Algebra homework

1. Find the first four terms of the sequence whose nth term is 3n+1. (Ans. 4, 7, 10, 13)
2. Find the first three terms of the sequence whose nth term is  (Ans. 2,  
3. If  for a sequence then find the first three terms. 
4. Write the next three terms of the A.P. whose first term is 11 and the common difference is 1.5. (Ans. 12.5, 14, 15.5)
5. State whether the following sequence is an A.P. or not.  . (Not an A.P.)
6. The sum of the measures of angles of a triangle is 1800 , of a quadrilateral is 3600, of a pentagon is 5400 , of a hexagon is 7200 , and so on. Then find the sum of measures of angles of a dodecagon (i.e. polygon with 12 sides). (Ans. 18000).
7. Find tn and hence t15 of the following A.P.  3, 8, 13, 18, .... (Ans. tn  = 5n – 2 , t15 = 73)
8. Show that the nth  odd natural number is 2n – 1 .
9. Find n if the sequence is 5, 8, 11, 14 ....68. (Ans. n=22)
10 . If the 10th term and the 18th term of an A.P. are 25 and 41 respectively then find the following. (i) the 1st term and the common difference. (ii) the 38th term. (iii) n such that the nth term is 99. (Ans. (i) a = 7 , d = 2 (ii) t38 = 81, (iii) n = 47)
11. How many terms are there in the A.P. 187, 194, 201, …439? (Ans. 37)
12. Find the sum of 7, 11, 15, 19, … , up to 60 terms. (Ans. S60 = 7500) 
13.If for an A.P., the first term is 3 and the common difference is 6 then find Sn and hence find S10. (Ans. Sn = 3n2 , S10 = 300)
14. You wish to buy a pencil, a notebook and a pen. The prices of these are in A.P. such that the sum of prices of these three things is 27 and sum of the square of their prices is 341. Find the price of each thing. (Ans. 2, 9, 16 OR 16, 9, 2)
15. Find four consecutive terms in an A.P. such that the sum of the middle two terms is 18 and product of the two end terms is 45. (Ans. 3, 7, 11, 15 OR 15, 11, 7, 3)
16. Find three consecutive terms in an A.P. whose sum is 27 and their product is 504. (Ans. 4,9, 14 OR 14, 9, 4)
17. A man repays a loan of Rs. 3250 by paying Rs. 305 in the first month and then decreases the payment by Rs. 15 every month. How long will it take to clear his loan? (Ans. n= 20 MONTHS)
18. A farmer borrows Rs. 1000 and agrees to repay with a total interest of Rs. 140 in 12 instalments, each instalment being less than the preceding instalment by Rs. 10. What should be his first instalment? (Ans. Rs. 150) 
19. Solve the following quadratic equations by factorization method. 
(i). (Ans. 10 , 3)  (ii)  (Ans. 9, -4) (iii)  (Ans. 4, ) (iv)  (Ans. )  (v)  (Ans. 7,5)
20. Solve by completing square:  (Ans. -3, -5)
21. Solve by completing square:  (Ans. 
22. Solve by formula: (i)  (ii) (iii)  (iv)  (Ans. (i)  (ii)    (iii)     (iv) )
23. Find the value of discriminant for each of the following equation. 
(i) (Ans.   (ii) 
25. Determine the nature of roots of the following equations from their discriminating. 
(i)  (Ans. Not real) (ii)  (Ans. Real and equal) 
26. Find the value of  if the given equation has real and equal roots. (Ans.  or  ) 
27. Find , if the root of the equation  is 5 times the other. (Ans. 
28. Find if the roots of the equation  are in the ratio 3:4. (Ans. 
29. If  and   are the roots of the equation  find (i)  (ii)  (Ans. (i) = 14   (ii) = 14)
30. If one root of the quadratic equation  is   , find  (Ans. 
31. If the roots of the quadratic equation are 6 and 7 find the quadratic equation. (Ans. 
32. If one of the root of the quadratic equation is .   Find the quadratic equation. (Ans. 
33. If  and  Find a quadratic equation whose roots are   and  . (Ans. 
34. If   and  Find a quadratic equation whose roots are   and  . (Ans. 
35. Solve:  (Ans.   )
36.   (Ans.   , )
37.  (Ans. -3, 2, Not real roots)
38. (Ans. 
39. . (Ans. 
40. Without actually solving the simultaneous equations given below, examine which simultaneous equations have unique solution, no solution or infinitely many solutions. 
(i)  ; 2x+ 6 y – 3 = 0 (No solution) (ii) 4x – y – 6 = 0;  (Infinitely many solutions) (iii) y = 2x + 14 ; 7x = 2y + 5 (unique solution)
41. Find the value of k for which the given simultaneous equations have infinitely many solutions: k y + 3 y = 8 ; 6 x + 9 y = 24 . (Ans. = 2) 
42 . Find the value of k for which the given simultaneous equations have infinitely many solutions. k x + 2 y = k – 2 ;  8 x + k y  = k  (Ans. k = 4 which satisfies both the conditions.)
43. Find the value of m for which the given simultaneous equations have unique solution:  ( Ans. Therefore, the simultaneous equations will have a unique solution for all the values of m, except 10)
44. Solve the following simultaneous equations:  (Ans. x = 2 and y = -3) 
45. Solve:      (Ans.   and 
46.  Ans. and . 
47. Two coins are tossed simultaneously. Write the sample space 'S' and the number of sample points n(S). Write the following events using set notation and mention the total number of elements in each of them. 
A is the event of getting at least one head. 
B is the event of getting exactly one head. 
C is the event of getting at most one tail.
D is the event of getting at most one tail. 
D is the event of getting no head. 
Find the events among the events defined above which are complementary events, mutually exclusive events. 
Ans. n(S) = 4, n(A) = 3, n(B) = 2, n(C) = 3, n(D) = 1
Events A and D are complementary events and mutually exclusive also. 
Events C and D are complementary events and also mutually exclusive. 
Events B and D are mutually exclusive events but not complementary events. 
48. In a bag, there are fifty cards bearing number from 1 to 50, one card is drawn at random. Write the sample space S. Write the events A and B using set and mention number of sample points in them, where A is the event that the number on the card is divisible by 5. B is the event that the number on the card is a prime number. Also examine whether events A and B are complementary events or mutually exclusive events or both. 
Ans. n(S) = 50, n(A) = 10, n(B) = 15. A and B are not complementary events and also not mutually exclusive. 
49. There are three boys and three girls. An environment committee of two is to be formed. Write the sample space S, the number of sample points n(S). 
A is the event that the committee should contain atleast two girls. 
B is the event that the committee should contain both boys. 
C is the event that there is only one girl in the committee. 
D is the event that there is atmost one boy in the committee. 
Find the events among the events defined above which are:
Complementary events, mutually exclusive events and exhaustive events. 
Ans. n(S) = 15, n(A) = 5, n(B) = 3, n(D) =12, 
A and B are mutually exclusive events. 
B and C are mutually exclusive events. 
B and D are mutually exclusive events. 
B and D are complementary events and exhaustive events. 
50. A coin is tossed. Find the probability of the events. (i) Getting a head (ii) Getting a tail. (Ans. (i) ½  (ii) ½ )
51. A card is drawn at random from a pack of well shuffled 52 cards. Find the probability that the card drawn is (i) a king (ii) an ace.  (Ans. (i) 1/13 , (ii) 1/13.
52. If a card is drawn from a pack of 52 cards then find the probability of getting (i)  a red card (ii) a face card. Ans. (i) ½  (ii) 3/13 
53.