Bank Math Master

Question: Tickets numbered 1 to 20 are mixed up and then a ticket is drawn at random. What is the probability that the ticket drawn has a number which is a multiple of 4 or 5?

Solution: 
Here, S = {1, 2, 3, 4, ...., 19, 20}. {4, 8, 12, 16, 20}{5,10,15,20}
Let E = event of getting a multiple of 3 or 5 = {4, 5, 8 , 10, 12, 15, 16, 20}

P(E) = n(E)/n(S) = 8/20 = 2/5

Question: If x > 7 and y > - 3 then which of the following is true?

Solution:
Given, 
x > 7 and y > - 3
Or, xy > 7 × (- 3)
Or, xy > - 21

Question: If set A = {5, 15, 20, 30} and B = {5, 15, 20} which of the following respresents (A ∩ B) ?

Solution:
দেওয়া আছে,
A = {5, 15, 20, 30}
B = {5, 15, 20}
এখন,
A ∩ B = {5, 15, 20, 30} ∩ {5, 15, 20}
= {5, 15, 20}

Question: If x ≥ 10 and y ≤ 7 which of the following must be true?

Solution:
দেওয়া আছে,
x ≥ 10
এবং
y ≤ 7
বা, - y ≥ - 7 [- 1 দ্বারা গুণ করে পাই]

এখন,
x - y ≥ 10 - 7 [অসমতাদ্বয় যোগ করে পাই]
∴ x - y ≥ 3

Question: What is probability of getting a sum 9 from two throws of a dice?

Solution: 
Two dice were thrown.
Total outcomes: 36
Let E = event of getting a sum ={(3, 6), (4, 5), (5, 4), (6, 3)}.
P(E) = n(E)/n(S)
= 4/36
= 1/9

Question: If x + 1 > 1 - 2x then -

Solution:
x + 1 > 1 - 2x
⇒ x + 2x > 1 - 1
⇒ 3x > 0
∴ x > 0

Question: The universal set of U = {1, 2, 3, 4, 5, 6} and A = {1, 3, 5}, B = {3, 5, 6} then determine (A′ ∩ B′)

Solution:
দেওয়া আছে,
U = {1, 2, 3, 4, 5, 6}
A = {1, 3, 5}
B = {3, 5, 6}

A‘ = U - A
= {1, 2, 3, 4, 5, 6} - {1, 3, 5}
= {2, 4, 6}

B‘ = U - B
= {1, 2, 3, 4, 5, 6} - {3, 5, 6}
= {1, 2, 4}

এখন,
(A‘ ∩ B‘) = {2, 4, 6} ∩ {1, 2, 4}
= {2, 4}

Question: Solve

Solution:
- 1 ≤ (3x - 4)/7 ≤ 5
⇒ - 7 ≤ (3x - 4) ≤ 35
⇒ - 7 + 4 ≤ 3x - 4 + 4 ≤ 35 + 4
⇒ - 3 ≤ 3x ≤ 39
⇒ - 1 ≤ x ≤ 13

Question: What will be the next number of the series?
4, 7, 13, 25, 49, 97, ..........

Solution: 
এখানে,
4 + 3 = 7
7 + 6 = 13
13 + 12 = 25
25 + 24 = 49
49 + 48 = 97
97 + 96 = 193

Question: If x > y, z < y and w < x which of the following is always true?

Solution:
দেওয়া আছে,
x > y, z < y
∴ x > y > z

∴ w < x অর্থাৎ x বড় w ছোট।
কিন্তু y ও z এর সাথে w এর কোন সম্পর্ক নেই।

∴ x বড় z ছোট।
∴ x > z অথবা, z < x এটাই সত্য। 

Question: If P = {x ∈ N : 2 < x ≤ 6} and Q = {x ∈ N : x even number and x ≤ 8} then what is the value of (P ∩ Q)?

