Centre of mass of two uniform rods of same length but made up of different materials & kept as shown, if the meeting point is the origin of co–ordinates

Statement I : The inverse of the matrix   does not exist.
Statement II : The matrix   is singular. [ = 0, since R₂ = 2R₁]

Two blocks A and B of masses m and 2m respectively are connected by a spring of spring constant k. The masses are moving to the right with a uniform velocity each, the heavier mass leading the lighter one. The spring is of natural length during this motion. Block B collides head on with a third block C of mass 2m. at rest, the collision being completely inelastic. The maximum compression of the spring after collision is –

Two blocks A and B of masses m and 2m respectively are connected by a spring of spring constant k. The masses are moving to the right with a uniform velocity each, the heavier mass leading the lighter one. The spring is of natural length during this motion. Block B collides head on with a third block C of mass 2m. at rest, the collision being completely inelastic. The velocity of centre of mass of system of block A, B & C is-

Two blocks A and B of masses m and 2m respectively are connected by a spring of spring constant k. The masses are moving to the right with a uniform velocity each, the heavier mass leading the lighter one. The spring is of natural length during this motion. Block B collides head on with a third block C of mass 2m. at rest, the collision being completely inelastic. The velocity of block B just after collision is-

Two blocks A and B are joined together with a compressed spring. When the system is released, the two blocks appear to be moving with unequal speeds in the opposite directions as shown in figure. Select incorrect statement(s)

Statement I : The order of the matrix A is 4 × 5 and that of B is 3 × 4. Then the matrix AB is not possible.
Statement II : AB is defined if number of columns of A = number of rows of B

The product of two matrices A and B is defined if the number of columns of A is equal to the number of rows of B. If both A and B are square matrices of the same order, then both AB and BA are defined. If AB and BA are both defined, it is not necessary that AB = BA.

Statement I : The determinant of matrix  
Statement II : The determinant of a skew symmetric matrix of odd order is zero.

A spring lies along an x axis attached to a wall at one end and a block at the other end. The block rests on a frictionless surface at x = 0. A force of constant magnitude F is applied to the block that begins to compress the spring, until the block comes to a maximum displacement During the first half of the motion, applied force transfers more energy to the

A spring lies along an x axis attached to a wall at one end and a block at the other end. The block rests on a frictionless surface at x = 0. A force of constant magnitude F is applied to the block that begins to compress the spring, until the block comes to a maximum displacement During the displacement, which of the curves shown in the graph best represents the work done on the spring block system by the applied force.

A spring lies along an x axis attached to a wall at one end and a block at the other end. The block rests on a frictionless surface at x = 0. A force of constant magnitude F is applied to the block that begins to compress the spring, until the block comes to a maximum displacement . During the displacement, which of the curves shown in the graph best represents the kinetic energy of the block?

A block of mass m slides down a wedge of mass m as shown. The whole system is at rest, when the height of the block is h = 10 m. Above the ground. The wedge surface is smooth and gradually flattens. There is no friction between wedge and ground. If there is no friction anywhere, the speed of the wedge, as the block leaves the wedge is :

A block of mass m slides down a wedge of mass m as shown. The whole system is at rest, when the height of the block is h = 10 m. Above the ground. The wedge surface is smooth and gradually flattens. There is no friction between wedge and ground. If there would have been friction between wedge and block, which of the following quantities would still remain conserved?

A block of mass m slides down a wedge of mass m as shown. The whole system is at rest, when the height of the block is h = 10 m. Above the ground. The wedge surface is smooth and gradually flattens. There is no friction between wedge and ground. As the block slides down, which of the following quantities associated with the system remains conserved?

STATEMENT-1 : One end of ideal massless spring is connected to fixed vertical wall and other end to a block of mass m initially at rest on smooth horizontal surface. The spring is initially in natural length. Now a horizontal force F acts on block as shown. Then the maximum extension in spring is equal to maximum compression in spring.

STATEMENT-2 : To compress and to expand an ideal unstretched spring by equal amount, same work is to be done on spring

Statement I : Trace of matrix  A =  is equal to a₁₁ + a₂₂ + a₃₃
Statement II : Trace of a matrix is equal to sum of its diagonal elements.

The trace has several properties that are used to prove important results in matrix algebra and its applications.
 Let A and B be two K X K matrices. Then,