Number System, Problems on Number

PrepBank · পাতা ৪ / ১৮ · ৩০১–৪০০ / ১,৭৩৬

৩০১.

P, Q, and R are three consecutive even integers. If P + R = Q + 14, what is the value of P?

ক
10
খ
8
গ
14
ঘ
12

ব্যাখ্যা

Question: P, Q, and R are three consecutive even integers. If P + R = Q + 14, what is the value of P?

Solution:
ধরি, P, Q, এবং R হলো তিনটি ক্রমিক জোড় পূর্ণসংখ্যা। যেখানে,
P = n (জোড় পূর্ণসংখ্যা)
Q = n + 2 (পরবর্তী ক্রমিক জোড় পূর্ণসংখ্যা)
R = n + 4 (তৃতীয় ক্রমিক জোড় পূর্ণসংখ্যা)

দেয়া আছে,
P + R = Q + 14
⇒ n + (n + 4) = (n + 2) + 14
⇒ 2n + 4 = n + 16
⇒ 2n - n = 16 - 4
∴ n = 12

অতএব, P = 12, Q = 14, R = 16
সুতরাং, P এর মান হলো 12.

৩০২.

What is the number of divisor of 1008?

ক
ক) 12
খ
খ) 20
গ
গ) 30
ঘ
ঘ) 32

ব্যাখ্যা

1008 = 2 × 2 × 2 × 2 × 3 × 3 × 7
= 2⁴ × 3² × 7
নির্ণেয় ভাজক সংখ্যা = (4 + 1)(2 + 1)(1 + 1)
= 5 × 3 × 2 = 30

৩০৩.

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

ক
36
খ
48
গ
54
ঘ
72

ব্যাখ্যা

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

৩০৪.

How many positive integers less than 100 have a remainder of 2 when divided by 13?

ক
ক) 6
খ
খ) 7
গ
গ) 8
ঘ
ঘ) 9

ব্যাখ্যা

13 x 1 = 13 --> অর্থাৎ, 13 + 2 = 15 কে ১৩ দিয়ে ভাগ করলে ২ ভাগ শেষ থাকবে।

একইভাবে,
13 x 2 = 26 --> 28 কে ১৩ দিয়ে ভাগ করলে ২ ভাগ শেষ থাকবে।
13 x 3 = 39 --> 41 কে ১৩ দিয়ে ভাগ করলে ২ ভাগ শেষ থাকবে।
13 x 4 = 52 --> 54 কে ১৩ দিয়ে ভাগ করলে ২ ভাগ শেষ থাকবে।
13 x 5 = 65 --> 67 কে ১৩ দিয়ে ভাগ করলে ২ ভাগ শেষ থাকবে।
13 x 6 = 78 --> 80 কে ১৩ দিয়ে ভাগ করলে ২ ভাগ শেষ থাকবে।
13 x 7 = 91 --> 93 কে ১৩ দিয়ে ভাগ করলে ২ ভাগ শেষ থাকবে।

13 x 8 = 104 --> যেহেতু ১০০ এর চেয়ে ছোট চেয়েছে তাই এটা হবে না।

অর্থাৎ, ১৫, ২৮, ৪১, ৫৪, ৬৭, ৮০, ৯৩ কে ১৩ দিয়ে ভাগ করলে ভাগশেষ ২ থাকে।
তাহলে, মোট সংখ্যা হল ৭টি।

কিন্তু আমরা জানি,
যেকোনো সংখ্যা দ্বারা তার চেয়ে ক্ষুদ্রতম কোন পূর্ণ সংখ্যাকে ভাগ করলে ভাগশেষ ক্ষুদ্র সংখ্যাটিই হবে।

উক্ত নিয়মানুসারে, ২ কে ১৩ দিয়ে ভাগ করলে ভাগশেষ ২ হবে। এবং, ২ অবশ্যই ১০০ এর চেয়ে ছোট ধনাত্মক পূর্ণ সংখ্যা (Positive Integer)।

সুতরাং, ১০০ এর চেয়ে ছোট ধনাত্মক পূর্ণ সংখ্যা যাদেরকে ১৩ দিয়ে ভাগ করলে ভাগশেষ ২ হবে এমন সর্বমোট সংখ্যা হল - ২, ১৫, ২৮, ৪১, ৫৪, ৬৭, ৮০, ৯৩ = ৮টি।

৩০৫.

√(441 – 41) × 42 ÷ 7 =?

ক
80
খ
120
গ
110
ঘ
20

ব্যাখ্যা

Question: √(441 – 41) × 42 ÷ 7 =?

Solution:
√(441– 41) × 42 ÷ 7
= √400 × 42 ÷ 7
= 20 × 42 ÷ 7
= 840 ÷ 7
= 120

৩০৬.

The H C F of x² - 1, x⁴ - 1 and x⁴ - x³ + x - 1 is -

ক
ক) x⁶ - 1
খ
খ) x³ + 1
গ
গ) x² - 1
ঘ
ঘ) x³ - 1

ব্যাখ্যা

১ম রাশি = x² - 1 = (x + 1)(x - 1)
২য় রাশি = x⁴ - 1 = (x²)² - (1²)² = (x² + 1)(x² - 1) = (x² + 1)(x + 1)(x - 1)
এবং ৩য় রাশি = x⁴ - x³ + x - 1 = x³(x - 1) + 1(x - 1) = (x³ + 1)(x - 1) = (x + 1)(x² - x + 1)(x - 1)

So, HCF is (x + 1)(x - 1) = x² - 1

৩০৭.

How many multiples of both 3 or 4 are there from 1 to 100 in total?

ক
33
খ
50
গ
55
ঘ
58

ব্যাখ্যা

Question: How many multiples of both 3 or 4 are there from 1 to 100 in total?

Solution:
On dividing 100 by 3 we get a quotient of 33 The number of multiple of 3, n(A) = 33
On dividing 100 by 4 we get a quotient of 25 The number of multiple of 4, n(B) = 25
LCM of 3 and 4 is 12
On dividing 100 by 12 we get a quotient of 8 The number of multiple of 12, n(A∩B) = 8

The number which is multiple of 3 or 4 = n(A∪B)
Now,
n(A∪B) = n(A) + n(B) - n(A∩B)
= 33 + 25 - 8
= 50

∴ The total number multiple of 3 or 4 from 1 to 100 is 50

৩০৮.

The difference between three times and seven times of a number comes to 36. What is the number?

ক
7
খ
9
গ
12
ঘ
15

ব্যাখ্যা

Question: The difference between three times and seven times of a number comes to 36. What is the number?

Solution:
let, the number be x

ATQ,
7x - 3x = 36
⇒ 4x = 36
⇒ x =36/4 = 9

৩০৯.

What is the least number of soldiers that can be drawn up in troops of 12, 15, 18 and 20 soldiers and also in form of a solid square?

ক
ক) 1600
খ
খ) 2500
গ
গ) 900
ঘ
ঘ) 3600

ব্যাখ্যা

Question: What is the least number of soldiers that can be drawn up in troops of 12, 15, 18 and 20 soldiers and also in form of a solid square?

Solution:
In this type of question, We need to find out the LCM of the given numbers.
LCM of 12, 15, 18 and 20;
12 = 2 × 2 × 3;
15 = 3 × 5;
18 = 2 × 3 × 3;
20 = 2 × 2 × 5;
Hence, LCM = 2 × 2 × 3 × 3 × 5

Since, the soldiers are in the form of a solid square.

The required number of soldiers
= 2 × 2 × 3 × 3 × 5 × 5
= 900

৩১০.

The sum of five consecutive multiples of 6 is 150. What is the second largest number?

ক
48
খ
36
গ
42
ঘ
30

ব্যাখ্যা

Question: The sum of five consecutive multiples of 6 is 150. What is the second largest number?

