Part -III

Mathematics paper -II(A)

(English version)

**Time:** 3 Hours **Max.Marks:** 75

**Note:-** This question paper consists of three sections – A, B, and C.

**SECTION – A**

**I. Very Short Answer Type Question**

(i) Attempt all the questions.

(ii) Each question carries two marks.

1. If (3 + i)100= 299(a + ib), then show that a2+b2= 4.

2. If z= 2 – 3i, then show that z2- 4z + 13 =0.

3. If 1, w, w2are the **cube roots** of unity, then find the value of ( 1 – w + w2)5 + ( 1 + w + w2)5 +

4. Find the value of m, for which the equation x2-15-m(2x-8)=0have equal roots.

5. If the product of the roots 4×3 + 16×2 -9x -a=0 is 9,then find a.

6. Find the number of ways of arranging the letters of the word

7. If nC5 =nC6,then find 13C

8. Find the 6th term in the expansion of (2×3+ 3y2)9.

9. Find the mean and variance of a binomial distribution are 4 and 3 respectively, fix the distribution and find P(X>1).

**SECTION – B**

**II. Short Answer Type Questions**

(i) Attempt any five questions.

(ii) Each question carries four marks.

11. If x +iy = 11 + cos 0 +i sin 0, then show that 4×2 – 1=0

12. If x is real, prove that x1+5x+9 lies between -11and 1.

13. If the letters of the words MASTER are permuted in all possible ways and the words thus formed are arranged in the dictionary order, then find the rank of the word MASTER.

14. Simplify:34C5+ r=04(38- r)C4

15. Resolve: x3(x-1)(x+2)into **partial fractions**.

16. A, B, C are three horses in a race. If the probability of A to win the race is twice that of B, and the probability of B is twice that of C. What are the probabilities of A, B, and C to win the race?

17. A speaks truth in 75% of the cases and B in 80% cases. What is the probability that their statements about an incident which do not match?

**SECTION – C**

**III. Long Answer Type questions**

(i) Attempt any five questions.

(ii) Each question carries seven marks.

18. If n+n = 2n+1.cose(nx3)

19. **Solve the problem** 18×3 + 81×2 + 121x + 60 =0, give that one root is equal to half ,the sum of the remaining roots are.

20. If the coefficient of x10in the expansion of (ax2+1bx)11is equal to the coefficient of x-10in the expansion of (ax-1bx2)11, then find the relation between a and b where a and b are **real numbers**.

21. If x=1.33.6+1.3.5.3.6.9+1.3.5.73.6.9.12+…,then prove that 9×2+24x=11.

22. Calculate the **variance and standard deviation** for the following discrete frequency

distribution:

xi 4 8 11 17 20 24 32

fi 3 5 9 5 4 3 1

23. State and prove Bayes Theorem.

24. The probability of distribution in the random variable is X shown below :

X=xi 1 2 3 4 5

p(X=xi) k 2k 3k 4k 5k

Find the value of k and the mean and variance of X.

SD