Quant Boosters  Hemant Malhotra  Set 14

Q3) A set S of positive integers is called perfect if any two integers in S have no common divisors greater than 1. Candy wants to build a perfect set of numbers between 1 and 20 inclusive, in such a way that her set contains as many numbers as possible. How many elements will her set have?

Q4) If a = 3^x, b = 3^y , c = 3^z , and d = 3^w and if x, y, z, and w are positive integers, determine the smallest value of x + y + z + w such that a^2 + b^3 + c^5 = d^7
a) 17
b) 31
c) 106
d) 247
e) 353

If you know basic base rule think of base 3
If 3^m + 3^n + 3^p = 3^q, then
m = n = p is the only possibility
3^(2x)+3^(3y)+(3^5z)=3^7z
2x=3y=5z
3^(2x+1)=3^7z
so 2x=7z1
2x=3y=5z=7z1=k
so x=k/2 , y=k/3 , z=k/5 , z=(k+1)/7
k=90
so x+y+z+w=106

Q5) In a group of people, there are 13 who like apples, 9 who like blueberries, 15 who like cantaloupe,and 6 who like dates. (A person can like more than 1 kind of fruit.) Each person who likes blueberries also likes exactly one of apples and cantaloupe. Each person who likes cantaloupe also likes exactly one of blueberries and dates. Find the minimum possible number of people in the group.

Q6) f(x) = min(1  3x, 2x  1), find the maximum value of f(x)

Q7) What will be the minimum value of n for which 48!/12ⁿ is not divisible by 12 ?
a) 44
b) 43
c) 23
d) 22
e) 5

Q8) When a certain number 'x' is multiplied by 18 the product 'y' has all of its digits as 4. What is the minimum number of digits 'x' can have ?

OA=8 , 18 = 2 * 9
for a number consisting of 4s to be divisible by 9
the 444.... has to have 9 4s
so 444444444 / 18 = 24691358= 8 digits

Q9) Find the minimum value of the expression: √((x + 2)^2 + 16) + √((x  14)^2 + 64)

Q10) The sum of the squares of three numbers, x, y and z is 138, while the sum of their products taken two at a time is 131. Their sum is:
a) 30
b) 40
c) 50
d) None of the above

Q11) a1, a2, ...., a40 are 40 natural numbers such that 0 < a1 < a2 < ... < a40 < 100, then find the difference between the maximum and minimum possible sum of all the differences di, where di = a(i + 1)  a(i)

Q12) A function f(k) is defined on natural numbers as follows
f(k) = 2k if k is odd
= k/2 if k is even.
if f(f(f(k))) = 22 then find the sum of all possible values of k.

f(f(f(k))=22
f((f(k))=11 or 44 (( 11 odd so we will get 22 or 44 even so we will get 44/2=22
case= f(f(k)=11
then f(k)=22 (K if even then k/2 ,)
when f(k)=22
then k=44 or 11
When f(f(k)=44
then f(k)=88 or 22
when f(k)=22 then k=44 or 11
f(k)=88 then k=44 or 176
so k=11,44,176
so sum 11+44+176=231

Q13) If [ x ] + [ 2x ] + [ 3x ] = n, where [ x ] is the greatest integer less than or equal to x, then how many different values n can take if 1 ≤ n ≤ 2007 ?

Q14) Given that log 3 = 0.477, log 7 = 0.845, log 2 = 0.301. Find the number of digits in y if y = 252^10

Q15) For what values of k does the equation log(kx) = 2 log(x + 1) have only one real root?

Q16) A loan of Rs. 48,000 is borrowed at an interest rate of 10% per annum compounded annually and repaid in 3 equal yearly installments of Rs. X starting after one year. What is the approximate value of X?
a) Rs. 19300.50
b) Rs. 19300.75
c) Rs. 19301
d) Rs. 19301.25
e) Rs. 19301.50

Q17) What is the units digit of 1^1 + 2^2 + 3^3 + … + 2008^2008?

Q18) 10 students are going to perform 10 different items in school’s annual program. What is the number of ways in which these 10 items can be arranged, if Amar will perform before Bindu, Bindu will perform before Chetan, and Chetan will perform before David?

Q19) Four friends P, Q, R and S have Rs. 200 among them. They made the following statements about the money with them.
P: Q has Rs. 60 and if I give Rs. 10 to R, then Q and R will have equal amounts of money.
Q: I have more money than R and two people have the same amount of money
R: The person(s) with the lowest amount has (ve) Rs. 30 and S does not have the highest amount.
S: Q and I have the same amount of money and if R gives Rs. 10 to Q, he would have Rs. 5 less than Q.Only one of the statements is completely incorrect.
How much money does S have?a) Rs. 50
b) Rs. 60
c) Rs. 20
d) Rs. 30
e) Cannot be determined