Solution:
ধরি, ৬ এর পাঁচটি ক্রমিক গুণিতক হলো যথাক্রমে (x - 12), (x - 6), x, (x + 6) এবং (x + 12)

প্রশ্নমতে,
(x - 12) + (x - 6) + x + (x + 6) + (x + 12) = 150
⇒ 5x = 150
⇒ x = 150/5
⇒ x = 30

সুতরাং, সংখ্যাগুলো হলো 18, 24, 30, 36, 42।
এদের মধ্যে দ্বিতীয় বৃহত্তম সংখ্যাটি হলো 36।

৩১১.

If the sum of three consecutive odd integers is 141, what is the largest number?

ক
41
খ
43
গ
49
ঘ
None

ব্যাখ্যা

Question: If the sum of three consecutive odd integers is 141, what is the largest number?

Solution:
Let the three consecutive odd integers be:
x, x + 2, x + 4

Accordingly:
x + (x + 2) + (x + 4) = 141
⇒ 3x + 6 = 141
⇒ 3x = 141 - 6
⇒ 3x = 135
⇒ x = 135/3
⇒ x = 45

So the numbers are:
45, 47, 49(Largest)

∴ The largest number is 49.

৩১২.

A number when divided by the sum of 555 and 445 gives two times their difference as quotient and 30 as remainder. The number is:

ক
ক) 22030
খ
খ) 220030
গ
গ) 23030
ঘ
ঘ) 24030

ব্যাখ্যা

(550 + 445) × 2 × 110 + 30 = 220030

৩১৩.

How many pairs of natural numbers is there the difference of whose squares are 45?

ক
6
খ
3
গ
1
ঘ
None of these

ব্যাখ্যা

Question: How many pairs of natural numbers is there the difference of whose squares are 45?

Solution:
ধরি, 45 কে x ও y এর বর্গের বিয়োগফল আকারে প্রকাশ করা যায়।
45 = x² - y² = (x - y) (x + y)

45
= 1 × 45
= 5 × 9
= 3 × 15

♦ x + y = 45, x - y = 1 হলে,
x + y + x - y = 45 + 1
⇒ 2x = 46
⇒ x = 23∴ y = 45 - 23 = 22

♦ x + y = 9, x - y = 5 হলে,
x + y + x - y = 5 + 9
⇒ 2x = 14
⇒ x = 7∴ y = 9 - 7 = 2

♦ x + y = 15, x - y = 3 হলে,
x + y + x - y = 15 + 3
⇒ 2x = 18
⇒ x = 9

∴ y = 15 - 9 = 6

অতএব, 45 কে দুটি স্বাভাবিক সংখ্যার বর্গের বিয়োগফল আকারে প্রকাশ করা যায় এমন 3 টি ক্রমজোড় হল (x, y) = (23, 22), (7, 2), (9, 6)

৩১৪.

There are 6 true-false questions in an examination. In how many ways can these questions be answered?

ক
64 ways
খ
24 ways
গ
56 ways
ঘ
42 ways

ব্যাখ্যা

Question: There are 6 true-false questions in an examination. In how many ways can these questions be answered?

Solution:
Total number of questions = 6
Each question has 2 possible answers (True or False).

∴ Total number of ways = 2⁶ = 64 ways

৩১৫.

A sport club has 50 members. Of these, 35 play golf, 30 play soccer and 18 play both golf and soccer. How members do play neither golf nor soccer?

ক
ক) 0
খ
খ) 5
গ
গ) 3
ঘ
ঘ) 17

ব্যাখ্যা

Players who play at least one sport = 35 + 30 - 18 = 47
Players who play neither golf nor soccer = 50 - 47 = 3

৩১৬.

If 10⁵⁰ - 74 is written as an integer in base 10 notation, what is the sum of the digits in that integer?

ক
433
খ
467
গ
424
ঘ
440
ঙ
449

ব্যাখ্যা

Question: If 10⁵⁰ - 74 is written as an integer in base 10 notation, what is the sum of the digits in that integer?

Solution:
10⁵⁰ has 51 digits (1 followed by 50 zeros).
10⁵⁰ - 74 has 50 digits: the last 2 digits are 2 and 6 [100 - 74 = 26] and the first 48 digits are 9's
So the sum of the digits is (9 × 48) + 2 + 6 = 440

৩১৭.

On dividing a number by 56, we get 29 as remainder. On dividing the same number by 8, what will be the remainder?

ক
3
খ
4
গ
5
ঘ
6

ব্যাখ্যা

Question: On dividing a number by 56, we get 29 as remainder. On dividing the same number by 8, what will be the remainder?

Solution:
If a number divided by 56 leaves a remainder of 29, we can express it as:
Number = 56 × Quotient + 29

Now, to find the remainder when this number is divided by 8, let's try to simplify the expression.
Number = 56 × Quotient + 29

We know that 56 is divisible by 8, so let's express 56 in terms of 8:
56 = 8 × 7

Therefore, we can rewrite the expression for the number:
Number = 8 × 7 × Quotient + 29
Number = 8 × (7 × Quotient) + 29

Now, when we divide this expression by 8, we'll focus on the remainder:

8 × (7 × Quotient) leaves no remainder when divided by 8 because it's a multiple of 8.

So, the remainder when 29 is divided by 8 is 5.

৩১৮.

What reminder of any perfect square is divided by 3?

ক
0
খ
1
গ
0 or 1
ঘ
0 and 1
ঙ
None

ব্যাখ্যা

When a perfect square is divided by 3, the remainder is always 0 or 1.
It is zero when the base number (a in case of a^2) is divisible by 3.
For example - 3² is divided by 3. So, we get 0 as a remainder.
For all other numbers, the square when divided by 3, the remainder is always 1.
When the number is divisible by 3, the remainder is obviously going to be zero.
Let’s say the number is not divisible by 3.
Such a number is always in the form of 3a + 1 or 3a - 1,
for example 4 = 3 × 2 + 1 and 5 = 3× 2 - 1.
Now, when you square 3a + 1,
you get 9a² + 6a + 1.
Now 9a² + 6a = 3×(3a²+2),
therefore that portion is divisible by 3 and what remains (remainder) is 1.
Similarly, when you square 3a - 1, you get 9a² - 6a + 1.
Now 9a² - 6a = 3×(3a²-2),
therefore that portion is divisible by 3 and what remains (remainder) is 1.
Thus, in all situations, we find that the remainder is 1 when the starting number is not divisible by 3.

৩১৯.

Sum of two numbers is thrice their difference. Their ratio is :

ক
ক) 2 : 1
খ
খ) 3 : 1
গ
গ) 4 : 1
ঘ
ঘ) 2 : 3

ব্যাখ্যা

Let the numbers = a, b
According to the question,
a + b = 3 (a - b)
⇒ a/b = 2
⇒ a : b = 2 : 1

৩২০.

(52.02² - 34.01²)÷17.99 × √? = 1,720

ক
ক) 25
খ
খ) 20
গ
গ) 400
ঘ
ঘ) 625

ব্যাখ্যা

(52.02² - 34.01²)÷17.99 ×√x =1720
(2706.0804 - 1156.6801) ÷ 17.99 ×√x = 1720
1549.4 ÷ 17.99 ×√x = 1720
86.1256253 ×√x = 1720
√x =1720÷86.1256253
√x =19.9708
(√x)² = (19.9708)²x= 398.83465
≈400

৩২১.

If a number is in the form of 8¹⁰ × 9⁷ × 7⁸, find the total number of prime factors of the given number.

ক
52
খ
560
গ
3360
ঘ
25

ব্যাখ্যা

Question: If a number is in the form of 8¹⁰ × 9⁷ × 7⁸, find the total number of prime factors of the given number.

Solution:
If a number of the form x^a × y^b × z^c ...... and so on,
then total prime factors = a + b + c ..... and so on Where x, y, z, ... are prime numbers

The number 8¹⁰ × 9⁷ × 7⁸ can be written as (2³)¹⁰ × (3²)⁷ × 7⁸
= 2³⁰ × 3¹⁴ × 7⁸

Total number of prime factors = 30 + 14 + 8 = 52

∴ The total number of prime factors are 52

৩২২.

