**1.** If a vector making angles α, β, and γ respectively with the X, Y and Z axes respectively. Then sin^{2}α + sin^{2}β+ sin^{2}γ =

**Ans: **2

**Explanation: **

**2.**If the resultant of n forces of different magnitudes acting at a point is zero, then the minimum value of n is

**Ans: **3

**Explanation: **If vectors are of equal magnitude then two vectors can give zero resultant, if they works in opposite direction. But if the vectors are of different magnitudes then minimum three vectors are required to give zero resultant.

**3. **Can the resultant of 2 vectors be zero

**Ans: **Yes, when the 2 vectors are same in magnitude but opposite in sense

**4.** The sum of the magnitudes of two forces acting at point is 18 and the magnitude of their resultant is 12. If the resultant is at 90^{o} with the force of smaller magnitude, what are the, magnitudes of forces

**Ans: **5, 13

**Explanation:** Let P be the smaller force and Q be the greater force then according to problem –

**5. **A vector is turned without a change in its length through a small angle dθ. The value of _{ }and Δa are respectively

**Ans: **a dθ, O

**Explanation:**

**6. **Find the resultant of three vectors * *and shown in the following figure. Radius of the circle is R.

**Ans: **^{ }

**Explanation: **

**7.** Figure shows ABCDEF as a regular hexagon. What is the value of

**Ans:**

^{ }

**8.** The length of second's hand in watch is 1 cm. The change in velocity of its tip in 15 seconds is**Ans:** ^{ }**Explanation: **^{ }

**9.** A particle moves towards east with velocity 5 m/s. After 10 seconds its direction changes towards north with same velocity. The average acceleration of the particle is**Ans:**

**Explanation: **^{ }

**10.** A force

_{ }(where K is a positive constant) acts on a particle moving in the x-y plane. Starting from the origin, the particle is taken along the positive x- axis to the point (a, 0) and then parallel to the y-axis to the point (a, a). The total work done by the forces on the particle is

**Ans: **− Ka^{2}

**Explanation: **

**11.** The vectors from origin to the points A and B are _{ }and respectively. The area of the triangle OAB be

**Ans: ** _{ }sq.unit

**Explanation:**

**12.** A metal sphere is hung by a string fixed to a wall. The sphere is pushed away from the wall by a stick. The forces acting on the sphere are shown in the second diagram. Which of the following statements is wrong

**Ans:** T = P + W

**Explanation: **

**13.** The speed of a boat is 5 km/h in still water. It crosses a river of width 1 km along the shortest possible path in 15 minutes. The velocity of the river water is

**Ans: ** 3 km/h

**14.** A man crosses a 320 m wide river perpendicular to the current in 4 minutes. If in still water he can swim with a speed 5/3 times that of the current, then the speed of the current, in m/min is

**Ans:** 60

**15. **** _{ }**represents a unit vector when c is

**Ans:**

**Explanation:**

**16.** The angles which a vector _{ }makes with X, Y and Z axes respectively are

**Ans:** 60^{o}, 60^{o}, 45^{o}

**Explanation:**

**17.** The value of a unit vector in the direction of vector

**Ans: **

**Explanation: **

**18. **Which of the following is independent of the choice of coordinate system

**Ans: **

**19.** A car travels 6 km towards north at an angle of 45° to the east and then travels distance of 4 km towards north at an angle of 135° to the east. How far is the point from the starting point. What angle does the straight line joining its initial and final position makes with the east

**Ans: **:_{ }and tan^{−1}(5)

**Explanation:**

**20.** Given that * _{ }*out of three vectors two are equal in magnitude and the magnitude of third vector is times that of either of the two having equal magnitude. Then the angles between vectors are given by

**Ans:**90°, 135°, 135°

From polygon law, three vectors having summation zero should form a closed polygon. (Triangle) since the two vectors are having same magnitude and the third vector is times that of either of two having equal magnitude. i.e. the triangle should be right angled triangle

Angle between A and B, α = 90

^{o}

Angle between B and C, β = 135

^{o}

Angle between A and C, γ = 135

^{o}

**21.** Two forces F_{1} = 1 N and F_{2} = 2 N act along the lines x = 0 and y = 0 respectively. Then the resultant of forces would be**Ans: **

