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.

Find the minors and cofactors of all the elements of the following determinants. \(\begin{bmatrix} 1&-3&2\\4&-1&2\\3&5&2 \end{bmatrix}\)

Find |AB| if \(A=\begin{bmatrix} 3&-1\\2&1 \end{bmatrix} \) and \(B =\begin{bmatrix} 3&0\\1&-2 \end{bmatrix}\)

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

Using the property of determinant, evaluate \(\begin{vmatrix} 6 &5 &12 \\ 2 & 4 &4 \\2 & 1 & 4 \end{vmatrix}.\)

Show that \(\left[ \begin{matrix} 8 & 2 \\ 4 & 3 \end{matrix} \right] \)is non – singular.

Expand the following by using binomial theorem. (2a - 3b)⁴

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

Show that 10P₃ = 9 P₃ + 3. 9P₂

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

From a class of 32 students, 4 students are to be chosen for a competition. In how many ways can this be done?

Find the center and radius of the circle  x² + y² - 22x - 4y + 25 = 0

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

Find the equation of the circle with centre at (3, –1) and radius is 4 units.

Prove that \(2\tan^{-1}(x)=\sin^{-1}\left(\frac{2x}{1+x^2}\right)\)

Find the values of each of the following trigonometric ratios. \(\sin { { 300 }^{ o } } \)

Prove that \(\tan^{-1}\left(\frac mn\right)-\tan^{-1}\left(\frac{m-n}{m+n}\right)=\frac{\pi}{4}\)

Prove that \(sin^2\left(\frac{\pi}{8}+\frac x2\right)-sin^2\left(\frac{\pi}{8}-\frac x2\right)=\frac{1}{\sqrt2}\sin x.\)

Prove that \(\frac{\tan 69^o+\tan 66^o}{1-\tan 69^o\tan 66^o}=-1\)

Evaluate: cos 20° + cos 100° + cos 140°

Show that \(\frac { sin2\theta }{ 1+cos2\theta } =tan\theta \)

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

If \(f(x)={ x }^{ 3 }-\frac { 1 }{ { x }^{ 3 } } \), x \(\neq\) 0, then show that \(f(x)+f\left( \frac { 1 }{ x } \right) =0\)

Differentiate \(\frac { { x }^{ 2 }cos\frac { \pi }{ 4 } }{ sinx } \)

Evaluate: \(\underset { x\rightarrow 2 }{ lim } \frac { { x }^{ 2 }-4x+6 }{ x+2 } \)

Find the second order derivative of the following functions with respect to x, 3 cos x + 4 sin x

The cost function of a firm is \(C={1\over3}x^3-3x^2+9x\). Find the level of output (x > 0) when average cost is minimum.

The profit function of a firm in producing x units of a product is given by\(p(x)=\frac { { x }^{ 3 } }{ 3 } +{ x }^{ 2 }+x\). Check whether the firm is running a profitable business or not. 

Let u = x cos y + y cos x. Verify \(\frac { { \partial }^{ 2 }u }{ { \partial x }\partial y } +\frac { { \partial }^{ 2 }u }{ { \partial y }{ \partial x } } \)

A certain manufacturing concern has total cost function C = 15 + 9x - 6x² + x³ . find Find x, when the total cost is minimum 

Prove that 75-12x+6x²-x³ always decreases as x increases.

What is the amount of perpetual annuity of Rs. 50 at 5% compound interest per year?

A cash prize of Rs. 1,500 is given to the student standing first in examination of Business Mathematics by a person every year. Find out the sum that the person has to deposit to meet this expense. Rate of interest is 12% p.a

The price of a commodity increased by 5% from 2004 to 2005, 8% from 2005 to 2006 and 77% from 2006 to 2007. Calculate the average increase from 2004 to 2007?

A die is thrown. Find the probability of getting
(i) a prime number
(ii) a number greater than or equal to 3

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

Draw a network diagram for the project whose activities and their predecessor relationships are given below:

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

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?

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

Prove that \(\begin{vmatrix} -a^{ 2 } & ab & ac \\ ab & -b^{ 2 } & bc \\ ac & bc & -c^{ 2 } \end{vmatrix}=4a^{ 2 }b^{ 2 }{ c }^{ 2 }\)

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.

Prove that \(\left| \begin{matrix} x & sin\theta & cos\theta \\ -sin\theta & -x & 1 \\ cos\theta & 1 & x \end{matrix} \right| \) is independent of \(\theta\)

Find the inverse of \(\begin{bmatrix}-1 & 5 \\-3 & 2 \end{bmatrix}\).

