Number System, Problems on Number

PrepBank · পাতা ২ / ১৮ · ১০১–২০০ / ১,৭৩৬

১০১.

If x > 2 and x < 5, then which of the following expressions is positive?
I. (x - 2)(x - 5)
II. (2 - x)(x - 5)
III. (2 - x)(5 - x)

ক
I only
খ
II only
গ
III only
ঘ
I and III
ঙ
None of these

ব্যাখ্যা

Question: If x > 2 and x < 5, then which of the following expressions is positive?
I. (x - 2)(x - 5)
II. (2 - x)(x - 5)
III. (2 - x)(5 - x)

Solution:
Given,
x > 2 and x < 5

For expression I: (x - 2)(x - 5)
Since x > 2, (x - 2) will be positive.
Since x < 5, (x - 5) will be negative.
(x - 2)(x - 5) = positive × negative = negative

For expression II: (2 - x)(x - 5)
Since x > 2, (2 - x) will be negative.
Since x < 5, (x - 5) will be negative.
(2 - x)(x - 5) = negative × negative = positive

For expression III: (2 - x)(5 - x)
Since x > 2, (2 - x) will be negative.
Since x < 5, (5 - x) will be positive.
(2 - x)(5 - x) = negative × positive = negative

∴ Only expression II is positive.

১০২.

Two number are in the ratio 3 : 5. If 9 is subtracted from each, the new numbers are in the ratio 12 : 23. The smaller number is

ক
ক) 27
খ
খ) 33
গ
গ) 49
ঘ
ঘ) 55

ব্যাখ্যা

Let the numbers be 3x and 5x.
Then, (3x-9)/(5x - 9) = 12/23
⇒ 23(3x - 9) = 12(5x - 9)
⇒ 69x - 207 = 60x - 108
⇒ 9x = 99
⇒ x = 11.
∴ The smaller number = (3×11) = 33

১০৩.

The sum of the digits of a two-digit number is 12 and the difference between the two digits of the two-digit number is 6. What is the two-digit number?

ক
75
খ
95
গ
39
ঘ
84

ব্যাখ্যা

Question: The sum of the digits of a two-digit number is 12 and the difference between the two digits of the two-digit number is 6. What is the two-digit number?

Solution:
Let, the two-digit number be 10a + b where a > b.

ATQ,
a + b = 12 ------- (1)
a - b = 6 --------- (2)

On adding equation (1) & (2)
21 =18
∴ a = 9

Putting this value in (1) we get,
9 + b = 12
∴ b = 3

So the number is 10a + b = 9 . 10 + 3 = 93

When, a < b, Then the required number is 39

১০৪.

The LCM and the HCF of the numbers 28 and 42 are in the ratio-

ক
ক) 5 : 1
খ
খ) 2 : 3
গ
গ) 3 : 2
ঘ
ঘ) 6 : 1

ব্যাখ্যা

প্রশ্ন: 28 এবং 42 এর ল.সা.গু ও গ.সা.গু এর অনুপাত কত?

সমাধান:
28 ও 42 এর ল.সা.গু= 84
28 ও 42 এর গ.সা.গু = 14

∴ ল.সা.গু : গ.সা.গু = 84 : 14 = 6 : 1

১০৫.

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

ক
150
খ
190
গ
210
ঘ
250

ব্যাখ্যা

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

Solution:
Let the number be x

ATQ,
(2/5) × (1/4) × (3/7) × x = 15
⇒ (1/10) × (3/7) × x = 15
⇒ (3/70) × x = 15
⇒ x = (15 × 70)/3
∴ x = 350

The Three-fifths of the number = (3 × 350)/5
= 210

১০৬.

n is an integer between 40 and 90, then any of the following could be n + 8 except -

ক
52
খ
65
গ
75
ঘ
99

ব্যাখ্যা

Question: n is an integer between 40 and 90, then any of the following could be n + 8 except -

Solution:
if n is a positive integer, then 40 < n < 90
from option (4)
n + 8 = 99
n = 91

its not possible

১০৭.

A circle has a number of tangents equal to-

ক
0
খ
1
গ
2
ঘ
3
ঙ
Infinite

ব্যাখ্যা

A circle has infinitely many tangents, touching the circle at infinite points on its circumference.

১০৮.

When 17 is divided by k, where K is a positive integer less than 17, the remainder is 3. What is the remainder when the sum of possible values of it is divided by 17?

ক
ক) 2
খ
খ) 3
গ
গ) 4
ঘ
ঘ) 5

ব্যাখ্যা

Question: When 17 is divided by k, where K is a positive integer less than 17, the remainder is 3. What is the remainder when the sum of possible values of it is divided by 17?

Solution:
যেহেতু k < 17, এটা বোঝায় যে k এর সম্ভাব্য মান অবশ্যই 1 থেকে 16 এর মধ্যে হতে হবে।
1 এবং 16 এর মধ্যে, শুধুমাত্র 7 বা 14 দ্বারা 17 কে ভাগ করতে ব্যবহৃত হয় অবশিষ্ট 3 দেবে
তাই, 7 এবং 14 হল k এর সম্ভাব্য মান।
k এর সম্ভাব্য মানের সমষ্টি = 7 + 14 = 21
k-এর সম্ভাব্য মানের সমষ্টি 21 কে 17 দ্বারা ভাগ করলে 4 ভাগশেষ থাকবে।

১০৯.

If you divide 30 by half and add 10 with the resulting figure, then what is the final result?

ক
ক) 25
খ
খ) 70
গ
গ) 45
ঘ
ঘ) 55

ব্যাখ্যা

Question: If you divide 30 by half and add 10 with the resulting figure, then what is the final result?

Solution:
dividing 30 by half we get = 30/(1/2) = 60

adding 10 we get = 60 + 10 = 70

১১০.

The sum of four numbers is 64. If you add 3 to the first number, 3 is subtracted from the second number, the third is multiplied by 3 and the fourth is divided by 3, then all the results are equal. What is the difference between the largest and the smallest of the original numbers?

ক
ক) 21
খ
খ) 27
গ
গ) 32
ঘ
ঘ) 36

ব্যাখ্যা

Let the four numbers be, A, B, C and D
Let A + 3 = B - 3 = 3C = D/3 = x
Then,
A = x - 3
B = x + 3
C = x/3
D = 3x
⇒ (A + B + C + D) = 64
⇒ (x - 3) + (x + 3) + x/3 + 3x) = 64
⇒ 5x + x/3 = 64
⇒ 16x = 192
⇒ x = 12.
Thus the numbers are 9, 15, 4 and 36.
Required difference
= 36 - 4
= 32.

১১১.

A number consists of two digits such that the digit in the ten's place is less by 2 than the digit in the unit's place. Three times the number added to 6/7 times the number obtained by reversing the digits 108. The sum of the digits in the number is -

ক
ক) 4
খ
খ) 5
গ
গ) 8
ঘ
ঘ) 6

ব্যাখ্যা

Question: A number consists of two digits such that the digit in the ten's place is less by 2 than the digit in the unit's place. Three times the number added to 6/7 times the number obtained by reversing the digits 108. The sum of the digits in the number is -

Solution:
ধরি,
একক স্থানীয় অঙ্কটি x এবং দশক স্থানীয় অঙ্কটি (x - 2)

সংখ্যাটি = 10((x - 2) + x
= 11x - 20

স্থান বিনিময়কৃত সংখ্যাটি = 10x + x - 2
= 11x - 2

ATQ,
3(11x - 20) + (6/7) (11x - 2) = 108
⇒ 33x - 60 + (6/7) (11x - 2) = 108
⇒ 231x - 420 + 66x - 12 = 756
⇒ 297x = 756 + 432
⇒ 297x = 1188
⇒ x = 1188/297
∴ x = 4

∴ অঙ্কগুলোর সমষ্টি = x + x - 2
= 2x - 2
= (2 × 4) - 2
= 6

১১২.

If you divide 40 by half and add 5 with the resulting figure, then what is the final result?

