WBCHSE Class 11 Physics Vector Multiple Choice Questions And Answers

Vector Multiple Choice Questions And Answers

Question 1. What is the condition for \(\vec{A}+\vec{B}=\vec{A}-\vec{B}\) to be valid?

1. \(\vec{A}=0\)
2. \(\vec{B}=0\)
3. \(\vec{A}=\vec{B}\)
4. \(\vec{A}=-\vec{B}\)

Answer: 2. \(\vec{B}=0\)

Question 2. The magnitudes of the sum and difference of the two vectors are equal. The angle between the vectors is

1. 0°
2. 90°
3. 120°
4. 60°

Answer: 2. 90°

Question 3. The magnitude of each of the two vectors is P. The Magnitude of the resultant of the two is also P. The Angle between the vectors is

1. 0°
2. 60°
3. 120°
4. 90°

Answer: 3. 120°

Question 4.Two forces of equal magnitude act simultaneously on a particle. If the resultant of the forces is equal to the magnitude of each of them then the angle between the forces is

1. An acute angle
2. An obtuse angle
3. A right angle
4. Of any value

Answer: 2. An obtuse angle

Question 5. A man travels 30 m to the north, then 20 m to the east, and after that 30√2 m to the south-west. His displacement from the starting point is

1. 15 m to the east
2. 28 m to the south
3. 10 m to the west
4. 15 m to the southwest

Answer: 3. 10 m to the west

Question 6. \(0.2 \hat{i}+0.6 \hat{j}+a \hat{k}\) is a unit vector. The value of a should be

1. 0.3
2. 0.4
3. 0.6
4. 0.8

Answer: 3. 0.6

Question 7. \(\vec{R}\) is the resultant of the vectors \(\vec{A}\) and \(\vec{B}\). If R = \(\frac{B}{\sqrt{2}}\), then the angle θ is

1. 30°
2. 45°
3. 60°
4. 75°

Answer: 2. 45°

Question 8. The position vector of a particle is related to time t as \(\vec{r}=\left(t^2-1\right) \hat{i}+2 t \hat{j}\). The locus of the particle on the x-y plane is

1. Parabolic
2. Circular
3. Straight line
4. Elliptical

Answer: 1. Parabolic

Question 9. Two forces of the same magnitude F are at right angles to each other. The magnitude of the net force (total force) acting on the object is

1. F
2. 2 F
3. Between F and 2F
4. More than 2 F

Answer: 3. Between F and 2 F

Question 10. If a and b are two unit vectors inclined at an angle of 60° to each other, then

1. \(|a+b|>1\)
2. \(|a+b|<1\)
3. \(|a-b|>1\)
4. \(|a-b|<1\)

Answer: 1. \(|a+b|>1\)

Question 11. The condition (a-b)² = a²b² is satisfied when

1. a is parallel to b
2. a ≠ b
3. a-b = 1
4. a ⊥ b

Answer: 3. a-b = 1

Question 12. If \(\vec{P}+\vec{Q}=\vec{R} \text { and }|\vec{P}|=|\vec{Q}|=\sqrt{3} \text { and }|\vec{R}|=3\), then the angle between \(\vec{P}\) and \(\vec{Q}\) is

1. \(\frac{\pi}{4}\)
2. \(\frac{\pi}{6}\)
3. \(\frac{\pi}{3}\)
4. \(\frac{\pi}{2}\)

Answer: 3. \(\frac{\pi}{3}\)

Question 13. The angles which the vector \(\vec{A}=3 \hat{i}+6 \hat{j}+2 \hat{k}\) makes with the coordinate axes are

1. \(\cos ^{-1} \frac{3}{7}, \cos ^{-1} \frac{6}{7}\) and \(\cos ^{-1} \frac{2}{7}\)
2. \(\cos ^{-1} \frac{4}{7}, \cos ^{-1} \frac{5}{7} \)and \(\cos ^{-1} \frac{3}{7}\)
3. \(\cos ^{-1} \frac{3}{7}, \cos ^{-1} \frac{4}{7}\) and \(\cos ^{-1} \frac{1}{7}\)
4. None of the above