What is the difference between the smallest six-digit number and the largest five-digit number?

ক
ক) - 1
খ
খ) 1
গ
গ) 10
ঘ
ঘ) 99

ব্যাখ্যা

Question: What is the difference between the smallest six-digit number and the largest five-digit number?

Solution:
ছয় অঙ্কের ক্ষুদ্রতম সংখ্যা = ১০০০০০
পাঁচ অঙ্কের বৃহত্তম সংখ্যা = ৯৯৯৯৯

∴ অন্তর = ১০০০০০ - ৯৯৯৯৯ = ১

৩২৩.

3x² + kx + 4 is divisible by x - 1. The expression is also divisible by-

ক
ক) 4x - 3
খ
খ) 2x - 3
গ
গ) 3x - 2
ঘ
ঘ) 3x - 4

ব্যাখ্যা

Question: 3x² + kx + 4 is divisible by x - 1. The expression is also divisible by-

Solution:
Given that
The expression 3x² + kx + 4 divisible by x - 1,
⇒ 3x² + kx + 4 = 0............(1)

Put x = 1 in equation (1)
⇒ 3 × 1² + k × 1 + 4 = 0
⇒ 3 + k + 4 = 0
⇒ k = - 7

Put the value k in equation (1)
⇒ 3x² - 7x + 4 = 0
⇒ 3x² - 3x - 4x + 4 = 0
⇒ 3x (x - 1) - 4(x – 1) = 0
⇒ (x - 1) (3x - 4) = 0

So, expression divisible by (x - 1) and (3x - 4).

৩২৪.

If n is an even integer, which of the following must be an odd integer?

ক
n² - n
খ
n + 2
গ
3n - 1
ঘ
3n³

ব্যাখ্যা

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

Solution:
ধরি
n = 2

ক) n² - n = 2² - 2 = 4 - 2 = 2 [যা জোড়]
খ) n + 2 = 2 + 2 = 4 [যা জোড়]
গ) 3n - 1 = 3 × 2 - 1 = 5 [যা বিজোড়]
ঘ) 3n³ = 3 × 2³ = 24 [যা জোড়]

৩২৫.

2% of 2 = ?

ক
ক) 0.0004
খ
খ) 0.4
গ
গ) 0.02
ঘ
ঘ) 0.04

ব্যাখ্যা

2 এর 2% = 2 এর 2/100 = 4/100 = 0.04

৩২৬.

The product of two numbers is 9375 and the quotient, when the larger one is divided by the smaller, is 15. The sum of the numbers is:

ক
372
খ
388
গ
400
ঘ
420
ঙ
None of these

ব্যাখ্যা

Question: The product of two numbers is 9375 and the quotient, when the larger one is divided by the smaller, is 15. The sum of the numbers is:

Solution:
Let, the numbers be x and y and x is greater than y.

As given,
xy = 9375 ....(i)
And, x/y = 15
⇒ x = 15y

∴ From equation (i),
15y × y = 9375
⇒ y² = 9375/15
⇒ y² = 625
⇒ y = √625
⇒ y = 25

∴ x = (15 × 25) = 375

So, the requried solution is = x + y
= 375 + 25
= 400

৩২৭.

How many prime numbers are there between 110 and 120?

ক
0
খ
1
গ
3
ঘ
2

ব্যাখ্যা

Question: How many prime numbers are there between 110 and 120?

Solution:
The numbers from 110 to 120 are,
111, 112, 113, 114, 115, 116, 117, 118, 119

Now let's check which of these are prime numbers (divisible only by 1 and themselves).
111 = 3 × 37 = not prime
112 = even = not prime
113 = prime (only divisible by 1 and 113)
114 = even = not prime
115 = ends with 5 = divisible by 5 = not prime
116 = even = not prime
117 = 9 × 13 = not prime
118 = even = not prime
119 = 7 × 17 = not prime

∴ Only one number is prime 113

৩২৮.

A number between 1000 and 2000 which when divided by 30, 36 and 80 gives a remainder 11 in each case is = ?

ক
ক) 1523
খ
খ) 1451
গ
গ) 1641
ঘ
ঘ) 1712

ব্যাখ্যা

LCM of 30, 36, 80 = 720
Number = 720 × K + 11 (K = 2)
Then the number = 720 × 2 + 11
= 1440 + 11
= 1451

৩২৯.

(0.01 × 0.001 × 0.1 × 10⁶) is equal to:

ক
1
খ
10
গ
20
ঘ
100

ব্যাখ্যা

Question: (0.01 × 0.001 × 0.1 × 10⁶) is equal to:

Solution:
Given expression,
(0.01 × 0.001 × 0.1 × 10⁶)
= (1/100) × (1/1000) × (1/10) × 10⁶
= 10⁶/10⁶
= 10^{(6 - 6)}
= 10⁰
= 1

৩৩০.

If (x - 1) is an odd number, what are the two other odd numbers nearest to it?

ক
x, x - 1
খ
x - 3, x + 5
গ
x - 3, x + 1
ঘ
x, x - 2

ব্যাখ্যা

Question: If (x - 1) is an odd number, what are the two other odd numbers nearest to it?

Solution:
(x - 1) is an odd number

The previous odd number = x - 1 - 2 = x - 3
The next odd number = x - 1 + 2 = x + 1

So, the two required odd numbers are: x - 3, x + 1

৩৩১.

The sum of two numbers is 21 and the sum of their squares is 333. The product of two numbers is -

ক
ক) 40
খ
খ) 49
গ
গ) 54
ঘ
ঘ) 58

ব্যাখ্যা

Let, the numbers be x and y
According to the question, x + y = 21 ......(i) and x² + y² = 333 ......(ii)
∴ (x + y)² = x² + y² + 2xy
⇒ (21)² = 333 + 2xy
⇒ 441 = 333 + 2xy
⇒ 2xy = 108
⇒ xy = 54

৩৩২.

The G.C.D. of 1.08, 0.36 and 0.9 is -

ক
ক) 0.36
খ
খ) 0.24
গ
গ) 0.09
ঘ
ঘ) 0.18

ব্যাখ্যা

Question: The G.C.D. of 1.08, 0.36 and 0.9 is -

Solution:
১.০৮ = ১০৮/১০০ = ২৭/২৫
০.৩৬ = ৩৬/১০০ = ৯/২৫
০.৯ = ৯/১০

ভগ্নাংশের গ.সা.গু = ( লব সমূহের গ.সা.গু ) / ( হর সমূহের ল.সা.গু)
= (২৭,৯, ৯ এর গ.সা.গু)/(২৫,২৫,১০ এর ল.সা.গু)
= ৯/৫০
= ০.১৮

৩৩৩.

4,7,8,4 = 26

ক
ক) +,−,×
খ
খ) ×,−,÷
গ
গ) ×,+,−
ঘ
ঘ) +,÷,×
ঙ
ঙ) None of these

ব্যাখ্যা

4×7−8÷4 = 26

৩৩৪.

The sum of three consecutive multiples of 3 is 72. What is the largest number?

ক
ক) 27
খ
খ) 24
গ
গ) 21
ঘ
ঘ) 36

ব্যাখ্যা

Question: The sum of three consecutive multiples of 3 is 72. What is the largest number?

Solution:
মনে করি,
3 এর তিনটি ধারাবাহিক গুণিতক যথাক্রমে 3x, 3(x + 1) ও 3(x + 2)

প্রশ্নমতে,
3x + 3(x + 1) + 3(x + 2) = 72
বা, 3x+ 3x + 3 + 3x + 6 = 72
বা, 9x + 9 = 72
বা, 9x = 72 - 9
বা, 9x = 63
বা, x = 63/9
∴ x = 7

∴ বৃহত্তম গুণিতক = 3(7 + 2) = 3 × 9 = 27

৩৩৫.

