The technology matrix of an economic system of two industries is\(\begin{bmatrix} 0.50 & 0.30 \\ 0.41 & 0.33 \end{bmatrix}\). Test whether the system is viable as per Hawkins Simon conditions.

If A \(=\begin{bmatrix} 1 \\ -4\\3 \end{bmatrix}\) and  B = [-1 2 1], verify that (AB)^T = B^T. A^T

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

Evaluate: \(\left| \begin{matrix} 1 & 2 & 4 \\ -1 & 3 & 0 \\ 4 & 1 & 0 \end{matrix} \right| \)

Resolve into partial fractions for the following : \(\frac{4 x+1}{(x-2)(x+1)}\)

ResoIve into partial fractions :\(\frac { 12x-17 }{ (x-2)(x-1) } \)

Find the number of diagonals that can be drawn by joining the angular points of octagon ?

Find the values of A and B if \(\frac { 1 }{ \left( { x }^{ 2 }-1 \right) } =\frac { A }{ x-1 } +\frac { B }{ x+1 } \)

Find the cartesian equation of the circle whose parametric equations are x = 3 cos\(\theta\), y = 3 sin\(\theta,\) \(0\le \theta \le 2\pi \) 

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

Find the values of the following sin 76^o cos 16^o + cos 76^o sin 16^o

Prove that : \(\frac { \cos 2A-\cos 3A }{ \sin 2A-\sin 3A } =\tan\frac { A }{ 12 } \)

Using multiple angle identity, find tan60^o

If three angles A, B and C are in arithmetic progression, Prove that \(cotB=\frac { sinA-sinC }{ cosC-cosA } \)

Determine whether the following functions are odd or even?
\(f(x)=\left( \frac { { a }^{ x }-1 }{ { a }^{ x }+1 } \right) \)

Differentiate the following with respect to x  \(\frac { 5 }{ { x }^{ 4 } } -\frac { 2 }{ { x }^{ 3 } } +\frac { 5 }{ x } \)

Prove that the function given by f(x) = \(\left| x-1 \right| \), x \(\in\) R is not differentiable at x =1

If x = \(a\ \theta\) and \(y=\frac{a}{\theta}\), then prove that \(\frac{dy}{dx}+\frac{y}{x}=0\)

The total cost function for the production of x units of an item is given by c = 10 - 4x³ + 3x⁴ find the (i) average cost function (ii) marginal cost function (iii) marginal average cost fuction.

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

Find the first quartile and third quartile for the given observations
2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22

The following table shows the sales and advertisement expenditure of a form

Title	Sales	Advertisement expenditure(Rs.Cross)
Mean	40	6
SD	10	1.5

Coefficient of correlation r = 0.9. Estimate the likely sales for a proposed advertisement expenditure of Rs. 10 crores.

Draw the network for the project whose activities with their relationships are given below:
Activities A, D, E can start simultaneously; B, C > A; G, F > D, C; H > E, F.

A toy company manufactures two types of dolls A and B. Market tests and available resources have indicated that the combined production level should not exceed 1200 dolls per week and the demand for dolls of type B is atmost half of that for dolls of type A. Further, the production level of dolls of type A can exceed three times the production of dolls of other type by at most 600 units. If the company makes profit of n2 and n6 per doll, how many of each should be produced weekly in order to maximize the profit. Formulate the above as mathematical LPP.

A retired person has Rs. 70,000 to invest and two types of bonds are available in the market for investment. First type of bond yields an annual income of 8% on the amount invested and the second type yields 10% per annum. As per norms, he has to invest a minimum of Rs. 10,000 in the first type and not more than Rs.30,000 in the second type. How should he plan his investment, so as to get maximum returns after one year of investment? Formulate the above as LPP.