Evaluate: \(\begin{bmatrix} 3&-2&4\\2&0&1\\1&2&3 \end{bmatrix}\)

Without actual expansion show that the value of the determinant \(\begin{vmatrix}5 &5^2 &5^3 \\5^2 & 5^3 & 5^4\\5^4&5^5&5^6 \end{vmatrix}\)is zero.

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

Using co-factors of elements of  second column evaluate \(\left| \begin{matrix} 6 & -1 & 5 \\ 3 & 0 & 4 \\ -2 & 7 & -3 \end{matrix} \right| \)

The technology matrix of an economic system of two industries is \(\left[ \begin{matrix} 0.8 & 0.2 \\ 0.9 & 0.7 \end{matrix} \right] \) Test whether the system is viable as per Hawkins – Simon conditions.

Let p(n) be the statement "n² + n is even". If P(k) is true, then show that P(k+1) is true.

Find the equation of the circle which touches the line x = 0, y = 0 and x = a.

Differentiate: \(\sin ^{ -1 }{ \left( \sqrt { \cos { x } } \right) } \)

A bank pays 8% per annum interest compounded quarterly. Find the equal deposits to be made at the end of each quarter for 10 years to have Rs. 30,200? [(1.02)⁴⁰ = 2.2080]

If I deposit Rs.500 every year for a period of 10 years in a bank which gives C.I. 5% per year, find out the amount I will receive at the end of 10 years.

Calculate the value of Q₁, Q₃, D₆ and P₅₀ from the following data

Roll No	1	2	3	4	5	6	7
Marks	20	28	40	12	30	15	50

From the following data compute the value of Harmonic Mean.

Marks	10	20	30	40	50
No. of students	20	30	50	15	5

X speaks truth 4 out of 5 times. A die is thrown. He reports that there is a six. What is the chance that actually there was a six?

Calculate Quartile deviation and Coefficient of Quartile deviation of the following data.

Marks:	0	10	20	30	40	50	60	70
No. of students:	150	142	130	120	72	30	12	4

Find the probability of drawing a one-rupee coin from a purse with two compartments one of which contains 3 fifty-paise coins and 2 one-rupee coins and other contains 2 fifty paise coins and 3 one-rupee coins.

Find the mean deviation about the mean of the following observations 2, 3, 5, 6, 9, 10, 12, 17, 20, 26 and n = 10

Calculate the co-efficient of correlation between x and y on the basis of the following observations. \(\sum\)x=125, \(\sum\)x²=1650, \(\sum\)y=100, \(\sum\)y²=1460, \(\sum\)xy=50, n=25.

A company is producing three products P₁, P₂ and P₃, with profit contribution of Rs.20, Rs.25 and Rs.15 per unit respectively. The resource requirements per unit of each of the products and total availability are given below.

Product	P1	P2	P3	Total availability
Man hours/unit	6	3	12	200
Machine hours/unit	2	5	4	350
Material/unit	1kg	2kg	1kg	100kg

Formulate the above as a linear programming model.

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

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