Three small villages with nearly equal population are situated at the position (1,2),(5,4) and (3,8), The block development officer wants to set up a community centre at a place which is at the same distance from the three villages.
Find the position at which the community centre should be set up.

Shown in the figure a spring fixed at the bottomm end of all incline of inclination 37^@ . A small block of mass 2 kg starts slipping down the incline form a point 4.8 m aaway from the spring. The block compresses the spring by 20 cm, stops momentarily and then rebounds through a distance of 1 m up the incline. Find the friction coefficient between the palne

A vessel ABCD of 10 cm width has two small slits S_(1) and S_(2) sealed with idebtical glass plates of equal thickness. The distance between the slits is 0.8 mm. POQ is the line perpendicular to the plane AB and passing through O, the middle point of S_(1) and S_(2) . A monochromatic light source is kept at S, 40 cm below P and 2 m from the vessel, to illuminate the slits as shown in the figure. Calculate the position of the central bright fringe on the other wall CD with respect of the line OQ. Now, a liquid is poured into the vessel and filled up to OQ. The central bright fringe is fiund to be at Q. Calculate the refractive index of the liquid.

Find the equation of the set of points P such that its distances from the points A (3, 4, - 5) and B (- 2, 1, 4) are equal.

Shown in the figure a spring fixed at the bottomm end of all incline of inclination 37^@ . A small block of mass 2 kg starts slipping down the incline form a point 4.8 m aaway from the spring. The block compresses the spring by 20 cm, stops momentarily and then rebounds through a distance of 1 m up the incline. Find the spring constant of the spring Take g = 10 ms^-2

Two schools A and B want to award their selected students on the values of sincerity, truthfulness and helpfulness. The school A wants to award Rs. X and Rs y each and Rs Z each of the three respective values to 3,2 and 1 students respectively with a total award money of Rs 1600. School B wants to spend Rs 2300 to award its 4,1 and 3 students on respective values (b giving the same award money for the three values as before). If the total amount of award for one prize on each value is Rs 900 , using matrices find the award money for each value. Apart from these three values, suggest one more value which should be considered for award.

A small sphere rolls down without slipping from the top of a track in a vertical plane. The track has an elevated section and a horizontal part is 1.0 m above the ground level and the top of the track is 2.4 m above the ground. Find the distance on the ground with respect to the point B (which is vertically ) below the end of the track as shown in figure. Where the sphere lands. During its flight as a projectile does the sphere continue to rotate about its centre of mass? Explain.

Point masses m_1 and m_2 are placed at the opposite ends of a rigid rod of length L, and nelgligible mass. The rod is to be set rotating about an axis perpendicular to it. Find the position on this rod through which the axis should pass in order that the work required to set the rod rotating with angular velocity omega_0 should be minimum.

A small retangular coil ABCD contains 140 turns of wire. The sides AB and BC of the coil are of length 4.5 and 2.8 cm respectively, as shown in the figure The coil is held between the poles of a large magnet so that the coil can rotate about an axis thorugh its centre. The magnet produces a uniform magnetic field of flux density B between its poles. When the current in the coil is 170 mA, the maximum torque produced in the coil is 2.1 xx 10^-3 N m . For the coil in the position shown in the figure. calculate the magnitude of the force on (i) side AB of the coil and (ii) side BC of the coil.

A small retangular coil ABCD contains 140 turns of wire. The sides AB and BC of the coil are of length 4.5 and 2.8 cm respectively, as shown in the figure The coil is held between the poles of a large magnet so that the coil can rotate about an axis thorugh its centre. The magnet produces a uniform magnetic field of flux density B between its poles. When the current in the coil is 170 mA, the maximum torque produced in the coil is 2.1 xx 10^-3 N m . For the coil in the position for maximum torque, state whether the plane of the coil is parallel to, or normal to , the direction of the magnetic field.

A small retangular coil ABCD contains 140 turns of wire. The sides AB and BC of the coil are of length 4.5 and 2.8 cm respectively, as shown in the figure The coil is held between the poles of a large magnet so that the coil can rotate about an axis thorugh its centre. The magnet produces a uniform magnetic field of flux density B between its poles. When the current in the coil is 170 mA, the maximum torque produced in the coil is 2.1 xx 10^-3 N m . The current in the coil in (a) is switched off and the coil s positioned as shown in the figure. The coil is then turned thorugh an angle of 90^@ in a time of 0.14 s. Calculate the average e.m.f. induced in the coil.