Quant Boosters  Sagar Gupta  Set 2

Total work =36 units
A > +9 units/day
B > +6 units/day
C > 4.5 units/day
D > 3 units/day
24 (x+y+z) = 36
x+y+z =1.5
94.53 =1.5
Hence B did not work

Q13) Three persons A, B and C rent the grazing of a park for Rs. 570. A put 126 oxen in the park for 3 months, B puts in 162 oxen for 5 months and C puts in 216 oxen for 4 months. What part of the rent should each person pay?
a. 105, 220, 245
b. 125, 205, 245
c. 105, 225, 240
d. data inadequate
d. None of these

126 * x * 3 + 162 * 5 * x + 216 * 4 * x = 570
x = 5/18
A= 126 * 3 * 5/18 =105
B=162 * 5 * 5/18 =225
C=216 * 4 * 5/18=240
Hence option C

Q14) a,b and c are three positive real numbers. The minimum value that the expression [(a+b)/2 * (b+c)/2 * (c+a)/2] can take when the product of the three numbers is 3/2 is?

abc = 3/2
a+b+c > 3 [ 1.5 ]^1/3
(a+b)/2 > 1.5[1.5]^1/3  c/2 = 1.5^4/3  c/2
(b+c)/2> 1.5 [1.5]^1/3  a/2 = 1.5^4/3  a/2
(a+c)/2 >1.5[1.5)^1/3 b/2 = 1.5^4/3  b/2
a = b = c = (1.5)^1/3
1.5^4/3  0.5* 1.5^1/3
1.5^1/3 [ 1.5  0.5 ] =1*1.5^1/3 =1.5^1/3
(1.5^1/3) ^3 = 1.5 = 3/2

Q15) The number 'm' and 'n' are reciprocals of each other. Both m and n are positive real numbers. If both m^3 + n^3 = 65/8, determine (m+n).

m = 1/n
mn = 1
m^3 + n^3 +3mn [ m+n ] = (m+n)^3
65/8 + 3 x = x^3
5/2 = x

Q16) The median of five positive integers is 7. If the only mode is greater than the median and the mean is greater than 9, what is the lowest possible value of the mode of these 5 integers?

a < b < c < d < e
c = 7
d and e are same
a + b + c + d + e > 45
5 = a
b = 6
c = 7
18 + 2d > 45
2d > 27
d > 13.5
d = 14
e = 14
Mode =14

Q17) The number of factors of the number N = 4^6 + 6^8 is
a. 18
b. 36
c. 54
d. 72

2^12 + 2^8 * 3^8
2^8 [ 16 + 3^8 ]
2^8 [ 16 + 81 * 81 ]
2^8 [ 16 + 6561 ]
2^8 [6577]
9 * 2 = 18 factors

Q18) When a natural number, N is divided by D, the remainder is 35. When 50N is divided by D, the remainder is 11. Find D
a. 1739
b. 43
c. 47
d. Cannot be determined

n=ad +35
50n=bd +11
50n=50ad+1750
50n=bd+11
50ad+1750 = bd +11
1739= (b50a)d
d can be 1739 or 47
CBD

Q19) In a row at a bus stop, A is 9th from the right and B is 7th from the left. They both interchange their positions. If there are 20 people in the row, what will be the new position of B from the left'?
(1) 11th
(2) 12th
(3) 13th
(4) 10th

1,2,3,4,5,6,B,8,9,10,11,A,13,14,15,16,17,18,19,20
So, 12'th from left

Q20) Find the sum of all positive twodigit integers that are divisible by each of their digits

{11,22...99}
10a+b mod a=0 > b mod a=0
10a+b mod b=0 >10a mod b=0
when a=b , the above 2 conditions will be satisfied > {11,22...99}
also : {12,15,24,36,48 } will satisfy the above conditions
sum =630

Q21) If a + b + c + d + e = 8 & a^2 + b^2 + c^2 + d^2 + e^2 = 16, where a,b,c,d& e are real numbers then
maximum (a,b,c,d,e) = ?
a. 4
b. 2
c. 16/5
d. 6/5
e. None of these

Application of cauchyschwarz inequality
a+b+c+d = 8e
a^2+b^2+c^2+d^2 = 16e^2
now
(a^2+b^2+c^2+d^2)(1+1+1+1) >= (a+b+c+d)^2
or 4*(16e^2) >= (8e)^2
or 644e^2 >= 64+e^216e
or 5e^2 < = 16e
or e(5e16) < = 0
so 0 < = e < = 16/5

Q22) Let f(x) be a function such that f(x).f(y)  f(xy) = 3(x+y+2). Then f(4)=?
(1) can not be determined
(2) 7
(3) 8
(4) either 7 or 8
(5) none of these