In a two–digit number, the digit at the unit’s place is 1 less than twice the digit at the ten’s place. If the digits at unit’s and ten’s place are interchanged, the difference between the new and the original number is less than the original number by 20. The original number is-

ক
35
খ
41
গ
47
ঘ
55
ঙ
31

ব্যাখ্যা

Question: In a two–digit number, the digit at the unit’s place is 1 less than twice the digit at the ten’s place. If the digits at unit’s and ten’s place are interchanged, the difference between the new and the original number is less than the original number by 20. The original number is-

Solution:
Let, Ten’s digit = x
Unit’s digit = 2x - 1

∴ Original number = 10x + (2x - 1) = 12x - 1

So, New number = 10 (2x - 1) + x
= 20x - 10 + x
= 21x - 10

ATQ,
(21x - 10) - (12x + 1) = 12x - 1 - 20
⇒ 9x - 9 = 12x - 21
⇒ 3x = 12
⇒ x = 4

∴ Original number = 12x - 1
= 12 × 4 - 1
= 47

৩৩৬.

7 is added to a certain number; the sum is multiplied by 5; the product is divided by 9 and 3 is subtracted from the quotient. Thus, if the remainder left is 12, what was the original number?

ক
20
খ
37
গ
30
ঘ
27

ব্যাখ্যা

Question: 7 is added to a certain number; the sum is multiplied by 5; the product is divided by 9 and 3 is subtracted from the quotient. Thus, if the remainder left is 12, what was the original number?

Solution:
let the number be x.

ATQ,
[{5(x + 7)}/9] - 3 = 12
or, 5(x + 7) = 15 × 9
or, x + 7 = 135/5
or, x + 7 = 27
or, x = 27 - 7
∴ x = 20

৩৩৭.

If √5ⁿ = 625, then the value of n is?

ক
8
খ
18
গ
20
ঘ
22

ব্যাখ্যা

Question: If √5ⁿ = 625, then the value of n is?

Solution:
Given that, √5ⁿ = 625
⇒ √5ⁿ = 5⁴
⇒ (√5ⁿ)² = (5⁴)²
⇒ 5ⁿ = 5⁸
∴ n = 8

৩৩৮.

If the sum of five consecutive odd integers is 165, what is the middle number?

ক
31
খ
33
গ
35
ঘ
37

ব্যাখ্যা

Question: If the sum of five consecutive odd integers is 165, what is the middle number?

Solution:
ধরি,
মাঝের সংখ্যাটি = x
সুতরাং, ৫ টি ক্রমিক বিজোড় সংখ্যা হবে যথাক্রমে-
x - 4, x - 2, x, x + 2 এবং x + 4 [যেখানে মাঝের সংখ্যা x]

প্রশ্নমতে,
(x - 4) + (x - 2) + x + (x + 2) + (x + 4) = 165
⇒ 5x = 165
⇒ x = 165/5
⇒ x = 33

অর্থাৎ 5 টি ক্রমিক বিজোড় সংখ্যার মধ্যে মাঝের সংখ্যাটি = 33

৩৩৯.

What is the cube root of 0.008?

ক
ক) 0.2
খ
খ) 0.02
গ
গ) 0.002
ঘ
ঘ) 1.2

ব্যাখ্যা

প্রশ্ন: What is the cube root of 0.008?

সমাধান:

৩৪০.

a, b, c, d and e are five consecutive integers in increasing order of size. Which one of the following expression is not odd?

ক
a + b + c
খ
ab + c
গ
ac + e
ঘ
ac + d

ব্যাখ্যা

Question: a, b, c, d and e are five consecutive integers in increasing order of size. Which one of the following expression is not odd?

Solution:
ধরি
a = 1, b = 2, c = 3, d = 4, and e = 5,

অপশন (ক) a + b + c = 1 + 2 + 3 = 6
অপশন (খ) ab + c = 1 × 2 + 3 = 2 + 3 = 5
অপশন (গ) ac + e = 1 × 3 + 5 = 3 + 5 = 8
অপশন (ঘ) ac + d = 1 × 3 + 4 = 3 + 4 = 7

আবার
ধরি
a = 2, b = 3, c = 4, d = 5, and e = 6,

অপশন (ক) a + b + c = 2 + 3 + 4 = 9
অপশন (খ) ab + c = 2 × 3 + 4 = 6 + 4 = 10
অপশন (গ) ac + e = 2 × 4 + 6 = 8 + 6 = 14
অপশন (ঘ) ac + d = 2 × 4 + 5 = 8 + 5 = 13

উভয় ক্ষেত্রে অপশন (গ) জোড় সংখ্যা। তাই সঠিক উত্তর হিসেবে অপশন (গ) নেওয়া হয়েছে।

৩৪১.

One million two thousand and two is written as-

ক
ক) 102002
খ
খ) 1002002
গ
গ) 100202
ঘ
ঘ) None

ব্যাখ্যা

Question: One million two thousand and two is written as-

Solution:
১ মিলিয়ন = ১০ লাখ

১ মিলিয়িন ২ হাজার ২
= ১০ লাখ ২ হাজার ২
= ১০০২০০২

৩৪২.

The sum of three consecutive even integers is 42 more than the first number. What is the middle number?

ক
18
খ
20
গ
22
ঘ
None

ব্যাখ্যা

Question: The sum of three consecutive even integers is 42 more than the first number. What is the middle number?

Solution:
Let,
denote the three consecutive even integers as x, x + 2 and x + 4

ATQ,
x + (x + 2) + (x + 4) = x + 42
⇒ 3x + 6 = x + 42
⇒ 2x + 6 = 42
⇒ 2x = 36
∴ x = 18

The three consecutive even integers are 18, 20, and 22
∴ The middle number is 20

৩৪৩.

The number 2 - 0.5 is how many times the number 1 - 0.5?

ক
2
খ
2.5
গ
3
ঘ
3.5

ব্যাখ্যা

Question: The number 2 - 0.5 is how many times the number 1 - 0.5?

Solution:
Let,
2 - 0.5 is x times the number 1 - 0.5

So,
(2 - 0.5) = x × (1 - 0.5)
⇒ 1.5 = x × 0.5
⇒ x = 1.5/0.5
∴ x = 3

৩৪৪.

19.99 × 9.9 + 99.9 =?

ক
ক) 129.79
খ
খ) 1009
গ
গ) 297.801
ঘ
ঘ) 296.910

ব্যাখ্যা

Question: 19.99 × 9.9 + 99.9 =?

Solution:
19.99 × 9.9 + 99.9
= 197.901+ 99.9
= 297.801

৩৪৫.

Find the number of prime factors in expression (35)⁶ × (38)⁴?

ক
15
খ
18
গ
20
ঘ
22

ব্যাখ্যা

Question: Find the number of prime factors in expression (35)⁶ × (38)⁴?

Solution:
35 = 5 × 7
∴ 35⁶ = 5⁶ × 7⁶

38 = 2 × 19
∴ 38⁴ = 2⁴ × 19⁴

∴ Total number of prime factors in expression: 6 + 6 + 4 + 4 = 20

৩৪৬.

The square root of (5 + 2√7)(5 - 2√7) is:

ক
√3
খ
- √3
গ
i√5
ঘ
i√3

ব্যাখ্যা

Question: The square root of (5 + 2√7)(5 - 2√7) is:

Solution:
√{(5 + 2√7)(5 - 2√7)
= √{5² - (2√7)²}
= √{25 - (4 × 7)}
= √(25 - 28)
= √(- 3)
= √{3 × (- 1)}
= √(3 × i²) [ i² = - 1]
= i√3

৩৪৭.

A garden of 100 meter length and 60 meter width has a walkway of 2 meter width on every side. What is the area of the garden, in square meters, excluding the walkway?

ক
ক) 5376
খ
খ) 5576
গ
গ) 2556
ঘ
ঘ) 7874

ব্যাখ্যা

The area of the garden, excluding the walkway, is
= {(100 - 2×2) × (60 - 2×2)}
= 96×56
= 5376 m²

৩৪৮.

If P is the sum of positive odd numbers which are less than 84 and Q is the sum of the positive even integers that are less than 84, what is the value of Q – P?

