Bank Math Master

Question: In a simultaneous throw of a pair of dice, find the probability of getting a total more than 9.

Solution:
When two fair six-sided dice are thrown together, the total number of possible outcomes = 6 × 6 = 36.
We need the cases where the sum is more than 9, i.e., 10, 11, or 12.
Here are all favorable outcomes, (4, 6), (5, 5), (6, 4), (5, 6), (6, 5), (6, 6) = 6

∴ Probability = (number of favorable outcomes)/(total possible outcomes)
= 6/36
= 1/6

So the probability of getting a total more than 9 is 1/6.

Question: Look at this series: 53, 53, 40, 40, 27, 27, ... What number should come next?

Solution:
Given that, the series is: 53, 53, 40, 40, 27, 27, ...

Pattern - Each number is repeated twice, and the values themselves decrease by 13 each time.
1^st = 53
2^nd = 53 - 13 = 40 ⇒ 40
3^rd = 40 - 13 = 27 ⇒ 27
4^th = 27 - 13 = 14 ⇒ 14

So the next two numbers should be 14, 14.

So the number that should come next is 14.

Question: If a set has 5 elements, then the power set of that set has ______ elements.

Solution:
The number of elements in the power set of a set with n elements is 2ⁿ.
Here, n = 5
∴ Number of elements in the power set = 2⁵ = 32

So the power set has 32 elements.

Question: Solve the inequality, (x/4) < (4x - 1)/15

Solution: 
Given that, 
x/4 < (4x - 1)/15
⇒ 15x < 16x - 4
⇒ 15x - 16x < - 4
⇒ - x < - 4
⇒ x > 4 

Question: Three unbiased coins are tossed. What is the probability of getting at least 2 tails?

Solution:
When three fair (unbiased) coins are tossed, the total number of possible outcomes = 2 × 2 × 2 = 8.
{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT} 

And all possible outcomes are, {HTT, THT, TTH, TTT} = 4

∴ Probability = (number of favorable outcomes)/(total possible outcomes)
= 4/8
= 1/2

So the probability of getting at least 2 tails is 1/2.

Question: What will come at the place of question mark ?
6, 13, 28, 59, ?, 249.

Solution: 
First term ⇒ 6
Second term ⇒ (6 × 2 + 1) = 13
Third term ⇒ (13 × 2 + 2) = 28
Fourth term ⇒ (28 × 2 + 3) = 59
Fifth term ⇒ (59 × 2 + 4) = 122
Sixth term ⇒ (122 × 2 + 5) = 249

So, the required term = 122.

Question: If Set A = {1, 2, 3} and Set B = {1}, which of the following is true?

Solution:
A = {1, 2, 3}, B = {1}
∴ A - B = {1, 2, 3} - {1}
= {2, 3} ; true

খ)
A ∩ B ; common elements
A ∩ B = {1}
not equal to {1, 2, 3} ;  false

গ) 
A × B ; set of all ordered pairs (a, b) where a ∈ A and b ∈ B
A × B = {(1, 1), (2, 1), (3, 1)}
This is not equal to {1, 2, 3} ; false

ঘ) 
A ∪ B ; all elements from A or B
A ∪ B = {1, 2, 3}
not equal to {1} ; false

Final answer ক) A - B = {2, 3}

Question: If {1/|2q - 7|} > 1/5, then what is the value of q?

Solution:
Given that, 
{1/|2q - 7|} > 1/5
⇒ |2q - 7| < 5
⇒ - 5 < 2q - 7 < 5
⇒ - 5 + 7 < 2q - 7 + 7 < 5 + 7
⇒ 2 < 2q < 12
∴ 1 < q < 6

Question: There are 5 red, 4 white, and 3 blue balls in a box. If 3 balls are drawn at random from the box, what is the probability that all three are red?

Solution:
Total number of balls = 5 + 4 + 3 = 12 balls
We are drawing 3 balls at random without replacement.
Total number of ways to choose 3 balls from 12
= ¹²C₃ 
= 12!/(3! × 9!)
= (12 × 11 × 10)/(3 × 2 × 1)
= 220

And
Number of favorable ways (all 3 balls are red)
= Number of ways to choose 3 red balls from 5 red balls
= ⁵C₃
= 5!/(3! × 2!)
= (5 × 4 × 3)/(3 × 2 × 1)
= 10

∴ Probability = (favorable outcomes)/(total outcomes)
= 10/220
= 1/22
So the probability that all three balls are red is 1/22.

Alternatively, using sequential probability (without replacement):
P(1st Red) = 5/12
P(2nd Red) = 4/11 (4 red left out of 11 total)
P(3rd Red) = 3/10 (3 red left out of 10 total)

∴ Total Probability = (5/12) × (4/11) × (3/10) 
= 60/1320
= 1/22. 

Question: Find the next number in the series: 8, 27, 64, 125, 216, ?

Solution:
The series represents the cubes of consecutive natural numbers:
2³ = 8
3³ = 27
4³ = 64
5³ = 125
6³ = 216
7³ = 343

So, the next number will be 343.

Question: If B = {x : x ∈ N such that x² + 11x + 30 = 0} then B is ?

