Bank Math Master

Question: If m is an odd integer, which of the following must be an even integer?

সমাধান:
ধরি, m = 3 (একটি বিজোড় সংখ্যা)

ক) m² + m = 3³ + 3 = 12 → জোড় 
খ) 2m + 1 = 2(3) + 1 = 6 + 1 = 7 → বিজোড়
গ) 5m - 2 = 5(3) - 2 = 15 - 2 = 13 → বিজোড় 
ঘ) m³ + 2 = 3³ + 2 = 27 + 2 = 29 → বিজোড়

Question: The ratio of two numbers is 5 : 6 and their H.C.F is 7. Their L.C.M is -

Solution:
ধরি, সংখ্যা দুটি হলো 5x এবং 6x
∴গসাগু (H.C.F) = x = 7

∴ সংখ্যা দুটি: হলো 5 × 7 = 35 এবং 6 × 7 = 42

∴ সংখ্যাদ্বয়ের গুণফল = 35 × 42 = 1470
এবং H.C.F = 7

আমরা জানি,
L.C.M = (Product of two numbers)/H.C.F
= 1470/7
= 210

∴ সংখ্যা দুটির লসাগু (L.C.M) = 210

Question: Which of the following fractions is smaller than 7/8 and greater than 3/7?

Solution:
7/8 = 0.875
3/7 = 0.4286

ক) 11/12 = 0.917  (> 0.875)
খ) 5/9 = 0.5556   (satisfies the condition)
গ) 4/11 = 0.3636 (< 0.4286)
ঘ) 1/3 = 0.3333  (< 0.4286))

∴ অপশন (খ) সঠিক উত্তর।

Question: If 

Solution:

Question:  = ?

Solution:

Question: 4 log 2 + 2 log 3 - 1/2 log 81 = ?

Solution:
4 log 2 + 2 log 3 - 1/2 log 81
= log 2⁴ + log 3² - log(81)^1/2   
= log 16 + log 9 - log √81
= log 16 + log 9 - log 9
= log(16 × 9) - log 9
= log 144 - log 9
= log(144/9)
= log 16

Question: A father is five times as old as his son. After eight years, the father will be three times as old as his son will be then. What is the father's present age?

Solution:
Let the son's present age be x years.
Then, the father's present age = 5x years.

After 8 years, 
Son’s age = x + 8
Father’s age = 5x + 8

According to the question,
5x + 8 = 3(x + 8)
⇒ 5x + 8 = 3x + 24
⇒ 5x - 3x = 24 - 8
⇒ 2x = 16
∴ x = 8

So, father’s present age = 5 × 8 = 40 years.

Question: The average of nine numbers is 18. The average of six of these numbers is 16. What is the average of the remaining three numbers?

Solution:
৯ টি সংখ্যার গড় = ১৮
৯ টি সংখ্যার সমষ্টি = ১৮ × ৯ = ১৬২

৬ টি সংখ্যার গড় = ১৬
৬ টি সংখ্যার সমষ্টি = ১৬ × ৬ = ৯৬

∴ বাকী ৩ টি সংখ্যার সমষ্টি = ১৬২ - ৯৬ = ৬৬
∴ ৩ টি সংখ্যার গড় = ৬৬/৩ = ২২

Question: Bipul ate 2/3 of a cake. His friend Rafi ate 2/3 of what was left. Then Rafi's sister ate 2/3 of what was still left. What fraction of the cake remains uneaten?

Solution:
বিপুল খেয়েছে = 2/3
∴ অবশিষ্ট = 1 - 2/3 = 1/3

রাফি খেয়েছে = 1/3 × 2/3 = 2/9
∴ অবশিষ্ট = 1/3 - 2/9 = (3 - 2)/9 = 1/9

রাফির বোন খেয়েছে = 1/9 × 2/3 = 2/27
∴ অবশিষ্ট = 1/9 - 2/27 = (3 - 2)/27 = 1/27

Question: Find the value of x, if 9^{2x + 1} = 81³.

Solution:
9^{2x + 1} = 81³
⇒ (3²)^{2x + 1} = (3⁴)³
⇒ 3^{2(2x + 1)} = 3¹²
⇒ 4x + 2 = 12
⇒ 4x = 10
⇒ x = 10/4
∴ x = 2.5

Question: What is the least number which, when tripled, becomes exactly divisible by 15, 20, 24, and 30?

Solution:
Let the required number be x.
Then, 3x must be divisible by 15, 20, 24, and 30.

Find the LCM of the numbers:
15 = 3 × 5
20 = 2² × 5
24 = 2³ × 3
30 = 2 × 3 × 5

∴ LCM = 2³ × 3 × 5 = 8 × 3 × 5 = 120

∴ 3x = 120
∴ x = 120/3 = 40

Question: How many prime numbers are there between 70 and 80?

Solution:
যে সংখ্যা ১ এবং সেই সংখ্যা ছাড়া অন্য কোনো সংখ্যা দ্বারা বিভাজ্য নয়, তাকে মৌলিক সংখ্যা বলে।

৭০ থেকে ৮০ এর মধ্যে সংখ্যাগুলো হলো:
৭১, ৭২, ৭৩, ৭৪, ৭৫, ৭৬, ৭৭, ৭৮, ৭৯

এদের মধ্যে মৌলিক সংখ্যা:
৭১, ৭৩, ৭৯

অতএব, মোট মৌলিক সংখ্যা = ৩টি

Question: A mother was 27 years old when her daughter was born. Currently, the mother's age is 9 years more than three times her daughter's age. How old will the daughter be in 6 years?

Solution:
Let the daughter's current age be x years.
∴ Mother's current age = 3x + 9 years.

According to the question,
3x + 9 - x = 27
⇒ 2x = 27 - 9
⇒ 2x = 18
∴ x  = 9

∴ Daughter's current age 9 years.
∴ Daughter's age after 6 years = 9 + 6 = 15 years.

Question: If one fifth of one third of a number is 12, then what is 2/5 of the number?

Solution:
Let the number be x.
According to the question,
(1/5) of (1/3) of x = 12
⇒ (1/5) × (1/3) × x = 12
⇒ (1/15)x = 12
⇒ x = 12 × 15
∴ x = 180

Now,
(2/5) of x = (2/5) × 180
= 360/5
= 72

Question: If log_m 128 = 7, then find the value of m.

Solution:
log_m 128 = 7
⇒ m⁷ = 128  [log_b A = C, then b^C = A]
⇒ m⁷ = 2⁷
∴ m = 2

Question: A cricketer scores 60 runs in the 10th innings and increases his average by 2 runs. What is his average after the 10th innings?

Solution:
ধরি, প্রথম 9 ইনিংসের গড় রান = x
∴ 9 ইনিংসের মোট রান = 9x

10 তম ইনিংসে 60 রান করার পর নতুন গড় = x + 2
সুতরাং, 10 ইনিংসের মোট রান = 10 × (x + 2)

এখানে, 9 তম ইনিংসের মোট রান এবং 10 তম ইনিংসের মোট রানের পার্থক্য = 60.
অর্থাৎ,
10(x + 2) - 9x = 60
⇒ 10x + 20 - 9x = 60
⇒ x + 20 = 60
∴ x = 40

অতএব, ১০ তম ইনিংসের পর গড় = 40 + 2 = 42