ক
ক) 25
খ
খ) 30
গ
গ) 35
ঘ
ঘ) 85

ব্যাখ্যা

Question: If you divide 40 by half and add 5 with the resulting figure, then what is the final result?

Solution:
40/(1/2)
= 80

adding 5 = 80 + 5
= 85

১১৩.

Sum of three different positive integers is the same as their product. What is the largest of these 3 integers?

ক
ক) 1
খ
খ) - 1
গ
গ) 2
ঘ
ঘ) 3

ব্যাখ্যা

ধরি,
প্রথম সংখ্যা = 1 দ্বিতীয় সংখ্যা = 2 এবং তৃতীয় সংখ্যা = 3

সংখ্যা তিনটির যোগফল = 1 + 2 + 3 = 6
সংখ্যা তিনটির গুণফল = 1 × 2 × 3 = 6
সবচেয়ে বড় সংখ্যা = 3

১১৪.

The sum of four consecutive two-digit odd numbers, when divided by 10, become a perfect square. Which of the following can possibly be one of these four numbers?

ক
ক) 21
খ
খ) 41
গ
গ) 25
ঘ
ঘ) 67

ব্যাখ্যা

Suppose the numbers are x, x+2, x+4, x+6
Their sum will be 4x +12, which when divided by 10 gives us (4x+12)/10 becoming a perfect square.
The expression 4x+12 ends in 0 to be divisible by 10, which means 4x has to end at 8.
Any odd number multiplied by 4, giving us product ending at 8 means that odd number ends at 7.
Let the first number now be n7 then the 4 numbers would be n7, n9, (n+1)1 and (n+1)3.
The expression of sum of the 4 numbers divided by 10 can be expressed now as (10*n+7)+(10n+9)+{(n+1)*10+1}+{(n+1)*10+3}/10.
This can be simplified to (40n+40)/10 or can be expressed as 4n+4.
Value of 4n+4 for all values of n from 2 to 8 is 12(for n=2), 16(for n=3), 20(for n=4), 24(for n=5), 28(for n=6), 32(for n=7) and 36(for n=8).
Out of these, only 16 and 36 are perfect squares
Therefore the two possible sets of 4 such numbers will be 37, 39, 41 & 43 and 87, 89, 91 & 93.
---------------------------------------------
---------------------------------------------
বিকল্প - ১:
Using options,
We find that four consecutive odd numbers are 37, 39, 41 and 43
The sum of these 4 numbers is 160, when divided by 10 we get 16 which is a perfect square.
Thus, 41 is one of the odd numbers

---------------------------------------------
---------------------------------------------
বিকল্প - ২:
দুই অঙ্কবিশিষ্ট চারটি ক্রমিক বিজোড় সংখ্যার যোগফল ১০ দ্বারা বিভাজ্য হলে, পূর্ণ বর্গ সংখ্যা পাওয়া যাবে এমন সংখ্যা চারটির একটি হচ্ছে ৪১
দুই অঙ্কবিশিষ্ট চারটি ক্রমিক বিজোড় সংখ্যা ৩৭, ৩৯, ৪১ ও ৪৩
এদের যোগফল = ৩৭ + ৩৯ + ৪১ + ৪৩ = ১৬০
১৬০ কে ১০ দ্বারা ভাগ করলে ১৬ পাওয়া যায় যা পূর্ণ বর্গ সংখ্যা
অপশনে ৪১ থাকায় সঠিক উত্তর ৪১

১১৫.

How many perfect squares lie from 100 to 300?

ক
ক) 6
খ
খ) 7
গ
গ) 8
ঘ
ঘ) 9

ব্যাখ্যা

Question: How many perfect squares lie from 100 to 300?

Solution:
(10)²= 100 And
(17)²= 289

So, the perfect squares between 100 and 300 are the squares of numbers from 10 to 17.
Clearly, there are 8 in number.

১১৬.

A student multiplied 765 by a certain number and got 448835 as their answer. If in the answer both 8s are wrong, but the other digits are correct, then what will be the correct answer?

ক
446435
খ
445935
গ
444635
ঘ
442935
ঙ
442560

ব্যাখ্যা

Question: A student multiplied 765 by a certain number and got 448835 as their answer. If in the answer both 8s are wrong, but the other digits are correct, then what will be the correct answer?

Solution:
The correct answer must be divisible by 765.

By checking possible corrections:

446435 ÷ 765 → remainder ≠ 0
445935 ÷ 765 → remainder ≠ 0
444635 ÷ 765 → remainder ≠ 0
442935 ÷ 765 → remainder = 0 → quotient = 579

Hence, the correct answer = 442935

১১৭.

The average of three times a number and its square is nine times the number. The number is -

ক
ক) 9
খ
খ) 12
গ
গ) 15
ঘ
ঘ) 18

ব্যাখ্যা

Question: The average of three times a number and its square is nine times the number. The number is -

Solution:
Let, the number be x

Now,
(3x + x²)/2 = 9x
⇒ 3x + x² = 18x
⇒ x² - 15x = 0
⇒ x(x - 15) = 0
Now, x = 0 or 15

১১৮.

When 4 is added to 1/2 of a number, the result is 14. What is the number?

ক
ক) 20
খ
খ) 21
গ
গ) 27
ঘ
ঘ) 35

ব্যাখ্যা

Let,
The number is x,
therefore, (1/2) x + 4 = 14,
so, x = 20

১১৯.

If p and q are odd numbers, which of the following is always odd?

ক
p + q + 2
খ
pq + 2
গ
2p + q + 1
ঘ
p² + q

ব্যাখ্যা

Question: If p and q are odd numbers, which of the following is always odd?

Solution:
Let p = 1 and q = 3 (both are odd numbers)

a) p + q + 2 = 1 + 3 + 2 = 6 ............. Even

b) pq + 2 = (1 × 3) + 2 = 5 ......... Odd

c) 2p + q + 1 = (2 × 1) + 3 + 1 = 2 + 4 = 6 ......... Even

d) p² + q = (1)² + 3 = 1 + 3 = 4 .......... Even

১২০.

A jar contains red balls and green balls in the ratio 3 : 1. If the jar contains only the two types of balls, which of the following cannot be the number of balls in the jar?

ক
ক) 96
খ
খ) 80
গ
গ) 72
ঘ
ঘ) 54

ব্যাখ্যা

Question: A jar contains red balls and green balls in the ratio 3: 1. If the jar contains only the two types of balls, which of the following cannot be the number of balls in the jar?

Solution:
পাত্রে লাল বল ও সবুজ বলের অনুপাত 3: 1
লাল বল = 3x
সবুজ বল = x

মোট বল = 3x + x = 4x

যদি মোট বলের সংখ্যা 96 টি হয়,
4x = 96
x = 24 ; যা একটি পূর্ণসংখ্যা

যদি মোট বলের সংখ্যা 80 টি হয়,
4x = 80
x = 20 ; যা একটি পূর্ণসংখ্যা

যদি মোট বলের সংখ্যা 72 টি হয়,
4x = 72
x = 18 ; যা একটি পূর্ণসংখ্যা

যদি মোট বলের সংখ্যা 54 টি হয়,
4x = 54
x = 13.5 ; যা একটি পূর্ণসংখ্যা নয়।

অতএব, মোট বলের সংখ্যা ৫৪ হতে পারে না।

১২১.

If one-fourth of one-sixth of a number is 5, then what is 3/4 of the number?

ক
56
খ
26
গ
70
ঘ
90

ব্যাখ্যা

Question: If one-fourth of one-sixth of a number is 5, then what is 3/4 of the number?

Solution:
Let the number be x.
According to the question,
(1/4) of (1/6) of x = 5
⇒ (1/4) × (1/6) × x = 5
⇒ (1/24)x = 5
⇒ x = 5 × 24
∴ x = 120

Now,
(3/4) of x = (3/4) × 120
= 360/4
= 90

১২২.

What is the least number which when divided by the numbers 3, 5, 6, 8, 10, and 12 leaves no remainder?

