# Question Bank - Modern Math - Hemant Malhotra

• a + b + c + d = 100 - - - - - (i)
a + 2b + 3c + 4d = 245 - - - -(ii)
solve above equations we get
b + 2c + 3d = 145
Max value of d can be 48
Now if d = 48 then b will be 1
Now Put this in second equation
a + 2 + 4(48) = 245
a + 194 = 245
a = 51
So d will be 48

• Q32) A word is a combination of 8 letters, each either A or B. Let x and y be 2 words differing in exactly 3 places. How many words differ from each of x and y in atleast 5 places?
(a) 38
(b) 48
(c) 58
(d) 68
(e) 78

• Q33) His name is HUNNYBUNNY. If the position of two letters of his name are interchanged with each other, it is called an op. What is the probability that his name remains same after an op, if no two consecutive letters are interchanged ?

• Q34) Find rank of the word MOTHER

• Q35) 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.

• All those who likes blueberries also likes cantaloupe
None of them likes apples
To minimize no of persons, we need to assume that all those who likes dates also likes apples
There are atleast 15 + (13 - 6) = 22 persons

• Q36) Gold is 19 times as heavy as water and copper is 9 times as heavy as water in what ratio should these be mixed to get an alloy 15 times as heavy as water.

• Q37) A committee of 5 is to be chosen from a group of 9 people. How many ways it can be done if Vikram and kaizen must serve together or none at all, and Sunit and Pratyush refuse to serve with each other?

• Q38) How many four digit numbers are divisible by 3, such that digit 4 appears only once in the number

• case1=4abc
a+b+c should be 3k+2 form
means a+b+c mod 3=2
so (0,0,2) or (0,1,1) or (1,2,2) this could be remainder when a ,b,c divided by 3
now (0,0,2) means a mod 3=0 b mod 3=0 and c mod 3=2
so a could be 0,3,6,9 so 4 possibilities same for b and for c =2,5,8 so 3 possibilities
and check for others
so 3 * (4 * 4 * 3 + 4 * 2 * 2 + 2 * 3 * 3)=246

case2= a4bc
again a+b+c mod 3=2
so total in this case =218
and for ab4c and abc4 218 cases

so total 246 + 3 * 218= 246+654=900 numbers

• Q39) Find the 625th term of a series where each term is made of even digits only and term is starting from 2 .

• Q40) How many 3-digit number from 100-999 inclusive have one digit which is the average of the other two?

• All numbers in the form of 111, 222, 333 ... 999 satisfies. So 9 possibilities

if digits a,b and c constitutes the numbers, then we should be able to write a = (b+c)/2
for a to be a proper digit, (b+c) should be divisible by 2.
Means either a & b are both even or a & b are both odd

if a and b are both odd, a can take 5 values and b can take 4 values making a total of 20 possibilities and with 6 permutations we have a total of 120 cases.
after accounting for double counting (like, 567 and 765) we have 120/2 = 60 cases

if a and b are both even (without zero), a can take 4 values and b can take 3 values making a total 12 possibilities. Again 6 permutations so we have 72 cases.
After accounting for double counting, we have 72/2 = 36 cases

Now we will consider cases having 0.
We can have even number as a digit (4 cases), and the third digit is fixed as 1/2 the even digit. (eg: 204, 603)
So 4 cases and 4 permutation = 16 cases.

Total = 9 + 60 + 36 + 16 = 121

• case1- when vikram and kaizen are there so we have to select 3 out of 7
there will be 3 cases in this
a) when sunit is there and pratyush is not there
so we have to select 2 out of 5 so 5c2 =10
b) when pratyush is included but sunit is nt
same 5c2=10
c) when both are not included then 5c3 =10
10 + 10 + 10 = 30 cases

case2- when vikram and kaizen are not included so we have to select 5 out o 7
there will again 3 cases
a) when sunit is there and pratyush is not there
so 5c4 =5
b) when pratyush is included but sunit is not
so 5c4=5
c) when both are not included so 5c5=1
5 + 5 + 1 = 11 cases

Total = 30 + 11 = 41

• I + 2II + 3III = 80 + 54 + 66 = 200
let number of students using all = x
so II=12 + 6 + 22 - x = 40 - x
III = x
I + II + III = 10x - 6
I + 40 - x + x = 10x - 16
so I = 10x - 56
now 10x - 56 + 2(40 - x) + 3x = 200
11x + 24 = 200
so 11x = 176
so x = 16
so 10x = 160

• Q41) Akshay kumar worked in his next movie "SPECIAL 12" ,with a team of six pairs of twins. He wishes to set up teams for robbery in CP , but he wants to avoid putting any pair of twins into the same team.
i) In how many ways can he split them into two teams of six?
ii) In how many ways can he split them into three teams of four?

• Q42) A shop sells two kinds of rolls - egg roll and mutton roll. Onion, tomato, carrot, chilli sauce and tomato sauce are the additional ingredients. You can have any combination of additional ingredients, or have standard rolls without any additional ingredients subject to the following constraints:

a) You can have tomato sauce if you have an egg, but not if you have a mutton roll.
b) If you have onion or tomato or both you can have chilli sauce but bot otherwise.

In how many ways can a roll be made ?

58

89

33

106

42

136

207

212