Suppose the inter-industry flow of the product of two sectors X and Y are given as under.

Production Sector	Consumption Sector		Domestic demand	Gross output
	X	Y
X	15	10	10	35
Y	20	30	15	65

Find the gross output when the domestic demand changes to 12 for X and 18 for Y.

If \(A=\begin{bmatrix} 1 & 2 \\ 4 & 2 \end{bmatrix}\) then show that |2A| = 4 |A|.

Find the values of x if \(\begin{vmatrix} 2 & 4 \\5 & 1 \end{vmatrix}=\begin{vmatrix} 2x & 4\\6 & x \end{vmatrix}.\)

Show that \(\left[ \begin{matrix} 1 & 2 \\ 2 & 4 \end{matrix} \right] \)is a singular matrix.

Resolve into partial fractions for the following : \(\frac{3 x+7}{x^2-3 x+2}\)

Evaluate the following : \(\frac { 7! }{ 6! } \)

Show that 10P₃ = 9 P₃ + 3. 9P₂

In the expansion of \({ \left( x+\frac { 1 }{ x } \right) }^{ 6 }\), find the third term.

Convert the equation of the parabola x²+y=6x-14 into the standard form.

Find the angle between the pair of lines represented by the equation 3x²+10xy+8y²+14x+22y+15=0.

Find the locus of the point which is equidistant from (2, –3) and (3, –4).

Find the equation of the circle when the end points of the diameter are (2, 4) and (3, –2).

Evaluate : \(\cos\left[\frac{\pi}{3}-\cos^{-1}\left(\frac{1}{2}\right)\right]\)

Find the principal value of \(\cos^{-1}\left(\frac{-1}{\sqrt2}\right)\)

Find the principal value of sin^-1(1/2)

Evaluate the following tan\(\left(\cos ^{-1} \frac{8}{17}\right)\)

Differentiate the following with respect to x.  (x² - 3x + 2)(x + 1)

For what value of k, the following function is continous at x =0?
f(x) = \(\begin{cases} \frac { 1-cos4x }{ 8{ x }^{ 2 } } \quad ifx\neq 0 \\ k\quad \quad \quad ifx=0 \end{cases}\)

Show that the function f(x) = 5x -3 is continous at x = +3

Find the domain for which the functions f(x) = 2x² - 1 and g(x) = 1 - 3x are equal.

Find the elasticity of supply for the supply function x = 2p² - 5p + 1, p > 3.

A demand function is given by xpⁿ = k where n and k are constants. Prove that elasticity of demand is always constant.

If f(x,y) = 3x² + 4y³ + 6xy - x²y³ + 6. Find f_yy(1,1)

If f(x,y) = 3x² + 4y³ + 6xy - x²y³ + 6. Find f_xy(2,1)

A person pays Rs 64,000 per annum for 12 years at the rate of 10% per year. Find the annuity [(1.1)¹² = 3.3184]

The chairman of a society wishes to award a gold medal to a student getting highest marks in Business Mathematics. If this medal costs Rs. 9,000 every year and the rate of compound interest is 15% what amount is to be deposited now.

Find D₂ and D₆ for the following series 22, 4, 2, 12, 16, 6, 10, 18, 14, 20, 8

A person purchases tomatoes from each of the 4 places at the rate of 1kg., 2kg., 3kg., and 4kg. per rupee respectively. On the average, how many kilograms has he purchased per rupee?

From the following data calculate the correlation coefficient Σxy = 120, Σx² = 90, Σy² = 640

Calculate the coefficient of correlation from the following data:
ΣX = 50, ΣY = –30, ΣX² = 290, ΣY² = 300, ΣXY = –115, N = 10

Construct the network for the projects consisting of various activities and their precedence relationships are as given below:
A, B, C can start simultaneously A < F, E; B < D, C; E, D < G

Draw a network diagram for the following activities.

Activity code	A	B	C	D	E	F	G	H	I	J	K
Predecessor activity	-	A	A	A	B	C	C	C,D	E,F	G,H	I,J

A producer has 30 and 17 units of labour and capital respectively which he can use to produce two types of goods X and Y. To produce one unit of X, 2 unit of labour and 3 units of capital are required. Similarly, 3 units of labour and 1 unit of capital is required to produce one unit of Y. If X and Yare priced at HOO and H20 per unit respectively, how should the producer use his resources to maximize the total revenue? Formulate the LPP for the above.

A dealer whises to purchase a number of fans and sewing machines. He has only Rs.5760 to invest and has a space for atmost 20 items. A fan costs him Rs.360 and a sewing machine Rs. 240. His expectation is he can sell a fan at a profit of Rs.22  and a sewing machine at a profit of ns. Formulate this as an LPP to maximize his profit?