Quant Boosters  Hemant Yadav  Set 3

Q14) If a and b are twodigit prime numbers such that a^2  b^2 = 2a + 14b + 48. Find the largest possible value of a + b.
[ OA : 89 + 97 = 186]

Q15) Find the greatest positive integer n so that 3^n divides 70! + 71! + 72!

Q16) A rectangular billiards table is 3 meters by 4 meters. A ballis on the long edge at distance (1/3)√5 from a corner. Amit hit it in such a way that it strikes each of the other walls once and returns to its starting point. How many meters did the ball travel?
(Ball can be considered to have radius 0.)

Q17) A rectangular box having dimensions a X b X c, where a, b, c are integers and 1 ≤ a ≤ b ≤ c.If volume and surface area of this box are equal then how many such rectangular boxes are possible.

Q18) What would be the highest power of 2 in 15^4096  1
a. 16
b. 15
c. 17
d. 14

Q19) A and B are running on a circular track. A runs in anticlockwise and completes a lap every 90 seconds while B runs in clockwise direction and completes a lap every 80 seconds. Both start from the start line at the same time. At some random time between 10 minutes and 11 minutes after they begin to run, a photographer standing inside the track takes a picture that shows onefourth of the track, centered on the starting line. What is the probability that both A and B are in the picture?
[OA : 3/16]

Q20) A number N when divided by 20, remainder is k. How many different possible remainders can we get if N is divided by 200?
[OA : 10]

Q21) Solve for integers x and y such that x^2  3y = 1008, where 0 ≤ x, y ≤ 100.
[OA : 2 solutions. (33,27), (36,96)]

Q22) For a natural number 'p', f(p) represents the number of natural numbers 'q' for which the equation x² + qx + p = 0 has integral roots. What is the smallest value of 'p' for which f(p) = 6?

Q23) What is the unit digit of 2^78  3^80 ?
[OA : 7]

Q24) 3 candles which can burn for 30, 40 and 50 minutes respectively are lit at different times. All three are burning simultaneously for 10 minutes and for 20 minutes exactly one of them is burning. The number of minutes in which exactly two of them are burning
(a) 35
(b) 45
(c) 70
(d) 90
(e) none of these

i + 2 * ii + 3 * iii = 120
10 + 2 * ii + 3 * 10 = 120.
so ii = 35

Q25) In how many ways can 2016 be written as the sum of an increasing sequence of two or more consecutive positive integers

Let's say the numbers are a+1, a+2, ..., a+n, then
a+1 + a+2 + ... + a+n = 2016
=> n(n+2a+1) = 4032We can see that one of n and (n+2a+1) is even and other odd. So we need to write 4032 as product of one even and one odd number.
4032 has 6 odd factors (4032 = 32*63)
But in one of these 6 cases n = 1, so we need to remove this case
=> 5 possibilities

Q26) All the positive divisors of 2010 are arranged in a straight line not in a particular order. In how many ways they can be arranged so that a pair of any two consecutive numbers, selected at random, is co prime?

Q27) There are 4 noncoplanar points in the space. How many distinct planes can be there which are equidistant from the given 4 noncoplanar points?
[OA : 7]

Q28) Find the least possible value of the expression x^2 + 6xy + 11y^2 + 4y + 5 given that x and y are real numbers
[OA : 3]

Q29) If a, b, and c are integers satisfying abc +4(a + b + c) = 2(ab + bc + ca) + 7 then how many ordered triple (a, b, c) are possible?

2(ab + bc + ac)  4(a + b + c)  abc = 7
2(ab + bc + ac)  4(a + b + c)  abc + 8 = 1
(2  a)(2  b)(2  c) = 1 = 1 * 1 * 1, 1 * 1 * 1, 1 * 1 * 1 and 1 * 1 * 1
so 4 cases
[@swetabh_kumar ]

Q30) A number is interesting if all of its digits are same. How many interesting numbers are there less than 10000 which are prime also