Answer: 1. \(\cos ^{-1} \frac{3}{7}, \cos ^{-1} \frac{6}{7}\) and \(\cos ^{-1} \frac{2}{7}\)

Question 14. If  \(\vec{a}+\vec{b}=\vec{c}\), then the angle between \(\vec{a}\) and \(\vec{b}\) is

1. 90°
2. 180°
3. 120°
4. Zero

Answer: 4. Zero

Question 15. If at and are two non-collinear unit vectors and if \(\left|a_1+a_2\right|=\sqrt{3}\) then the value of \(\left(a_1-a_2\right) \cdot\left(2 a_1+a_2\right)\) is

1. 2
2. 3/2
3. 1/2
4. 1

Answer: 3. 1/2

Question 16. Which of the following is a vector quantity?

1. Temperature
2. Impulse
3. Gravitational potential
4. Power

Answer: 2. Impulse

Question 17. The resultant of two forces of magnitude (x+y) and (x-y) is \(\sqrt{x^2+y^2}\). The angle between them is

1. \(\cos ^{-1}\left[-\frac{\left(x^2+y^2\right)}{2\left(x^2-y^2\right)}\right]\)
2. \(\cos ^{-1}\left[-\frac{2\left(x^2-y^2\right)}{x^2+y^2}\right]\)
3. \(\cos ^{-1}\left[-\frac{x^2+y^2}{x^2-y^2}\right]\)
4. \(\cos ^{-1}\left[-\frac{x^2-y^2}{x^2+y^2}\right]\)

Answer: 1. \(\cos ^{-1}\left[-\frac{\left(x^2+y^2\right)}{2\left(x^2-y^2\right)}\right]\)

Question 18. ABC is an equilateral triangle and O is its centre of mass. Find n if \(\overrightarrow{A B}+\overrightarrow{A C}=n \overrightarrow{A O}\)

1. 1
2. 2
3. 3
4. 4

Hint: \(\overrightarrow{A B}+\overrightarrow{A C}=(\overrightarrow{A O}+\overrightarrow{O B})+(\overrightarrow{A O}+\overrightarrow{O C})\)

= \(2 \overrightarrow{A O}+(\overrightarrow{O B}+\overrightarrow{O C})=2 \overrightarrow{A O}+\overrightarrow{A O}\)

= \(3 \overrightarrow{A O}\)

Answer: 3. 3

Question 19. The sum of the magnitudes of two forces acting at a point is 16 N. The resultant has a magnitude of 8N and is perpendicular to the force of lower magnitude. The two forces are

1. 6N and 10N
2. 8N and 8N
3. 4N and 12N
4. 2N and 14N

Answer: 1. 6N and 10N

Question 20. Two men have velocities of 4 m/s towards the east and 3 m/s towards west, respectively. The velocity of the first man relative to the second is

1. \((4 \hat{i}+3 \hat{j})\)
2. \((3 \hat{i}+4 \hat{j})\)
3. \((4 \hat{i}-3 \hat{j})\)
4. \((3 \hat{i}-4 \hat{j})\)

Answer: 1. \((4 \hat{i}+3 \hat{j})\)

Question 21. Two forces in the ratio 1:2 act simultaneously on a particle. The result of these forces is three times the first force. The angle between them is

1. 0°
2. 60°
3. 90°
4. 45°

Answer: 1. 0°

Question 22. Given \(\vec{A}=2 \hat{i}+3 \hat{j} \text { and } \vec{B}=\hat{i}+\hat{j}\). The component of vector \(\vec{A}\) along vector \(\vec{B}\) is

1. \(\frac{1}{\sqrt{2}}\)
2. \(\frac{3}{\sqrt{2}}\)
3. \(\frac{5}{\sqrt{2}}\)
4. \(\frac{7}{\sqrt{2}}\)

Answer: 3. \(\frac{5}{\sqrt{2}}\)

