Bank Math Master

Question: If a * b = 2a - 3b + ab, then 3 * 5 + 5 * 3 is equal to?

Solution:
3 ∗ 5 + 5 ∗ 3

3 ∗ 5 = 2 × 3 - 3 × 5 + 3 × 5
= 6 - 15 + 15
= 6

5 ∗ 3 = 2 × 5 - 3 × 3 + 3 × 5
= 10 - 9 +15
= 16

∴ 3 ∗ 5 + 5 ∗ 3
= 6 + 16
= 22

Question:
 
Solution:

Question: In a group of 60 people, 27 like coffee and 42 like tea and each person likes at least one of the two drinks. How many like both coffee and tea? 

Solution: 
Let,
A = Set of people who like coffee
B = Set of people who like tea

Given,
n(A ∪ B) = 60 
n(A) = 27
n(B) = 42

then;
n(A ∩ B) = n(A) + n(B) - n(A ∪ B) 
= 27 + 42 - 60 
= 69 - 60
= 9 

Therefore, 9 people like both tea and coffee. 

Question: What is the probability of getting a number greater than 6 on dice?

Solution:
Since number greater than 6 cannot appear on dice so probability of getting a number greater than 6 is zero.

Question: If the mode of the following data is 7, then the value of k in the data set 3, 8, 6, 7, 1, 6, 10, 6, 7, 2k + 5, 9, 7, and 13 is-

Solution:
Mode is the value that occurs most often in the data set of values.

Given data values are 3, 8, 6, 7, 1, 6, 10, 6, 7, 2k + 5, 9, 7, and 13
In the above data set, values 6, and 7 have occurred more times i.e., 3 times
But given that mode is 7.
So, 7 should occur more times than 6.
Hence the variable 2k + 5 must be 7
2k + 5 = 7
⇒ 2k = 2
∴ k = 1

Question: Which number will complete the series :
0, 1, 4, 15, ?, 325, 1956

Solution:
0
0 × 1 + 1 = 1
1 × 2 + 2 = 4
4 × 3 + 3 = 15
15 × 4 + 4 = 64
64 × 5 + 5 = 325
325 × 6 + 6 = 1956

Question: Solve |5x + 5| - 8 ≤ 17.

Solution:
|5x + 5| - 8 ≤ 17
⇒ |5x + 5| ≤ 25
⇒ - 25 < 5x + 5 ≤ 25
⇒ - 30 < 5x ≤ 20
⇒ - 6 ≤ x ≤ 4

Question: If A = {x : x is a natural number and a factor of 18} and B = {x : x is a natural number and less than 6}, then find A ∩ B. 

Solution: 
Here,
A = {1, 2, 3, 6, 9, 18} 
B = {1, 2, 3, 4, 5} 

Therefore,
A ∩ B = {1, 2, 3, 6, 9, 18} ∩ {1, 2, 3, 4, 5}
= {1, 2, 3}

Question: A number is selected from the first 20 natural numbers. Find the probability that it would be divisible by 3 or 7?

Solution:
Let X be the event that the number selected would be divisible by 3
and Y be the event that the selected number would be divisible by 7.
Then X ∪ Y denotes the event that the number would be divisible by 3 or 7.

Now,
X = {3, 6, 9, 12, 15, 18}
and Y = {7, 14}
∴ X ∪ Y = {3, 6, 7, 9, 12, 14, 15, 18}

∴ n(X ∪ Y) = 8

∴ The probability that it would be divisible by 3 or 7 = 8/20
= 2/5

Question: Find the mode and median of the given numbers 12, 7, 8, 14, 21, 23, 27, 7, 11.

Solution:
Mode = The value that appears most frequent = 7 which is repeated twice

Arrane the data in ascending order: 7, 7, 8, 11, 12, 14, 21, 23, 27
Where n = number of terms = 9
the middle one is (9 + 1)/2 = 5^th number
 
so, the 5^th number is 12

Question: If in a certain language, PLAYER Is coded as QNDCJX, then how SINGER will be coded in the same language?

Solution:

PLAYER
P + 1 = Q
L + 2 = N
A + 3 = D
Y + 4 = C
E + 5 = J
R + 6 = X
PLAYER = QNDCJX

SINGER
S + 1 = T
I + 2 = K
N + 3 = Q
G + 4 = K
E + 5 = J
R + 6 = X
SINGER = TKQKJX

Question: The sum of the first 16 terms of an AP whose first term and third term are 5 and 15 respectively is-

Solution:
1st term = 5;
3rd term =15
∴ 5 + d + d = 15
⇒ 2d = 10
∴ d = 5

16th term = a + 15d
= 5 + 15 × 5
= 80

The sum of the first 16 terms = (n/2)[2a + (n - 1)d]
= (16/2)[2 × 5 + (16 - 1)5]
= 8 × (10 + 75)
= 8 × 85
= 680

Question: If P = {multiples of 3 between 1 and 20} and Q = {even natural numbers up to 15}, then find P - Q.

Solution:
P = {multiples of 3 between 1 and 20}
∴ P = {3, 6, 9, 12, 15, 18}

Q = {even natural numbers upto 15}
∴ Q = {2, 4, 6, 8, 10, 12, 14}
  

Hence,
P - Q = {3, 6, 9, 12, 15, 18} - {2, 4, 6, 8, 10, 12, 14}
= {3, 9, 15, 18}

Question: In a bucket there are 5 purple, 15 grey and 25 green balls. If the ball is picked up randomly, find the probability that it is neither grey nor purple?

Solution:
If the ball is neither grey nor purple then it must be green.
There are 45 balls in total of which 25 are green 
so the probability of picking a green ball is 25/45 = 5/9

Question: Find the arithmetic mean of the set of data: 6, 1, 5, 8, and 10.

Solution:
In the given question, total number = 5

Arithmetic mean = (6 + 1 + 5 + 8 + 10)/5
= 30/5
= 6

Question: Six people Nitro, Nemesis, Nick Fury, Viper, Kronos and Hulk are attending a hexagonal table conference. All the sides of the hexagon table so formed are of same length. Nitro is not adjacent to Nemesis or Nick Fury, Viper is not adjacent to Nick Fury or Kronos. Nemesis and Nick Fury are adjacent, Hulk is in the middle of Viper and Nick Fury. Who is placed exactly opposite to Kronos?

Solution:
Hulk is in the middle of Viper and Nick Fury.
Nemesis and Nick Fury are adjacent.
Nitro is not adjacent to Nick Fury.

Thus, two arrangements are possible.

Hulk is exactly opposite to Kronos.