ক
ক) 96
খ
খ) 120
গ
গ) 150
ঘ
ঘ) 180

ব্যাখ্যা

Question: What is the least number which when divided by the numbers 3, 5, 6, 8, 10, and 12 leaves no remainder?

Solution:
LCM of 3, 5, 6, 8, 10, and 12 = 120

১২৩.

If x and y are integers and |x - y| = 12, what is the minimum possible value of xy?

ক
- 12
খ
- 24
গ
- 36
ঘ
- 48
ঙ
- 52

ব্যাখ্যা

Question: If x and y are integers and |x - y| = 12, what is the minimum possible value of xy?

Solution:
Squaring both sides, we get (x - y)² = 144
⇒ x² + y² - 2xy = 144
⇒ x² + y² - 2xy + 4xy = 144 + 4xy [By adding, 4xy to both sides of the equation]
⇒ x² + y² + 2xy = 144 + 4xy
⇒ (x + y)² = 144 + 4xy
⇒ (x + y)² ≥ 0 [(x + y)² will not be negative for real values of x and y]

∴ 144 + 4xy ≥ 0
⇒ 4xy ≥ -144
∴ xy ≥ -36

The least value that xy can take is - 36.

১২৪.

A two-digit number has 3 in its unit digit. The sum of its digits is one seventh of the number itself. What is the number?

ক
ক) 73
খ
খ) 53
গ
গ) 63
ঘ
ঘ) 83

ব্যাখ্যা

Let the number be 10x + 3
ATQ,
7(x + 3) = 10x + 3
⇒ 7x + 21 = 10x + 3
⇒ 21 - 3 = 10x – 7x
⇒ 3x = 18
⇒ x = 6
∴ The number is, 10×6 + 3 = 63

১২৫.

The LCM and HCF of two numbers is 1820 and 26. If one number is 130 then the other number is?

ক
ক) 364
খ
খ) 1690
গ
গ) 70
ঘ
ঘ) 1264

ব্যাখ্যা

প্রশ্ন: দুটি সংখ্যার ল.সা.গু এবং গ.সা.গু যথাক্রমে 1820 এবং 26 । একটি সংখ্যা 130 হলে অপর সংখ্যাটি কত?

সমাধান:
ধরি,
অপর সংখ্যাটি = p
আমরা জানি,
দুটি সংখ্যার গুণফল = ল.সা.গু × গ.সা.গু
130p = 1820 × 26
p = (1820 × 26)/130
= 364

১২৬.

The number of terms between 11 and 200 which are divisible by 7 but not by 3 is -

ক
ক) 18
খ
খ) 19
গ
গ) 27
ঘ
ঘ) 28

ব্যাখ্যা

Multiple of 7 between 11 and 200 are 14, 21, 28, 35, 42,....., 189, 196
T_m = 196 ⇒ 14 + (m - 1) × 7 = 196
⇒ (m - 1) × 7 = 182
⇒ (m - 1) = 26
⇒ m = 27.
Multiples of 7 and 3 both i.e. that of 21 are 21, 42, 63,......., 189
T_n = 189
⇒ 21 + (n - 1) × 21 = 189
⇒ (n - 1) × 21 = 168
⇒ (n - 1) = 8
⇒ n = 9.
Required number of terms = (27 - 9) = 18
Answer : 18

১২৭.

ক
ক) 17
খ
খ) 19
গ
গ) 361
ঘ
ঘ) 21

ব্যাখ্যা

Question:

Solution:
এখানে,
√27225 = 165
√38416 = 196

এখন,

= √(165 + 196)
= √361
= 19

১২৮.

P and Q are two positive integers such that PQ = 60. Which of the following cannot be the value of P + Q?

ক
16
খ
19
গ
20
ঘ
23

ব্যাখ্যা

Question: P and Q are two positive integers such that PQ = 60. Which of the following cannot be the value of P + Q?

Solution:
60-এর উৎপাদক জোড়াগুলো হল:
1 × 60 = 60 ⇒ P + Q = 1 + 60 = 61
2 × 30 = 60 ⇒ P + Q = 2 + 30 = 32
3 × 20 = 60 ⇒ P + Q = 3 + 20 = 23
4 × 15 = 60 ⇒ P + Q = 4 + 15 = 19
5 × 12 = 60 ⇒ P + Q = 5 + 12 = 17
6 × 10 = 60 ⇒ P + Q = 6 + 10 = 16

সুতরাং, P + Q এর সম্ভাব্য মানগুলো: 16, 17, 19, 23, 32, 61.

তাই, P + Q = 20 হতে পারে না।

১২৯.

Two numbers are in the ratio 5:4 and their difference is 10. What is the smallest number?

ক
ক) 30
খ
খ) 40
গ
গ) 50
ঘ
ঘ) 60
ঙ
ঙ) None

ব্যাখ্যা

Given ratio = 5 : 4
Let the numbers are 5x and 4x
ATQ, 5x - 4x = 10
So, x = 10
∴ Smallest number = 4×10 = 40

১৩০.

০.৩৫ কে ভগ্নাংশে প্রকাশ করলে কত হবে?

ক
ক) ৭/২০
খ
খ) ৭/২৫
গ
গ) ৫/২০
ঘ
ঘ) ৫/২৫

ব্যাখ্যা

প্রশ্ন: ০.৩৫ কে ভগ্নাংশে প্রকাশ করলে কত হবে?

সমাধান:
০.৩৫
= ৩৫/১০০
= ৭/২০

১৩১.

A cake is divided into 18 pieces. If Niloy takes 1/3^rd of the cake and Nihal takes 1/3^rd of the cake left, how many pieces are left?

ক
4
খ
6
গ
8
ঘ
10

ব্যাখ্যা

Question: A cake is divided into 18 pieces. If Niloy takes 1/3^rd of the cake and Nihal takes 1/3^rd of the cake left, how many pieces are left?

Solution:
Niloy কেক নেয় = 18 × (1/3) = 6 টুকরা
বাকি রইল = 18 - 6 = 12 টুকরা
Nihal কেক নেয় = 12 × (1/3) = 4 টুকরা

অবশিষ্ট রইল = 12 - 4 = 8 টুকরা

১৩২.

The number of numbers from 1 to 200 which are divisible by neither 3 nor 7 is-

ক
103
খ
85
গ
106
ঘ
115

ব্যাখ্যা

Question: The number of numbers from 1 to 200 which are divisible by neither 3 nor 7 is-

Solution:
The required number = Number of numbers, which are (divisible by 3 + divisible by 7 - divisible by 21)

Now,
Number of number divisible by 3,
= {(198 - 3)/3} + 1
= 65 + 1 = 66

Number of number divisible by 7
= {(196 - 7)/7} + 1
= 27 + 1 = 28

And,
Number of number divisible by 21,
= {(189 - 21)/21} + 1
= 8 + 1 = 9

Thus, the divisible value = 66 + 28 - 9 = 85

Thus, number of numbers which are not divisible by 3 or 7
= 200 - 85 = 115

১৩৩.

A car manufacturer has 2,992 forklifts, which is approximately one forklift for every 48.9 employees. Which of the following is the closest approximation in thousands, of the number of employees employed by the manufacturer?

ক
ক) 60
খ
খ) 100
গ
গ) 150
ঘ
ঘ) 175

ব্যাখ্যা

Question :A car manufacturer has 2,992 forklifts, which is approximately one forklift for every 48.9 employees. Which of the following is the closest approximation in thousands, of the number of employees employed by the manufacturer?

Solution:

একটি প্রতিষ্ঠানে ট্রলি রয়েছে 2,992।

প্রতিষ্ঠানে একটি ট্রলির জন্য কর্মীসংখ্যা বরাদ্ধ রয়েছে 48.9 জন।

Approx 2992 = 3000
and approx 48.9 = 50

সুতরাং, কর্মীসংখ্যা হতে পারে = 3,000 × 50 = 150,000 = 150 thousands

১৩৪.

If 8826P is divisible by 9, what is the value of P?