Question 23. One of the components of a velocity vector of magnitude 50 m · s^-1 Is 30 m · s^-1, Its other orthogonal component Is

1. 15 m · s^-1
2. 20 m · s^-1
3. 25 m · s^-1
4. 40 m · s^-1

Answer: 4. 40 m · s^-1

Question 24. The times of flight of the two projectiles are t₁ and t₂ If R is the horizontal range of each of them, then

1. \(t_1 t_2 \propto R\)
2. \(t_1 t_2 \propto R^2\)
3. \(t_1 t_2 \propto R^3\)
4. \(t_1 t_2 \propto R^{\frac{1}{2}}\)

Hint: The angles of projection are \(\theta\) and \(\left(90^{\circ}-\theta\right)\).

∴ \(t_1 t_2=\frac{2 u \sin \theta}{g} \cdot \frac{2 u \sin \left(90^{\circ}-\theta\right)}{g}\)

= \(\frac{4 u^2}{g^2} \sin \theta \cos \theta=\frac{2}{g} \cdot \frac{u^2 \sin 2 \theta}{g}=\frac{2}{g} R\)

∴ \(t_1 t_2\propto R\)

Answer: 1. \(t_1 t_2 \propto R\)

Question 25. For angles θ and (90° – θ) of projection, a projectile has the same horizontal range R. The maximum heights attained are H₁ and H₂ respectively. Then the relation among R, H_1, and H₂ is

1. \(R=\sqrt{H_1 H_2}\)
2. \(R=\sqrt{H_1^2+H_2^2}\)
3. \(R=H_1+H_2\)
4. \(R=4 \sqrt{H_1 H_2}\)

Answer: 4. \(R=4 \sqrt{H_1 H_2}\)

Question 26. An airplane is flying with a velocity of 216 km/h at an altitude of 1960 m relative to the ground. It drops a bomb when it is just above point A on the ground. The bomb hits the ground at B. The distance AB is

1. 1.2 km
2. 0.33 km
3. 3.33 km
4. 33 km

Answer: 1. 1.2 km

Question 27. The initial velocity and the acceleration of a particle are \(\vec{u}\) = 3\(\hat{i}\)+4\(\hat{j}\) and \(\vec{a}\) = 0.3 \(\hat{i}\)+0.4\(\hat{j}\). The magnitude of its velocity after 10 s is

1. 10 units
2. 8.5 units
3. 77√2 units
4. 7 units

Answer: 1. 10 units

Question 28. The equations of motion of a projectile are x = 36t and 2y = 96t- 9.8 t². The angle of projection is

1. \(\sin ^{-1}\left(\frac{4}{5}\right)\)
2. \(\sin ^{-1}\left(\frac{3}{5}\right)\)
3. \(\sin ^{-1}\left(\frac{3}{4}\right)\)
4. \(\sin ^{-1}\left(\frac{4}{3}\right)\)

Hint: x= 36t and y = 48t – \(\frac{1}{2}\) x 9.8t²—they suggest that the horizontal and vertical components of initial velocity are 36 and 48 units, respectively.

∴ \(\tan \theta=\frac{48}{36}=\frac{4}{3} \text { or, } \sin \theta=\frac{4}{5}\)

Answer: 1. \(\sin ^{-1}\left(\frac{4}{5}\right)\)

Question 29. Two projectiles, projected with angles (45°-θ) and (45° +θ) respectively, have their horizontal ranges in the ratio

1. 2:1
2. 1:1
3. 2:3
4. 1:2

Answer: 2. 1:1

Question 30. For a projectile, (horizontal range)² = 48 x (maximum height)². The angle of projection is

1. 45°
2. 60°
3. 75°
4. 30°

Answer: 4. 30°

Question 31. Two railway tracks are parallel to the west-east direction. Along one track train A moves with a speed of 30 m · s^-1 from west to east, while along the second track, train B moves with a speed of 48 m · s^-1 from east to west. The relative speed of B with respect to A is

