Differentiate the function (sin x)^x + sin^–1 √xwith respect to x.

12.

If x^m yⁿ = (x + y)^{m + n}, prove that 

Evaluate: .

Find the vector equation of the line passing through the point A(1, 2, –1) and parallel to the line 5x – 25 = 14 – 7y = 35z.

Prove that if E and F are independent events, then the events E and F' are also independent.

A small firm manufactures necklaces and bracelets. The total number of necklaces and bracelets that it can handle per day is at most 24. It takes one hour to make a bracelet and half an hour to make a necklace. The maximum number of hours available per day is 16. If the profit on a necklace is ` 100 and that on a bracelet is ` 300. Formulate on L.P.P. for finding how many of each should be produced daily to maximize the profit? It is being given that at least one of each must be produced.

Find 

Evaluate: 

Evaluate: