Bank Math

PrepBank · পাতা ৬৩ / ১৬১ · ৬,২০১–৬,৩০০ / ১৬,১২৪

৬,২০১.

A train 216 m long moving at a speed of 50 km/hr crosses a train 224 m long coming from opposite direction in 12 seconds. The speed of the second train is-

ক
48 km/hr
খ
54 km/hr
গ
82 km/hr
ঘ
66 km/hr

ব্যাখ্যা

Question: A train 216 m long moving at a speed of 50 km/hr crosses a train 224 m long coming from opposite direction in 12 seconds. The speed of the second train is-

Solution:
Distance covered = (216 + 224) meter
= 440 meter.

Time = 12 seconds.

Relative speed = 440/12 = 110/3 m/s.
= (110 × 3600)/(3 × 1000) km/hr
= 132 km/hr.

Now,
50 + Speed of second train = 132 km/hr.
∴ Speed of second train = (132 - 50) km/hr.
= 82 km/hr.

৬,২০২.

The base of a right-angled triangle is 16 and hypotenuse is 20. What is its area?

ক
96 sq. meters
খ
58 sq. meters
গ
68 sq. meters
ঘ
60 sq. meters
ঙ
None of these

ব্যাখ্যা

Question: The base of a right-angled triangle is 16 and hypotenuse is 20. What is its area?

Solution:
The area of a right angled triangle = (1/2) × base × height

Base = 16, Hypotenuse = 20
Height² = Hypotenuse² - Base²
= 20² - 16²
= 400 - 256
Height² = 144
∴ Height = 12

Area = (1/2) × base × height
= (1/2) × 16 × 12
= 96 sq. meters

৬,২০৩.

9 men bind 1800 books in 20 days. Find how many binders will be required to bind 1200 books in 24 days?

ক
6 men
খ
5 men
গ
7 men
ঘ
8 men

ব্যাখ্যা

Question: 9 men bind 1800 books in 20 days. Find how many binders will be required to bind 1200 books in 24 days?

Solution:
To bind 1800 books in 20 days binders needed 9 men
To bind 1 books in 1 days binders needed 180/1800 men
To bind 1200 books in 24 days binders needed (180 × 1200)/(1800 × 24) men
= 5 men

৬,২০৪.

A square sheet of paper is converted into a cylinder by rolling it along its length. What is the ratio of the base radius to the side of the square?

ক
ক) 1 : π
খ
খ) 1 : 2π
গ
গ) 1 : √2π
ঘ
ঘ) 1 : 2√π

ব্যাখ্যা

Question: A square sheet of paper is converted into a cylinder by rolling it along its length. What is the ratio of the base radius to the side of the square?

Solution:
ধরি, বর্গক্ষেত্রের বাহুর দৈর্ঘ্য x মিটার

সিলিন্ডারের ভূমির পরিধি = বর্গের বাহু
= x মিটার
ব্যাসার্ধ = x/2π
= x/2π

সিলিন্ডারের ব্যাসার্ধ ও বর্গের বাহুর দৈর্ঘ্যের অনুপাত = x/2π : x
= 1/2π : 1
= 1 : 2π

৬,২০৫.

If A = {x ∈ N : 4x < 16} then, how many subsets does set "A" have?

ক
3
খ
8
গ
16
ঘ
32

ব্যাখ্যা

Question: If A = {x ∈ N : 4x < 16} then, how many subsets does set "A" have?

Solution:
Given,
A = {x ∈ N : 4x < 16}
∴ A = {1, 2, 3}

We know,
the number of subsets = 2ⁿ
Here, n=3

the number of subsets A have = 2³ = 8

৬,২০৬.

Rahman spends 40% of his monthly income on household items, 25% of his monthly income on buying clothes, 10% of his monthly income on medicines and saves the remaining Tk. 9,000. What is Rahman’s monthly income?

ক
Tk. 32,000
খ
Tk. 30,000
গ
Tk. 35,000
ঘ
Tk. 36,000

ব্যাখ্যা

Question: Rahman spends 40% of his monthly income on household items, 25% of his monthly income on buying clothes, 10% of his monthly income on medicines and saves the remaining Tk. 9,000. What is Rahman’s monthly income?

Solution:
Total spends = (40 + 25 + 10) = 75%

∴ Saves = (100 - 75) = 25%

Let total income of Rahman be x. Then,
⇒ x × 25% = 9000
⇒ 25x/100 = 9000
⇒ x/4 = 9000
⇒ x = 9000 × 4
∴ x = 36000

∴ Rahman’s monthly income is Tk. 36,000

৬,২০৭.

