QBM006
M46. One pendulum ticks 57 times in 58 seconds and another 608 times in 607 seconds. If they start simultaneously, find the time after which they will tick together? M47. Find the last digit of the number 12 + 22 + 32 + . . . . . . . 992 M48. Prove that 10n-1 is divisible by 11 if n is even and 10n+1 is divisible by 11 if n is odd. M49. Find the greatest number of five digits, which is exactly divisible by 7, 10, 3, 11 and 28. M50. Find the remainder when 1010+ 10100+ 101000 + . . . +1010000000000 is divided by 7. M51. 2! + 4! + 6! + 8! + . . . + 100! When divided by 3 would leave the remainder M52. 6n + 8n is divisible by 7 iff n is odd. Even integer prime M3. How many numbers are there between 500 and 600 in which 9 occurs only once? M54. Let n be the number of different 5 digit numbers, divisible by 4 with the digits 1, 2, 3, 4, 5 and 6, no digit being repeated in the numbers. What is the value of n? M55. If x = 174 and y= 14 x 16 x 18 x 20 , then which of the following is true x >y , x < y , x =y M56. How many different pairs of natural numbers add to make one-thousand? M57. Which among the following is greatest √7 + √3 , √5 + √5 , √6 + 2? M58. In deciding who should pay for lunch, Jane challenges John and James to a game of chance, "I shall take two ordinary 6-sided dice, roll them, and add their scores together. Then one of you shall do the same. If the second total is higher, John pays for lunch, if it is lower, James pays, or if it is same, I will pay. As there are three equally likely outcomes, the game is fair." Is the game fair? M59. 461 + 462 + 463 + 464 + 465 4 is divisible by 3 , 5, 11 or 17. M60. A 6 digit number 200ABC is divisible by 13, 11 and 7. Find A, B and C? END
Some answers
M46. One pendulum ticks 57 times in 58 seconds and another 608 times in 607 seconds. If they start simultaneously, find the time after which they will tick together?
Ans. Time between ticks = 58/57 and for other the time b/w ticks =607/608 thus LCM is = 58x607/19 sec =1853 second = 30 mts and 48 sec
M47. Find the last digit of the number 1^2 + 2^2 + 3^2 + . . . . . . . 99^2
Ans. 1 =1 mod 10 and 2=2mod10 and ..
Term 1 : 1 , Term 2 : 4 , Term 3 : 9, Term 4 : 6, Term 5 : 5, Term 6 : 6
Term 7 : 9, Term 8 = 4 , term 9 =1 term 10 = 0 thus remainder for 10 terms = 5
Total terms are 99 in no. and remainder of is same as 100 term =5x5=25 thus remainder is 5
M49. Find the greatest number of five digits, which is exactly divisible by 7, 10, 3, 11 and 28.
Ans. LCM =4620 thus the greatest number is 97020 Ans
M50. Find the remainder when 10^10+ 10^100+ 10^1000 + . . . +10^10000000000 is divided by 7.
Ans. Total terms =11
Sum of remainder is 2+6+4+5+1 = 4
2+6+4+5+1 = 4
and 2 thus remainder is 3
M3. How many numbers are there between 500 and 600 in which 9 occurs only once?
Ans. 9 in unit place and ten place = 9 numbers and similarly 9 in ten place and no. of 9 thus total numbers are 18
M54. Let n be the number of different 5 digit numbers, divisible by 4 with the digits 1, 2, 3, 4, 5 and 6, no digit being repeated in the numbers. What is the value of n?
Ans. Here the unit and ten diugits are ={12,16,24,32,36,52,56,64 } For each the number of ways of selecting other 3 digits is 4x3x2 =24
Thus n = 24x8 = 192 Ans.
M55. If x = 17^4 and y= 14 x 16 x 18 x 20 , then which of the following is true
x >y , x < y , x =y
Ans. Here y = 2^8 x3^2x35 thus 4(3)^0.5(35)^0.25 =16.85 thus x > y ans
M56. How many different pairs of natural numbers add to make one-thousand?
Ans. x+y = 1000 the solution is C(1001,1) -2
M57. Which among the following is greatest √7 + √3 , √5 + √5 , √6 + 2?
Ans. Approximately (9-2)^1/2+(4-1)^1/2 = 3(1-1/9)+2(1-1/8) = 4.41 and 2(1+1/8)x2= 4.5 => 4.471 and third term is (9-3)^.5 + 2 = 3(1-1/6)+2 = 4.5 => 4.449
Thus greatest term is middle term.
M58. In deciding who should pay for lunch, Jane challenges John and James to a game of chance, "I shall take two ordinary 6-sided dice, roll them, and add their scores together. Then one of you shall do the same. If the second total is higher, John pays for lunch, if it is lower, James pays, or if it is same, I will pay. As there are three equally likely outcomes, the game is fair." Is the game fair?
Ans. Let first dice rolled out the values x and y and their sum is N = { 2,3,4,5,6,7,8,9,10,11,12} Possible sample space has total possible combination is 11
The game is not fair
M60. A 6 digit number 200ABC is divisible by 13, 11 and 7. Find A, B and C?
Ans. 143-1-4-3 Thus 2-A-4B-3C is divisible by 13 thus
A+4B+3C is multiple of 13
and 2+B-A-C is multiple of 11
and 1-2-3,-1,2,3 => 2 -A+2B+3C is multiple of 7 solving these will give solution.