ক
0
খ
42
গ
- 42
ঘ
84

ব্যাখ্যা

Question: If P is the sum of positive odd numbers which are less than 84 and Q is the sum of the positive even integers that are less than 84, what is the value of Q – P?

Solution:
The sum of first n even numbers, Q = n(n+1)
Number of terms, n = {(82−2)/2} + 1= 41
Hence, the sum of the first 41 even numbers is, Q = (41 × 42)

The sum of the first n odd numbers, P = n²
Number of terms, n = {(83−1)/2} + 1= 42
Hence, the sum of the first 42 odd numbers, P = (42 × 42)

Q - P = (41 × 42) - (42 × 42) = - 42

৩৪৯.

P and Q are two positive integers such that PQ = 64. Which of the following cannot be the value of P + Q?

ক
16
খ
20
গ
35
ঘ
65

ব্যাখ্যা

Question: P and Q are two positive integers such that PQ = 64. Which of the following cannot be the value of P + Q?

Solution:
Given,
P and Q are two positive integers and PQ = 64

64 = 1 × 64
= 2 × 32
= 4 × 16
= 8 × 8
The possible combinations of (P, Q)
So that, the product is 64 are (1, 64), (2, 32), (4, 16) and (8, 8)
∴ P + Q cannot be 35.

৩৫০.

What number should be divided by √0.25 to give the result as 25?

ক
ক) 13.5
খ
খ) 12.5
গ
গ) 15.5
ঘ
ঘ) 16.5

ব্যাখ্যা

Question: What number should be divided by √0.25 to give the result as 25

Solution:
x/√0.25 = 25
⇒ x/0.5 = 25
⇒ x = 25 × 0.5
⇒ x = 12.5

৩৫১.

What is the smallest number of apples that can be distributed equally (without cutting any apple) among 5,10,15 and 75 boys?

ক
ক) 350
খ
খ) 125
গ
গ) 150
ঘ
ঘ) 630

ব্যাখ্যা

প্রশ্ন : What is the smallest number of apples that can be distributed equally (without cutting any apple) among 5,10,15 and 75 boys?

সমাধান :
5,10,15, 75 এর লসাগু = 150
অতএব, সর্বনিম্ন 150টি আপেল ৬, ১০, ১৪ এবং ১৮ জন বালককে সমানভাবে ভাগ করে দেয়া যাবে।

৩৫২.

If the 10th number in a series of 10 consecutive integers has the value n + 5, find the 1st number in the series expressed in terms of n?

ক
n - 4
খ
n + 4
গ
n - 5
ঘ
n + 1
ঙ
n

ব্যাখ্যা

Question: If the 10th number in a series of 10 consecutive integers has the value n + 5, find the 1st number in the series expressed in terms of n?

Solution:
এখানে, একটি সিরিজে পরপর 10টি পূর্ণসংখ্যা আছে।
ধরা যাক, সিরিজের প্রথম সংখ্যাটি হলো x
তাহলে, সিরিজটি হবে: x, (x + 1), (x + 2), ..., (x + 9)
প্রশ্ন অনুযায়ী, সিরিজের 10ম সংখ্যাটি হলো n + 5
সুতরাং,
x + 9 = n + 5
⇒ x = n + 5 - 9
⇒ x = n - 4
সুতরাং, সিরিজের প্রথম সংখ্যাটি হলো n - 4

৩৫৩.

Find the largest number of 5-digits which, when divided by 16, 24, 30 or 36, leaves the same remainder 10 each case.

ক
99370
খ
99360
গ
99350
ঘ
99340

ব্যাখ্যা

Question: Find the largest number of 5-digits which, when divided by 16, 24, 30 or 36, leaves the same remainder 10 each case.

Solution:
১৬ = ২ × ২ × ২ × ২
২৪ = ২ × ২ × ২ × ৩
৩০ = ২ × ৩ × ৫
৩৬ = ৩ × ৩ × ২ × ২

১৬, ২৪, ৩০, ৩৬ এর ল সা গু = ২ × ২ × ২ × ২ × ৩ × ৩ × ৫
= ১৬ × ৯ ×৫
= ৭২০

পাঁচ অঙ্কের বৃহত্তম সংখ্যা ৯৯৯৯৯ কে ৭২০ দ্বারা ভাগ করলে ৬৩৯ অবশিষ্ট থাকে।
অতএব ৭২০ দ্বারা বিভাজ্য ৫ অঙ্কের বৃহত্তম সংখ্যা = ৯৯৯৯৯ - ৬৩৯
= ৯৯৩৬০

∴ নির্ণেয় সংখ্যাটি হবে = ৯৯৩৬০ + ১০
= ৯৯৩৭০

৩৫৪.

If 100 is divided into two parts in such a way that the sum of their reciprocals is 1/21, the difference between the two parts is -

ক
ক) 30
খ
খ) 40
গ
গ) 45
ঘ
ঘ) 50
ঙ
ঙ) None

ব্যাখ্যা

Question: If 100 is divided into two parts in such a way that the sum of their reciprocals is 1/21, the difference between the two parts is -

Solution:
বড় সংখ্যাটি = x
ছোট সংখ্যাটি = y

প্রশ্নমতে
x + y = 100 ...................(1)

আবার
(1/x) + (1/y) = 1/21
(y + x)/xy = 1/21
100/xy = 1/21
xy = 2100

আমরা জানি
(x - y)² = (x + y)² - 4xy
(x - y)² = (100)²- 4 × 2100
(x - y)² = 10000 - 8400
(x - y)² =1600
(x - y)² =40²
(x - y) = 40

৩৫৫.

A number when divided by 44, gives 432 as quotient and 0 as remainder. What will be the remainder when dividing the same number by 31?

ক
8
খ
5
গ
4
ঘ
0

ব্যাখ্যা

Let p be the number
p ÷ 44 = 432, remainder = 0
⇒ p = 432 × 44 + 0 = 19008
p ÷ 31
= 19008 ÷ 31
= 613, remainder = 5.

৩৫৬.

The six digit number 54321A is divisible by 9 where A is a single digit whole number. Find A.

ক
0
খ
1
গ
2
ঘ
3
ঙ
4

ব্যাখ্যা

Question: The six digit number 54321A is divisible by 9 where A is a single digit whole number. Find A.

Solution:
A number is divisible by 9, when the sum of its digits is divisible by 9.
Here, 5 + 4 + 3 + 2 + 1 + A = 15 + A should be divisible by 9.
Therefore, A = 3 gives 15 + 3 = 18 as the sum of digits, which is divisible by 9.

৩৫৭.

What is the value of

ক
4
খ
6
গ
1
ঘ
2

ব্যাখ্যা

Question: What is the value of

Solution:

৩৫৮.

What smallest number should be subtracted from 9805 so that it is divisible by 8?

ক
5
খ
6
গ
7
ঘ
8

ব্যাখ্যা

Question: What smallest number should be subtracted from 9805 so that it is divisible by 8?

Solution:
On dividing 9805 by 8, the remainder is 5. So, 5 is the smallest number which should be subtracted from 9805 to make it divisible by 8.

৩৫৯.

If the sum of 3 consecutive integer is 312, then the sum of the two smaller integer is-

ক
205
খ
206
গ
207
ঘ
208

ব্যাখ্যা

Question: If the sum of 3 consecutive integer is 312, then the sum of the two smaller integer is-

Solution:
Let,
Three consecutive integer is, x - 1, x, x + 1.

ATQ,
x - 1 + x + x + 1 = 312
⇒ 3x = 312
∴ x = 104

The sum of the two smaller integer is : x - 1 + x
= 104 - 1 + 104
= 208 - 1
= 207

৩৬০.

If 5374Q is divisible by 9, what is the value of Q?

ক
6
খ
5
গ
7
ঘ
8

ব্যাখ্যা

Question: If 5374Q is divisible by 9, what is the value of Q?

