Quant Boosters  Gaurav Sharma  Set 4

gaurav_sharma last edited by
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
Number of Questions  30
Solved ?  Not yet
Topic  Quant Mixed Bag
Source  Genius Tutorial Preparation Forum

gaurav_sharma last edited by
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
Q1) Compute 1/10^0 + 1/10^1 + 2/10^2 + 3/10^3 + 5/10^4 + 8/10^5 + 13/10^6 + …
a) 100/89
b) 100/81
c) 100/99
d) None of these

gaurav_sharma last edited by
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
S = 1/10^0 + 1/10^1 + 2/10^2 + 3/10^3 + 5/10^4 + 8/10^5 + 13/10^6 + …
S/10 = 1/10^1 + 1/10^2 + 2/10^3 + 3/10^4 + 5/10^5 + 8/10^6 + 13/10^7 + …
S – S/10 = 1/10^0 + 0 + 1/10^2 + 1/10^3 + 2/10^4 + 3/10^5 + 5/10^6
S – S/10 = 1/10^0 + 1/10^2 (1/10^0 + 1/10^1 + 2/10^2 + 3/10^3 + 5/10^6)
S – S/10 = 1 + S/10^2
S (1 – 1/10 – 1/10^2) = 1
S = 100/89

gaurav_sharma last edited by gaurav_sharma
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
Q2) In the given figure what is the radius of the inscribed circle
a) 3/2
b) 5/2
c) 7/5
d) None of these

gaurav_sharma last edited by gaurav_sharma
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
Ar(ACD) + Ar(CDB ) = Ar(ABC)
5r/2 + 3r/2 = 12/2
8r = 12
r = 3/2

gaurav_sharma last edited by
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
Q3) How many numbers from 1 – 2000 are such that at least two of their digits are same?
a) 800
b) 757
c) 758
d) 750

gaurav_sharma last edited by
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
Single digit number with no digit repeated = 9
Two digit numbers with no digit repeated = 9C1 x 9C1 = 81
Three digit numbers with no digit repeated = 9C1 x 9C1 x 8C1 = 9 x 9 x 8 = 648
Four digit numbers less than 2000, with no digit repeated = 9C1 x 8C1 x 7C1 x 1 = 9 x 8 x 7 = 504
Total numbers with no digit repeated = 9 + 81 + 684 + 504 = 1242
Numbers with at least two digits same = 2000 – Numbers with no repetition
2000 – 1242 = 758

gaurav_sharma last edited by
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
Q4) A triangle has sides 6, 7 and 8. The line through it’s incenter parallel to the shortest side is drawn to meet other two sides at P and Q. Then find the length of the line segment PQ
a) 4
b) 30/7
c) 15/7
d) Cannot be determined

gaurav_sharma last edited by
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
Area of triangle = r x s
21r/2 = 6h/2 = 3h
r/h = 2/7
APQ and ABC are similar thus,
( h – r)/h = PQ/6
Or 1 – r/h = PQ/6 = > 1 – 2/7 = PQ/6
PQ = 30/7

gaurav_sharma last edited by
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
Q5) N = 99^3 – 36^3 – 63^3. How many factors does N have?
a) 48
b) 84
c) 96
d) 134

gaurav_sharma last edited by
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
N = 99^3 – 36^3 – 63^3
N = 99^3 + (– 36) ^3 + (– 63) ^3
N = 3 x 99 x 36 x 63 [if a + b + c = 0, then a^3 + b^3 + c^3 = 3abc]
N = 2^2 x 3^7 x 7 x 11
Number of factors of N = (2 + 1) (7 + 1) (1 + 1) (1 + 1) = 96

gaurav_sharma last edited by
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
Q6) A person is travelling on the given grid and has to go from A to B through C – D. in each move he can take a step in either the north direction of the east direction. Following the given instructions, in how many different ways can he travel from A to B

gaurav_sharma last edited by
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
Number of ways to travel from A to C = 6! / (3! X 3!) = 20
Number of ways to travel from C to D = 1
Number of ways to travel from D to B = 6!/(4! X 2!) = 15
Total ways = 20 x 15 = 300

gaurav_sharma last edited by
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
Q7) AC is the diameter of a circle with center O. OE and OF are perpendicular to AD and AB respectively such that A – F – B and A – E – D. If perimeter of ABCD is x, then what will be the perimeter of AEOF?
a) x/2
b) 2x/3
c) x/3
d) root(2)x/3

gaurav_sharma last edited by
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
Triangle AEO ~ Triangle ADC ( AA Similarity)
EO/DC = AE/AD = AO/AC = 1/2
Similarly, Triangle AFO ~ Triangle ABC (by AA Similarity)
FO/BC = AF/AB = AO/AC = 1/2
AD + DC + CB + AB = x
1/2 (AD + DC + CB + AB ) = x/2
AE + OE + OF + AF = x/2

gaurav_sharma last edited by
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
Q8) How many positive divisors does 7^12 + 7^13 + 7^14 + 7^15 have
a) 200
b) 199
c) 195
d) 197

gaurav_sharma last edited by
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
7^12 + 7^13 + 7^14 + 7^15 = 7^12 (1 + 7 + 49 + 343)
= 7^12 x 400
= 7^12 x 2^4 x 5^2Number of factors:
(12 + 1) (4 + 1) (2 + 1) = 195

gaurav_sharma last edited by
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
Q9) There are 10 numbers and none of them are divisible by 3. What will be the remainder when sum of their squares is divided by 3?
a) 0
b) 1
c) 2
d) 1 or 2

gaurav_sharma last edited by
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
The Square of a natural number other than one is either a multiple of 3 or exceeds a multiple of 3 by 1. In other words, a perfect square leaves remainder 0 or 1 on division by 3.
Here it is said that none of them is divisible by 3. Hence each square term will give a reminder of 1
So for 10 terms, 1 + 1 + 1 … + 1 (10 times) = 10
10 mod 3 = 1

gaurav_sharma last edited by
Director, Genius Tutorials, Karnal ( Haryana ) & Delhi  MSc (Mathematics)
Q10) Find the value of [root (1)] x [root (2)] x [root (3)] x … x [root (100)]
a) 2^87 x 3^59 x 5^13 x 7^14
b) 2^88 x 3^58 x 5^12 x 7^15
c) 2^81 x 3^59 x 5^12 x 7^17
d) 2^79 x 3^61 x 5^21 x 7^12