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## Thursday, 28 January 2021

### CBSE Class 12 Maths - MCQ and Online Tests - Unit 10 - Vector Algebra

#### CBSE Class 12 Maths – MCQ and Online Tests – Unit 10 – Vector Algebra

Every year CBSE conducts board exams for 12th standard. These exams are very competitive to all the students. So our website provides online tests for all the 12th subjects. These tests are also very effective and useful for those who preparing for competitive exams like NEET, JEE, CA etc. It can boost their preparation level and confidence level by attempting these chapter wise online tests.

These online tests are based on latest CBSE Class 12 syllabus. While attempting these our students can identify the weak lessons and continuously practice those lessons for attaining high marks. It also helps to revise the NCERT textbooks thoroughly.

#### CBSE Class 12 Maths – MCQ and Online Tests – Unit 10 – Vector Algebra

Question 1.
$$\vec{a}$$. $$\vec{a}$$ =
(a) 0
(b) 1
(c) |$$\vec{a}$$|²
(d) |$$\vec{a}$$|

Answer: (c) |$$\vec{a}$$|²

Question 2.
$$\vec{k}$$ × $$\vec{j}$$ =
(a) 0
(b) 1
(c) $$\vec{i}$$
(d) –$$\vec{i}$$

Answer: (d) –$$\vec{i}$$

Question 3.
The projection of the vector 2$$\hat{i}$$ – $$\hat{j}$$ + $$\hat{k}$$ on the vector $$\hat{i}$$ – 2$$\hat{j}$$ + $$\hat{k}$$ is
(a) $$\frac{4}{√6}$$
(b) $$\frac{5}{√6}$$
(c) $$\frac{4}{√3}$$
(d) $$\frac{7}{√6}$$

Answer: (b) $$\frac{5}{√6}$$

Question 4.
If $$\vec{a}$$ = $$\vec{i}$$ – $$\vec{j}$$ + 2$$\vec{k}$$ and b = 3$$\vec{i}$$ + 2$$\vec{j}$$ – $$\vec{k}$$ then the value of ($$\vec{a}$$ + 3$$\vec{b}$$)(2$$\vec{a}$$ – $$\vec{b}$$)=.
(a) 15
(b) -15
(c) 18
(d) -18

Question 5.
If |$$\vec{a}$$|= $$\sqrt{26}$$, |b| = 7 and |$$\vec{a}$$ × $$\vec{b}$$| = 35, then $$\vec{a}$$.$$\vec{b}$$ =
(a) 8
(b) 7
(c) 9
(d) 12

Question 6.
$$\vec{i}$$ – $$\vec{j}$$ =
(a) 0
(b) 1
(c) $$\vec{k}$$
(d) –$$\vec{k}$$

Question 7.
The position vector of the point (1, 0, 2) is
(a) $$\vec{i}$$ +$$\vec{j}$$ + 2$$\vec{k}$$
(b) $$\vec{i}$$ + 2$$\vec{j}$$
(c) $$\vec{2}$$ + 3$$\vec{k}$$
(d) $$\vec{i}$$ + 2$$\vec{K}$$

Answer: (d) $$\vec{i}$$ + 2$$\vec{K}$$

Question 8.
If $$\vec{a}$$ = 2$$\vec{i}$$ – 3$$\vec{j}$$ + 4$$\vec{k}$$ and $$\vec{b}$$ = $$\vec{i}$$ + 2$$\vec{j}$$ + $$\vec{k}$$ then $$\vec{a}$$ + $$\vec{b}$$ =
(a) $$\vec{i}$$ + $$\vec{j}$$ + 3$$\vec{k}$$
(b) 3$$\vec{i}$$ – $$\vec{j}$$ + 5$$\vec{k}$$
(c) $$\vec{i}$$ – $$\vec{j}$$ – 3$$\vec{k}$$
(d) 2$$\vec{i}$$ + $$\vec{j}$$ + $$\vec{k}$$

