Quant Boosters  Gaurav Sharma  Set 1

gaurav_sharma last edited by gaurav_sharma
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
Number of Questions  30
Solved ?  Yes
Topic  Quant Mixed Bag
Source  Genius Tutorial Preparation Forum

gaurav_sharma last edited by
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
Q1) How many natural numbers between 1 and 700 are not divisible by 7 and leave a remainder of 3 when divided by 4 ?
a) 125
b) 150
c) 100
d) 126

gaurav_sharma last edited by
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
Numbers of the form 4k + 3 between 1 to 700 are:
3,7,11,15,19,23,27 … 699 ( 175 terms)
Number of the form 7p are 7, 35, … 679 ( 25 terms)
Hence the total number of terms of the form 4k + 3 and not divisible by 7 are 175 – 25 = 150

gaurav_sharma last edited by
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
Q2) How many points with the integer coordinates lie inside the triangle whose vertices are (0,3), (5,2), (5,2)?
a) 14
b) 15
c) 16
d) 17

gaurav_sharma last edited by
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
We can draw the triangle on coordinates axes as shown
Now from the graph it is clear that the number of points with integer coordinates inside the triangle on line:
Y = 2 is 1
Y = 1 is 3
Y = 0 is 5
Y = 1 is 7
Hence the number of points is 16

gaurav_sharma last edited by
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
Q3) If a, b and c are nonzero real numbers such that a^2/(b * c) + b^2/(a * c) + c^2/(b * a) = 3, which of the following is necessarily true?
a) a + b + c = 0
b) a^2 + b^2 + c^2 = ab + bc + ca
c) a = b = c = 1
d) Either (a) or (b)

gaurav_sharma last edited by
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
Given a^2/(b * c) + b^2/(a * c) + c^2/(b * a) = 3
a^3/abc + b^3/abc + c^3/abc = 3
a^3 + b^3 + c^3 – 3abc = (a + b + c) (a^2 + b^2 + c^2 – ab – ac – bc)
Hence either a + b + c = 0 or a^2 + b^2 + c^2 = ab + bc + ca
Hence option (d) is correct

gaurav_sharma last edited by
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
Q4) Find the right most nonzero digit in 17!

gaurav_sharma last edited by
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
Method 1
We have 17! = 2^15 x 3^6 x 5^3 x 7^2 x 11 x 13 x 17
17! = (2^3 x 5^3)(2^12 x 3^6 x 7^2 X 11 x 13 x 17)
Since (2^3 x 5^3) ends in zero the right most nonzero digit in 17! Will be the unit digit of (2^12 x 3^6 x 7^2 X 11 x 13 x 17) i.e, 6Method 2
If number is of the form 5a + b then it’s rightmost nonzero digit will be 2^a x a! x b!
Here 17 = (5 x 3) + 2
A = 3 and b = 2
So, rightmost nonzero digit is 2^3 x 3! X 2! = 6

gaurav_sharma last edited by
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
Q5) How many natural numbers less than 30 have a composite number of factors?
a) 13
b) 11
c) 14
d) 10

gaurav_sharma last edited by
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
Number of natural numbers less than 30 = 29
Number with only 1 factor = 1
Numbers with 2 factors = all primes below 30 = 10
Numbers with 3 factors = Numbers of the form a^2 where a is prime, so 2^2 = 4, 3^2 = 9, 5^2 = 25 = 3 numbers
Numbers with 5 factors = numbers of the form a^4 where a is prime. Only 2^4 is in this range
So, numbers less than 30 have a composite number of factors are = 29 – 1 – 10 – 3 – 1 = 14

gaurav_sharma last edited by
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
Q6) Aman and eight of his friends took a test of 100 marks. Each of them got a different integer score and the average of their scores was 86. The score of Aman was 91 and it was more than that of exactly three of his friends. What could have been the maximum possible absolute difference between the scores of two of his friends?
a) 83
b) 73
c) 86
d) 44

gaurav_sharma last edited by
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
Aman’s score is 91. In order to maximize the absolute different thake the score of the friend who scored highest as 100 marks. Now, 5 of his friends scored marks greater than 91, one of those is the one who scored 100 marks. Let the marks of other 4 friends be 99, 98, 97 and 96 (Maximum values)
And 3 of his friends scored less than 91 marks
Assume two of them scored 90 and 89 marks (maximum value)
Their average marks are 86
Now the lowest score will be 86 x 9 – ( 100 + 99 + 98 + 97 + 96 + 91 + 80 + 89) = 14
Hence absolute difference = 100 – 14 = 86

gaurav_sharma last edited by
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
Q7) 4x + 2y + z = 28 where x, y and z are real numbers. Find the maximum value of x^1/2 * y^1/4 * z ^1/8
a) 5^3/4
b) 4^7/8
c) 5^3/4
d) 4^8/7

gaurav_sharma last edited by gaurav_sharma
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
Using AM ≥ GM
(x + x + x +x + y + y + z) / 7 ≥ (x^4 * y^2 * z)^1/7
(4x + 2y + z) / 7 ≥ (x^4 * y^2 * z)^1/7
28/7 ≥ (x^4 * y^2 * z)^1/7
4^7 ≥ (x^4 * y^2 * z)
4^(7/8 ) ≥ (x^4 * y^2 * z)^1/8
Maximum value of (x^4 * y^2 * z)^1/8 is 4^(7/8 )

gaurav_sharma last edited by
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
Q8) F(a,b) = HCF(a,b)/LCM(a,b)
F(1/F(a,b), C) = 1/12
If a, b and c are distinct positive integers such that any tow of them are coprime, then what is the sum of a, b and c ?
a) 6
b) 7
c) 8
d) 9

gaurav_sharma last edited by
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
a, b and c are distinct positive integers such that any two of them are coprime
HCF(a,b) = 1 and LCM(a,b) = ab
F(a,b) = 1/ab
Now F(1/F(a,b),c) = F(ab,c) = 1/abc = 1/12
abc = 12 = 1 x 3 x 4
Hence sum = 1 + 3 + 4 = 8

gaurav_sharma last edited by
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
Q9) The sum of the roots of the quadratic equation ax^2 + bx + c = 0 is equal to the sum of squares of their reciprocals. If a, b and c are real numbers and a # 0 then bc^2, ca^2 and ab^2 are in
a) G.P
b) A.P
c) H.P
d) None of these

gaurav_sharma last edited by
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
Let the roots of the equation be p and q
Then, p + q = b/a and pq = c/a
Given p + q = 1/p^2 + 1/q^2 = (p^2 + q^2)/p^2q^2
P + q = [(p + q)^2 – 2pq]/(pq)^2
b/a = [ (b/a)^2 – 2c/a] / (c/a)^2
2ca^2 = ab^2 + bc^2
bc^2, ca^2 and ab^2 are in A.P

gaurav_sharma last edited by
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
Q10) Two horses are tethered at the mid points of two adjacent sides of a square field. Each of them is tied with a rope that does not allow it to go beyond the centre of the field for grazing. If the length of a side of the field is 8m what is the ratio of areas grazed by both of them together to that of the area grazed by them individually?