ক
3
খ
4
গ
6
ঘ
7

ব্যাখ্যা

Question: If 8826P is divisible by 9, what is the value of P?

Solution:
একটি সংখ্যা ৯ দ্বারা বিভাজ্য হবে যদি সংখ্যাটির অঙ্কগুলোর সমষ্টি ৯ দ্বারা বিভাজ্য হয়।

৮ + ৮ + ২ + ৬ = ২৪; এর সাথে ৩ যোগ করলে ২৭ হয়, যা ৯ দ্বারা বিভাজ্য।
∴ P = ৩

১৩৫.

Three numbers are in the ratio 1 : 2 : 3, and the sum of their cubes is 4500. The smallest number will be -

ক
ক) 4
খ
খ) 5
গ
গ) 6
ঘ
ঘ) 10

ব্যাখ্যা

x:2x:3x
x³+8x³+27x³=4500
36x³=4500
x³= 4500/ 36 =125
x= 5
Smallest number is 5

১৩৬.

Which number should replace both the x in the following equation?
x /1776 = 111/ x

ক
ক) 354
খ
খ) 544
গ
গ) 644
ঘ
ঘ) 444

ব্যাখ্যা

Let x /1776 = 111/ x
Then,
⇔x²=111×1776
⇔x²=111×111×16
⇔x= √ {(111)²×(4)²}
⇔x=111×4
⇔x=444

১৩৭.

Which number is divisible by 2, 3, 4 and 6 but is not divisible by 5?

ক
ক) 138
খ
খ) 644
গ
গ) 1020
ঘ
ঘ) 1428

ব্যাখ্যা

Question: Which number is divisible by 2, 3, 4 and 6 but is not divisible by 5?

Solution:
অপশন টেস্ট:
অপশন ক) 138 সংখ্যাটি 4 দ্বারা বিভাজ্য নয়
অপশন খ) 644 সংখ্যাটি 3 দ্বারা বিভাজ্য নয়
অপশন গ) 1020সংখ্যাটি 5 দ্বারা বিভাজ্য
অপশন ঘ) 1428 সংখ্যাটি 2, 3, 4 এবং 6 দ্বারা বিভাজ্য 5 দ্বারা বিভাজ্য নয়

১৩৮.

What should be added to 2x² + 3x - 5 to make x² - x + 1?

ক
ক) -x² - 4x + 6
খ
খ) x² - 4x + 6
গ
গ) x² - 4x + 6
ঘ
ঘ) x² - 4x + 6

ব্যাখ্যা

Let A is to be added them 2x² + 3x - 5 + A = x² - x + 1
A = x² - x + 1 - (2x² + 3x - 5)
A = x² - x + 1 - 2x² - 3x + 5
A = -x² - 4x + 6.

১৩৯.

28√? + 1,426 = 3/4 of 2,872

ক
ক) 576
খ
খ) 1,296
গ
গ) 676
ঘ
ঘ) 1,444

ব্যাখ্যা

28√? + 1,426 = 3/4 of 2872
⇒ 28√? = 3/4 × 2872 - 1,426 = 728
⇒ √? = 728/28 = 26
⇒ ? = 26² = 676

১৪০.

যদি কখ <০ এবং খ > ০ হয়, তাহলে নিচের কোনটি সত্য নয়?

ক
(২খ + ৩) × (ক + ২) > ৬
খ
(২খ + ৩খ) × (২ - ক) > ৬
গ
(২ক + ১) × (২ - ক) < ৬
ঘ
(২ক - ১) × (২ + ক) < ৬
ঙ
খ ও গ উভয়ই

ব্যাখ্যা

প্রশ্ন: যদি কখ <০ এবং খ > ০ হয়, তাহলে নিচের কোনটি সত্য নয়?

সমাধান:
দেওয়া আছে,
কখ <০ এবং খ > ০ হলে, খ এর মান ০ অপেক্ষা বড় এবং ক এর মান ০ অপেক্ষা ছোট হবে।

ধরি,
ক = - ১
খ = ১

এখন,
(২খ + ৩খ) × (২ - ক) > ৬
⇒ (২ + ৩) × {২ - (- ১)} > ৬
⇒ ৫ × ৩ > ৬
∴ ১৫ > ৬, ইহা সত্য

(২খ + ৩) × (ক + ২) > ৬
⇒ (২ + ৩) × {(- ১) + ২} > ৬
⇒ ৫ × ১ > ৬
∴ ৫ > ৬, ইহা সত্য নয়

(২ক + ১) × (২ - ক) < ৬
⇒ {২ (- ১)} + ১} × {২ - (- ১)} < ৬
⇒ (- ২ + ১) × (২ + ১) < ৬
⇒ - ১ × ৩ < ৬
∴ - ৩ < ৬, ইহা সত্য

(২ক - ১) × (২ + ক) < ৬
⇒ {২ × (- ১) - ১} × (২ + (- ১)} < ৬
⇒ (- ২ - ১) × ১ < ৬
⇒ - ৪ < ৬, ইহা সত্য

১৪১.

If x² + 4x + 2 is even, then which one of the following could be the value of x?

ক
ক) 3
খ
খ) 4
গ
গ) 9
ঘ
ঘ) 7

ব্যাখ্যা

x = 3 হলে x² + 4x + 2 = 3² + 4 × 3 + 2 = 9 + 12 + 2 = 23
x = 4 হলে x² + 4x + 2 = 4² + 4 × 4 + 2 = 16 + 16 + 2 = 34
x = 9 হলে x² + 4x + 2 = 9² + 4 × 9 + 2 = 81 + 36 + 2 = 119
x = 7 হলে x² + 4x + 2 = 7² + 4 × 7+ 2 = 49 + 28 + 2 = 79

১৪২.

The greatest number that can be subtracted from 10000 so that the remainder may be divisible by 32, 36, 48 and 54 is-

ক
9136
খ
9316
গ
9216
ঘ
9236
ঙ
None of these

ব্যাখ্যা

Question: The greatest number that can be subtracted from 10000 so that the remainder may be divisible by 32, 36, 48 and 54 is-

Solution:
LCM of 32, 36, 48 and 54 = 864
Required greatest number = 10000 - 864 = 9136
∴ Required greatest number = 9136

১৪৩.

Three boys agree to divide a bag of marbles in the following manner. The first boy takes one more than half the marbles. The second takes a third of the number remaining. The third boy finds that he is left with twice as many marbles as the second boy. The original number of marbles is -

ক
38
খ
36
গ
32
ঘ
Cannot be determined

ব্যাখ্যা

Question: Three boys agree to divide a bag of marbles in the following manner. The first boy takes one more than half the marbles. The second takes a third of the number remaining. The third boy finds that he is left with twice as many marbles as the second boy. The original number of marbles is -

Solution:
মোট মার্বেল সংখ্যা ৩৮ হলে,
প্রথম জন পাবে = (৩৮/২) + ১ = ১৯ + ১ = ২০
দ্বিতীয় জন পাবে = (৩৮ - ২০)/৩ = ১৮/৩ = ৬
তৃতীয় জন পায় = ৩৮ - (২০ + ৬) = ৩৮ - ২৬ = ১২ ; যা দ্বিতীয় জনের দ্বিগুণ।

মোট মার্বেল সংখ্যা ৩৬ হলে,
প্রথম জন পাবে = (৩৬/২) + ১ = ১৮ + ১ = ১৯
দ্বিতীয় জন পাবে = (৩৬ - ১৯)/৩ = ১৭/৩
তৃতীয় জন পায় = ৩৮ - (১৯ + ১৭/৩) = ৩৪/৩ ; যা দ্বিতীয় জনের দ্বিগুণ।

মোট মার্বেল সংখ্যা ৩২ হলে,
প্রথম জন পাবে = (৩২/২) + ১ = ১৬ + ১ = ১৭
দ্বিতীয় জন পাবে = (৩২ - ১৭)/৩ = ১৫/৩ = ৫
তৃতীয় জন পায় = ৩২ - (১৭ + ৫) = ৩২ - ২২ = ১০ ; যা দ্বিতীয় জনের দ্বিগুণ।

অতএব, প্রশ্নের প্রদত্ত তথ্য অনুযায়ী তিনটিই উত্তর হিসেবে গ্রহণযোগ্য। মূলত কয়টি মার্বেল ছিল তা নির্ণয় করার জন্য পর্যাপ্ত তথ্য প্রশ্নে দেয়া নেই।

১৪৪.