1. 48 m · s^-1
2. -78 m · s^-1
3. 30 m · s^-1
4. Zero

Answer: 2. 48 m · s^-1

Question 32. Angle between the vectors \(\hat{i}+\hat{j} \text { and } \hat{i}-\hat{k}\) is

1. 60°
2. 30°
3. 45°
4. 90°

Answer: 1. 60°

Question 33. A vector is multiplied by (-2). As a result

1. The magnitude of the vector is doubled and the direction is unaltered
2. The magnitude of the vector remains the same and the direction is reversed
3. The magnitude of the vector is doubled and the direction is reversed
4. No change in the magnitude or direction of the vector

Answer: 3. The Magnitude of the vector is doubled and the direction is reversed

Question 34. In a clockwise system

1. \(\hat{j} \times \hat{j}=1\)
2. \(\hat{k} \cdot \hat{i}=1\)
3. \(\hat{j} \times \hat{k}=\hat{i}\)
4. \(\hat{i} \cdot \hat{i}=0\)

Answer: 3. \(\hat{j} \times \hat{k}=\hat{i}\)

Question 35. A vector \(\vec{P}=3 \hat{i}-2 \hat{j}+a \hat{k}\) is perpendicular to the vector \(\vec{Q}=2 \hat{i}+\hat{j}-\hat{k}\). The value of a is

1. 2
2. 1
3. 4
4. 3

Answer: 3. 4

Question 36. For any two vectors \(\vec{A} \text { and } \vec{B} \text {, if } \vec{A} \cdot \vec{B}=|\vec{A} \times \vec{B}|\), the magnitude of \(\vec{C}=\vec{A}+\vec{B}\) is equal to

1. \(\sqrt{A^2+B^2}\)
2. \(A+B\)
3. \(\sqrt{A^2+B^2+\frac{A B}{\sqrt{2}}}\)
4. \(\sqrt{A^2+B^2+\sqrt{2} A B}\)

Answer: 4. \(\sqrt{A^2+B^2+\sqrt{2} A B}\)

Question 38. A vector perpendicular to both of \((3 \hat{i}+\hat{j}+2 \hat{k})\) and \((2 \hat{i}- 2 \hat{j}+4 \hat{k})\) is

1. \(\frac{1}{\sqrt{3}}(\hat{i}+\hat{j}-\hat{k})\)
2. \(\hat{i}-\hat{j}-\hat{k}\)
3. \(\frac{1}{\sqrt{3}}(\hat{i}+\hat{j}+\hat{k})\)
4. \((\sqrt{3} \hat{i}-\hat{j}-\hat{k})\)

Answer: 2. \(\hat{i}-\hat{j}-\hat{k}\)

Question 39. A vector normal to \(a \cos \theta \hat{i}+b \sin \theta \hat{j}\) is

1. \(b \sin \theta \hat{i}-a \cos \theta \hat{j}\)
2. \(\frac{1}{a} \sin \theta \hat{i}-\frac{1}{b} \cos \theta \hat{j}\)
3. \(5 \hat{k}\)
4. All of the above

Answer: 4. All of the above

Question 40. A vector \(\vec{A}\) of magnitude 2 units is inclined at angles 30° and 60° with positive x- and y-axes, respectively. Another vector of magnitude 5 units is aligned along the positive x-axis. Then \(\vec{A}\) • \(\vec{B}\) is

1. 5√3
2. 3√5
3. 2√3
4. 3√2

Answer: 1. 5√3

Question 41. For what values of a and b, the vector \((a \hat{i}+b \hat{j})\) will be a unit vector perpendicular to the vector \((\hat{i}+\hat{j})\)?

1. 1,0
2. 0,1
3. \(\frac{1}{\sqrt{3}},-\frac{2}{\sqrt{3}}\)
4. \(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}\)

Answer: 4. \(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}\)

Question 42. Two billiard balls, starting from the same point, have velocities \((\hat{i}+\sqrt{3} \hat{j}) \text { and }(2 \hat{i}+2 \hat{j})\), respectively. The angle between them is

1. 60°
2. 15°
3. 45°
4. 30°

Answer: 2. 15°

Question 43. If \(|\vec{A} \times \vec{B}|=\sqrt{3} \vec{A} \cdot \vec{B}\), then \(|\vec{A}+\vec{B}|=\)?