If x, y, z and w are all integers greater than 2, which of the following is the greatest?

ক
x + yz + w
খ
x + y( z + w)
গ
(x + y) ( z + w)
ঘ
(x + y) + z
ঙ
x + (yz + w)

ব্যাখ্যা

If x, y, z and w are all integers greater than 2,
then sum of two numbers is multiplied by sum of other two numbers is the greatest in above mentioned option.

৬,২০৮.

A goods train runs at the speed of 72 kmph and crosses a 250 m long platform in 26 seconds. What is the length of the goods train?

ক
250 m
খ
270 m
গ
220 m
ঘ
200 m

ব্যাখ্যা

Question: A goods train runs at the speed of 72 kmph and crosses a 250 m long platform in 26 seconds. What is the length of the goods train?

Solution:
Let the length of the train is x m
speed = 72 kmph
= 72/3.6 mps
= 20 mps

As we know,
Distance = Speed × Time
x + 250 = 20 × 26
x = 520 - 250
x = 270 m

৬,২০৯.

A sum of Taka 50,000 is invested at 6% simple interest for the first 3 years and 8% compound interest for the next 2 years. What is the total amount after 5 years?

ক
Taka 68,000.50
খ
Taka 68,817.60
গ
Taka 70,200.20
ঘ
Taka 69,500.25

ব্যাখ্যা

Question: A sum of Taka 50,000 is invested at 6% simple interest for the first 3 years and 8% compound interest for the next 2 years. What is the total amount after 5 years?

Solution:
Given,
Tk. 50,000 is invested.
6% simple interest for the first 3 years.
8% compound interest for the next 2 years.

Simple Interest, I = Pnr
= 50000 × 3 × (6/100)
= 9000

New Principal = 50,000 + 9,000 = 59,000

Compound Principal = P × (1 + r)ⁿ
= 59000 × (1 + (8/100))²
= 59000 × (1.08)²
= 68,817.60

৬,২১০.

Three boys have marbles in the ratio of 19 : 5 : 3. If the boy with the least number has 9 marbles, how many marbles does the boy with the highest number have?

ক
ক) 57
খ
খ) 15
গ
গ) 76
ঘ
ঘ) 38

ব্যাখ্যা

Question: Three boys have marbles in the ratio of 19 : 5 : 3. If the boy with the least number has 9 marbles, how many marbles does the boy with the highest number have?

Solution:
ধরি,
তিন জনের মারবেলের পরিমান যথাক্রমে 19x, 5x, 3x

প্রশ্নমতে,
3x = 9
x = 3

প্রথম জনের মারবেল / সর্বোচ্চ মারবেল = 19x
= 19 × 3
= 57

৬,২১১.

If four fair coins are flipped, what is the probability that they all will come up heads?

ক
ক) 3/16
খ
খ) 1/4
গ
গ) 1/8
ঘ
ঘ) 1/16

ব্যাখ্যা

4টি নিরপেক্ষ মুদ্রা একবার নিক্ষেপ করলে মোট নমুনা বিন্দু হবে = {HHHH, HHTH, HTHH, HTTH, THHH, THTH, TTHH, TTTH,HHHT, HHTT, HTHT, HTTT, THHT, THTT, TTHT, TTTT} = 16টি

4টিতেই Head পাওয়ার সম্ভাবনা = 1/16

বিকল্প

১টি মুদ্রায় Head উঠার সম্ভাবনা = 1/2
৪টিমুদ্রায় Head উঠার সম্ভাবনা = (1/2) × (1/2) × (1/2) × (1/2)
= 1/16

৬,২১২.

The area of a square is equal to the area of a parallelogram. If the base of the parallelogram is 81 meters and its height is 4 meters, what is the length of one side of the square?

ক
16 meters
খ
18 meters
গ
24 meters
ঘ
12 meters

ব্যাখ্যা

Question: The area of a square is equal to the area of a parallelogram. If the base of the parallelogram is 81 meters and its height is 4 meters, what is the length of one side of the square?

Solution:
Area of the parallelogram = Base × Height
= 81 × 4
= 324 square meters

Let,
Side of the square = k meters
∴ Area of the square = k² square meters

ATQ,
k² = 324
⇒ k = √324
∴ k = 18

Therefore, the length of one side of the square = 18 meters

৬,২১৩.

