Evaluate:\(\begin{vmatrix} 1&a&a^2-bc\\1&b&b^2-ca\\1&c&c^2-ab \end{vmatrix}\)

If A = \(\begin{bmatrix}1 & 1 & 1 \\ 3 & 4 & 7\\1 & -1 & 1 \end{bmatrix}\) verify that A ( adj A ) = ( adj A ) A = |A| I₃.

Solve by using matrix inversion method: x - y + z = 2; 2x - y = 0 , 2y - z = 1.

Without expanding show that \(\Delta =\left| \begin{matrix} { cosec }^{ 2 }\theta & { cot }^{ 2 }\theta & 1 \\ { cot }^{ 2 }\theta & { cosec }^{ 2 }\theta & -1 \\ 42 & 40 & 2 \end{matrix} \right| =0\)

If \(A=\left[ \begin{matrix} 1 & tan\quad x \\ -tan\quad x & \quad \quad \quad 1 \end{matrix} \right] \), then show that A^TA^-1 = \(\left[ \begin{matrix} cos\quad 2x & -sin2x \\ sin\quad 2x & cos2x \end{matrix} \right] .\)

Solve by matrix inversion method: 3x - y + 2z = 13 ; 2x + Y - z = 3 ; x + 3y - 5z = - 8.

Show that the matrices A =\(\left[ \begin{matrix} 2 & 2 & 1 \\ 1 & 3 & 1 \\ 1 & 2 & 2 \end{matrix} \right] \)and B =\(\left[ \begin{matrix} \frac { 4 }{ 5 } & -\frac { 2 }{ 5 } & -\frac { 1 }{ 5 } \\ -\frac { 1 }{ 5 } & \frac { 3 }{ 5 } & -\frac { 1 }{ 5 } \\ -\frac { 1 }{ 5 } & -\frac { 2 }{ 5 } & \frac { 4 }{ 5 } \end{matrix} \right] \) are inverses of each other.

The sum of three numbers is 20. If we multiply the first by 2 and add the second number and subtract the third we get 23. If we multiply the first by 3 and add second and third to it, we get 46. By using matrix inversion method find the numbers.

Find adjoint of \(A=\left[ \begin{matrix} 1 & -2 & -3 \\ 0 & 1 & 0 \\ -4 & 1 & 0 \end{matrix} \right] \)

Solve by using matrix inversion method:
\(3 x-2 y+3 z=8 ; 2 x+y-z=1\)
\(4 x-3 y+2 z=4\)

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

Prove that the term independent of x in the expansion of \({ \left( x+\frac { 1 }{ x } \right) }^{ 2n }is\quad \frac { 1.3.5.....,(2n-1){ 2 }^{ n } }{ n! } \)

By the principle of mathematical induction, prove the following.
1.2 + 2.3 + 3.4 + ..... + n(n + 1) = \(\frac { n(n+1)(n+2) }{ 3 } \), for all \(n\in N\).

Resolve into partial factors : \(\frac { { x }^{ 2 }+x+1 }{ { x }^{ 2 }+2x+1 } \)

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

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

Show that the equation 12x² - 10xy + 2y² + 14x - 5y + 2 = 0 represents a pair of straight lines and also find the separate equations of the straight lines.

Show that the given lines 3x - 4y - 13 = 0, 8x - 11y = 33 and 2x - 3y - 7 = 0 are concurrent and find the concurrent point.

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 \(\frac { 4tan\ x(1-{ tan }^{ 2 }x) }{ 1-6{ tan }^{ 2 } x+{ tan }^{ 4 } x } =tanx\)

Prove that cos 6x = 32 cos⁶x - 48 cos⁴x + 18 cos²x - 1.

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

Draw the graph of the following function f(x) = e^-2x

Evaluate \(\begin{matrix} \underset { x\rightarrow 1 }{ lim } & \frac { { x }^{ 7 }-2{ x }^{ 5 }+1 }{ { x }^{ 3 }-{ 3x }^{ 2 }+2 } \end{matrix}\)

If \(y={ e }^{ a\cos ^{ -1 }{ x } }\) , show that \(\left( 1-{ x }^{ 2 } \right) \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } -x\frac { dy }{ dx } -{ a }^{ 2 }y=0\)

Evaluate the left hand and right hand limits of the function \(f(x)=\left\{\begin{aligned} \frac{|x-3|}{x-3} & \text { if } x \neq 3 \\ 0 & \text { if } x=3 \end{aligned} \right.\) at x = 3.