1. \(\left(A^2+B^2+A B\right)^{\frac{1}{2}}\)
2. (\(\left(A^2+B^2+\frac{A B}{\sqrt{3}}\right)^{\frac{1}{2}}\)
3. A+B
4. \(\left(A^2+B^2+\sqrt{3} A B\right)^{\frac{1}{2}}\)

Answer: 1. \(\left(A^2+B^2+A B\right)^{\frac{1}{2}}\)

Question 44. The vector product of \(\vec{A}\) and \(\vec{B}\) is zero. The scalar product of \(\vec{A}\) and \((\vec{A}+\vec{B})\) is

1. 0
2. \(A^2\)
3. \(A B\)
4. \(A^2+A B\)

Answer: 4. \(A^2+A B\)

Question 45. If \(\vec{A} \cdot \vec{B}=\vec{A} \cdot \vec{C}=0\), then the vector parallel to \(\vec{A}\) would be

1. \(\vec{C}\)
2. \(\vec{B}\)
3. \(\vec{B} \times \vec{C}\)
4. \(\vec{A} \times(\vec{B} \times \vec{C})\)

Answer: 3. \(\vec{B} \times \vec{C}\)

In this type of question, more than one option is correct.

Question 46. In the following figure which of the statements is correct?

1. The sign of x -component of \(\vec{l}_1\) and \(\vec{l}_2\) is negative
2. The signs of the y -component of \(\vec{l}_1 \text { and } \vec{l}_2\) are positive and negative respectively
3. The signs of x and y-components of \(\vec{l}_1+\vec{l}_2\) are both positive
4. None of these

Answer:

1. The sign of x -component of \(\vec{l}_1\) and \(\vec{l}_2\) is negative

3. The signs of x and y-components of \(\vec{l}_1+\vec{l}_2\) are both positive

Question 47. Two particles are projected in the air with speed v₀ at angles θ₁ and θ₂ (both acute) to the horizontal, respectively. If the height reached by the first particle is greater than that of the second, then which are the correct choices?

1. The angle of projection:  θ₁ > θ₂
2. Time of flight: T₁ > T₂
3. Horizontal range: R₁ > R₂
4. Total energy: U₁ > U₂

Answer:

Question 48. For two vectors \(\vec{A} \text { and } \vec{B},|\vec{A}+\vec{B}|=|\vec{A}-\vec{B}|\) is always true when

1. \(|\vec{A}|=|\vec{B}| \neq 0\)
2. \(\vec{A} \perp \vec{B}\)
3. \(|\vec{A}|=|\vec{B}| \neq 0\) and \(\vec{A}\) and \(\vec{B}\) are parallel or antiparallel
4. When either \(|\vec{A}|\) or \(|\vec{B}|\) is zero

Answer:

2. \(\vec{A} \perp \vec{B}\)

4. When either \(|\vec{A}|\) or \(|\vec{B}|\) is zero

Question 49. Given \(\vec{a}+\vec{b}+\vec{c}+\vec{d}=\overrightarrow{0}\). Which of the following statements is correct?

1. \(\vec{a}, \vec{b}, \vec{c}\) and \(\vec{d}\) must each be a null vector
2. The magnitude of \((\vec{a}+\vec{c})\) equals the magnitude of \((\vec{b}+\vec{d})\)
3. The magnitude of \(\vec{a}\) can never be greater than the sum of the magnitudes of \(\vec{b}, \vec{c}\), and \(\vec{d}\)
4. \((\vec{b}+\vec{c})\) must lie on the plane of \(\vec{a}\) and \(\vec{d}\) if \(\vec{a}\) and \(\vec{d}\) are not collinear and in the line of \(\vec{a}\) and \(\vec{d}\), if they are collinear

Answer:

