If the vectors  and  form a triangle then λ=

Let  is greatest integer function. If  and  are parallel vectors then =

For such questions, we should know the concept of greatest integer number. We should also know the properties of parallel vectors.

The points with position vectors  and  are collinear for

For such questions, we should know how to find the vector joining two points. Also, we should know the condition for two vectors to be collinear.

A= (1,1,1) B=(1,2,3) C=(2,-1,1) then the length of the internal bisector of is

A vector a has component in a right-handed rectangular Cartesian coordinate system OXYZ. The coordinate system is rotated about y-axis through an angle . The components of a in the new system are

If ,m,n are the d.c’s of a vector if = , then the maximum value of mn is

A body performs simple harmonic oscillations along the straight line ABCDE with C as the midpoint of AE. Its kinetic energies at B and D are each one fourth of its maximum value. If AE = 2R, the distance between B and D is

A system consists of a ball of mass M₂ and a uniform thin rod of mass M₁ and length d. The rod attached to a frictionless horizontal table by a pivot at point P and initially rotates at an angular speed as shown in figure. The rod strikes the ball, which is initially at rest. As a result just after collision, the rod stops and ball moves in the direction shown. If collision is elastic, the ratio M₁ /M₂ is

In the system shown all the surfaces are frictionless while pulley and string are massless. Mass of block A is 2m and that of block B is m. Acceleration of block B immediately after system is released from rest is

In the arrangement shown, end A of light inextensible string is pulled with constant velocity v. The velocity of block B is

A right circular cone with semi-vertical angle rests on a rough incline plane is increased, the cone will slide before it topples over, ifqangle of inclination coefficient of friction

The Atwood machine shown is suspended from a spring balance. The mass on one hanger is M, that on the other is (M + m). Suppose the heavier side (right side) hanger is fastened to the top pulley by a thread. The scale reads (2M + m)g The thread is burned and the system accelerates. While the masses on the Atwood machine accelerates the spring balance reads.

A uniform disc of mass M and radius R is supported vertically by a pivot at its periphery as shown. A particle of mass M is fixed to the rim and raised to the highest point above the centre. The system is then released from rest and it can rotate about its pivot freely. The angular speed of the system when the attached object is directly beneath the pivot is

Two trolleys 1 and 2 are moving with accelerations a₁ and a₂ respectively in the same direction. A block of mass ‘’m’’ on trolley 1 is in equilibrium from the frame of observer stationary w.r.t. trolley 2. The magnitude of friction force on block due to trolley is (assume that no horizontal force other than friction force is acting on block) 

Acceleration versus time curve for a particle moving in a straight line is shown in the figure. If particle starts from rest at t = 0, then which of the following curve is true for the same particle

In the figure shown a ring A is initially rolling without sliding with a velocity v on the horizontal surface of the body B (of same mass as A). All surfaces are smooth. B has no initial velocity. What will be the maximum height reached by A on B.