Prove that \(\begin{vmatrix} {1\over a}&bc&b+c\\{1\over b}&ca&c+a\\{1\over c}&ab&a+b \end{vmatrix}=0\)

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.

Use the product \(\left[ \begin{matrix} 1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{matrix} \right] \left[ \begin{matrix} -2 & 0 & 1 \\ 9 & 2 & -3 \\ 6 & 1 & -2 \end{matrix} \right] \)to solve the system of equations x - 2y + 2z = 1, 2y - 3z = 1, 3x - 2y + 4z = 2.

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 : \(\frac{1}{\left(x^2+4\right)(x+1)}\)

By the principle of mathematical induction, prove the following.
1 + 4 + 7 + ..... + (3n - 2) = \(\frac { n(3n-1) }{ 2 } \) , for all \(n\in N\).

Using the principle of mathematical induction, prove that 1.3 + 2.3² + 3.3³ + ... + n.3ⁿ =\(\frac { (2n-1){ 3 }^{ n+1 }+3 }{ 4 } for\ all\ n\in N\)

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

Find the equation of the circle on the line joining the points (1,0), (0,1) and having its centre on the line x + y = 1

As the number of units produced increases from 500 to 1000 and the total cost of production increases from. Rs 6000 to Rs 9000. Find the relationship between the cost (y) and the number of units produced (x) if the relationship is linear.

Prove that the tangents to the circle x² + y²  = 169 at (5,12) and (12,-5) are perpendicular to each other.

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:  \(\sin { \theta } \cos { \theta } \left\{ \sin { \left( \frac { \pi }{ 2 } -\theta \right) } \csc { \theta } +\cos { \left( \frac { \pi }{ 2 } -\theta \right) \sec { \theta } } \right\} =1\)

Prove that sin(n+1)x sin(n+2) x+cos(n+1)xcos(n+2)x=cosx.

Prove that cos 20° cos 40° cos 60° cos 80⁰ = \(\frac { 1 }{ 16 } \)

If  \(sin\left( { sin }^{ -1 }\left( \frac { 1 }{ 5 } \right) +{ cos }^{ -1 }(x) \right) =1\)  then find the value of x

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: xy + y² = tan x + y.

Find the derivate of (x³-27)from first principles.

Show that \( f(x)= \begin{cases}5 x-4, & \text { if } 0< x \leq 1 \\ 4 x^3-3 x, & \text { if } 1< x< 2\end{cases}\) is continuous at x = 1

The demand for a commodity A is q = 80 - \({ p }_{ 1 }^{ 2}\) + 5p₂ - p₁p₂. Find the partial elasticities \(\frac { { E }q }{ { E }p_{ 1 } } \) and \(\frac { { E }q }{ { E }p_{ 2 } } \) when p₁ = 2, p₂ = 1.

For the cost function C = 2000 + 1800 x - 75x² + x³  find when the total cost (C) is increasing and when it is decreasing

If  z = 4x⁶ - 8x³ - 7x + 6xy + 8y + x³y⁵, find 
(i) \({\partial^2z\over \partial y^2}\) (ii)\(\partial^2 z\over \partial x\partial y\)(iii) \(\partial^2z\over \partial y\partial x\)

Show that the maximum value of the function f(x) = x³ - 27x + 108 is 108 more than the minimum value.

A person deposits Rs. 2,000 at the end of every month from his salary towards his contributory pension scheme. The same amount is credited by his employer also. If 8% rate of compound interest is paid, then find the maturity amount at end of 20 years of service. [(1.0067)²⁴⁰ = 4.9661]

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.

Equal amounts are invested in 12% stock at 95 (brokerage). If 12% stock brought at Rs.120 more by way of dividend income than the other, find the amount invested in each stock?

A company has a total capital of Rs.5,00,000 divide into 1000 preference shares of 6% dividend with par value RS.100 each and 4000 ordinary shares of per value Rs.100 each. The company delares an annual dividend of Rs.40,000. Find the dividend received by Sundar having 100 preference shares and 200 ordinary shares.

Find out the coefficient of mean deviation about median in the following series.

Age in years	0-10	10-20	20-30	30-40	40-50	50-60	60-70	70-80
No. of persons	20	25	32	40	42	35	10	8

Calculations have to be made correct to two places of decimals.

Three coins are tossed simultaneously. Consider the events A ‘three heads or three tails’, B ‘atleast two heads’ and C ‘at most two heads’ of the pairs (A, B), (A, C) and (B, C), which are independent? Which are dependent?

Find D₆,D₈,P₇ and P₂₀ for the data 57, 58, 61, 42, 38, 65, 72, 66. 

In a Shooting test, the probabilities of hitting the target are \(\frac { 1 }{ 2 } \) for A, \(\frac { 2 }{ 3 } \) for B and \(\frac { 3 }{ 4 } \)  for C. If all of them fire at the same target, calculate the probabilities that only one of them hit the target.

For the given lines of regression 3X – 2Y = 5 and X – 4Y = 7. Find
(i) Regression coefficients
(ii) Coefficient of correlation

The heights ( in cm.) of a group of fathers and sons are given below

Heights of fathers:	158	166	163	165	167	170	167	172	177	181
Heights of Sons:	163	158	167	170	160	180	170	175	172	175

Find the lines of regression and estimate the height of son when the height of the father is 164 cm.

The equations of two regression lines are 4x+3y+7=0 and 3x+4y+8=0.
Find (i) the mean of x and the mean of y
(ii) the regression co-efficient b_xy and b_yx 
(iii) the correlation co-efficient between x and y.

For the following observations, find the regression co-efficient b_yx and b_xy and hence find the correlation co-efficient (4,2)(2,3)(3,2)(4,4)(2,4).

Solve the following LPP by graphical method Minimize z = 5x₁ + 4x₂ Subject to constraints 4x₁ + x₂ ≥ 40 ; 2x₁ + 3x₂ ≥ 90 and x₁, x₂ > 0.

A Project has the following time schedule

Activity	1-2	2-3	2-4	3-5	4-6	5-6
Duration (in days)	6	8	4	9	2	7

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.

The following table use the activities in a building project.

Activity	1-2	1-3	2-3	2-4	3-4	4-5
Duration (days)	21	26	11	13	5	11

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.