If A = \(\begin{bmatrix}3 & -1 & 1 \\ -15 & 6 & -5\\5 & -2 & 2 \end{bmatrix}\) then, find the Inverse of A.

If\(A=\begin{bmatrix}1&3&3\\1&4&3\\1&3&4 \end{bmatrix}\)then verify that A (adj A) = |A| I and also find A^-1.

Let a, b and c denote the sides BC, CA and AB repectively of \(\Delta\) ABC. If \(\left| \begin{matrix} 1 & a & b \\ 1 & c & a \\ 1 & b & c \end{matrix} \right| =0\), then find the value of sin² A + sin²B + sin²C.

The sum of three numbers is 20. If we multiply the first by 2 and add the second number and subtract the third we get 23. If we multiply the first by 3 and add second and third to it, we get 46. By using matrix inversion method find the numbers.

Show that the matrices \(A=\left[ \begin{matrix} 1 & 3 & 7 \\ 4 & 2 & 3 \\ 1 & 2 & 1 \end{matrix} \right] \)and \(B=\left[ \begin{matrix} \frac { -4 }{ 35 } & \frac { 11 }{ 35 } & \frac { -5 }{ 35 } \\ \frac { -1 }{ 35 } & \frac { -6 }{ 35 } & \frac { 25 }{ 35 } \\ \frac { 6 }{ 35 } & \frac { 1 }{ 35 } & \frac { -10 }{ 35 } \end{matrix} \right] \)are inverses of each other.

Resolve into partial fractions for the following : \(\frac{x+2}{(x-1)(x+3)^2}\)

Prove that cot x cot 2x - cot 2x cot 3x - cot 3x cot x = 1.

If \(y={ e }^{ a\cos ^{ -1 }{ x } }\) , show that \(\left( 1-{ x }^{ 2 } \right) \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } -x\frac { dy }{ dx } -{ a }^{ 2 }y=0\)

A company uses 48000 units/year of a raw material costing Rs. 2.5 per unit. Placing each order costs Rs. 45 and the carrying cost is 10.8 % per year of the average inventory. Find the EOQ, total number of orders  per year and time between each order. Also verify that at EOQ carrying cost is equal to ordering cost.

The total revenue (TR) for commodity x is \(TR=12x+{x^2\over2}-{x^3\over 3}\)S.T. at the highest point of average revenue (AR), AR = MR

Kamal sold Rs.9000 worth 7% stock at 80 and invested the proceeds in 15% stock at 120. Find the change in his income?

Bag I contains 3 Red and 4 Black balls while another Bag II contains 5 Red and 6 Black balls. One ball is drawn at random from one of the bags and it is found to be red. Find the probability that it was drawn from Bag I.

In a Shooting test, the probabilities of hitting the target are \(\frac { 1 }{ 2 } \) for A, \(\frac { 2 }{ 3 } \) for B and \(\frac { 3 }{ 4 } \)  for C. If all of them fire at the same target, calculate the probabilities that only one of them hit the target.

Calculate correlation coefficient for the following data.

X	25	18	21	24	27	30	36	39	42	48
Y	26	35	48	28	20	36	25	40	43	39

The equations of two lines of regression obtained in a correlation analysis are the following 2X = 8 – 3Y and 2Y = 5 – X. Obtain the value of the regression coefficients and correlation coefficient.

For the data on price (in rupees) and demand (in tonnes) for a commodity, calculate the co-efficient of correlations.

Price(X)	22	24	26	28	30	3	34	36	38	40
Demand(Y)	60	58	58	50	48	48	48	42	36	32

Compute the earliest start time, earliest finish time, latest start time and latest finish time of each activity of the project given below:

Activity	1-2	1-3	2-4	2-5	3-4	4-5
Duration( in days)	8	4	10	2	5	3

Solve the following linear programming problem graphically.
Maximize Z = 3x₁ + 5x₂ subject to the constraints x₁ + x₂ ≤ 6, x₁ ≤ 4; x₂ ≤ 5, and x₁, x₂ ≥ 0

One kind of the cake requires 200 g of flour and 25 g of fat, and another kind of cake requires 100 g of flour and 50 g of fat. Find the maximum number of cakes which can be made from 5 kg of flour and 1 kg of fat assuming that there is no shortage of other ingredients used in making the cakes?

The following table use the activities in a building project.

Activity	1-2	1-3	2-3	2-4	3-4	4-5
Duration (days)	21	26	11	13	5	11

Draw the network for the project, calculate the earliest start time, earliest finish time, latest start time and latest finish time of each activity and find the critical path. Compute the project duration.