Solution:
একটি সংখ্যা 9 দ্বারা বিভাজ্য হবে যদি সংখ্যাটির অঙ্কগুলোর সমষ্টি 9 দ্বারা বিভাজ্য হয়।

5 + 3 + 7 + 4 = 19; এর সাথে Q যোগ করলে (19 + Q) হবে, যা 9 দ্বারা বিভাজ্য হতে হবে।

19 + Q = 27 (যা 9 দ্বারা বিভাজ্য নিকটতম সংখ্যা)
⇒ Q = 27 - 19
⇒ Q = 8
∴ Q = 8

৩৬১.

What is the difference of largest and smallest prime number between 60 and 80?

ক
ক) 8
খ
খ) 12
গ
গ) 18
ঘ
ঘ) 140

ব্যাখ্যা

Question: What is the difference between the largest and smallest prime number between 60 and 80?

Solution:
The primes between 60 and 80 is = 61, 67, 71, 73 and 79.

∴ the difference between the smallest and the largest prime between 60 to 80 is = (79 - 61) = 18

৩৬২.

For which of the following value of s is (50 +S)/S an integer?

ক
ক) 3
খ
খ) 4
গ
গ) 9
ঘ
ঘ) 50

ব্যাখ্যা

S = 3; (50 + 3)/3 = 17.67 which is not an integer
S = 4; (50 + 4)/4 = 13.5 which is not an integer
S = 9; (50 + 9)/9 = 6.56 which is not an integer
S = 50; (50 + 50)/50 = 2 which is an integer

৩৬৩.

A, B, C, D and E are five consecutive odd numbers. The sum of A and C is 146. What is the value of E?

ক
ক) 71
খ
খ) 75
গ
গ) 79
ঘ
ঘ) 81

ব্যাখ্যা

Let A = x
B = x + 2
C = x + 4
D = x + 6 and
E = x + 8
Then,
⇒A+C=146
⇒x+(x+4)=146
⇒2x=142
⇒x=71
∴ E=x+8=71+8=79

৩৬৪.

What number is 200 less than the greatest number you can form if you use digits 4,0,5,6 once?

ক
ক) 6140
খ
খ) 6340
গ
গ) 6540
ঘ
ঘ) 6240

ব্যাখ্যা

Question: What number is 200 less than the greatest number you can form if you use digits 4,0,5,6 once?
Solution:
0, 4, 5, and 6 দ্বারা গঠিত বৃহত্তম সংখ্যা 6540

6540 থেকে 200 বিয়োগ করে পাই:
6540 - 200 = 6340

৩৬৫.

Which one of the following equals the number of multiples of 3 between 102 and 210, inclusive?

ক
ক) 32
খ
খ) 33
গ
গ) 36
ঘ
ঘ) 37

ব্যাখ্যা

The numbers 102 and 210 are themselves multiples of 3.
also a multiple of 3 exists once in every three consecutive integers.
Counting the multiples of 3 starting with 1 for 102,
2 {= 1 + (105 - 102)/3 = 1+ 1 = 2 } for 105,
3 { = 1 +(108 - 102)/3 = 1 + 2 = 3} for 108,
and so on, the count we get for 210 equals 1+(210 - 102)/3 = 1 + 36 = 37,
hence the answer is 37

৩৬৬.

Of the following which is the closet to (6.01 × 501) ÷ (25.05 × 19.97)?

ক
6
খ
8
গ
10
ঘ
15

ব্যাখ্যা

Question: Of the following which is the closet to (6.01 × 501) ÷ (25.05 × 19.97)?

Solution:
(6.01 × 501) ÷ (25.05 × 19.97)
= (601 × 501)/100 ÷ (2505 × 1997)/10000
= (601 × 501)/100 × 10000/(2505 × 1997)
= 3011.01 × 0.00199906
= 6.0192
∴ নিকটতম মান 6

বিকল্প সমাধান:
(6.01 × 501) ÷ (25.05 × 19.97)
= (6 × 500) ÷ (25 × 20) [ যেহেতু নিকটতম মান চাওয়া হয়েছে তাই সংখ্যাগুলোকে নিকটতম পূর্ণ সংখ্যায় রূপান্তর করা হয়েছে]
= 3000 ÷ 500
= 6

৩৬৭.

Which one of the following fractions is greater than 1/2?

ক
ক) 2/5
খ
খ) 7/13
গ
গ) 4/9
ঘ
ঘ) 5/11

ব্যাখ্যা

প্রশ্ন : Which one of the following fractions is greater than 1/2?

সমাধান :

1/2 = 0.5
2/5 = 0.4
7/13 = 0.54
4/9 = 0.44
5/11 = 0.45

৩৬৮.

The product of two positive numbers is m. If each number is increased by 3, the new product is how much greater than thrice the sum of the two original number?

ক
ক) m
খ
খ) m + 4
গ
গ) m + 9
ঘ
ঘ) m + 11

ব্যাখ্যা

Question: The product of two positive numbers is m. If each number is increased by 3, the new product is how much greater than thrice the sum of the two original number?

Solution:
let, two positive number x, y
xy = m

After increasing 3, their product = (x + 3) (y + 3)
= xy + 3 (x + y) + 9
= m + 3 (x + y) + 9

thrice of sum of two original number = 3 (x + y)

∴ m + 3 (x + y) + 9 - 3 (x + y)
= m + 9

৩৬৯.

The difference between a number and its square is 56. What is the number?

ক
ক) 6
খ
খ) 7
গ
গ) 8
ঘ
ঘ) None of the above

ব্যাখ্যা

Question: The difference between a number and its square is 56. What is the number?

Solution:
ধরি,
সংখ্যাটি x
প্রশ্নমতে,
x² - x = 56
x² - x - 56 = 0
x² - 8x + 7x - 56 = 0
x(x - 8) + 7(x - 8) = 0
(x - 8)(x + 7) = 0

হয়
x - 8 = 0
x = 8

অথবা
x + 7 = 0
x = - 7 [গ্রহণযোগ্য নয় ]

৩৭০.

If p is an even integer and q is an odd integer, which of the following must be an odd integer?

ক
11pq
খ
4(p + q)
গ
3p + q
ঘ
4p - q + 1

ব্যাখ্যা

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

Solution:
3 × even integer = even integer
So, 3p is an even integer

even integer + odd integer = odd integer

So,
3p + q must be an odd integer.

৩৭১.

48.2 × 2.5 × 2.2 + ? = 270

ক
ক) 2.9
খ
খ) 4.8
গ
গ) 4.9
ঘ
ঘ) 5.0

ব্যাখ্যা

Let the missing number be x
Given, 48.2 × 2.5 × 2.2 + x = 270
⇒ x = 270 - 48.2 × 2.5 × 2.2
⇒ x = 270 - 265.1
⇒ x = 4.9
Hence, the number is 4.9

৩৭২.

The sum of four consecutive multiples of 2 is 52. What is the largest number?

ক
ক) 14
খ
খ) 16
গ
গ) 18
ঘ
ঘ) 20

ব্যাখ্যা

Question: The sum of four consecutive multiples of 2 is 52. What is the largest number?

Solution:
মনে করি,
2 এর চারটি ধারাবাহিক গুণিতক যথাক্রমে 2x, 2(x + 1), 2(x + 2) ও 2(x + 3)

প্রশ্নমতে,
2x + 2(x + 1) + 2(x + 2) + 2(x + 3) = 52
বা, 2x + 2x + 2 + 2x + 4 + 2x + 6 = 52
বা, 8x + 12 = 52
বা, 8x = 52 - 12
বা, 8x = 40
বা, x = 40/8
∴ x = 5

∴ বৃহত্তম সংখ্যা = 2(5 + 3) = 2 × 8 = 16

৩৭৩.

