Quant Boosters by VP  Set 1

Odd is of the form (2k+1)
(2a+1) + (2b+1) ... + (2e+1) = 35
2a + 2b +....+ 2e = 30
a + b + ... + e = 15
Whole number Solution = 19c4

Q2) If N = 2^3 * 3^2 then how many sets of 2 distinct factors of N are coprime to each other

N => 2^3 * 3^2
No. Of co primes => [(2 * 3 + 1) ( 2 * 2 + 1)  1]/2
(7 * 5 1)/2
=> 17If N is of the form a^x * b^y, Number of coprimes = [(2x +1) (2y+1)  1]/2

Q3) 116 people participated in a singles tennis tournament of knock out format. The players are paired up in the first round, the winners of the first round are paired up in second round, and so on till the final is played between two players. If after any round, there is odd number of players, one player is given a bye, i.e. he skips that round and plays the next round with the winners. Find the total number of matches played in the tournament.

116 ppl.
1 winner 115 loser.
In 1 match 1 loser is decided. So 115 matches to decide 115 losers and 1 winner.

Q4) Find the one hundredth smallest positive integer that can be written using the digits 1,3 and 5 in base 7?

1,3,5
1 digit : 3
2 digit: _ _ =>9
3 digit : _ _ _ =>27
Total 39 so far
1 _ _ _ => 27
3 _ _ _ => 27
Total 93 so far
5 1 1 _ => 3
5 1 3 _ => 3
So 99 done
100th will be 5 1 5 1 Ans

Q5) If ‘x’ and ‘y’ are real numbers such that 3x + 4y = 48. Then, what is the maximum possible value of xy?

x + x + x + 4y/3 + 4y/3 + 4y/3 /6 > = (x^3y^3 * 64/27)^1/6
8^6 * 27/64 >= (xy)^3
2^12 * 3^3 >= (xy)^3
xy = 2^4 * 3 = 48

Q6) N! is ending with m zeroes.(N+2)! is ending with (m+2) zeroes. Also 90 < = N < = 190.Find possible values of N?

Increment of 2 zeroes as result of multiple of 25
Means if you check for 24! =>
24/5 => 4
25/5 => 5
5/5 => 1
So 5 +1 => 6
Same for 26!So
98! And 100!
99! And 101!
Similarly
148! > 150!
149! > 151!
And
173! > 175!
174! > 176!Remember you don't have to consider 125! (As it results into increment of 3 zeroes )
So 6 possibilities

Q7) Which of the following statements is false?
a) (23!)^2 > 23^23
b) (20!∗19!∗18!) < 57!
c) (33!)^4 < 33^60
d) None of these

B is clearly right since 20 + 19 + 18 = 57
A is right coz
Lhs=> 23!^2 = (23 * 1)(22 * 2)(21 * 3)...(2 * 22)(3 * 23)However in rhs each terms is =>23 * 23 * 23...
Certainly lhs > rhsNow in 3rd
33!^2 > 33^33
So 33!^2 * 33!^2 > 33^33 * 33^33
33!^4 > 33^66 hence 3rd is falseOA:C

Q8) In a plane there are 37 straight lines of which 13 passes through Point A and 11 passes through Point B. Besides, no line passes through 3 points and no line passes through both A and B and no two are parallel. Find the total number of point of intersections of the straight lines

Total number of points of intersection of 37 lines is 37c2
But 13 straight lines out of the given 37 straight lines pass through the same point A.
Therefore instead of getting (13c2) points, we get only one point A.
Similarly 11 straight lines out of the given 37 straight lines intersect at point B.
Therefore instead of getting (11c2) points, we get only one point B.
Hence the number of intersection points of the lines is = 37c2  13c2 11c2 + 2 = 535

Q9) Find the number of ways of selection 4 letters from the word EXAMINATION

EXAMINATION has 11 letters in total
And out of which
E > 1
X > 1
M >1
T > 1
O > 1
(A) > 2
I > 2
N > 28 distinct letters.
 4 letters selected, which are all distinct: 8C4 = 70
 2 letters alike, and 2 distinct
(eg: AAEM)
= 3c1 x 7c2 = 63  2 letters alike, and 2 letters alike
(eg: NNII)
= 3c2 = 3
So answer is, 70 + 63 + 3 = 136

Q10) Find the 1000th term of the sequence : 1,3,4,7,8,9,10,11,13,14,... in which there is no number which contain digit 2,5 or 6.

Method:1 (Not recommended)
_ : 6 numbers ; _ _ : 6 * 7 = 42 numbers ; _ _ _ : 6 * 7 * 7 = 294 numbers
342 numbers added till 999
1 _ _ _ : 7 * 7 * 7 = 343 added = 685 numbers till now. Remaining 1000  685 = 315
3 0 _ _ : 49 added
3 1 _ _ : 49 added
3 3 _ _ : 49 added
3 4 _ _ , 3 7 _ _ , 3 8 _ _ = 49*3 = 147
Total 294 more. Remaining 21
3 9 0 _ : 7
3 9 1 _ : 7
3 9 3 _ : 7
3939Method:2
Base method approach
0 > 0
1 > 1
2 > 3
3 > 4
4 > 7
5 > 8
6 > 9
1000th term : 1000 in base 7 = 2626 <  > 3939

Q11) One day Vikram was out bicycling. After entering a oneway tunnel and after having ridden onefourth of the distance through it, he looked back over his shoulder and saw a bus approaching the tunnel entrance at a speed of 80 miles/hr. Doing a quick mental exercise, Vikram realized that if he accelerated immediately to his top speed, he could just escape with his life, whichever direction he rode. What is Vikram's top biking speed in miles/hr?