2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise

Students can Download Maths Chapter 10 Vector Algebra Miscellaneous Exercise Questions and Answers, Notes Pdf, 2nd PUC Maths Question Bank with Answers helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations.

Karnataka 2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise

Question 1.
Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of x-axis.
Answer:
Let OP be the required vector
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise .1

Question 2.
Find the scalar components and magnitude of the vector joining the points P(x1,y1,z1) and Q (x2, y2, z2).
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise .2

KSEEB Solutions

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.
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.3
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.4

Question 4.
\(\overrightarrow{\mathbf{a}}=\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}, \text { then is it true that }|\overrightarrow{\mathbf{a}}|=|\overrightarrow{\mathbf{b}}|+|\overrightarrow{\mathbf{c}}|\) justify your answer.
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.5

Question 5.
Find the value of x for which \(\mathbf{x}(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}) \)is a unit vector.
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.6

Question 6.
Find a vector of magnitude 5 units, and parallel to the resultant of the vectors
\(\overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-\hat{\mathrm{k}} \text { and } \overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}}\)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.7

KSEEB Solutions

Question 7.
If\(\overrightarrow{\mathrm{a}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \overrightarrow{\mathbf{b}}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+3 \hat{\mathbf{k}} \text { and }\) and \(\overrightarrow{\mathbf{c}}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}\)find a unit vector parellel to the vector
\(2 \vec{a}-\vec{b}+3 \vec{c}\)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.8

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.
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.9
∴ hence AC and AB are coimear. Since A s
common A, B, C arc collinear
Let B divide AC in the ratio k: I
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.10

Question 9.
Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are (2a + b) and \((\vec{a}-3 \vec{b})\)externally in the ratio 1:2. Also, show that P is the mid point of the line segment RQ.
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.11
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.12

KSEEB Solutions

Question 10.
The two adjacent sides of a parallelogram are \(2 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}+5 \hat{\mathbf{k}} \text { and } \hat{\mathbf{i}}-2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}\) .Find the unit vector parallel to its diagonal. Also,find its area.
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.13

Question 11.
Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ
\(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.14

Question 12.
Let \(\overrightarrow{\mathrm{a}}=\hat{\mathbf{i}}+4 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}, \overrightarrow{\mathbf{b}}=3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+7 \hat{\mathbf{k}} \text { and }\overrightarrow{\mathrm{c}}=2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+4 \hat{\mathrm{k}}\) .find the vector \(\overrightarrow{\mathrm{d}}\) which is perpendicular to both \(\overrightarrow{\mathbf{a}} \text { and } \overrightarrow{\mathbf{b}} \text { and } \overrightarrow{\mathbf{c}} . \overrightarrow{\mathbf{d}}\) = 15
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.15

Question 13.
The scalar product of the vector \(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}\) a unit vector along the sum of vectors \(2 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-5 \hat{\mathbf{k}} \text { and } \quad \lambda \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\) is equal to ane. find the value of λ.
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.16

KSEEB Solutions

Question 14.
If \(\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}, \overrightarrow{\mathbf{c}}\) are mutually perpendicular vectors of equal magnitudes, show that the vector is equally inclined to \(\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}, \overrightarrow{\mathbf{c}}\)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.17
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.18

Question 15.
Prove that
\((\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}) \cdot(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}})=|\overrightarrow{\mathbf{a}}|^{2}+|\overrightarrow{\mathbf{b}}|^{2}\) if and only if \(\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}\) are perpendicular ,givens \(\overrightarrow{\mathrm{a}} \neq \overrightarrow{0}, \overrightarrow{\mathrm{b}} \neq \overrightarrow{0}\)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.19

Choose the correct answer in Exercises 16 to 19.

Question 16.
If θ is the angle between two vectors \(\overrightarrow { { a } } \) and \(\overrightarrow { { b } } \), then 3.6 > 0 only when then \(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}} \geq \mathbf{0}\) only when
(A)\(0<\theta<\frac{\pi}{2}\)
(B)\(0 \leq \theta \leq \frac{\pi}{2}\)
(C) 0<θ<π
(D) 0≤θ≤π
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.20

Question 17.
let \(\overrightarrow{\mathbf{a}} \text { and } \overrightarrow{\mathbf{b}}\) be two unit vector and θ is the angle between them. Then 3 + b is a unit vector if
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.21
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.22

Question 18.
The value\(\hat{\mathbf{i}} \cdot(\hat{\mathbf{j}} \times \hat{\mathbf{k}})+\hat{\mathbf{j}} \cdot(\hat{\mathbf{i}} \times \hat{\mathbf{k}})+\hat{\mathbf{k}} \cdot(\hat{\mathbf{i}} \times \hat{\mathbf{j}})\)
(A) 0
(B) -1
(C) 1
(D) 3
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.23

KSEEB Solutions

Question 19.
If θ is the angle between any two vectors \(\overrightarrow{\mathbf{a}} \text { and } \overrightarrow{\mathbf{b}}, \text { then }|\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}|=|\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}| \)when θ is equal to
(A) 0
(B) \(\frac{\pi}{4}\)
(C) \(\frac{\pi}{2}\)
(D) π
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.24