Answer: (b) 3$$\vec{i}$$ – $$\vec{j}$$ + 5$$\vec{k}$$

Question 9.
If $$\vec{a}$$ = $$\vec{i}$$ + 2$$\vec{j}$$ + 3$$\vec{k}$$ and $$\vec{b}$$ = 3$$\vec{i}$$ + 2$$\vec{j}$$ + $$\vec{k}$$, then cos θ =
(a) $$\frac{6}{7}$$
(b) $$\frac{5}{7}$$
(c) $$\frac{4}{7}$$
(d) $$\frac{1}{2}$$

Answer: (b) $$\frac{5}{7}$$

Question 10.
The modulus of 7$$\vec{i}$$ – 2$$\vec{J}$$ + $$\vec{K}$$
(a) $$\sqrt{10}$$
(b) $$\sqrt{55}$$
(c) 3$$\sqrt{6}$$
(d) 6

Answer: (c) 3$$\sqrt{6}$$

Question 11.
If |$$\vec{a}$$ + $$\vec{b}$$| = |$$\vec{a}$$ – $$\vec{b}$$|, then
(a) $$\vec{a}$$ || $$\vec{a}$$
(b) $$\vec{a}$$ ⊥ $$\vec{b}$$
(c) |$$\vec{a}$$| = |$$\vec{b}$$|
(d) None of these

Answer: (b) $$\vec{a}$$ ⊥ $$\vec{b}$$

Question 12.
The projection of the vector 2$$\hat{i}$$ + 3$$\hat{j}$$ – 6$$\hat{k}$$ on the line joining the points (3, 4, 2) and (5, 6,3) is
(a) $$\frac{2}{3}$$
(b) $$\frac{4}{3}$$
(c) –$$\frac{4}{3}$$
(d) $$\frac{5}{3}$$

Answer: (b) $$\frac{4}{3}$$

Question 13.
If |$$\vec{a}$$ × $$\vec{b}$$| – |$$\vec{a}$$.$$\vec{b}$$|, then the angle between $$\vec{a}$$ and $$\vec{b}$$, is
(a) 0
(b) $$\frac{π}{2}$$
(c) $$\frac{π}{4}$$
(d) π

Answer: (c) $$\frac{π}{4}$$

Question 14.
The angle between two vector $$\vec{a}$$ and $$\vec{b}$$ with magnitude √3 and 4, respectively and $$\vec{a}$$.$$\vec{b}$$ = 2√3 is
(a) $$\frac{π}{6}$$
(b) $$\frac{π}{3}$$
(c) $$\frac{π}{2}$$
(d) $$\frac{5π}{2}$$

Answer: (b) $$\frac{π}{3}$$

Question 15.
The scalar product of 5$$\hat{i}$$ + $$\hat{j}$$ – 3$$\hat{k}$$ and 3$$\hat{i}$$ – 4$$\hat{j}$$ + 7$$\hat{k}$$ is
(a) 10
(b) -10
(c) 15
(d) -15

Question 16.
If $$\vec{a}$$.$$\vec{b}$$ = 0, then
(a) a ⊥ b
(b) $$\vec{a}$$ || $$\vec{b}$$
(c) $$\vec{a}$$ + $$\vec{b}$$ = 0
(d) $$\vec{a}$$ – $$\vec{b}$$ = 0

Question 17.
Unit vector perpendicular to each of the vector 3$$\hat{i}$$ + $$\hat{j}$$ + 2$$\hat{k}$$ and 2$$\hat{i}$$ – 2$$\hat{j}$$ + 4$$\hat{k}$$ is
(a) $$\frac{\hat{i}+\hat{j}+\hat{k}}{√3}$$
(b) $$\frac{\hat{i}-\hat{j}+\hat{k}}{√3}$$
(c) $$\frac{\hat{i}-\hat{j}-\hat{k}}{√3}$$
(d) $$\frac{\hat{i}+\hat{j}-\hat{k}}{√3}$$

