# Quant Boosters - Hemant Malhotra - Set 12

• Q10) Find the largest value of t such that x^t + 1 divides 1 + x + x^2 + x^3 + ... + x^143

• Q11) If x + 1/x = 3 then find the value of x^4 + 1/x^4

• Q12) A father and his son are waiting at a bus stop in the evening. There is a lamp post behind them. The lamp post, the father and his son stand on the same straight line. The father observes that the shadows of his head and his son's head are incident at the same point on the ground. If the heights of the lamp post, the father and his son are 6 metres, 1.8 metres and 0.9 metres respectively, and the father is standing 2.1 metres away from the post then how far (in metres) is son standing form his father?

• Q13) For how many values of y, y(y + 4)(y + 6)(y + 8) < 300 is satisfied

• Q14) What is the minimum value of (p + q)(q + r)(r + p) if p, q and r are positive numbers

• Q15) Find the sum of the first 10 terms of the series 1, 2, 5, 12, 27 ...

• The given term can be written as (2-1), (4-2), (8 - 3), (16 - 4) and so on
so sum of first 10 terms will be = (2 + 4 + 8 + 16 + - - - - - ) - (1 + 2 +3 +4 +5 +6 +7 - - -)
2046 - 55 = 1991

• Q16) If 2x + 9y ≤ 90, x ≥ 0, y ≥ 0 and x & y are integers, find the total number of solutions

• Q17) x, y and z are real numbers such that -6 ≤ x ≤ -2, -4 ≤ y ≤ 4 and 3 ≤ z ≤ 7. If w = y/xz then which of the below options is necessarily true ?
a) 0 ≤ w ≤ 2
b) -1/2 ≤ w ≤ 1/2
c) -2/3 ≤ w ≤ 1/3
d) -2/3 ≤ w ≤ 2/3

• Q18) What is the maximum value of p for which the below equations have a unique value of x
(2x - p)^2 + y^2 + 32 = 0
3x^2 + y^2 - 16 = 0

• Q19) Sum of two natural numbers is 600 and their hcf is 15. How many such pairs are possible?

• Let numbers be 15a and 15b
so 15a + 15b = 600
so a + b = 40
so E(40)/2 will be number of pairs

Why E(40)/2 ?
E(40) means all numbers below 40 which are coprime to 40

Example- E(10) = 10 * 1/2 * 4/5 = 4
1, 3, 7, 9 - there are 4 numbers below 10 which are coprime to 10
2 pairs 1,9 and 3,7 whose sum is 10

• Q20) Which one of the following is a prime number?
(a) 999,991
(b) 999,973
(c) 999,983
(d) 1,000,001
(e) 7,999,973

• a) (10^3)^2 - 3^2 = (1003)(997)
b) (10^2)^3 - 3^3 divisible by (10^2 -3)= 97
d) (10^2)^3 + 1^3 divisible by 101
e) (2*10^2)^3 - 3^3 divisible by 200 -3 =197
so c) is prime

• Q21) N = 373839404142 ... 9192 (numbers from 37 to 92 written in order to form a number). Find the maximum value of k if the number N is perfectly divisible by 3^k

• N is not divisible by 9 but divisible by 3. As it is not divisible by 9, no need to check for higher powers
so OA = 1

• Q22) In a 4 digit no. having non zero and distinct digits, the sums of the digits at the unit place and the tens place is equal to the sum of other 2 digits. The sum of the digits at the tens and the hundreds places is three times the sum of the remaining 2 digits. If the sum of the digits is atmost 20, then how many such 4 digit numbers are possible ?

• abcd
a+b=c+d --(1)
b+c= 3 (a+d) => 4(a+d) a+d a=1, d=2 => b+c= 9
to fulfill (1), 1+b= c+2 => b-c=1 => b=5, c=4.
=> 1542

or a=2, d=1 => b+c= 9 and 2+b=c+1 => c-b=1 => c=5, b=4
=> 2451

if a+d=4 =>a=1 d=3 or a=3, d=1
this gives b+c=12 and b-c=2 or c-b=2 => b,c= (7,5) or (5,7)
=> 1753 or 3571

if a+d=5 => a=1, d=4 or a=4, d=1
b+c=15 and b-c=3 or c-b=3 => (b,c)= (9,6) or (6,9)
=> 1964 or 4691
similarly 3782 and 2873 for a=2, d=3 and a=3, d=2

=> 8 cases

(bruce wayne)

• Q23) A dishonest ration guy cheats 20% while buying and 25% while selling. If he buys ration for 176 Rs/kg and manages to earn a profit of 10% even after making an offer of buy 2 get 3 free. Find the Marked price per unit kg.

61

48

61

61

61

61

61