Quant Boosters  Hemant Yadav  Set 3

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

Here,
16/12 =8/x
X=6

N should be 36?

Is p+q 16?

2,3,5,7,11?

10(41)
=7

N =16..can anyone confirm?

P might be 12...
As 12 has 6 factors ..which gives 3 pairs of neg and pairs of positive