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

Find the extremum values of the function f(x) = 2x³ + 3x² – 12x.

Let u = log\(\frac { { x }^{ 4 }+{ y }^{ 4 } }{ x+y } \). By using Euler’s theorem show that \(x.\frac { \partial u }{ \partial x } +y.\frac { \partial u }{ \partial y } =3\) .

The relationship between Profit P and advertising cost x is given by \(P={4000x\over 500+x}-x\) . Find x which maximises P.

For the cost function C= 2000 + 1800x - 75x² + x³, discuss the behaviour of the marginal cost function.

If \(u=tan^{-1} \left(x^2+y^2\over x+y\right)\), then using Euler's theorem, prove that \(x{∂u\over ∂x}+y{∂u\over ∂y}={1\over 2}sin2u\)

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?

A man, deposits Rs.75 at the end of 6 months in a bank which pays interest at 8% compounded semiannually. How much is to his credit at the end of 10 years?

Compute coefficient of quartile deviation from the following data

Marks	10	20	30	40	50	60
No. of Students	4	7	15	8	7	2

Bag I contains 3 Red and 4 Black balls while another Bag II contains 5 Red and 6 Black balls. One ball is drawn at random from one of the bags and it is found to be red. Find the probability that it was drawn from Bag I.

A factory has 3 machines A₁, A₂, A₃ producing 1000, 2000, 3000 bolts per day respectively. A₁ produces 1% defectives, A₂ produces 1.5% and A₃ produces 2% defectives. A bolt is chosen at random and found defective. What is the probability that it comes from machine A₁?

Calculate the two regression equations of X on Y and Y on X from the data given below, taking deviations from a actual means of X and Y.

Price (Rs)	10	12	13	12	16	15
Amount demanded	40	38	43	45	37	43

Estimate the likely demand when the price is Rs.20.

Find the equation of the regression line of Y on X, if the observations ( X_i, Y_i) are the following (1, 4) (2, 8) (3, 2) ( 4, 12) (5, 10) (6, 14) (7, 16) ( 8, 6) (9, 18).

A computer while calculating the correlation co-efficient between two variables x and y from 25 pairs of observations, obtained the following results. \(\sum\)x=125, \(\sum\)x²=650, \(\sum\)y=100, \(\sum\)y²=460, xy=508. It was later found out that it had copied down two pairs as while the correct values are 

x	y
6	14
8	6

 

x	y
8	12
6	8

Obtain the correlation co-efficient for the correct value.

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.

The following table use the activities in a construction projects and relevant information.

Activity	1-2	1-3	2-3	2-4	3-4	4-5
Duration (in days)	22	27	12	14	6	12

Draw the network for the project, calculate the earliest start time, earliest finish time, latest start time and latest finish time of each activity and find the critical path. Compute the project duration.

Reshma wishes to mix two types of food P and Q in such a way that the Vitamin contents of the mixture contain at least 8 units of vitamin A and 11 units of vitamin B. Food P costs Rs.60/kg and Food Q costs Rs.80/kg. Food P contains 3 units 1 kg of vitamin A and 5 units 1 kg of vitamin B while food Q contains 4 units 1 kg of vitamin A and 2 units 1 kg of vitamin B. Determine the minimum cost of the mixture.

Every gram of wheat provides 0.1 g of proteins and 0.25 g of carbohydrates. The corresponding values of rice are 0.05 g and 0.5 g respectively. Wheat cost Rs.4 per kg and rice cost Rs.6 per kg. The minimum daily requirements of proteins and carbohydrate for an average child are 50 g and 200 g respectively. In what quantities should wheat and rice be mixed in the daily diet to provide minimum daily requirements of proteins and carbohydrate at minimum cost. Frame an LPP and solve it graphically. 

Calculate the earliest start time, earliest finish time, latest start time and latest finish time of each activity of the project given below and determine the critical path of the project and duration to complete the project.

Activity	1-2	1-3	1-5	2-3	2-4	3-4	3-5	3-6	4-6	5-6
Duration (in week)	7	6	11	3	9	2	4	9	6	3