# Quant Boosters - Hemant Yadav - Set 1

• a² + b² + c² + d² = a(b + c + d)
=> a²/4 + (a²/4 + b² - ab) + (a²/4 + c² - ac) + (a²/4 + d² - ad) = 0
=> a²/4 + (a/2 - b)² + (a/2 - c)² + (a/2 - d)² = 0
=> a = b = c = d = 0 is the only solution

• Q15) There are 3 ants at 3 corners of a triangle, they randomly start moving towards another corner. What is the probability that they don't collide?

[OA : 1/4]

• Each ants can go towards two direction. so total different possibilities are 2 * 2 * 2 = 8. Out of which 2 are favorable where they all goes clockwise or anticlockwise . Hence required probability = 1/4

• Q16) A motor cyclist is travelling on a straight road at 70 kmph towards his house. His pet bird travels at 80 kmph from the motorcylist to the house and back in 1.6 hours. Find the distance between the house and the place from where the bird starts its journey towards the house?

[OA : 120 km]

• In 1.6 hours the bird flies 128 km and in the same time the motor cyclist goes 112 km and they travel a distance twice the the distance of the house. So distance is (128 + 112)/2 = 120 km

• Let x = a, y + z = b

we get (x + y + z)^2006 + (x - y - z)^2006 = k1 * a^2006 * b^0 + k2 * a^2004 * b^2 + .... + k1004 * b^2006

=> number of terms = sum of the number of terms in b^0, b^2, .... , b^2006
= 1 + 3 + 5 + ... + 2007
= 1004² = 1008016

• a) boxes and balls are all distinct and only 1 ball in one box
Its like permuting a 5 digit number, so 5! Ways

b) boxes are same and balls are distict and only 1 ball per box
Since all the boxes are same and one ball per box, so just one way

c) balls are same and boxes are different and only 1 ball per box
Here all the balls are same and every box has one ball, so again one way

d) balls and boxes are same and only 1 ball per box
Here also only only way

e) repeat all above questions with any number of balls in any box

i) 5^5 ways, as all the balls and all the boxes are different.

ii) Here boxes are same but balls are different.
(5, 0, 0, 0, 0) – 1 way
(4, 1, 0, 0, 0) – C(5, 4) = 5 ways
(3, 2, 0, 0, 0) – C(5, 3) = 10 ways
(3, 1, 1, 0, 0) – C(5, 3) = 10 ways, we don’t have to choose one of other two as they will automatically choose any of the two boxes (all the boxes are similar)
(2, 2, 1, 0, 0) – C(5, 2)*C(3, 2)/2 = 15, here we need to divide by 2 as as it doesn’t matter if I put the first pair in box 1 and second pair in box 2 or vice-versa
(2, 1, 1, 1, 0) – C(5, 2) = 10 ways
(1, 1, 1, 1, 1) – 1 way
So, total = 1 + 5 + 10 + 10 + 15 + 10 + 1 = 52 ways

iii) x1 + x2 + x3 + x4 + x5 = 5
So, C(9, 5) = 126 ways

iv) Since all balls and boxes are same, we just need to find what different combinations are possible:-
(5, 0, 0, 0, 0)
(4, 1, 0, 0, 0)
(3, 2, 0, 0, 0)
(3, 1, 1, 0, 0)
(2, 2, 1, 0, 0)
(2, 1, 1, 1, 0)
(1, 1, 1, 1, 1)
So, 7 ways

• Q17) Find the largest possible value of k for which 3^11 is expressible as the sum of k consecutive positive integers. What will be the answer if 3^11 is expressed as sum of k consecutive integers.

