If y = e^x(sin x + cos x), then show that

Form the differential equation representing the family of curves given by (x - a)² + 2y² = a², where a is an arbitrary constant.

Using properties of determinants, prove the following:

Find the area of the region bounded by the curves y² = 4ax and x² = 4ay.

A man is known to speak the truth 3 out of 5 times. He throws a die and reports that it is a number greater than 4. Find the probability that it is actually a number greater than 4.