A number is as less than 480 as is (3/2)times greater than 320. What is the number?

ক
ক) 64
খ
খ) 66
গ
গ) 62
ঘ
ঘ) 60

ব্যাখ্যা

Question: A number is as less than 480 as is (3/2)times greater than 320. What is the number?

Solution:
ধরি,
সংখ্যাটি x

প্রশ্নমতে,
480 - x = 320 + (3x/2)
বা, 960 - 2x = 640 + 3x
বা, 5x = 960 - 640
বা, 5x = 320
∴ x = 64

∴ সংখ্যাটি = 64

১৪৫.

If X is an odd integer and Y is an even integer. Which of the following statements is (are) always true?
(I) (X + Y) is odd
(II) XY is odd
(III) (2X + Y) is even

ক
I only
খ
II and III only
গ
III only
ঘ
I and III only
ঙ
None of these

ব্যাখ্যা

odd + even = odd (4 + 5 = 9)
odd × even = even (4 × 5 = 20)
even × odd = even × even = even (2 × 5 + 4 = 14).

১৪৬.

Divide 60 by half and deduct twenty. What do you get?

ক
ক) 120
খ
খ) 100
গ
গ) 60
ঘ
ঘ) 30
ঙ
ঙ) 10

ব্যাখ্যা

Question: Divide 60 by half and deduct twenty. What do you get?

Solution:
60/(1/2) - 20
= (60 × 2) - 20
= 120 - 20
= 100

১৪৭.

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

ক
ক) n² - n
খ
খ) 5n - 1
গ
গ) 3n³
ঘ
ঘ) n + 2

ব্যাখ্যা

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

Solution:
let, n = 4

n² - n
= 4² - 4
= 16 - 4
= 12

5n - 1
= 5 × 2 - 1
= 10 - 1
= 9

3n³
= 3 × 4³
= 3 × 64
= 192

n + 2
= 4 + 2
= 6

১৪৮.

The difference between 4/5 of a number and 40% of the number is 30. what is the 2/5 of that number?

ক
40
খ
45
গ
30
ঘ
35

ব্যাখ্যা

Question: The difference between 4/5 of a number and 40% of the number is 30. what is the 2/5 of that number?

Solution:
let the number be x.

ATQ,
4x/5 - (40% of x) = 30
or, 4x/5 - 2x/5 = 30
or, 2x/5 = 30
∴ x = 75

∴ 2/5 of 75 is = (75 × 2)/5
= 30

১৪৯.

The average of a natural number and its cube is 13 times the number. The number is-

ক
5
খ
- 5
গ
±5
ঘ
25

ব্যাখ্যা

Question: The average of a natural number and Its cube Is 13 times the number. The number is-

Solution:
let the natural number be = x

According to the Question,
(x + x³)/2 = 13x
⇒ x + x³ = 26x
⇒ x³ = 26x - x
⇒ x³ = 25x
⇒ x³/x = 25
⇒ x² = 25
⇒ x = ± 5

But since x is a natural number, the value of x must be positive.

Therefore, x = 5.

১৫০.

Which of the following fractions is the largest?

ক
7/4
খ
6/5
গ
3/2
ঘ
5/3

ব্যাখ্যা

Question: Which of the following fractions is the largest?

Solution:
7/4 = 1.75
6/5 = 1.2
3/2 = 1.5
5/3 = 1.66

Hence the largest fraction is 7/4

১৫১.

A number whose fifth part increased by 4 is equal to its fourth part diminished by 10, is -

ক
ক) 240
খ
খ) 260
গ
গ) 270
ঘ
ঘ) 280

ব্যাখ্যা

Let the number be x.
Then, x/5 + 4 = x/4 - 10
⇔ x/4 - x/5 = 14
⇔ x/20 = 14
⇔ x = 20 × 14 = 280.
Answer: 280.

১৫২.

If x = 10, which of the following has the minimum value?

ক
ক) 2 - x
খ
খ) x/2
গ
গ) 2/x
ঘ
ঘ) (2 - x)(2 - x)

ব্যাখ্যা

Question: If x = 10, which of the following has the minimum value?

Solution:
2 - x
= 2 - 10
= - 8

x/2
= 10/2
= 5

2/x
= 2/10
= 1/5

(2 - x) (2 - x)
= (2 - 10) (2 - 10)
= - 8 ×- 8
= 64

So, 2 - x has the minimum value.

১৫৩.

The ratio of two numbers is 3 : 4 and their L.C.M is 84, find the number-

ক
ক) 28
খ
খ) 34
গ
গ) 82
ঘ
ঘ) 54

ব্যাখ্যা

মনে করি,
একটি সংখ্যা ৩ক এবং অপর সংখ্যাটি ৪ক।
সুতরাং সংখ্যা দুটির গ.সা.গু = ক এবং ল.সা.গু = ১২ক।
শর্তমতে,
১২ক = ৮৪
ক = ৭

একটি সংখ্যা ৩ × ৭ = ২১ এবং
অপর সংখ্যাটি ৪× ৭ =২৮

১৫৪.

If a = 0.67 then which one of the following is smaller than a?

ক
ক) √a
খ
খ) 1/a
গ
গ) a²
ঘ
ঘ) 1/a²

ব্যাখ্যা

√a = √0.67 = 0.81
1/a = 1/0.67 = 1.49
a² = 0.67² = 0.4489
1/a² = 1/0.67² = 1/0.4489 = 2.23

So, a² is the smaller than a

১৫৫.

The sum of two numbers is 14 and their difference is 10. Find the product of these two numbers.

ক
24
খ
26
গ
28
ঘ
20

ব্যাখ্যা

Question: The sum of two numbers is 14 and their difference is 10. Find the product of these two numbers.

Solution:
let, the numbers be x and y
∴ x + y = 14.......(i)
x - y = 10......(ii)

adding (i) and (ii) we get,
x + y + x - y = 14 + 10
2x = 24
x = 12

from (i) we get,
12 + y = 14
y = 2

product = xy = 12 × 2 = 24

১৫৬.

Find the largest number -

ক
ক) 13/15
খ
খ) 16/17
গ
গ) 17/21
ঘ
ঘ) 21/25

ব্যাখ্যা

• 13/15 = 0.87
• 16/17 = 0.94
• 17/21 = 0.81
• 21/25 = 0.84

১৫৭.

The next number in the sequence 1, 2, 3, 5, 8, 13, 21, ____ is-

ক
ক) 34
খ
খ) 36
গ
গ) 38
ঘ
ঘ) 40

ব্যাখ্যা

আমরা জানি,
Fibonacci সংখ্যা = 0, 1, 1, 2, 3, 5, 8, 13, 21 ... পরপর দুটি সংখ্যার যােগফল পরবর্তী সংখ্যার সমান।
সুতরাং, এই ধারার পরবর্তী সংখ্যা হবে = 21 + 13 = 34

১৫৮.

It is being given that (2³² + 1) is completely divisible by a whole number. Which of the following numbers is also completely divisible by that whole number?

ক
7 × 2²³
খ
2¹⁶ + 1
গ
2⁹⁶ + 1
ঘ
2¹⁶ - 1

ব্যাখ্যা

Question: It is being given that (2³² + 1) is completely divisible by a whole number. Which of the following numbers is also completely divisible by that whole number?

Solution:
2⁹⁶ + 1
= (2³²)³ + 1
= (2³² + 1) {(2³²)² - 2³² + 1}
= (2³² + 1) (2⁶⁴ - 2³² + 1)

যেহেতু (2³² + 1) একটি পূর্ণসংখ্যা দ্বারা বিভাজ্য, 2⁹⁶ + 1 ও ঐ পূর্ণ সংখ্যাটি দ্বারা বিভাজ্য হবে।

১৫৯.

