Suppose the inter-industry flow of the product of two industries are given as under.

Production sector	Consumption sector		Domestic demand	Total output
	X	Y
X	30	40	50	120
Y	20	10	30	60

Determine the technology matrix and test Hawkin's -Simon conditions for the viability of the system. If the domestic demand changes to 80 and 40 units respectively, what should be the gross output of each sector in order to meet the new demands.

An amount of Rs. 5000 is put into three investment at the rate of interest of 6%, 7% and 8% per annum respectively. The total annual income is Rs. 358. If the combined income from the first two investment is Rs. 70 more than the income from the third, find the amount of each investment by matrix method.

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

Using binomial theorem, find the value of \({ \left( \sqrt { 2 } +1 \right) }^{ 5 }+{ \left( \sqrt { 2 } -1 \right) }^{ 5 }\)

Show by the principle of mathematical induction that 2³ⁿ–1 is a divisible by 7, for all \(n \in N\).

The profit Rs.y accumulated in thousand in x months is given by y = -x²  + 10x - 15. Find the best time to end the project.

An arch is in the form of a parabola with its axis vertical. The arch is 10 m high and 5 m wide at the base. How side is 2 m from the vertex of the parabola?

Prove that  \(\frac { \sin { \left( { 180 }^{ o }+A \right) \cos { \left( { 90 }^{ o }-A \right) \tan { \left( { 270 }^{ o }-A \right) } } } \quad \quad }{ \sec { \left( { 540 }^{ o }-A \right) \cos { \left( { 360 }^{ o }+A \right) \ cosec { \left( { 270 }^{ o }+A \right) } } } } =-\sin { A } \cos ^{ 2 }{ A } \)

Prove that cos 4x = 1 - 8 sin²x cos²x.

If cosA = \(\frac{4}{5}\)and cosB = \(\frac{12}{13}\),\(\frac{3\pi}{3}<(A, B)<2 \pi,\) find the value of cos(A+B).

Examine the following functions for continuity at indicated points
\(f(x)=\left\{\begin{array}{cl} \frac{x^2-4}{x-2}, & \text { if } x \neq 2 \\ 0, & \text { if } x=2 \end{array} \right.\) at x = 2

Differentiate the following with respect to x.
(i) x^x
(ii) (log x)^{cos x}

Verify the relationship of elasticity of demand, average revenue and marginal revenue for the demand law p = 50 - 3x.

The following table gives the annual demand and unit price of 3 items

Items	Annual Demand (units)	Unit Price Rs
A	800	0.02
B	400	1.00
C	13,800	0.20

Ordering cost is Rs. 5 per order and holding cost is 10% of unit price.
Determine the following:
(i) EOQ in units
(ii) Minimum average cost
(iii) EOQ in rupees
(iv) EOQ in years of supply
(v) Number of orders per year.

A man invests Rs. 13,500 partly in 6% of Rs. 100 shares at Rs. 140 and the remaining in 5% of Rs. 100 shares at Rs 125. If his total income is Rs. 560, how much has he invested in each?

The first of three urns contains 7 White and 10 Black balls, the second contains 5 White and 12 Black balls and third contains 17 White balls and no Black ball. A person chooses an urn at random and draws a ball from it. And the ball is found to be White. Find the probabilities that the ball comes from
(i) the first urn
(ii) the second urn
(iii) the third urn

Find the Quartile deviation.

Wages (Rs)	20-30	30-40	40-50	50-60	60-70
No. of labourers	3	5	20	10	5

Calculate coefficient of correlation for the ages of husbands and their respective wives:

Age of husbands	23	27	28	29	30	31	33	35	36	39
Age of wives	18	22	23	24	25	26	28	29	30	32

You are given the following data:

Details	X	Y
Arithmetic Mean	36	85
Standard Deviation	11	8

If the Correlation coefficient between X and Y is 0.66, then find (i) the two regression coefficients, (ii) the most likely value of Y when X = 10

Solve the following LPP. Maximize Z = 2 x₁ + 5x₂ subject to the conditions x₁+ 4x₂ ≤ 24. 3x₁+x₂ ≤ 21, x₁+x₂ ≤ 9 and x₁, x₂ ≥ 0.