 1: 2. Then, on using the section formula, we get:
Therefore, the position vector of point R is
.
Position vector of the mid-point of RQ =
Hence, P is the mid-point of the line segment RQ.
Question 10. Two adjacent sides of a parallelogram are
Find the unit vector parallel to its diagonal. Also, find its area.
Solution :
Adjacent sides of a parallelogram are given as:
Then, the diagonal of a parallelogram is given by
+
.
Hence, the area of the parallelogram is 11√5 square units.
Question 11. Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are
Solution :
Let a vector be equally inclined to axes OX, OY, and OZ at angle
α
.
Then, the direction cosines of the vector are cos
α
, cos
α
, and cos
α
.
Hence, the direction cosines of the vector which are equally inclined to the axes are
.
Question 12. Let
Find a vector
.png)
which is perpendicular to both
and
and
.
= 15
Solution :
Question 13. The scalar product of the vector
.png)
with a unit vector along the sum of vectors
is equal to one. Find the value of λ.
Solution :
Hence, the value of
λ
is 1.
Question 14. If are mutually perpendicular vectors of equal magnitudes, show that the vector
+
+
is equally inclined to
,
, and
.
Solution :
Since
,
, and
are mutually perpendicular vectors, we have
.
=
.
=
.
= 0
It is given that:
|
| = |
| = |
|
Let vector
+
+
be inclined to
,
,
and
at angles θ
1
,θ
2
and θ
3
respectively.
Then, we have:
Hence, the vector (
+
+
)
is equally inclined to
,
,
and
.
Question 15. Prove that
if and only if
,
are perpendicular given
.
Solution :
Question 16. Choose the correct answer:
If
θ
is the angle between two vectors
and
then
.
≥0
only when:
Solution :
Let
θ
be the angle between two vectors
and
.
Then, without loss of generality,
and
are non-zero vectors so that |
| and |
| are positives
It is known that
..
= |
||
|cosθ
Therefore, option (B) is correct.
Question 17. Choose the correct answer:
Let
and
be two unit vectors and
θ
is the angle between them. Then
+
is a unit vector if
(A) θ = π/4
(B) θ = π/3
(C) θ = π/2
(D) θ = π/3
Solution :
Let
and
be two unit vectors and
θ
be the angle between them.
Then, |
| = |
| = 1
Now,
+
is a unit vector if |
+
| = 1
Hence,
+
is a unit vector if θ = 2π/3.
Therefore, option (D) is correct.
Question 18. Choose the correct answer:
The value of
is:
(A) 0
(B) -1
(C) 1
(D) 3
Solution :
Therefore, option (C) is correct.
Question 19. If θ be the angle between any two vectors
and
, then
when
θ is equal to:
(A) 0
(B) π/4
(C) π/2
(D) π
Solution :
Let
θ
be the angle between two vectors
and
.
Then, without loss of generality,
and
are non-zero vectors, so that
|
| and |
| are positive.
Therefore, option (B) is correct.
NCERT Solutions for Class 12 Maths Chapter 10 Miscellaneous Exercise FAQs
What is the hardest chapter in 12 Maths?
For most of the students, the hardest chapter is integrals.
How many exercises are there in Chapter 10 Maths class 12?
NCERT Solutions Class 12 Maths Chapter 10 Vector Algebra has a total of 63 questions that have been spread across 5 exercises including a miscellaneous one.
Which is the most important chapter in Maths class 12?
Students must give priority to those chapters primarily that consist of major weightage of marks. Algebra, Calculus, Vectors, and Three-Dimensional Geometry are the most vital chapters from this perspective.
What is a collinear vector?
Vectors that lie along the same line or parallel lines are known to be collinear vectors. They are also known as parallel vectors. Two vectors are collinear if they are parallel to the same line irrespective of their magnitudes and direction.
What is a vector in 12th maths?
Vector Algebra for class 12 is all about the study of vectors and scalars. There are various quantities, which involves magnitude as well as direction. If the quantity that has magnitude, as well as direction, is known as vectors. Such quantities are known as Vector Quantities.
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