What will be the least number which when doubled will be exactly divisible by 12, 18, 21 and 30?

ক
ক) 630
খ
খ) 1260
গ
গ) 2520
ঘ
ঘ) 196

ব্যাখ্যা

The LCM of 12, 18, 21, 30 is 1260.
So, the number is 1260/2 = 630

১৬০.

(10.3×10.3×10.3+1) /(10.3×10.3−10.3+1) is equal to :

ক
ক) 10.3
খ
খ) 10.5
গ
গ) 11.3
ঘ
ঘ) 13.3

ব্যাখ্যা

Given expression :
= (10.3)³+1³/ (10.3)²−(10.3×1)+1²= (a³+b³/ a²−ab+b²)
=(a+b)
=(10.3+1)
=11.3

১৬১.

ক
ক) a
খ
খ) a + 1
গ
গ) 1
ঘ
ঘ) a²

ব্যাখ্যা

Question:

Solution:

১৬২.

A number when divided by 91 gives a remainder 17. When the same number is divided by 13, the remainder will be?

ক
2
খ
4
গ
5
ঘ
6

ব্যাখ্যা

Question: A number when divided by 91 gives a remainder 17. When the same number is divided by 13, the remainder will be?

Solution:
Here,
the first divisor (91) is a multiple of the second divisor (13).

∴ Required remainder = Remainder obtained on dividing 17 by 13
⇒ 17 = ( 13 × 1 ) + 4

∴ Hence Required remainder = 4

১৬৩.

The sum of four consecutive two-digit odd numbers, when divided by 10, becomes a perfect square. Which of the following can be the set of the four numbers?

ক
27, 29, 31 and 33
খ
37, 39, 41 and 43
গ
47, 49, 51 and 53
ঘ
None of the above

ব্যাখ্যা

Question: The sum of four consecutive two-digit odd numbers, when divided by 10, becomes a perfect square. Which of the following can be the set of the four numbers?

Solution:
Using options,
We find that four consecutive odd numbers are 37, 39, 41 and 43

Here, 37 + 39 + 41 + 43 = 160

The sum of these 4 numbers is 160, when divided by 10 we get 16 which is a perfect square.

Hence, the required set = 37, 39, 41 and 43

১৬৪.

If m/n = 1/4, then the value of (m² + n²)/(m² - n²) is-

ক
5/7
খ
- √7/√5
গ
15/17
ঘ
- 17/15

ব্যাখ্যা

Question: If m/n = 1/4, then the value of (m² + n²)/(m² - n²) is-

Solution:
Given that,
m/n = 1/4

Now,
(m/n) = (1/4)
⇒ (m/n)² = (1/4)²
⇒ m²/n² = 1/16
∴ (m² + n²)/(m² - n²) = (1 + 16)/(1 - 16); [যোজন-বিয়োজন]
= 17/(- 15)
= - 17/15

১৬৫.

What least number must be added to 1056, so that the sum is completely divisible by 23?

ক
2
খ
3
গ
18
ঘ
21

ব্যাখ্যা

Question: What least number must be added to 1056, so that the sum is completely divisible by 23?

Solution:
23 ) 1056 ( 45
92
_______
136
115
_______
21

∴ Required number = (23 - 21) = 2

১৬৬.

What is the least multiple of 7 which leaves a remainder of 4 when divided by 6, 9, 15 and 18?

ক
ক) 343
খ
খ) 350
গ
গ) 371
ঘ
ঘ) 364

ব্যাখ্যা

LCM of 6, 9, 15 and 18 = 90
Required Number = (90k + 4) which is a multiple of 7
Put k = 1. We get numbers as (90 × 1) + 4 = 94. But this is not a multiple of 7
Put k = 2. We get numbers as (90 × 2) + 4 = 184. But this is not a multiple of 7
Put k = 3. We get numbers as (90 × 3) + 4 = 274. But this is not a multiple of 7
Put k = 4. We get numbers as (90 × 4) + 4 = 364. This is a multiple of 7
Hence 364 is the answer.

১৬৭.

The sum of three consecutive multiples of 4 is 312. Find the largest number.

ক
108
খ
208
গ
123
ঘ
311

ব্যাখ্যা

Question: The sum of three consecutive multiples of 4 is 312. Find the largest number.

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

According to the question,
4x + (4x + 4) + (4x + 8) = 312
⇒ 12x + 12 = 312
⇒ 12x = 300
⇒ x = 25

∴ The largest number = 4x + 8
= 4 × 25 + 8
= 108

১৬৮.

ক
0.4
খ
0.5
গ
0.6
ঘ
0.8

ব্যাখ্যা

Question:

Solution:

১৬৯.

If a number is multiplied by two-thirds of itself the value obtained is 384. What is the number?

ক
ক) 36
খ
খ) 18
গ
গ) 30
ঘ
ঘ) 24

ব্যাখ্যা

Question: If a number is multiplied by two-thirds of itself the value obtained is 384. What is the number?

Solution:
Let, the number be x

ATQ,
x × (2x/3) = 384
⇒ 2x² = 1152
⇒ x² = 576
⇒ x = 24

১৭০.

What is the greatest number of boys among whom 100 pens and 165 ice cream can be divided equally so that no item remains left?

ক
4
খ
5
গ
6
ঘ
7

ব্যাখ্যা

Question: What is the greatest number of boys among whom 100 pens and 165 ice cream can be divided equally so that no item remains left?

Solution:
The H.C.F is the highest number of boys.
H.C.F of 100 and 165 is = 5

১৭১.

Three numbers are in ratio 1 : 3 : 4 and HCF is 12. The numbers are -

ক
ক) 12, 36, 54
খ
খ) 12, 36, 48
গ
গ) 12, 36, 44
ঘ
ঘ) 12, 32, 48

ব্যাখ্যা

Question: Three numbers are in ratio 1 : 3 : 4 and HCF is 12. The numbers are -

Solution:
ধরি,
সংখ্যা তিনটি x, 3x, 4x.
গ.সা.গু = x

∴ x = 12

সংখ্যাগুলো = 12, 36, 48.

১৭২.

Which of the following is a rational number?

ক
√(9/23)
খ
√12
গ
√(361/289)
ঘ
√11
ঙ
None of the above

ব্যাখ্যা

√(361/289) = 19/17

মূলদ সংখ্যাঃ যেসব সংখ্যাকে p/q আকারে প্রকাশ করা যায় যেখানে p,q স্বাভাবিক সংখ্যা এবং q≠0 তাদেরকে মূলদ সংখ্যা বলে।

১৭৩.

The H.C.F and L.C.M of two numbers are 12 and 336 respectively. If one of the numbers is 84, the other one is -

ক
28
খ
32
গ
64
ঘ
48

ব্যাখ্যা

Question: The H.C.F and L.C.M of two numbers are 12 and 336 respectively. If one of the numbers is 84, the other one is -

Solution:
দুইটি সংখ্যার ল.সা.গু ও গ.সা.গু এর গুণফল সংখ্যা ২টির গুণফলের সমান।
ধরি,
অপর সংখ্যাটি = ক

তাহলে,
ক × ৮৪ = ১২ × ৩৩৬
বা, ক = (১২ × ৩৩৬)/৮৪
∴ ক = ৪৮

১৭৪.

(০.২)^২ ÷ (০.১)^৩ = কত?

ক
ক) ৩০
খ
খ) ৪০
গ
গ) ৪৪
ঘ
ঘ) ৪২

ব্যাখ্যা

প্রশ্ন: (০.২)^২ ÷ (০.১)^৩ = কত?

সমাধান:
(০.২)^২ ÷ (০.১)^৩
= (০.২ × ০.২) ÷ (০.১ × ০.১ × ০.১)
= ০.০৪ ÷ ০.০০১
= ৪০

১৭৫.

