Use Properties of determinants, prove that:

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 x each, y each and z each for the three respective values to 3, 2 and 1 students respectively with a total award money of 1,600. School B wants to spend 2,300 to award its 4, 1 and 3 students on the respective values (by giving the same award money to the three values as before). If the total amount for one prize on each value is 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.

If the sum of the lengths of the hypotenuse and a side of a right triangle is given, show that the area of the triangle is maximum when the angle between them is 

15.

Find the value of 'p' for which the vectors  are parallel.

If the cartesian equations of a line are  write the vector equation for the line. 

If , find the value of a.

Solve the differential equation 

Show that the four points A, B, C and D with position vectors
 respectively are coplanar.