Answer: (c) $$\frac{\hat{i}-\hat{j}-\hat{k}}{√3}$$

Question 18.
Which one of the following can be written for ($$\vec{a}$$ – $$\vec{b}$$) × ($$\vec{a}$$ + $$\vec{b}$$)
(a) $$\vec{a}$$ × $$\vec{b}$$
(b) 2$$\vec{a}$$ × $$\vec{b}$$
(c) $$\vec{a}$$² – $$\vec{b}$$
(d) 2$$\vec{b}$$ × $$\vec{b}$$

Answer: (b) 2$$\vec{a}$$ × $$\vec{b}$$

Question 19.
If $$\vec{a}$$ = 2$$\vec{i}$$ – 5$$\vec{j}$$ + k and $$\vec{b}$$ = 4$$\vec{i}$$ + 2$$\vec{j}$$ + $$\vec{k}$$ then $$\vec{a}$$.$$\vec{b}$$ =
(a) 0
(b) -1
(c) 1
(d) 2

Question 20.
If 2$$\vec{i}$$ + $$\vec{j}$$ + $$\vec{k}$$, 6$$\vec{i}$$ – $$\vec{j}$$ + 2$$\vec{k}$$ and 14$$\vec{i}$$ – 5$$\vec{j}$$ + 4$$\vec{k}$$ be the position vector of the points A, B and C respectively, then
(a) The A, B and C are collinear
(b) A, B and C are not colinear
(c) $$\vec{AB}$$ ⊥ $$\vec{BC}$$
(d) None of these

Answer: (a) The A, B and C are collinear

Question 21.
According to the associative lass of addition of addition of s ector
($$\vec{a}$$ + …….) + $$\vec{c}$$ = …… + ($$\vec{b}$$ + $$\vec{c}$$)
(a) $$\vec{b}$$, $$\vec{a}$$
(b) $$\vec{a}$$, $$\vec{b}$$
(c) $$\vec{a}$$, 0
(d) $$\vec{b}$$, 0

Answer: (a) $$\vec{b}$$, $$\vec{a}$$

Question 22.
The points with position vectors (2. 6), (1, 2) and (a, 10) are collinear if the of a is
(a) -8
(b) 4
(c) 3
(d) 12

Question 23.
|$$\vec{a}$$ + $$\vec{b}$$| = |$$\vec{a}$$ – $$\vec{b}$$| then the angle between $$\vec{a}$$ and $$\vec{b}$$
(a) $$\frac{π}{2}$$
(b) 0
(c) $$\frac{π}{4}$$
(d) $$\frac{π}{6}$$

Answer: (a) $$\frac{π}{2}$$

Question 25.
If $$\vec{a}$$ and $$\vec{b}$$ are any two vector then ($$\vec{a}$$ × $$\vec{b}$$)² is equal to
(a) ($$\vec{a}$$)²($$\vec{b}$$)² – ($$\vec{a}$$.$$\vec{b}$$)²
(b) ($$\vec{a}$$)² ($$\vec{b}$$)² + ($$\vec{a}$$.$$\vec{b}$$)²
(c) ($$\vec{a}$$.$$\vec{b}$$)²
(d) ($$\vec{a}$$)²($$\vec{b}$$)²

Answer: (a) ($$\vec{a}$$)²($$\vec{b}$$)² – ($$\vec{a}$$.$$\vec{b}$$)²

Question 26.
|$$\vec{a}$$ × $$\vec{b}$$| = |$$\vec{a}$$.$$\vec{b}$$| then the angle between $$\vec{a}$$ and $$\vec{b}$$
(a) 0
(b) $$\frac{π}{2}$$
(c) $$\frac{π}{4}$$
(d) π

Question 27.
If ABCDEF is a regular hexagon then $$\vec{AB}$$ + $$\vec{EB}$$ + $$\vec{FC}$$ equals
(a) zero
(b) 2$$\vec{AB}$$
(c) 4$$\vec{AB}$$
(d) 3$$\vec{AB}$$

Answer: (d) 3$$\vec{AB}$$

Question 28.
Which one of the following is the modulus of x$$\hat{i}$$ + y$$\hat{j}$$ + z$$\hat{k}$$?
(a) $$\sqrt{x^2+y^2+z^2}$$
(b) $$\frac{1}{\sqrt{x^2+y^2+z^2}}$$
(c) x² + y² + z²
(d) none of these

