At an election, a voter may vote for any number of candidates, not greater than the number to be elected. There are 10 candidates and 4 are to be elected. If a voter votes for at least one candidate, then the number of ways in which he can vote is

Then number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty is

Then number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty is

How many ways are there to arrange the letters in the word GARDEN with the vowels in alphabetical order?

Total number of four digit odd numbers that can beformed using0,1,2,3,5,7 are

The sum of integers from 1 to 100 that are divisible by 2 or 5 is

Five digit number divisible by 3 is formed using 0, 1, 2, 3, 4, 6 and 7 without repetition. Total number of such numbers are

Number greater than 1000 but less than 4000 is formed using the digits 0, 2, 3, 4 repetition allowed is

## The real value of 'a' for which sum of square of the roots of the equation assume the least value is

The real value of 'a' for which sum of square of the roots of the equation

assume the least value is

A point oscillates along the x axis according to the law . Draw the approximate plots

(a) of displacement x, velocity projection vx, and acceleration projection wx as functions of time t;

(b) velocity projection vx and acceleration projection wx as func- tions of the coordinate x.

If a young man rides his motorcycle at 25 km/hour, he had to spend Rs. 2 per km on petrol. If he rides at a faster speed of 40 km/hour, the petrol cost increases at Rs. 5 per km. He has Rs. 100 to spend on petrol and wishes to find what is the maximum distance , he can travel within one hour. Express this as an LPP and solve it graphically

A dealer wishes to purchase a number of fans and sewing machines. He has only Rs. 5,760 to invest and has space for at most 20 items. A fan and sewing machine cost Rs. 360 and Rs. 240 respectively. He can sell a fan at a profit of Rs. 22 and sewing machine at a profit of Rs. 18. Assuming that he can sell whatever he buys, how should he invest his money in order to maximize his profit? Trans late the problem into LPP and solve it graphically

Two tailors A and B earn Rs.150 and Rs.200 per day respectively. A can stitch 6 shirts and 4 pants per day while B can stitch 10 shirts and 4 pants per day. Form a linear programming problem to minimize the labour cost to produce at lest 60 shirts and 32 pants

A company has two plants to manufacture motor cycles. 70% motor cycles are manufactured at the first plant, while 30% are manufactured at the second plant. At the first plant, 80% motor cycles are rated of the standard quality while at the second plant, 90% are rated of standard quality. A motor cycle, randomly picked up, Is found to be of standard quality. Find the probability that it has come out from the second plant?

An object starts at the origin in a straight line. Its velocity versus time graph is shown in the figure. Which one of the following choices best gives the proper intervals(s) of time for which the object is moving away from the origin?

Two identical discs of mass m and of radius R touch each other and move with the same velocity perpendicularly to the line segment which joins their centres of mass, along the surface of a horizontal smooth tabletop. There is a third disc of mass M and of radius R at rest, at a point on the perpendicular bisector of the line segment joining the centres of mass of the two moving discs as shown in the figure. The two moving discs collide elastically with the third one, which is at rest. There is no friction between the rims of the discs. What should the ratio of M/m be in order that after the collision the two discs of mass m move perpendicularly to their initial velocity?

A triangular frame consists of three identical rods, each of mass m and length *l*. It rests upright on a horizontal smooth surface with its lower right corner against a stop about which the frame could pivot. A horizontal force of magnitude F is applied to the upper corner of the

frame as shown in the figure. What is the largest value of F may have without causing the frame to pivot upward about the stop?

A particle is moving along straight line such that dot product of its acceleration and velocity is negative. How many times particle can be found at a distance d from the origin?

A uniform, solid cylinder having mass M and radius R is pulled by a horizontal force F acting through the center as shown. The cylinder rolls to the right without slipping. What is the magnitude of the force of friction between the cylinder and the ground?

An object of mass M starts from rest at the bottom of a fixed incline of height H. A person decides to push the object up the incline in one of two ways with an applied force shown in the diagram. In each of the trials, the object reaches the top of the incline with speed V. How would the work done by the person on the block compare for the two trials? Assume the same constant non-zero coefficient of kinetic friction between the incline and the object for both trials.

The position versus time graph of a particle moving along a straight line is shown. What is the total distance travelled by the particle from t = 0 s to t = 10s?

