#### Full Portion Three Marks Question Paper

11th Standard

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Time : 01:30:00 Hrs
Total Marks : 60
20 x 3 = 60
1. Evaluate: $\begin{bmatrix} 3&-2&4\\2&0&1\\1&2&3 \end{bmatrix}$

2. 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.

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

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

5. 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.

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

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

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

9. 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]

10. 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.

11. 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
12. From the following data compute the value of Harmonic Mean.

 Marks 10 20 30 40 50 No. of students 20 30 50 15 5
13. 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?

14. 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
15. 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.

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

17. 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.

18. 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.

19. 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
20. Solve the following LPP graphically.$Maximize\quad Z=-{ x }_{ 1 }+2{ x }_{ 2 }$
Subject to the constraints $-{ x }_{ 1 }+3{ x }_{ 2 }\le 10,\quad { x }_{ 1 }+{ x }_{ 2 }\le 6,\quad { x }_{ 1 }{ -x }_{ 2 }\le 2\quad and\quad { x }_{ 1 },{ x }_{ 2 }\ge 0$