Answer: (a) $$\sqrt{x^2+y^2+z^2}$$

Question 29.
If C is the mid point of AB and P is any point outside AB then,
(a) $$\vec{PA}$$ + $$\vec{PB}$$ = 2$$\vec{PC}$$
(b) $$\vec{PA}$$ + $$\vec{PB}$$ = $$\vec{PC}$$
(c) $$\vec{PA}$$ + $$\vec{PB}$$ = 2$$\vec{PC}$$ = 0
(d) None of these

Answer: (a) $$\vec{PA}$$ + $$\vec{PB}$$ = 2$$\vec{PC}$$

Question 30.
If $$\vec{OA}$$ = 2$$\vec{i}$$ – $$\vec{j}$$ + $$\vec{k}$$, $$\vec{OB}$$ = $$\vec{i}$$ – 3$$\vec{j}$$ – 5$$\vec{k}$$ then |$$\vec{OA}$$ × $$\vec{OB}$$| =
(a) 8$$\vec{i}$$ + 11$$\vec{j}$$ – 5$$\vec{k}$$
(b) $$\sqrt{210}$$
(c) sin θ
(d) $$\sqrt{40}$$

Answer: (b) $$\sqrt{210}$$

Question 31.
The distance of the point (- 3, 4, 5) from the origin
(a) 50
(b) 5√2
(c) 6
(d) None of these

Question 32.
If |a| = |b| = |$$\vec{a}$$ + $$\vec{b}$$| = 1 then |$$\vec{a}$$ – $$\vec{b}$$| is equal to
(a) 1
(b) √3
(c) 0
(d) None of these

Question 33.
If $$\hat{a}$$ and $$\hat{b}$$ be two unit vectors and 0 is the angle between them, then |$$\hat{a}$$ – $$\hat{b}$$| is equal to
(a) sin $$\frac{θ}{2}$$
(b) 2 sin $$\frac{θ}{2}$$
(c) cos $$\frac{θ}{2}$$
(d) 2 cos $$\frac{θ}{2}$$

Answer: (b) 2 sin $$\frac{θ}{2}$$

Question 34.
The angle between the vector 2$$\hat{i}$$ + 3$$\hat{j}$$ + $$\hat{k}$$ and 2$$\hat{i}$$ – $$\hat{j}$$ – $$\hat{k}$$ is
(a) $$\frac{π}{2}$$
(b) $$\frac{π}{4}$$
(c) $$\frac{π}{3}$$
(d) 0

Answer: (a) $$\frac{π}{2}$$

Question 34.
If O be the origin and $$\vec{OP}$$ = 2$$\hat{i}$$ + 3$$\hat{j}$$ – 4$$\hat{k}$$ and $$\vec{OQ}$$ = 5$$\hat{i}$$ + 4$$\hat{j}$$ -3$$\hat{k}$$, then $$\vec{PQ}$$ is equal to
(a) 7$$\hat{i}$$ + 7$$\hat{j}$$ – 7$$\hat{k}$$
(b) -3$$\hat{i}$$ + $$\hat{j}$$ – $$\hat{k}$$
(c) -7$$\hat{i}$$ – 7$$\hat{j}$$ + 7$$\hat{k}$$
(d) 3$$\hat{i}$$ + $$\hat{j}$$ + $$\hat{k}$$

Answer: (d) 3$$\hat{i}$$ + $$\hat{j}$$ + $$\hat{k}$$

Question 35.
If $$\vec{a}$$ = $$\hat{i}$$ – $$\hat{j}$$ + $$\hat{k}$$, $$\vec{b}$$ = $$\hat{i}$$ + 2$$\hat{j}$$ – $$\hat{k}$$, $$\vec{c}$$ = 3$$\hat{i}$$ – p$$\hat{j}$$ – 5$$\hat{k}$$ are coplanar then P =
(a) 6
(b) -6
(c) 2
(d) -2

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