Quant Boosters  Sagar Gupta  Set 2

Train X starts at 9:00 am @72 km/hr
at 11:00 am, it is 144 kms from A
A...144 kms...A'......d...........B
Now Y starts from B at 11:00am @90 km/hr
A' and B is the position at 11:00 am
:
They should have met at 1:30pm
72 * 2.5 + 90 * 2.5 = d =405 kms
:
But they meet at 4:30 pm
from 11:00 to 12:00 , they travel at normal speed
Distance reduced = 72+90=162
Distance between them at 12:00 = 243 kms
Now they reduce their speed by k
(72k) * 4.5 + (90k) * 4.5 =243
k=54
7254 =18km/hr

Q9) abcd is a 4 digit perfect square, then if each digit is increased by 3 the number is still a perfect square. Find abcd.

abcd = x^2
1000a +100b +10c +d = x^2
1000(a+3) +100(b+3) +10(c+3) +d+3 = y^2
y^2x^2=3000 +300 + 30 + 3
y^2  x^2 = 3 [ 1000 +100 +10 +1 ]
y^2  x^2 =3333
3333=101*33
y+x =101
yx =33
y = 67
x = 34
x^2 =1156

Q10) What is the remainder when 72 to the power 7202 is divided by 625?

E[625]=500
7202 mod 500 = 2
72^2 mod 625 = 5184 mod 625 = 184

Q11) If a^3 + b^3 = 10 and a^2 + b^2 = 60 then find a + b?

(a+b)^3 = 10 + 3ab (a+b)
(a+b)^2 = 60 + 2ab
a+b = p
(p^3  10 ) / 3p = ab
p^2 = 60 + 2 [ (p^3  10 ) / 3p ]
3p^3=180p + 2p^3  20
p^3 +20 = 180p
p=13.36

Q12) A and B can build a wall in 4 days and 6 days respectively working alone, while C and D can destroy the wall in 8 days and 12 days respectively working alone. If three of them start working together and the wall was built in 24 days. Which of the four persons did not work?

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