Solution:
Given that,
B = {x : x ∈ N such that x² + 11x + 30 = 0}
First let's solve the quadratic equation x² + 11x + 30 = 0
⇒ x² + 5x + 6x + 30 = 0
⇒ x(x + 5) + 6(x + 5) = 0
⇒ (x + 6)(x + 5) = 0
⇒ x = - 5 or - 6
Both solutions are negative numbers (- 5 and - 6). Natural numbers (N) are positive integers:

According to the definition of the given set x is a natural number but we know that neither x = - 5 nor x = - 6 is a natural number

So, the given set is an empty set i.e B = Ø

Question: What is the absolute value (modulus) of x if - 8 < x < 2?

Solution:
Given that, 
- 8 < x < 2
⇒ - 8 + 3 < x + 3 < 2 + 3
⇒ - 5 < x + 3 < 5
∴ |x + 3| < 5

Question: Tickets are numbered from 1 to 30. One ticket is drawn at random. What is the probability that the number is a multiple of 4 or 6?

Solution:
Total tickets = 30

Now, 
Multiples of 4 is-  4, 8, 12, 16, 20, 24, 28 = 7
multiples of 6 is- 6, 12, 18, 24, 30 = 5
And multiples of both (12, 24) = 2

∴ Favorable outcomes = 7 + 5 - 2 = 10

∴ Probability = favorable outcomes/total outcomes 
= 10/30
= 1/3

So the probability is 1/3.

Question: Find the next number in the series: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ?

Solution:
এই সিরিজটি হলো ফিবোনাচ্চি সিরিজ (Fibonacci sequence)। 
প্রতিটি সংখ্যা আগের দুটি সংখ্যার যোগফল।
0 + 1 = 1, 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8
5 + 8 = 13, 8 + 13 = 21, 13 + 21 = 34, 21 + 34 = 55

সুতরাং পরের সংখ্যা হবে 55।

∴ সিরিজটি হলো- 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...

Question: In a survey of 1,000 consumers it is found that 720 consumers liked product A and 450 liked product B. What is the least number that must have liked both the products?

Solution:
Given that,
Total consumers = 1000
Consumers who like product A = 720
Consumers who like product B = 450

We know, 
n(A U B) = n(A) + n(B) - n(A ∩ B)
⇒ 1000 = 720 + 450 - n(A ∩ B)  
⇒ 1000 = 1170 - n(A ∩ B)
⇒ n(A ∩ B) = 1170 - 1000
∴ n(A ∩ B) = 170

So 170 consumers like both the products A and B.

Question: If |2x - 2| ≤ 8, what is the maximum value of x?

Solution:
Given that,
|2x - 2| ≤ 8
⇒ - 8 ≤ 2x - 2 ≤ 8 
⇒ - 8 + 2 ≤ 2x - 2 + 2 ≤ 8 + 2 
⇒ - 6 ≤ 2x ≤ 10
⇒ (- 6/2) ≤ (2x/2) ≤ (10/2)
⇒ - 3 ≤ x ≤ 5

∴ The maximum value of x is 5

Question: One card is drawn at random from a pack of 52 cards. What is the probability that the card drawn is a face card (Jack, Queen and King) only?

Solution:
A standard deck has 52 cards.
Face cards are Jack, Queen, King
There are 4 suits (hearts, diamonds, clubs, spades)
So number of face cards = 3 × 4 = 12

Total possible outcomes (when drawing one card) = 52
Favorable outcomes (drawing a face card) = 12

∴ Probability = favorable outcomes/total outcomes
= 12/52
= 3/13

So the probability that the card drawn is a face card (Jack, Queen, or King) is 3/13. 

Question: Arrange the following numbers in ascending order: 105, 78, 92, 64, 119, 50, 83 

Solution: 
Given the series, 
105, 78, 92, 64, 119, 50, 83

Now, 
Ascending order(ছোট থেকে বড়)
50, 64, 78, 83, 92, 105, 119

Question: A committee of 3 members is to be selected out of 3 men and 2 women. What is the probability that the committee has at least one woman?

Solution:
Total member = 3 + 2 = 5

Committee can be form with at least 1 one woman:
1 woman, 2 men : ²C₁ ×  ³C₂ = 2 × 3 = 6
2 women, 1 man: ²C₂ × ³C₁ = 1 × 3 = 3
∴ The total number of ways to make committe with at least 1 one woman: 6 + 3 = 9

The total number of ways to make committe with all members = ⁵C₃ = 10

∴ The probability that the committee has at least woman = 9/10

Question: If |x - 3| < 4, then for which values of m and n will m < 3x + 5 < n hold?

Solution:
|x - 3| < 4
⇒ - 4 < x - 3 < 4
⇒ - 4 + 3 < x < 4 + 3
⇒ - 1 < x < 7
⇒ - 1 × 3 < 3x < 7 × 3  ; [Now multiply all parts by 3]
⇒ - 3 < 3x < 21
⇒ - 3 + 5 < 3x + 5 < 21 + 5 ; [Now add 5 to all parts]
⇒ 2 < 3x + 5 < 26

Comparing this with m < 3x + 5 < n, then we get,
m = 2 and n = 26.