Quant Boosters - Hemant Malhotra - Set 14

[hemant_malhotra]

Number of Questions - 30
Solved ? -
Topic - Quant Mixed Bag
Source - Elite's Grid CAT Preparation Forum

[hemant_malhotra]

Q1) In a test taken by 100 students, 60 cleared cut-off in DI, 44 cleared cut-off in mathematics, 38 cleared cut-off in English and 27 students cleared cut-off in GK. 20 students cleared cut-off in all 4 sections. How many maximum students could have failed to clear the cut-off in all four sections?

[hemant_malhotra]

Q2) What is the remainder when 2424...up to 300 times divided by 999 ?

[hemant_malhotra]

10^3-1 so pick last three digits & apply the concept
Answer is 333

[hemant_malhotra]

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?

[hemant_malhotra]

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

[hemant_malhotra]

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=7z-1
2x=3y=5z=7z-1=k
so x=k/2 , y=k/3 , z=k/5 , z=(k+1)/7
k=90
so x+y+z+w=106

[hemant_malhotra]

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.

[hemant_malhotra]

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

[hemant_malhotra]

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

[hemant_malhotra]

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 ?

[hemant_malhotra]

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

[hemant_malhotra]

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

[hemant_malhotra]

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

[hemant_malhotra]

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)

[hemant_malhotra]

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.

[hemant_malhotra]

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

[hemant_malhotra]

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 ?

[hemant_malhotra]

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

[hemant_malhotra]

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