If X \(=\begin{bmatrix} 8 &-1&-3 \\-5 &1&2\\10&-1&-4 \end{bmatrix}\)  and Y = \(\begin{bmatrix} 2 & 1 & -1\\0 & 2 & 1\\ 5& p & q \end{bmatrix}\) then, find p, q if Y =  X^-1

Resolve into partial fractions for the following : \(\frac{x^2-6 x+2}{x^2(x+2)}\)

Show that the middle term in the expansion of (1 + x)²ⁿ is \(\frac { 1.3.5....(2n-1){ 2 }^{ n }.{ x }^{ n } }{ n! } \)

Solve : \(\frac { (2x+1)! }{ (x+2)! } .\frac { (x-1)! }{ (2x-1)! } =\frac { 3 }{ 5 } \)

Using binomial theorem, expand \({ \left( { x }^{ 2 }+\frac { 1 }{ { x }^{ 2 } } \right) }^{ 4 }\)

Find the length of the tangent from (1,2) to the circle x² + y² - 2x + 4y + 9 = 0

Prove that the lines (b - c) x + (c - a) y + (a - b) = 0, (c - a) x + (a - b) y + (b - c) = 0, and (a - b) x + (b - c) y+c-a=0 are concurrent.

Show that \(\sin ^{-1}\left(-\frac{3}{5}\right)-\sin ^{-1}\left(-\frac{8}{17}\right)=\cos ^{-1} \frac{84}{85}\)

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

Prove that \(\frac{\sin(x+y)}{\sin(x-y)}=\frac{\tan x+\tan y}{\tan x-\tan y}\)

If tan A = m tanB, prove that \(\frac { sin(A+B) }{ sin(A-B) } =\frac { m+1 }{ m-1 } \)

Find y₂ of the following function x = a cos \(\theta\), y = a sin \(\theta\)

Differentiate: sin x.sin 2x. sin 3x with respect to 'x'.

Differentiate: \(\sqrt{\frac{(x-3)(x^2+4)}{3x^2+4x+5}}\)

If \(y={2x+1\over 3x+2}\) then, obtain the value of elasticity at x = 1.

If u = log(x²+y²), then show that \(\frac { { \partial }^{ 2 }u }{ { \partial x }^{ 2 } } +\frac { { \partial }^{ 2 }u }{ \partial { y }^{ 2 } } =0\)

Babu sold some Rs. 100 shares at 10% discount and invested his sales proceeds in 15% of Rs. 50 shares at Rs. 33. Had he sold his shares at 10% premium instead of 10% discount, he would have earned Rs. 450 more. Find the number of shares sold by him.

Find the amount of an ordinary annuity of 12 monthly payments of Rs.1000 that earn interset at 12% per year compounded monthly.

Find the harmonic mean of 6, 14, 21, 30

There are two series of index numbers P for price index and S for stock of the commodity. The mean and standard deviation of P are 100 and 8 and of S are 103 and 4 respectively. The correlation coefficient between the two series is 0.4. With these data obtain the regression lines of P on S and S on P.

Calculate the covariance of the following pairs of observation of two variates X and Y. (1, 5)(2, 4)(3, 3)(4, 2)(5, 1)

Two phychologist ranked 12 candidates in the selection list as below:

X	1	2	3	4	5	6	7	8	9	10	11	12
Y	12	9	6	10	3	5	4	7	8	2	11	1

Find the rank correlation co-efficient.

Maximize Z = 3x₁ + 4x₂ subject to x₁ – x₂ ≤ –1; –x₁ + x₂ ≤ 0 and x₁, x₂ ≥ 0

Solve the following LPP graphically. Maximize Z =−x₁ + 2x₂
Subject to the constraints −x₁ + 3x₂ ≤ 10, x₁ + x₂ ≤ 6,x₁ − x₂ ≤ 2 and x₁,x₂ ≥ 0

Solve the following LPP graphically. Minimize\(Z=-3{ x }_{ 1 }+4{ x }_{ 2 }\)
Subject to the constraints \({ x }_{ 1 }+2{ x }_{ 2 }\le 8\quad ,{ 3x }_{ 1 }+{ 2x }_{ 2 }\le 12\quad and\quad \quad { x }_{ 1 }\ge 0,{ x }_{ 2 }\ge 2.\)

Develop a network based on the following information.

Activity	A	B	C	D	B	E
Immediate Predecessor	-	-	A	C	E	F