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.

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

If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus.

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

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

The total revenue (TR) for commodity x is \(TR=12x+{x^2\over2}-{x^3\over 3}\)S.T. at the highest point of average revenue (AR), AR = MR

A bank pays interest at the rate of 8% p.a. compounded quarterly. Find how much should be deposited in the bank at the beginning of each of 3 months for 5 years in order to accumulate to Rs.10,000 at the of 5 years.

A box contains 4 red, 6 green balls. Two balls are picked out one by one at random without replacement. What is the probability that the second is green given that the first one is green?

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).

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