Solution:
দেওয়া আছে,
P = {x ∈ N : 2 < x ≤ 6}
∴ P = {3, 4, 5, 6}

Q = {x ∈ N : x জোড় সংখ্যা এবং x ≤ 8}
∴ Q = {2, 4, 6, 8}

এখন,
(P ∩ Q) = {3, 4, 5, 6} ∩ {2, 4, 6, 8}
= {4, 6}

Question: Find the median of the following data 2, 9, 15, 5, 20, 8, 25, 17, 21, 23, 11.

Solution:
উপাত্তগুলোকে মানের ক্রমানুসারে সাজিয়ে পাই,
2, 5, 8, 9, 11, 15, 17, 20, 21, 23, 25
এখানে মোট পদ 11টি
∴ নির্ণেয় মধ্যক = ৬ষ্ঠ পদ = 15

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

Solution:
Here S = {TTT, TTH, THT, HTT, THH, HTH, HHT, HHH}

Let E = event of getting at least two tails.
Then E = {TTT, TTH, THT, HTT}

∴ P(E) = n(E)/n(S)
= 4/8
= 1/2

Question: Solve |2x - 3| ≤ 1

Solution:
|2x - 3| ≤ 1
বা, - 1 ≤ 2x - 3 ≤ 1
বা, - 1 + 3 ≤ 2x - 3 + 3 ≤ 1 + 3
বা, 2 ≤ 2x ≤ 4
বা, 2/2 ≤ 2x/2 ≤ 4/2
∴ 1 ≤ x ≤ 2

Question: If A = {1, 3, 4, 5, 6} then the number of proper subsets of A is -

Solution:
কোনো সেটে সদস্য সংখ্যা n হলে ঐ সেটের প্রকৃত উপসেটের সংখ্যা = 2ⁿ - 1
এখানে,
A = {1, 3, 4, 5, 6}
যার সদস্য সংখ্যা, n = 5

∴ প্রকৃত উপসেটের সংখ্যা = 2⁵ - 1
= 32 - 1
= 31

Quesion: If |x + 2| = |x - 1| then what is the value of x?

Solution:
|x + 2| = |x - 1|
⇒ |x + 2|² = |x - 1|²
⇒ (x + 2)² = (x - 1)²
⇒ x² + 4x + 4 = x² - 2x + 1
⇒ 6x = - 3
∴ x = - 1/2

Question: In a simultaneous throw of two dice what is the probability of getting a total of 10 or 11?

Solution:
If two dices are thrown total events = (6 × 6) = 36
Event of getting a total of 10 or 11 = {(4, 6), (5, 5), (6, 4), (5, 6), (6, 5)}
Expected events = 5

∴ Probability = 5/36

Question: If n(A ∪ B) = 61, n(A) = 30, n(B) = 54 then what is the value of n(A ∩ B)?

Solution:
দেওয়া আছে,
n(A ∪ B) = 61
n(A) = 30
n(B) = 54

আমরা জানি,
n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
⇒ 61 = 30 + 54 - n(A ∩ B)
⇒ n(A ∩ B) = 84 - 61
∴ n(A ∩ B) = 23

Question: The mark obtained by 30 students in Mathematics in the annual examination are given below:
42, 31, 27, 60, 61, 42, 35, 58, 29, 48, 39, 52, 40, 47, 28, 78, 90, 52, 48, 36, 52, 39, 71, 32, 48, 56, 29, 33, 25, 48. Find the mode of the marks -

Solution:
উপাত্তগুলোর মধ্যে 48 মানটি সর্বাধিক 4 বার আছে।
∴ প্রচুরক = 48

Question: Solve ।3x - 4। < 5

Solution:
- 5 < 3x - 4 < 5
⇒ - 5 + 4 < 3x - 4 + 4 < 5 + 4 
⇒ - 1 < 3x < 9
⇒ -1/3 < 3x/3 < 9/3
∴ - 1/3 < x < 3

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 woman?

Solution:
Total Person = 3 + 2 = 5
Committee can be form

       Men (3)     -   Women (2)
1)        2                     1
2)        1                     2

∴ Expected events = (³C₂ × ²C₁) + (³C₁ × ²C₂)
= (3 × 2) + (3 × 1)
= 6 + 3
= 9

Total events = ⁵C₃ = 10

∴ Probability = 9/10