A wedge of mass M= 10kg, height h= 3m and angle of inclination is at rest at horizontal surface. There is a small point – like object (mass m = 0.5kg) next to the slope as shown in the figure. At what acceleration must wedge be moved in order that the point like object reaches its top in a time t = 5s (Neglect the friction between point like object and wedge)

Ram pushes eight identical blocks on the horizontal frictionless surface with horizontal force F. The force that block-1 exerts at block-2 has magnitude and the force that block-7 exerts on the block-8 is

Find velocity of piston A in the given situation if angular velocity of wheel of radius R is (constant), in the clockwise sense. (O is fixed point)

An ideal string-pulley system is shown in the figure and system is released from rest.

Choose correct option.

A two stage rocket is fired vertically upward from rest with acceleration as shown in a-t graphs. After 15 sec, the first stage burns out and second stage ignites. Choose the INCORRECT statement regarding motion of rocket in the time interval

One end of a heavy uniform rod AB can slide along rough horizontal guiding surface CD with the help of massless ring as shown in the figure. The dimension of ring is negligible. BE is the ideal string. If is right angle and is the angle between rod AB and horizontal when the rod is on the verge of sliding. Find the coefficient of friction between ring and horizontal guiding surface CD.

A man of mass 60 kg hangs himself from a massless spring balance, which itself suspended from an ideal string-pulley system as shown in the

figure. The string AB can bear maximum 900 N. Choose correct statement.

A uniform solid hemisphere of radius 10 m and mass 64 kg is placed with its curved surface on the smooth horizontal surface and a string AB of length 4m is attached to point A on its rim as shown in the figure. Find the tension in the string if hemisphere is in equilibrium.

A uniform rod of mass M and length L is free to rotate about a frictionless pivot located L/3 from one end. The rod is released from rest incrementally away from being perfectly vertical, resulting in the rod rotating clockwise about the pivot. When the rod is horizontal, what is the magnitude of the tangential acceleration of its center of mass?

An object is being pushed at constant speed on an inclined plane. The free body diagram of the object is shown with the gravitational force represented by W, the friction force by f, the applied external push parallel to the incline by F, and the normal force with surface by N. Which one of the following choices represents correct relationships between the forces?

A string of mass m (can be non uniform as well) is suspended through two points which are not in same horizontal level. Tension in the string at the end points are T1 and T2 and at the lowest point is T3. Mass of string in terms of T1, T2 and T3 can be represented a {uniform gravity ‘g’ exists downwards)

All the blocks are attached to an ideal rope which passes over an ideal pulley. If accelerations of blocks m1, m2, m3, and m4 are a1, a2, a3 and a4 respectively then choose the correct option.

A massless spring of stiffness 400 N/m is fastened at left end to a vertical wall as shown in the figure I. Initially block C of mass 2 kg and block D of mass 5 kg rest on horizontal surface with block C in contact with spring (But not compressing it.) and the block D in contact with block C. Block C is moved leftward, compressing spring by a distance of 0.5 m and held in place while block D remains at rest as shown in the figure.

Now Block C is released block D. The surface is rough and the coefficient of friction between each block and surface is 0.1. The block collide instantaneously stick together and move right. Find the velocity of combined system just after collision.

The mean of the numbers a ,b ,8 ,5 ,10 is 6 and the variance is 6.80. Then which one of the following gives possible values of a and b?

The average marks of boys in class is 52 and that of girls is 42. The average marks of boys and girls combined is 50. The percentage of boys in the class is ?

Suppose a population A has 100 observations 101,102, ..., 200, and another population B has 100 observations 151, 152, ..., 250. If VA and VB represent the variances of the two populations, respectively 1then VA /VB is

Let x1, x2, ...., xn be n observations such that and .Then a possible value

of n among the following is

In a frequency distribution, the mean and median are 21 and 22 respectively, then its mode is approximately

In a series of 2n observations, half of them equal a and remaining half equal – a. If the standard deviation of the observations is 2, then |a| equals

Consider the following statements :

(1) Mode can be computed from histogram

(2) Median is not independent of change of scale

(3) Variance is independent of change of origin and scale.

Which of these is/are correct?

The median of a set of 9 distinct observations is 20.5. If each of the largest 4 observations of the set is increased by 2, then the median of the new set

In an experiment with 15 observations on x, the following results were available.

One observation that was 20 was found to be wrong and was replaced by the correct value 30. Then the corrected variance is

In a class of 100 students there are 70 boys whose average marks in a subject are 75. If the average marks of the complete class is 72, then what is the average of the girls?

Two spherical bodies of mass *M* and 5*M* and radii *R* and 2*R* respectively are released in free space with initial separation between their centres equal to 12*R*. If they attract each other due to gravitational force only, then what will be the distance covered by the smaller body just before the collision?