The sum of four consecutive even numbers is 140. What is the sum of next two consecutive even numbers-

ক
76
খ
66
গ
82
ঘ
90

ব্যাখ্যা

Question: The sum of four consecutive even numbers is 140. What is the sum of next two consecutive even numbers-

Solution:
let, four consecutive even numbers is
x, x + 2, x + 4, x + 6

ATQ,
⇒ x + x + 2 + x + 4 + x + 6 = 140
⇒ 4x + 12 = 140
⇒ 4x = 140 - 12 = 128
⇒ x = 128/4
∴ x = 32

So four consecutive even numbers is- 32, 34, 36, 38

∴ next two consecutive even numbers- 40, 42
∴ sum = 40 + 42 = 82

৩৭৪.

The sum of three consecutive multiples of 4 is 300. What is the largest number?

ক
96
খ
100
গ
104
ঘ
108

ব্যাখ্যা

Question: The sum of three consecutive multiples of 4 is 300. What is the largest number?

Solution:
Let,
First multiple 4x
Second multiple 4(x + 1) = 4x + 4
Third multiple 4(x + 2) = 4x + 8

ATQ,
4x + 4x + 4 + 4x + 8 = 300
⇒ 12x + 12 = 300
⇒ 12x = 288
⇒ x = 288/12
∴ x = 24

So, largest number = 4x + 8
= (4 × 24) + 8
= 104

৩৭৫.

The ratio of two numbers is 15 : 28 and their HCF is 17. Find the numbers.

ক
51, 95
খ
85, 152
গ
170, 280
ঘ
255, 476

ব্যাখ্যা

Question: The ratio of two numbers is 15 : 28 and their HCF is 17. Find the numbers.

Solution:
Let the numbers be a and b.
Since the HCF is given as 17, we can write the numbers as:
a = 17 × 15 = 255,
b = 17 × 28 = 476

Check the ratio:
a : b = 255 : 476 =15 : 28 (justified)

৩৭৬.

If the number of boys in a class is 8 times the number of girls, which value can never be the total number of students?

ক
42
খ
81
গ
27
ঘ
45

ব্যাখ্যা

Let the number of girls = x
number of boys = 8x
Then, total number of students = x + 8x
= 9x
i.e., The total number of students must be a multiple of 9. In the given choices, 42 is not a multiple of 9
Hence, the total number of students cannot be 42.

৩৭৭.

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

ক
01469
খ
10469
গ
90146
ঘ
10000

ব্যাখ্যা

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.

৩৭৮.

(√1372 + √959) ÷ √292 × 19.003 = ?

ক
ক) 77
খ
খ) 97
গ
গ) 39
ঘ
ঘ) 19

ব্যাখ্যা

(√1372 + √959) ÷ √292 × 19.003
= (37.04 + 30.96) ÷ 17 × 19.003
= 68 ÷ 17 × 19.003
= 4 × 19.003 = 76.012 ≈ 77

৩৭৯.

What is the greatest prime factor of (2⁴)² - 1 ?

ক
ক) 3
খ
খ) 5
গ
গ) 11
ঘ
ঘ) 17

ব্যাখ্যা

Question: What is the greatest prime factor of (2⁴)² - 1 ?

Solution:
(2⁴)² - 1
= (2⁴- 1)(2⁴ + 1)
= (16 + 1)(16 - 1)
= 17 × 15
= 3 × 5 × 17

বৃহত্তম মৌলিক উৎপাদক হলো = 17

৩৮০.

The least number by which 150 must be multiplied to make it a perfect square is-

ক
2
খ
3
গ
5
ঘ
6
ঙ
9

ব্যাখ্যা

Question: The least number by which 150 must be multiplied to make it a perfect square is-

Solution:
একটি সংখ্যা পূর্ণবর্গ সংখ্যা হতে হলে তার মৌলিক গুণনীয়কগুলোকে অবশ্যই জোড় সংখ্যায় (even power) থাকতে হবে।

150 = 2 × 3 × 5 × 5 = 2¹ × 3¹ × 5²

জোড়া গঠন করে পাই, 2 × 3 × (5 × 5)
এখানে জোড়া বিহীন সংখ্যা 2 এবং 3

∴ 150 কে (2 × 3) = 6 দ্বারা গুণ করলে এটি পূর্ণবর্গ সংখ্যা হবে।

৩৮১.

Let U = {1,2,3,4,5,6,7,8}, A = {2,3,6}, and B = {1,4,5}. Find A^c∪B^c.

ক
{1,2,3,4,5,6,7,8}
খ
{ }
গ
{2,3,6}
ঘ
{1,4,5,7,8}
ঙ
{7,8}

ব্যাখ্যা

Question: Let U = {1,2,3,4,5,6,7,8}, A = {2,3,6}, and B = {1,4,5}. Find A^c∪ B^c.

Solution:
Complement of A:
A = {2,3,6}
U = {1,2,3,4,5,6,7,8}
A^c = U - A = {1,4,5,7,8}

Complement of B:
B = {1,4,5}
B^c = U - B = {2,3,6,7,8}

Union of Complements:
A^c ∪ B^c = {1,4,5,7,8} ∪ {2,3,6,7,8} = {1,2,3,4,5,6,7,8} = U

∴A^c ∪ B^c = {1,2,3,4,5,6,7,8}

৩৮২.

The product of two numbers is 120 and the sum of their squares is 289. The sum of the number is:

ক
7
খ
23
গ
9
ঘ
11

ব্যাখ্যা

Question: The product of two numbers is 120 and the sum of their squares is 289. The sum of the number is:

Solution:
let the two numbers be x, y

xy = 120
x² + y² = 289

We know
(x + y)² = x² + y² + 2xy
⇒ (x + y)² = 289 + 2 × 120
⇒ (x + y)² = 289 + 240
⇒ (x + y)² = 529
⇒ (x + y)² = 23²
∴ x + y = 23

৩৮৩.

Two-fifths of one-fourth of three-seventh of a number is 18. What is half of the number?

ক
ক) 175
খ
খ) 420
গ
গ) 350
ঘ
ঘ) 210

ব্যাখ্যা

প্রশ্ন : Two-fifths of one-fourth of three-seventh of a number is 15. What is half of the number?

সমাধান :

ধরি, সংখ্যাটি = x
ATQ,
2/5 × 1/4 × 3/7 × x = 18
Or, 6x / 140 = 18
Or, x = 420

∴ 1/2 of 420 = 210

৩৮৪.

There are 8 more men than women on board of directors of a company. If there are 20 members on the board, how many are men?

ক
ক) 6
খ
খ) 8
গ
গ) 12
ঘ
ঘ) 14
ঙ
ঙ) 16

ব্যাখ্যা

Question: There are 8 more men than women on board of directors of a company. If there are 20 members on the board, how many are men?

Solution:
Let,
Number of men = x
∴ Number of women = x - 8

ATQ,
x + x - 8 = 20
⇒ 2x = 28
∴ x = 14

∴ There are 14 men on the board.

৩৮৫.

The least number by which 294 must be multiplied to make it a perfect square is :

ক
2
খ
3
গ
6
ঘ
24

ব্যাখ্যা

Question: The least number by which 294 must be multiplied to make it a perfect square is :

Solution:
294 = 7 × 7 × 2 × 3

এখানে
2 এবং 3 জোড়াবিহীন
2 × 3 = 6 দ্বারা গুণ করলে 294 সংখ্যাটি পূর্ণবর্গ সংখ্যা হবে।

৩৮৬.

The sum of five consecutive multiples of 6 is 150. What is the second largest number?

ক
46
খ
35
গ
36
ঘ
30

ব্যাখ্যা

Question: The sum of five consecutive multiples of 6 is 150. What is the second largest number?

Solution:
ধরি, ৬ এর পাঁচটি ক্রমিক গুণিতক হলো যথাক্রমে (x - 12), (x - 6), x, (x + 6) এবং (x + 12)

প্রশ্নমতে,
(x - 12) + (x - 6) + x + (x + 6) + (x + 12) = 150
⇒ 5x = 150
⇒ x = 150/5
⇒ x = 30

সুতরাং, সংখ্যাগুলো হলো 18, 24, 30, 36, 42।
এদের মধ্যে দ্বিতীয় বৃহত্তম সংখ্যাটি হলো 36।

৩৮৭.

