If \(A=\left[ \begin{matrix} 2 & 4 \\ -3 & 2 \end{matrix} \right] \)then, find A ^-1.

How many distinct words can be formed using all the letters of the following words.
MATHEMATICS.

For what value of k does 2x² + 5xy + 2y² + 15x + 18y + k = 0 represent a pair of straight lines.

If tan A = m tanB, prove that \(\frac { sin(A+B) }{ sin(A-B) } =\frac { m+1 }{ m-1 } \)

Differentiate the following with respect to x. sin (x²)

A manufacturing company has a contract to supply 4000 units of an item per year at uniform rate. The storage cost per unit per year amounts to Rs. 50 and the set up cost per production run is Rs. 160. If the production run can be started instantaneously and shortages are not permitted, determine the number of units which should be produced per run to minimize the total cost.

Calculate the amount of an ordinary annuity of Rs. 10,000 per annum for 5 years at 10% per year compounded half-yearly.

A card is drawn from a pack of playing cards and then another card is drawn without the first being replaced. What is the probability of drawing
(i) two aces
(ii) two spades

The following information is given

Details	X(in Rs.)	Y(in Rs.)
Arithmetic Mean	6	8
Standard Deviation	5	\(\frac{40}{3}\)

Coefficient of correlation between X and Y is \(\frac{8}{15}\) . Find (i) The regression Coefficient of Y on X (ii) The most likely value of Y when X = Rs.100.

A firm manufactures pills in two sizes A and B. Size A contains 2 mgs of aspirin, 5 mgs of bicarbonate and 1 mg of codeine. Size B contains 1 mg. of aspirin, 8 mgs. of bicarbonate and 6 mgs. of codeine. It is found by users that it requires atleast 12 mgs. of aspirin, 74 mgs.of bicarbonate and 24 mgs. of codeine for providing immediate relief. It is required to determine the least number of pills a patient should take to get immediate relief. Formulate the problem as a standard LLP.