2. The magnitude of \((\vec{a}+\vec{c})\) equals the magnitude of \((\vec{b}+\vec{d})\)

3. The magnitude of \(\vec{a}\) can never be greater than the sum of the magnitudes of \(\vec{b}, \vec{c}\) and \(\vec{d}\)

4. \((\vec{b}+\vec{c})\) must lie on the plane of \(\vec{a}\) and \(\vec{d}\) if \(\vec{a}\) and \(\vec{d}\) are not collinear and in the line of \(\vec{a}\) and \(\vec{d}\), if they are collinear

Question 50. State which of the following statements are false. A scalar quantity is one that

1. Is conserved in a process
2. Can never take negative values
3. Must be dimensionless
4. Has the same value for observers with different orientations of axes

Answer:

1. Is conserved in a process
2. Can never take negative values
3. Must be dimensionless

Question 51. The resultant of \(\vec{P}\) and \(\vec{Q}\) is \(\vec{R}\). The angle between \(\vec{P}\) and \(\vec{Q}\) is

1. 90°, if R² = P² + Q²
2. Less than 90°, if R2 > (P² + Q²)
3. Greater than 90°, if R² < (P2 + Q²)
4. Greater than 90°, if R² > (P² + Q²)

Answer:

1. 90°, if R² = P² + Q²
2. Less than 90°, if R2 > (P² + Q²)
3. Greater than 90°, if R² < (P2 + Q²)

Question 52. The magnitude of the vector product of A and B may be

1. Greater than AB
2. Equal to AB
3. Less than AB
4. Zero

Answer:

2. Equal to AB

3. Less than AB

4. Zero

Question 53. If \(\vec{A}=5 \hat{i}+6 \hat{j}+3 \hat{k} \text { and } \vec{B}=6 \hat{i}-2 \hat{j}-6 \hat{k}\), then

1. \(\vec{A}\) and \(\vec{B}\) are perpendicular
2. \(\vec{A} \times \vec{B}=\vec{B} \times \vec{A}\)
3. \(\vec{A}\) and \(\vec{B}\) have the same magnitude
4. \(\vec{A} \cdot \vec{B}=0\)

Answer:

1. \(\vec{A}\) and \(\vec{B}\) are perpendicular

4. \(\vec{A} \cdot \vec{B}=0\)

Question 54. Two projectiles, thrown from the same point with the same magnitude of velocity, have angles 60° and 30° of their projection. Then their

1. Maximum heights attained are the same
2. Horizontal ranges are the same
3. Magnitudes of velocity at the instants of hitting the ground are the same
4. Times of flight are the same

Answer:

2. Horizontal ranges are the same

3. Magnitudes of velocity at the instants of hitting the ground are the same

Question 55. A body is projected with a velocity u at an angle θ with the horizontal. At t = 2s, the body makes an angle of 30° with the horizontal. 1 s later, it attains its maximum height. Then

1. u = 20√3 m/s
2. θ = 60°
3. θ = 45
4. u = \(\frac{20}{\sqrt{3}} \mathrm{~m} / \mathrm{s}\)

Answer:

1. u = 20√3 m/s
2. θ = 60°

Question 56. \(\vec{A} \perp \vec{B}\); \(\vec{C}\) is coplanar with \(\vec{A}\) and \(\vec{B}\). Therefore,

1. \(\vec{A}=x \vec{B}+y \vec{C}\), where x and y are scalars
2. \(\vec{A} \cdot(\vec{B} \times \vec{C})=0\)
3. \(|(\vec{A} \times \vec{B}) \times \vec{C}|=A B C\)
4. \(\vec{A} \cdot \vec{B}=0\)

Answer:

1. \(\vec{A}=x \vec{B}+y \vec{C}\), where x and y are scalars
2. \(\vec{A} \cdot(\vec{B} \times \vec{C})=0\)
3. \(|(\vec{A} \times \vec{B}) \times \vec{C}|=A B C\)
4. \(\vec{A} \cdot \vec{B}=0\)