[OA : 486, 2 * 3^11]

• Let the first of the consecutive integers be a + 1, then the consecutive integers will be a + 1, a + 2, ..., a + k
So, (a + 1) + (a + 2) + .... + (a + k) = 3^11
k(k + 2a + 1) = 2 * 3^11

Now for the first part all the integers are positive, so both (k + 2a + 1) > k
Hence maximum value of k will be 2 * 3^5 = 486

For the second part there is no such constraint, so maximum possible value of k will be 2 * 3^11

• Q18) Find the sum of 6 * 1! + 13 * 2! + 22 * 3! + 33 * 4! + 46 * 5! + ... n terms

• It is (n +2)! + 2[(n + 1)!] - 4 = (n + 4)[(n + 1)!] - 4.
Key lies in simply finding the nth term of the sequence which is (n + 1)[(n + 1)!] + 2n[n!].

• Q19) How many natural numbers N < 40000 such that cube of sum of digits of N gives back the same number N.
For eg, 8³ = 512, sum of digits is 8, same as cube root of 512.

[OA : 6]

• Since N < 40000, N < 35³

It is clear from the given condition that the sum of digits and cube of sum of digits should leave same remainder when divided by 9.

Hence sum of digits should be of form 9k + 1 or 9k - 1 or 9k

1³ = 1 (sum of digits is 1, so possible)
8³ = 512 (sum of digits is 8, so possible)
9³ = 729 (sum of digits is 18, so not possible)
10³ = 1000 (not possible)
19³ = 6859 (sum of digit is not 19, so not possible)
18³ = 5832 (sum of digits is 18, so possible)
17³ = 4913 (sum of digit is 17, so possible)
26³ = 17576 (possible)
27³ = 19683 (possible)
28³ = 21952 (not possible)

So, 1, 512, 4913, 5932, 17576 and 19683 are the only such numbers

• Q20) How many lines in a three dimensional rectangular coordinate system pass through four distinct points of the form i, j, k where i, j and k are positive integers not exceeding four?
a) 60
b) 72
c) 64
d) 76

[OA : 76]

• We can see that there are 12 planes (4 paralle to x-y plane, 4 parallel to y-z plane and 4 parallel to z-x plane) In every plane there would be 2 diagonals, so 24 lines. 4 diagonals of the cube and 16 lines parallel to every axis. So, total 4 + 24 + 48 = 76

• Q21) Circles A, B and C are externally tangent to each other, and internally tangent to a bigger circle D. Circles B and C are congruent. Circle A has radius 1 and passes through the center of D. What is the radius of circle B?

[OA : 8/9]

• O is the centre of the bigger circle. and A,B,C are the centres of the remaining circles. Circles with centres B and C have the same radius. Radius of circle with centre A = 1.
Obviously, the radius of the bigger circle is 2.
Let OD = x (D is the mid-point of BC)
Then we have: r^2 + x^2 = (2-r)^2
and (x+1)^2 + r^2 = (1+r)^2
Taking the diff of the above equations gives x = 3r - 2
using which we get r = 8/9

• Q22) For any positive integer n , f(n) is the highest power of 2 that divides n!
Find f(1) + f(2) + f(3) + ..... + f(1023)

[OA : 518656]

• It is similar to finding the highest power of 2 in 1! * 2! * 3! * ...... * 1023!

First we will count that how many numbers are divisible by 2 in every factorial, then by 4, then by 8, ...., and lastly by 512

So, S = (0 + 1 + 1 + 2 + 2 + ..... + 511 + 511) + (0 + 0 + 0 + 1 + 1 + 1 + 1 + .... + 255) + ..... + (0 + 0 + .... + 1 + 1 + 1 (512 times))
= 511 * 512 + 512 * 255 + 512 * 127 + 512 * 63 + 512 * 31 + 512 * 15 + 512 * 7 + 512 * 3 + 512
= 512(511 + 255 + 127 + 63 + 31 + 15 + 7 + 3 + 1) = 512 * 1013 = 518656

=> f(1) + f(2) + f(3) + ..... + f(1023) = 518656

• Q23) The sides of a triangle form an arithmetic progression. The altitudes also form an arithmetic progression. Then what can be said about the triangle
a) It's a rt angle triangle
b) It's an equilateral triangle
c) It's an obtuse angle triangle
d) Nothing can be said about it

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