**Explanation:**

**22.** At what angle must the two forces (x + y) and (x – y) act so that the resultant may be

**Ans:**

**Explanation:**

**23.** Following forces start acting on a particle at rest at the origin of the co-ordinate system simultaneously

**Ans:** In y – z plane**Explanation:**

**24.** The resultant of is On reversing the vector the resultant becomes

**Ans:**2(A

^{2}+ B

^{2})

**Explanation:**

**25.**Figure below shows a body of mass M moving with the uniform speed on a circular path of radius, R. What is the change in acceleration in going from P

_{1}to P

_{2}

**Ans:**

**Explanation: **

**26.** A particle is moving on a circular path of radius r with uniform velocity v. The change in velocity when the particle moves from P to Q is **Ans:** 2v sin 40°**Explanation:**

**27.** and are two vectors. The angle between them will be**Ans:** 60°**Explanation:**

**28.** and then projection of on will be**Ans:**

**Explanation:**

**29.** In above example a unit vector perpendicular to both and will be

(a) (b) (c) Both (a) and (b) (d) None of these**Ans: **Both (a) and (b)**Explanation:**

**30.** Two constant forces (N) and (N) act on a body and displace it from the position

**Ans:**9 J

**Explanation:**

**31.** For any two vectors

**Ans:**

**Explanation:**

**32.** Which of the following is the unit vector perpendicular to and

**Ans:**

**Explanation: **

**33.** Two vectors and will be parallel if**Ans:** b = 2**Explanation:**

**34.** Which of the following is not true ? If and where A and B are the magnitudes of and **Ans:**

**Explanation:**

**35.** The area of the triangle formed by is **Ans:**

**Explanation: **

**36.** Two trains along the same straight rails moving with constant speed 60 km/hr and 30 km/hr respectively towards each other. If at time t = 0 , the distance between them is 90 km, the time when they collide is**Ans:** 1 hr**Explanation: **

**37.** A steam boat goes across a lake and comes back (a) On a quite day when the water is still and (b) On a rough day when there is uniform air current so as to help the journey onward and to impede the journey back. If the speed of the launch on both days was same, in which case it will complete the journey in lesser time**Ans:** Case (b)**Explanation:**

**38.** To a person, going eastward in a car with a velocity of 25 km/hr, a train appears to move towards north with a velocity of km/hr. The actual velocity of the train will be**Ans: **25 km/hr**Explanation:**

**39.** A swimmer can swim in still water with speed υ and the river is flowing with velocity v/2. To cross the river in shortest distance, he should swim making angle θ with the upstream. What is the ratio of the time taken to swim across the shortest time to that is swimming across over shortest distance**Ans:** sin θ

**40.** A bus is moving with a velocity 10 m/s on a straight road. A scooterist wishes to overtake the bus in 100 s. If the bus is at a distance of 1 km from the scooterist, with what velocity should the scooterist chase the bus**Ans:** 20 m/s**Explanation:**

**Assertion Reason**

Read the assertion and reason carefully to mark the correct option out of the options given below:

(a) If both assertion and reason are true and the reason is the correct explanation of the assertion.

(b) If both assertion and reason are true but reason is not the correct explanation of the assertion.

(c) If assertion is true but reason is false.

(d) If the assertion and reason both are false.

(e) If assertion is false but reason is true.

**Reason : ** as well as lie in the plane containing

**Ans:**Both assertion and reason are true and the reason is the correct explanation of the assertion.

**Explanation:**Cross product of two vectors is perpendicular to the plane containing both the vectors.

**2. Assertion : ****Reason : ** is equally inclined to both and

**Ans:**Both assertion and reason are true and the reason is the correct explanation of the assertion.

**Explanation:**

**Reason : **

**Ans: **The assertion and reason both are false.**Explanation:**

**4. Assertion : ****Reason : ****Ans: **Both assertion and reason are true but reason is not the correct explanation of the assertion.**Explanation:**

**5. Assertion : **Vector product of two vectors is an axial vector**Reason : **If = instantaneous velocity, = radius vector and

**Ans:**Assertion is true but reason is false.

**Explanation:**

**6. Assertion : **Minimum number of non-equal vectors in a plane required to give zero resultant is three.**Reason : **If ,

**Ans:**Both assertion and reason are true but reason is not the correct explanation of the assertion.

**Explanation:**For giving a zero resultant, it should be possible to represent the given vectors along the sides of a closed polygon and minimum number of sides of a polygon is three.

