Solve:\(\begin{vmatrix}7&4&11\\-3&5&x\\-x&3&1 \end{vmatrix}=0\)

Find the 5^th term in the expansion of (x - 2y)^13.

Find the angle between the lines whose slopes are \(\frac { 1 }{ 2 } \) and 3

Prove that \(\sqrt3\) cosec 20^o- sec 20^o = 4

Differentiate sin³ x with respect to cos³x.

For the demand function \(x={20\over p+1}\) p > 0, find the elasticity of demand with respect to price at a point p = 3. Examine whether the demand is elastic at p = 3.

A person purchases a machine on 1^st January 2009 and agrees to pay 10 installments each of Rs. 12,000 at the end of every year inclusive of compound rate of 15%. Find the present value of the machine. [(1.15)¹⁰ = 4.016].

Compute the Geometric mean from the data given below

Marks	0-10	10-20	20-30	30-40	40-50
No. of Students	8	12	18	8	6

Calculate the correlation co-efficient for the following data.

X	5	10	5	11	12	4	3	2	7	1
Y	1	6	2	8	5	1	4	6	5	2

A soft drink company has two bottling plants C₁ and C₂. Each plant produces three different soft drinks S₁, S₂ and S₃. The production of the two plants in number of bottles per day are:

Product	Plant
	C1	C2
S1	3000	1000
S2	1000	1000
S3	2000	6000

A market survey indicates that during the month of April there will be a demand for 24000 bottles of S₁, 16000 bottles of S₂ and 48000 bottles of S₃. The operating costs, per day, of running plants C₁ and C₂ are respectively Rs.600 and Rs.400. How many days should the firm run each plant in April so that the production cost is minimized while still meeting the market demand? Formulate the above as a linear programming model.