ব্যাংক নিয়োগ প্রস্তুতি ⎯ লং কোর্স

Question: The sum of all even numbers between 1 and 11 is-

Solution:
The even numbers between 1 and 11 are:
2, 4, 6, 8, 10

∴ The sum of these numbers = (2 + 4 + 6 + 8 + 10) = 30 

Question: If two-third of one-fourth of a number is 14, then three seventh of that number is?

Solution:

Question: The product of two consecutive even numbers is 168. What are the numbers?

Solution:
Let the smaller number be x, 
so the next even number is = x + 2

According to the question,
 x(x + 2) = 168
⇒ x² + 2x = 168
⇒ x² + 2x -168 = 0 
⇒  x² + 14x - 12x - 168 = 0
⇒ x(x + 14) - 12(x + 14) = 0 
⇒ (x + 14)(x - 12) = 0 

⇒  x = - 14 and x = 12

If x = 12,
then the next even number =
12 + 2 = 14
The two numbers are: 12 and 14.

Again, If x = - 14, then the next even number = - 14 + 2 = - 12 
 So, the numbers are 12 and 14 or
- 14 and - 12

Question:  What is the value of 12 × 5 - 8 ÷ 2 = ?

Solution:
12 × 5 - 8 ÷ 2
= 60 - 4 [ According to the BODMAS rule ]
= 56

Question:  If two numbers are in a 3 to 4 ratio and their greatest common factor is 5, what's their least common multiple?

Solution:
Given:
The ratio of two numbers = 3 : 4
and HCF = 5

So the numbers are:
3 × 5 = 15 and 4 × 5 = 20

We know,
First number × Second number = LCM × HCF
⇒ 15 × 20 = LCM × 5
⇒ LCM × 5 = 300
⇒ LCM  = 300 ÷ 5 = 60

Question:  A number when multiplied by 3 and increased by 5 gives 20. Find the number.

Solution:
Let, the number be = x

According to the question,
3x + 5 = 20
⇒ 3x = 20 - 5 
⇒ 3x = 15 
⇒ x =15/3
⇒ x = 5

Therefore, the number is 5

সঠিক উত্তর: গ) - 8 ও ঘ) 8
অপশনে দ্বৈত উত্তর থাকায় প্রশ্নটি বাতিল করা হলো। 
----------------- 

Question: What is the answer in the square of a number subtracted from 100 is equal to 36?

Solution:
Let the number be = x

According to the question,
100 - x² = 36
⇒ x² = 100 - 36
⇒ x² = 64
⇒ x = √64 
⇒ x = + 8, - 8 

Question: Find the greatest number that exactly divides each of the numbers 36, 60, and 84.

Solution:
We know,
The HCF (Highest Common Factor) of two or more numbers is the greatest number that divides each of them exactly.

Now,
Prime factorization of 36 = 2 × 2 × 3 × 3

Prime factorization of 60 = 2 × 2 × 3 × 5

Prime factorization of 84 = 2 × 2 × 3 × 7

∴ HCF of 36, 60, and 84 = 2 × 2 × 3 = 12

Therefore, the greatest number is 12.

Question: The number of students in a school leaves a remainder of 4 when divided by 5, 8, and 20. Find the total number of students.

Solution:
Given,
When the number of students is divided by 5, 8, and 20, the remainder is always 4.
So, the number of students is 4 more than the LCM of 5, 8, and 20.

Now,
Prime factors of 5 = 5 × 1
Prime factors of 8 = 2 × 2 × 2
Prime factors of 20 = 2 × 2 × 5

LCM of 5, 8, and 20 = 2 × 2 × 2 × 5 = 40

Total number of students = (40 + 4) = 44 

Question: A two-digit number becomes 54 more than the original number when its digits are reversed. What is the difference between the digits of the number?

Solution:
ধরি,
সংখ্যাটির একক স্থানীয় অঙ্ক = x
দশক স্থানীয় অঙ্ক = y
∴ সংখ্যাটি = x + 10y 
এবং অঙ্কদ্বয় স্থান পরিবর্তন করলে সংখ্যাটি = 10x + y

সংখ্যাটির অঙ্কদ্বয়ের পার্থক্য = x - y

প্রশ্নমতে,
10x + y = x + 10y + 54
⇒ 10x - x + y - 10y = 54
⇒ 9x - 9y = 54
⇒ 9(x - y) = 54
⇒ x - y = 54/9
⇒ x - y = 6 

Question:  A number, when decreased by 175 and then increased by 130, equals 297. Find the number.

Solution:
Let the number be = x

According to the question,
(x - 175) + 130 = 297
⇒ x - 45 = 297
⇒ x = 297 + 45
⇒ x = 342 

So, the number is 342

Question: Find the value of ,


Solution:

Question: How many numbers from 1 to 225 inclusive are equal to the cube of an integer?