**7. Assertion : **Relative velocity of A w.r.t. B is greater than the velocity of either, when they are moving in opposite directions.**Reason : **Relative velocity of A w.r.t.

**Ans: **Both assertion and reason are true and the reason is the correct explanation of the assertion.**Explanation:**

**8. Assertion :**Vector addition of two vectors A and B is commutative.

**Reason :**

**Ans:**Both assertion and reason are true but reason is not the correct explanation of the assertion.

**Explanation:**

**9. Assertion : ****Reason : **Dot product of two vectors is commutative.**Ans: **Both assertion and reason are true and the reason is the correct explanation of the assertion.

**10. Assertion : ****Reason : **Cross product of vectors is commutative.

**Ans: **Assertion is true but reason is false.**Explanation: **

**11. Assertion : **A negative acceleration of a body is associated with a slowing down of a body.**Reason : **Acceleration is vector quantity.**Ans: **Both assertion and reason are true but reason is not the correct explanation of the assertion.

**12. Assertion : **A physical quantity cannot be called as a vector if its magnitude is zero.**Reason : **A vector has both, magnitude and direction.**Ans: **Assertion is false but reason is true.**Explanation: **If a vector quantity has zero magnitude then it is called a null vector. That quantity may have some direction even if its magnitude is zero.

**13. Assertion : **The sum of two vectors can be zero.**Reason : **The vector cancel each other, when they are equal and opposite.**Ans: **Both assertion and reason are true and the reason is the correct explanation of the assertion.

**Explanation: **

**14. Assertion : **Two vectors are said to be like vectors if they have same direction but different magnitude.**Reason : **Vector quantities do not have specific direction.**Ans: **If assertion is true but reason is false.**Explanation: **If two vectors are in opposite direction, then they cannot be like vectors.

**15. Assertion : **The scalar product of two vectors can be zero.**Reason : **If two vectors are perpendicular to each other, their scalar product will be zero.**Ans: **If both assertion and reason are true and the reason is the correct explanation of the assertion.**Explanation: **

**16. Assertion : **Multiplying any vector by an scalar is a meaningful operations.**Reason : **In uniform motion speed remains constant.**Ans: **Both assertion and reason are true but reason is not the correct explanation of the assertion.**Explanation: **We can multiply any vector by any scalar. For example, in equation mass is a scalar quantity, but acceleration is a vector quantity.

**17. Assertion : **A null vector is a vector whose magnitude is zero and direction is arbitrary.**Reason : **A null vector does not exist.**Ans: **Assertion is true but reason is false.**Explanation: **If two vectors equal in magnitude are in opposite direction, then their sum will be a null vector. A null vector has direction which is intermediate (or depends on direction of initial vectors) even its magnitude is zero.

**18. Assertion : **If dot product and cross product of and are zero, it implies that one of the vector and

**Reason :**Null vector is a vector with zero magnitude.

**Ans:**Both assertion and reason are true but reason is not the correct explanation of the assertion.

**Explanation:**

If and

**19. Assertion : **The cross product of a vector with itself is a null vector. A**Reason : **The cross-product of two vectors results in a vector quantity.**Ans: **Both assertion and reason are true but reason is not the correct explanation of the assertion.

**20. Assertion : **The minimum number of non coplanar vectors whose sum can be zero, is four.**Reason : **The resultant of two vectors of unequal magnitude can be zero.**Ans: **Assertion is true but reason is false.**Explanation: **The resultant of two vectors of unequal magnitude given by cannot be zero for any value of θ.

**21. Assertion : **

**Reason :**The dot product of two vectors involves cosine of the angle between the two vectors.

**Ans:**Both assertion and reason are true and the reason is the correct explanation of the assertion.

**Explanation:**

**22. Assertion : **Vector addition is commutative.**Reason : ****Ans: **Assertion is true but reason is false.**Explanation: **