Tk. 720 was divided among A, B, C, D, E. The sum received by them was in ascending order and in arithmetic progression. E received Tk. 40 more than A. How much did B receive?

ক
Tk. 143
খ
Tk. 134
গ
Tk. 104
ঘ
Tk. 152

ব্যাখ্যা

Question: Tk. 720 was divided among A, B, C, D, E. The sum received by them was in ascending order and in arithmetic progression. E received Tk. 40 more than A. How much did B receive?

Solution:
Given that,
A + B + C + D + E = Tk. 720
And, E - A = 40

Now,
Arithmatic progression-
a, a + d, a + 2d, a + 3d, a + 4d
∴ Amount of E is = (a + 4d) and Amount of A is= a

According to the question,
⇒ a + 4d - a = 40
⇒ 4d = 40
⇒ d = 10

Also,
a + (a + d) + (a + 2d) + (a + 3d) + (a + 4d) = 720
⇒ 5a + 10d = 720
⇒ 5a + 10 × 10 = 720
⇒ 5a = 720 - 100
⇒ a = 620/5 = 124

So, amount B = a + d = 124 + 10 = Tk. 134

১৭৬.

The difference between a two-digit number and the number obtained by interchanging the positions of its digits is 45. What is the difference between the two digits of that number?

ক
3
খ
4
গ
5
ঘ
6
ঙ
None

ব্যাখ্যা

Let the ten's digit be x and unit's digit be y.
Then, (10x + y) - (10y + x) = 45
⇒ 9(x - y) = 45
⇒ x - y = 5

১৭৭.

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

ক
ক) 2/5
খ
খ) 4/7
গ
গ) 4/9
ঘ
ঘ) 5/11

ব্যাখ্যা

1/2 = 0.5
2/5 = 0.4
4/7 = 0.57
4/9 = 0.44
5/11 = 0.45

১৭৮.

If the product of three consecutive integers is 120, then the sum of the integers is:

ক
5
খ
10
গ
15
ঘ
20

ব্যাখ্যা

Question: If the product of three consecutive integers is 120, then the sum of the integers is:

Solution:
120 = 2 × 2 × 2 × 3 × 5
= (2 × 2) × 5 × (2 × 3)
= 4 × 5 × 6

Clearly, the three consecutive integers whose product is 120 are 4, 5 and 6.

∴ Required sum
= 4 + 5 + 6
= 15

১৭৯.

How many pieces of 85 cm length stick can be cut from a 42.5 meters long stick?

ক
ক) 30
খ
খ) 40
গ
গ) 50
ঘ
ঘ) 60

ব্যাখ্যা

Question: How many pieces of 85 cm length stick can be cut from a 42.5 meters long stick?

Solution:
আমরা জানি,
1 মিটার= 100 সে.মি.
42.5 মিটার= (100 × 42.5) সে.মি.
= 4250 সে.মি.

টুকরার সংখ্যা হবে = 4250/85 টি
= 50 টি

১৮০.

Out of six consecutive natural numbers if the sum of the first three is 30, what is the sum of the other three?

ক
ক) 36
খ
খ) 39
গ
গ) 42
ঘ
ঘ) 45

ব্যাখ্যা

Let
the first three consecutive numbers be x, x + 1 and x + 2.
Then,
x + x + 1 + x + 2 = 30
⇒ 3x + 3 = 30
⇒ 3x = 27
⇒ x = 9
∴ The first three numbers are 9, 10,11
⇒ The next three numbers are 12, 13, 14.
Hence, the required sum =12 + 13 + 14
= 39

১৮১.

The greatest number of four digits that is divisible by 15, 25, 40, and 75 is:

ক
9600
খ
9200
গ
9400
ঘ
9800

ব্যাখ্যা

Question: The greatest number of four digits that is divisible by 15, 25, 40, and 75 is:

Solution:
The greatest number of 4 digits is 9999.
L.C.M. of 15, 25, 40 and 75 is 600.

On dividing 9999 by 600, the remainder is 399.
∴ Required number (9999 - 399) = 9600.

১৮২.

If (2^p + 1) is a prime number, which one of the following digits could be the value of p?

ক
3
খ
6
গ
5
ঘ
4

ব্যাখ্যা

Question: If (2^p + 1) is a prime number, which one of the following digits could be the value of p?

Solution:
In such questions, each alternative should be tried.

So, if P = 3, we will get; 8 + 1= 9

If P = 4, we will get; 16 + 1 = 17

If P = 5, we will get; 32 + 1 = 33

If P= 6, we will get; 64 + 1= 65

Out of the four results, only 17 is the prime number. So, the required value of the P is 4.

১৮৩.

What is the probability that an integer selected at random from those between 10 and 100 inclusive is a multiple of 5 or 9?

ক
ক) 27/89
খ
খ) 90/91
গ
গ) 27/91
ঘ
ঘ) 23/89

ব্যাখ্যা

প্রশ্ন: What is the probability that an integer selected at random from those between 10 and 100 inclusive is a multiple of 5 or 9?

সমাধান:
10 থেকে 100 এর মধ্যে 5 এর গুণিতক সংখ্যা গুলো হলো: 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100 = ১৯টি

10 থেকে 100 এর মধ্যে 9 এর গুণিতক সংখ্যা গুলো হলো: 18, 27, 36, 45, 54, 63, 72, 81, 90, 99 = 10টি
মোট গুণিতক = (19 + 10)টি = 29

45 ও 90 উভয়ের গুণিতক।
মোট অনুকূল ফলাফল = 29 - 2 = 27

10 থেকে 100 এর মধ্যে মোট সংখ্যা = 91টি

নির্ণেয় সম্ভাবনা = 27/91

১৮৪.

The value of 0.1 x 0.1 is-

ক
ক) 0.1
খ
খ) 1
গ
গ) 0.01
ঘ
ঘ) 0.001

ব্যাখ্যা

0.1 x 0.1= 0.01

১৮৫.

The sum of the two numbers is 22. Five times one number is equal to 6 times the other. The bigger of the two numbers is:

ক
10
খ
12
গ
15
ঘ
16

ব্যাখ্যা

Question: The sum of the two numbers is 22. Five times one number is equal to 6 times the other. The bigger of the two numbers is:

Solution:
Let,
The required number are x and y

Now
x + y = 22...........(1)

Five times one number is equal to 6 times the other,
5x = 6y
x = 6y/5..............(2)

(1)⇒
6y/5 + y = 22
(6y + 5y)/5 = 22
11y/5 = 22
y/5 = 2
y = 10

(2)⇒
x = (6 × 10)/5
x = 12

১৮৬.

A certain club has 237 local branches, one national office, and one social service office. If each local branch has 2 officers, and each of the two other offices has 4 officers, how many officers does the club have altogether?

ক
482
খ
476
গ
474
ঘ
239
ঙ
235

ব্যাখ্যা

Question: A certain club has 237 local branches, one national office, and one social service office. If each local branch has 2 officers, and each of the two other offices has 4 officers, how many officers does the club have altogether?

Solution:
237 × 2 + 1 × 4 + 1 × 4
= 474 + 4 + 4
= 482

১৮৭.

The LCM and ratio of four numbers are 630 and 2 : 3 : 5 : 7 respectively. The difference between the greatest and least numbers is = ?

ক
ক) 15
খ
খ) 18
গ
গ) 21
ঘ
ঘ) 24

ব্যাখ্যা

Let the numbers be 2x, 3x, 5x and 7x respectively
Then their L.C.M. = (2 × 3 × 5 × 7)x = 210x.[∴ 2, 3, 5, 7 are prime numbers]
So, 210x = 630 or x = 3
∴ The numbers are 6, 9, 15 and 21
Required difference = 21 - 6 = 15
======================
মনে করি, সংখ্যাগুলো ২ক, ৩ক, ৫ক ও ৭ক
তাদের লসাগু = (২ × ৩ × ৫ × ৭)ক = ২১০ক
∴ ২১০ক = ৬৩০
⇒ ক = ৩
অতএব, সংখ্যাগুলো ৬, ৯, ১৫ ও ২১
নির্ণেয় পার্থক্য = ২১ - ৬ = ১৫

১৮৮.