If p and n are integers such that p > n > 0 and p² - n²= 16, which of the following value of p - n?

ক
ক) 4
খ
খ) 2
গ
গ) 3
ঘ
ঘ) 1

ব্যাখ্যা

Question: If p and n are integers such that p > n > 0 and p² - n²= 16, which of the following value of p - n?

Solution:

p > n > 0 এবং p² - n² = 12

ধরি
p = 5 , n = 3

p² - n²= 16
5² - 3² = 16

সুতরাং,
p - n = 5 - 3 = 2

৩৮৮.

A number is as much greater than 31 as less than 61. Which of the follwing is that numbers?

ক
ক) 15
খ
খ) 46
গ
গ) 26
ঘ
ঘ) 30

ব্যাখ্যা

Question: A number is as much greater than 31 as less than 61. Which of the follwing is that numbers?

Solution:
let
the number be a
Now
a - 31 = 61 - a
a + a = 61 + 31
2a = 92
a = 46

৩৮৯.

If 3 less than twice the number is equal to 2 more than 3 times the number, then 5 less than 5 times the number is:

ক
- 25
খ
- 30
গ
46
ঘ
25

ব্যাখ্যা

Question: If 3 less than twice the number is equal to 2 more than 3 times the number, then 5 less than 5 times the number is:

Solution:
Let the number be x

ATQ,
2x - 3 = 3x + 2
∴ x = - 5

∴ Five times the number = 5 × (- 5)
= - 25
∴ Five less than this = - 25 - 5
= - 30

৩৯০.

Rakesh had to do a multiplication instead of taking 25 as one of the multipliers, she took 52. As a result, the product went up by 540. What is the new product?

ক
ক) 1050
খ
খ) 1060
গ
গ) 1040
ঘ
ঘ) 1080

ব্যাখ্যা

Question: Rakesh had to do a multiplication instead of taking 25 as one of the multipliers, she took 52. As a result, the product went up by 540. What is the new product?

Solution:
ধরি, সংখ্যাটি = x

প্রশ্নমতে
52x - 25x = 540
⇒27x = 540
⇒x = 540/27
⇒ x = 20

নতুন গুণফল = 52 × 20 = 1040

৩৯১.

If 1² + 2² + 3² + .....+ 10² = 385, then 2² + 4² + 6² + .... + 20² =?

ক
1155
খ
770
গ
385²
ঘ
1540

ব্যাখ্যা

Question: If 1² + 2² + 3² + .....+ 10² = 385, then 2² + 4² + 6² + .... + 20² =?

Solution:
2² + 4² + 6² + .... + 20²
= 2² (1 + 2² + 3² +....+10²)
= 4 × 385
= 1540

৩৯২.

If the sum of a number and its reciprocal be 2. What is the number?

ক
0
খ
- 1
গ
1
ঘ
- 3

ব্যাখ্যা

Question: If the sum of a number and its reciprocal be 2. What is the number?

Solution:
Let the number be = a
The reciprocal of the number is = 1/a

According to the question,
a + (1/a) = 2
⇒ a² + 1 = 2a
⇒ a² - 2a + 1 = 0
⇒ (a - 1)² = 0
∴ a = 1

Hence, the number = 1

৩৯৩.

A two-digit number becomes five-sixth of itself when its reversed. Two digits differ by one. The number is -

ক
ক) 45
খ
খ) 43
গ
গ) 54
ঘ
ঘ) 53

ব্যাখ্যা

Question: A two-digit number becomes five-sixth of itself when its reversed. Two digits differ by one. The number is -

Solution:
(যেহেতু অঙ্ক উল্টানোর পর সংখ্যাটি ছোট হয়, তাই মূল সংখ্যার একক স্থানীয় অঙ্কটি ছোট এবং দশক স্থানীয় অঙ্কটি বড়।

ধরি,
একক স্থানীয় অঙ্ক = x
প্রশ্নমতে, দশক অঙ্ক = x + 1

সংখ্যাটি = 10(x + 1) + x = 11x + 10
সংখ্যাদ্বয় স্থান বিনিময় করলে, নতুন সংখ্যা = 10x + (x + 1) = 11x +1

প্রশ্নমতে,
(5/6)(11x + 10) =11x + 1
⇒ 55x + 50 = 66x + 6
⇒ 11x = 44
⇒ x = 4

সংখ্যাটি = 11 × 4 + 10 = 54

৩৯৪.

Sum of a rational number and its reciprocal is 13/6. Find the number -

ক
ক) 2
খ
খ) 3/2
গ
গ) 4/2
ঘ
ঘ) 5/2

ব্যাখ্যা

⇒ x + 1/x = 13/6
⇒ (x² + 1)/x = 13/6
⇒ 6x² - 13x + 6 = 0
⇒ 6x² - 9x - 4x + 6 = 0
⇒ 3x(2x - 3) -2(2x - 3) = 0
⇒ (3x - 2)(2x - 3) = 0
⇒ x = 2/3 or 3/2

৩৯৫.

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

ক
m² + m
খ
m - 2
গ
5m + 3
ঘ
2m + 4

ব্যাখ্যা

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

Solution:
ধরি m = 2

ক) m² + m = 2² + 2 = 4 + 2 = 6 [যা জোড়]

খ) m - 2 = 2 - 2 = 0 [যা জোড়]

গ) 5m + 3 = 5 × 2 + 3 = 10 + 3 = 13 [যা বিজোড়]

ঘ) 2m + 4 = 2 × 2 + 4 = 8 [যা জোড়]

∴ উত্তর: গ) 5m + 3

৩৯৬.

The difference of two numbers is 20% of the larger number. If the smaller number is 12, the larger one is-

ক
15
খ
16
গ
18
ঘ
20

ব্যাখ্যা

Question: The difference of two numbers is 20% of the larger number. If the smaller number is 12, the larger one is-

Solution:
Let,
The larger number = x

According to the question,
x - 12 = 20% of x
⇒ x - 12 = 20x/100
⇒ x - 12 = x/5
⇒ 5x - 60 = x
⇒ 5x - x = 60
⇒ 4x = 60
⇒ x = 60/4
∴ x = 15

So, the larger number is 15

৩৯৭.

What will come at the place of the question mark?
5, 6, 8, 12, 20, ?

ক
36
খ
22
গ
24
ঘ
28

ব্যাখ্যা

Question: What will come at the place of the question mark?
5, 6, 8, 12, 20, ?

Solution:
5 + 1 =6
6 + 2 = 8
8 + 4 = 12
12 + 8 = 20
20 + 16 = 36

৩৯৮.

The difference between two numbers is 1365. When the larger number is divided by the smaller one, the quotient is 6 and the remainder is 15. What is the smaller number?

ক
ক) 240
খ
খ) 270
গ
গ) 295
ঘ
ঘ) 360

ব্যাখ্যা

Let the smaller number be x.
Then, a large number
= x + 1365.
∴ x + 1365 = 6x + 15
5x = 1350
x = 270.

৩৯৯.

A group of students decided to collect as many paise from each member of group as is the number of members. If the total collection amounts to Tk. 59.29, the number of the member is the group is:

ক
ক) 73
খ
খ) 75
গ
গ) 77
ঘ
ঘ) 79

ব্যাখ্যা

Money collected = (59.29 x 100) paise = 5929 paise.
Number of members = √5929 = 77.

৪০০.

6⁴ⁿ - 5⁴ⁿ, where n is an integer > 0, is divisible by-

ক
11
খ
15
গ
25
ঘ
31

ব্যাখ্যা

Question: 6⁴ⁿ - 5⁴ⁿ, where n is an integer > 0, is divisible by-

Solution:
6⁴ⁿ - 5⁴ⁿ

Assume that n = 1
= 6⁴ - 5⁴
= (6²)² - (5²)²
= (6² - 5²)(6² + 5²)
= (36 - 25) × (36 + 25)
= 11 × 61
The expression is divisible by 11 (and also 61 when n = 1)