2nd PUC Maths Vector Algebra Miscellaneous Exercise Extra Questions and Answers

Question 1.
Write a vector of magnitude 9 units in the direction of vector \(-2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}\) (CBSE 2019)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.25

Question 2.
find
\(\lambda \text { if }(2 i+6 j+4 k) \times(i-\lambda j+7 k)=0\)(CBSE 2010)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.26

Question 3.
If \(\overline{\mathbf{a}} \text { and } \overline{\mathbf{b}}\) are two vectors such that \(|\overline{\mathbf{a}} \cdot \overline{\mathbf{b}}|=|\overline{\mathbf{a}} \times \overline{\mathbf{b}}|\), what is the angle between \(\overline{\mathbf{a}} \text { and } \overline{\mathbf{b}}\)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.27

Question 4.
\(|\vec{a}|=\sqrt{3},|\vec{b}|=\frac{2}{3} \text { and } \vec{a} \times \vec{b}\)
is a vunit vector. Write the angle between \(\overline{\mathbf{a}} \text { and } \overline{\mathbf{b}}\) (CBSE 2010)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.28
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.29

KSEEB Solutions

Question 5.
Write the projection of i – j on i + j (CBSE 2011)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.30

Question 6.
Write the value of \((\mathbf{i} \times \mathbf{j}) \cdot \overline{\mathbf{k}}+\mathbf{i} \cdot \mathbf{j}\)(CBSE 2012)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.31

Question 7.
Find the scalar components of the vector \(\overrightarrow{\mathbf{A B}}\) with initial point A (2,1) and terminal point (-5,7).     (CBSE 2012)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.32

Question 8.
\(\bar { a } =\vec { i } +4\vec { j } +2\vec { k } ,\quad \bar { b } =3\vec { i } -2\vec { j } +7\bar { k } ,\bar { c } =3\vec { i } -\vec { j } +4\bar { k } \) .find a vector \(\overline{\mathbf{p}}\) which is perpendicular both \(\overline{\mathbf{a}} \text { and } \overline{\mathbf{b}} \text { and } \overline{\mathbf{p}} . \overline{\mathbf{c}}=18\)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.33
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.34

KSEEB Solutions

Question 9.
If \(\overline { { a } } =\overrightarrow { { i } } +\overrightarrow { { j } } +\overline { { k } } ,\quad \overline { { b } } =4\overrightarrow { { i } } -2\overrightarrow { { j } } +3\overline { { k } } ,\overline { { c } } =\overrightarrow { { i } } -2\overrightarrow { { j } } +\overline { { k } } \),find a vector of magnitude 6 units which is parallel to the vector \(2\overline { { a } } -\overline { b } +\overline { { 3c } } \)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.35

Question 10.
\(\overline { { a } } =\overrightarrow { { i } } +4\overrightarrow { { j } } +2\overline { { k } } ,\quad \overline { { b } } =3\overrightarrow { { i } } -2\overrightarrow { { j } } +7\overline { { k } } ,\overline { { c } } =2\overrightarrow { { i } } -\overrightarrow { { j } } +4\overline { { k } } \). find a vector d such that it is perpendiular to both \(\overline{\mathbf{a}} \text { and } \overline{\mathbf{b}} \text { and }\overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{d}}=18\)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.36
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.37

Question 11.
Using vectors find the area of the triangle with vectors A (1, 1, 2) , B (2, 3, 5), C(1, 5, 5) .   (CBSE 2011)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.38
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.39

Question 12.
\(\alpha=3 i – j, \bar{\beta}=2 i+j-3 \bar{k}\) Express β in the form the form
β = β1 + β2 where β1|| α and β2 ⊥α (CBSE 2012)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.40

KSEEB Solutions

Question 13.
\(\begin{aligned}&\overline{\mathbf{a}} \times \overline{\mathbf{b}}=\overline{\mathbf{c}} \times \overline{\mathbf{d}}, \quad \overline{\mathbf{a}} \times \overline{\mathbf{c}}=\overline{\mathbf{b}} \times \overline{\mathbf{d}} \text { Show that }\\&(\overline{\mathbf{a}}-\overline{\mathbf{d}}) \| \mathbf{t} \mathbf{o}-\overline{\mathbf{c}} \text { when } \overline{\mathbf{a}} \neq \overline{\mathbf{d}} \text { and } \overline{\mathbf{b}} \neq \overline{\mathbf{c}}\end{aligned}\)(ISE 2011)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.41
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.42

Question 14.
find
\(\overline{\mathbf{a}} \cdot \overline{\mathbf{b}} \text { of }|\overline{\mathbf{a}}|=\mathbf{3},|\overline{\mathbf{b}}|=\mathbf{4},|\overline{\mathbf{a}} \times \mathbf{b}|=6\) (CBSE 2009)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.43

Question 15.
If \(\overline{\mathbf{a}}, \overline{\mathbf{b}}, \overline{\mathbf{c}}\) represent the sides of Δ show that \(\overline{\mathbf{a}} \times \overline{\mathbf{b}}=\overline{\mathbf{b}} \times \overline{\mathbf{c}}=\overline{\mathbf{c}} \times \overline{\mathbf{a}}\) hence derive sine formula. (CBSE 2008)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise.44