In an examination, a student's average marks was 63. If he had obtained 20 more marks for Geography and 2 more marks for History, his average would have been 65. How many subjects were there in the examination?

ক
14
খ
12
গ
11
ঘ
13

ব্যাখ্যা

Let the number of subjects = x
Total marks = 63x
If he had obtained 20 more marks for Geography and 2 more marks for history, his average would have been 65. That is, in this case, the total marks would have been 65x
Now we have,
65x - 63x = 20 + 2
⇒ 2x = 22
⇒ x = 11.

১৮৯.

A rectangular block 6 cm by 12 cm by 15 cm is cut up into an exact number of equal cubes. Find the least possible number of cubes.

ক
ক) 30
খ
খ) 40
গ
গ) 50
ঘ
ঘ) 60

ব্যাখ্যা

Volume of the block = (6 × 12 × 15) cm³
= 1080 cm³
Side of the largest cube
= H.C.F of 6 cm, 12 cm, 15 cm
= 3 cm.
Volume of this cube = (3 × 3 × 3) cm³
= 27 cm³
Number of cubes = 1080/27
= 40.

১৯০.

The sum of three numbers is 132. If the first number be twice the second and third number be one third of the first, then the second number is:

ক
ক) 32
খ
খ) 36
গ
গ) 48
ঘ
ঘ) 60
ঙ
ঙ) None of the above

ব্যাখ্যা

Let 2nd = 3x,
therefore, 3x + 6x + 2x = 132, x = 12
so, 3x = 36

১৯১.

L.C.M. of two numbers is 2079 and their H.C.F. is 27. If one of the numbers is 189, the other number is :

ক
297
খ
584
গ
189
ঘ
216

ব্যাখ্যা

Given,
H.C.F. = 27
L.C.M. = 2079
one number = 189
Let another number be y

We know,
Product of numbers = L.C.M. × H.C.F.
∴ 189 x y = 27 × 2079
y = 297.

১৯২.

What is the greatest number of 3 digits which when divided by 6, 9 and 12 leaves a remainder of 3 in each case?

ক
903
খ
996
গ
975
ঘ
939

ব্যাখ্যা

Questions: What is the greatest number of 3 digits which when divided by 6, 9 and 12 leaves a remainder of 3 in each case?

Solution:
৬, ৯, ১২ এর ল.সা.গু = ৩৬

তিন অঙ্কের বৃহত্তম সংখ্যা = ৯৯৯
৯৯৯ কে ৩৬ দ্বারা ভাগ করলে অবশিষ্ট থাকে ২৭
৯৯৯ - ২৭ = ৯৭২ ; ৩৬ দ্বারা নিঃশেষে বিভাজ্য।

নির্ণেয় সংখ্যা = ৯৭২ + ৩ = ৯৭৫

১৯৩.

A monkey climbs a 12 meters-high slippery pillar. In his first minute, he climbs 2 meters, and in the next minute, he slip one meter down. In this way, how much time will he take to reach the top of the pillar?

ক
ক) 10 minutes
খ
খ) 12 minutes
গ
গ) 11 minutes
ঘ
ঘ) 21 minutes

ব্যাখ্যা

Question: A monkey climbs a 12 meters-high slippery pillar. In his first minute, he climbs 2 meters, and in the next minute, he slip one meter down. In this way, how much time will he take to reach the top of the pillar?

Solution:
On first minute monkey climb = 2 m
On the second minute it slips = 1 m
For every two minute, it climbs 1 m
So, average speed = 1 m/2 min For 10 m,
time is taken = 20 min
For the last 2 m jump add 1 min
So time taken = 20 + 1 = 21 min

∴ Monkey takes 21 minutes to reach the top of the pole.

১৯৪.

What is the least number when divided by 4, 5 and 6 leaves in each case a remainder of 3?

ক
ক) 33
খ
খ) 43
গ
গ) 53
ঘ
ঘ) 63
ঙ
ঙ) 72

ব্যাখ্যা

LCM of 4, 5, and 6 is 60.
So the number is = 60 + 3 = 63

১৯৫.

What is the largest number of mangoes not exceeding 440 that can be distributed among three persons in the proportions 5 : 6 : 7?

ক
ক) 420
খ
খ) 432
গ
গ) 436
ঘ
ঘ) 440

ব্যাখ্যা

Question: What is the largest number of mangoes not exceeding 440 that can be distributed among three persons in the proportions 5 : 6 : 7?

Solution:
ধরি,
তিনজন যথাক্রমে আম পেয়েছে 5x, 6x, 7x

প্রশ্নমতে,
5x + 6x + 7x ≤ 440
⇒ 18x ≤ 440
⇒ x ≤ 440/18
∴ x ≤ 24.44

যেহেতু, আমের সংখ্যা পূর্ণ সংখ্যা হয়, x এর সর্বোচ্চ মান = 24সর্বোচ্চ আমের সংখ্যা = 5x + 6x + 7x
= 18x
= 18 × 24
= 432 টি

১৯৬.

If today is Monday, what will be the day 350 days from now?

ক
ক) Monday
খ
খ) Tuesday
গ
গ) Wednesday
ঘ
ঘ) Thursday

ব্যাখ্যা

350 days, 350/7 = 50, no odd days, so it will be a Monday.

১৯৭.

The sum of two numbers is 40 and their difference is 4. The ratio of the number is

ক
ক) 12 : 13
খ
খ) 11 : 9
গ
গ) 11 : 10
ঘ
ঘ) 10 : 11

ব্যাখ্যা

Given:
The sum of two numbers is 40.
The difference of two number is 4.
Let
the larger number be x.
the smaller number be y.
According to the question,
(x + y)/(x - y) = 40/4
(x + y)/(x - y) =10/1
10(x - y) = x + y
10x - 10y = x + y
10x - x = 10y + y
9x = 11y
x/y = 11/9
x : y = 11 : 9

∴ The ratio of the number is 11 : 9

১৯৮.

The difference between two numbers is 16. If one-third of the smaller number is greater than one-seventh of the larger number by 4, then the two numbers are:

ক
9, 25
খ
12, 28
গ
33, 49
ঘ
None of the above

ব্যাখ্যা

Question: The difference between two numbers is 16. If one-third of the smaller number is greater than one-seventh of the larger number by 4, then the two numbers are:

Solution:
Let the numbers be p and (p + 16)
ATQ,
(p/3) - {(p + 16)/7} = 4
⇒ (7p - 3p - 48)/21 = 4
⇒ 4p = 81 + 48
⇒ p = 132/4
∴ p = 33

So, the numbers are 33 and (33 + 16) = 49

১৯৯.

The smallest value of natural number n for which 2n + 1 is not prime number?

ক
ক) 1
খ
খ) 2
গ
গ) 3
ঘ
ঘ) 4

ব্যাখ্যা

Given:
n is the natural number
Substitute values n = 1, 2, 3,4 ... in the expression 2n + 1

If n = 1, 2n + 1 = 2 × 1 + 1 ⇒ 3
If n = 2, 2n + 1 = 2 × 2 + 1 ⇒ 5
If n = 3, 2n + 1 = 2 × 3 + 1 ⇒ 7
If n = 4, 2n + 1 = 2 × 4 + 1 ⇒ 9

9 is not a prime number

∴ Smallest natural number is 4

২০০.

Which of the following numbers is divisible by 11?

ক
44212
খ
59403
গ
30217
ঘ
60411

ব্যাখ্যা

Question: Which of the following numbers is divisible by 11?

Solution:
Here,
30217, (3 + 2 +7) - (0 + 1) = 11, which is divisible by 11.
44212, (4 + 2 + 2) - (4 + 1) = 3, which is not divisible by 11.
59403, (5 + 4 + 3) - (9 + 0) = 3, which is not divisible by 11.
60411, (6 + 4 + 1) - (0 + 1) = 10, which is not divisible by 11.