Solution:
1³ = 1
2³= 8
3³= 27
4³= 64
5³= 125
6³ = 216
7³ = 343, যা 225 থেকে বড়।

সুতরাং,
1 থেকে 225 এর মধ্যবর্তী মোট 6 টি সংখ্যা (1, 8, 27, 64, 125, 216) আছে যেগুলো পূর্ণসংখ্যার ঘনফল।

Question:  For x = 10, determine which among the following expressions attains the minimum value.

Solution:
ক) x - 2 = 10 - 2 = 8

খ) x/2 = 10/2 = 5

গ) 2 - x = 2 - 10 = - 8

ঘ) x/5 = 10/5 = 2

সুতরাং,
2 - x  এর মান সবচেয়ে কম।

Question: Find the value of 2(x + 3) - 3(x - 1) + x.

Solution: 
2(x + 3) - 3(x - 1) + x 
= 2x + 6 - 3x + 3 + x
= 3x - 3x + 9
= 9

Question: What is the result when 60 is divided by 1/2 and then 20 is subtracted?

Solution: 

Question: Find the number that exceeds 36 by the same amount it falls short of 86.

Solution:
Let the number be = x

According to the question,
x - 36 = 86 - x
⇒ x + x = 86 + 36
⇒ 2x = 122
⇒ x = 122/2
⇒ x = 61 

Question: If a gas tank that is one-fifth full needs 32 additional gallons to reach three-sevenths of its capacity, find the total capacity of the tank.

Solution:

Question: A number is multiplied by 5 and then 7 is added. The result is 52. What is the number?

Solution:
Let the number be = x

According to the question,
5x + 7 = 52
⇒ 5x = 52 - 7
⇒ 5x = 45
⇒ x = 45/5
⇒ x = 9

Question: The ratio of two numbers is 2 : 3 and their least common multiple is 48. What is the sum of the two numbers?

Solution:
Given,
HCF of two numbers = x
the two numbers are 2x and 3x

According to the question,
2x × 3x = 48x
6x² = 48x
x²/x = 48/6
x = 8

So, the first  number is = 2 × 8 = 16
second number is = 3 × 8 = 24

Therefore,
Sum of the two numbers = 16 + 24 = 40

Question: What must be added to x/y to make 2y/x?

Solution:

Question: Which of the following numbers is divisible by 3?

Solution:
কোন সংখ্যা 3 দ্বারা বিভাজ্য হলে সংখ্যাটির অঙ্কগুলোর যোগফল 3 দ্বারা বিভাজ্য হবে।

প্রদত্ত সংখ্যা গুলোর মধ্যে,
177 → 1 + 7 + 7 = 15, যা 3 দ্বারা বিভাজ্য। 

182 → 1 + 8 + 2 = 11 , যা 3 দ্বারা বিভাজ্য নয়। 

220 → 2 + 2 + 0 = 4 , যা 3 দ্বারা বিভাজ্য নয়। 

331 → 3 +3 + 1 = 7 , যা 3 দ্বারা বিভাজ্য নয়। 

সুতরাং, 177 সংখ্যাটি 3 দ্বারা বিভাজ্য। 

Question: Given that x² is an odd number, determine the nature of x² − x.

Solution: 
যেহেতু x² বিজোড় তাই x ও বিজোড় 
এখন,
x² - x
= x(x - 1)
= (x - 1)x

(x - 1) এবং  x দুইটি ক্রমিক সংখ্যা।  

x বিজোড় সংখ্যা হলে (x - 1) অবশ্যই জোড় সংখ্যা হবে। 
কারণ দুইটি ক্রমিক সংখ্যার মধ্যে একটি বিজোড় হলে অন্যটি জোড় হবে। 

সুতরাং, x ও (x - 1) এর গুনফল = x² - x একটি জোড় সংখ্যা। [ জোড় × বিজোড় = জোড় ] 

Question: How many prime numbers are there between 56 and 100?

Solution:
56 থেকে  100 এর মধ্যবর্তী মৌলিক সংখ্যাগুলো হলো - 
59, 61, 67, 71, 73, 79, 83, 89, 97

সুতরাং ৫৬ এবং ১০০ এর মধ্যবর্তী মৌলিক সংখ্যা হলো = 9 টি 

Question: What is the smallest 5-digit number that can be formed using the digits 0, 1, 4, 6, and 9, where digits may repeat?

Solution:
If we arrange the digits 0, 1, 4, 6, 9 in ascending order, we get the smallest number.

The smallest non-zero digit is 1.
Digits like 4, 6, 9 are larger than 1 and 0.

By using 1 once and 0 multiple times, the smallest 5-digit number that can be formed is =  10000.