If the compound interest of a certain sum of money for two successive years be Tk. 225 and Tk. 238.50 . What is the rate of interest per annum

ক
ক) 5%
খ
খ) 6%
গ
গ) 9%
ঘ
ঘ) 10%

ব্যাখ্যা

225 টাকার 1 বছরের সরল সুদ = 238.50 - 225 = 13.50 টাকা

আমরা জানি,
হার = সুদ / (আসল × সময়)
= 13.50/(225 × 1) × 100%
= 6%

৬,২১৪.

The difference in Taka between simple and compound interest at 4% annually on a sum of Tk. 8,000 after 2 years is:

ক
Tk. 22
খ
Tk. 12.8
গ
Tk. 15
ঘ
Tk. 16.5

ব্যাখ্যা

Question: The difference in Taka between simple and compound interest at 4% annually on a sum of Tk. 8,000 after 2 years is:

Solution:
Given that, Principal, P = Tk. 8,000
Rate, r = 4%
Time, n = 2 years

We know that,
Simple Interest = Prn/100
= (8000 × 4 × 2)/100
= 640

And Compound Interest = P(1 + r/100)ⁿ - P
= 8000{1 + (4/100)}² - 8000
= 8000(104/100)² - 8000
= {8000 × (104/100) × (104/100)} - 8000
= 8652.8 - 8000
= 652.8

∴ Difference = 652.8 - 640 = 12.8

The difference between compound and simple interest is Tk. 12.8

৬,২১৫.

If a : b = 4 : 7 and b : c = 5 : 6 than a : b : c = ?

ক
ক) 8 : 15 : 12
খ
খ) 20 : 35 : 42
গ
গ) 20 : 35 : 24
ঘ
ঘ) None

ব্যাখ্যা

Question: If a : b = 4 : 7 and b : c = 5 : 6 than a : b : c = ?

Solution:
a : b = 4 : 7 = 20 : 35
b : c = 5 : 6 = 35 : 42

a : b : c = 20 : 35 : 42

৬,২১৬.

10, 25, 45, 54, 60, 75, 80
Which number doesn’t belong here?

ক
ক) 45
খ
খ) 54
গ
গ) 75
ঘ
ঘ) 60

ব্যাখ্যা

Question: 10, 25, 45, 54, 60, 75, 80
Which number doesn’t belong here?

Solution:
Each of the numbers except 54 is multiple of 5.

So, 54 doesn’t belong here.

৬,২১৭.

The area of a square and rectangle are equal. The length of the rectangle is greater than the length of any side of the square by 5 cm and the breadth is less 3 cm. Find the perimeter of the rectangle.

ক
30 cm
খ
34 cm
গ
17 cm
ঘ
26 cm

ব্যাখ্যা

Question: The area of a square and rectangle are equal. The length of the rectangle is greater than the length of any side of the square by 5 cm and the breadth is less 3 cm. Find the perimeter of the rectangle.

Solution:
Let, the length of each side of the square be x cm.
Then, the length of rectangle = (x + 5) cm
and its breadth = (x - 3) cm

ATQ,
(x + 5)(x - 3) = x²
⇒ x² + 5x - 3x - 15 = x²
⇒ 2x = 15
∴ x = 15/2

Length = (15/2) + 5 = 25/2 cm
Breadth = (15/2) - 3 = 9/2 cm

∴ Perimeter = 2(length + breadth) = 2 {(25/2) + (9/2)}
= 2 (34/2) = 34 cm

৬,২১৮.

A man buys Tk 20 shares paying 9% dividend. The man wants to have an interest of 12% on his money. The market value of each share is:

ক
ক) 12
খ
খ) 15
গ
গ) 18
ঘ
ঘ) 21
ঙ
ঙ) 23

ব্যাখ্যা

Dividend on Tk.20 = Tk. (9/100 × 20) = Tk. 9/5
Tk.12 is income of Tk.100.
∴Tk. 9/5 is an income of = Tk (100/12 × 9/5) = Tk.15

৬,২১৯.

A tank is filled in 6 hours by three pipes A, B and C. The pipe C is twice as fast as B and B is twice as fast as A. How much time will pipe A alone take to fill the tank?

ক
ক) 42 hours
খ
খ) 35 hours
গ
গ) 32 hours
ঘ
ঘ) 36 hours

ব্যাখ্যা

Question: A tank is filled in 6 hours by three pipes A, B and C. The pipe C is twice as fast as B and B is twice as fast as A. How much time will pipe A alone take to fill the tank?

Solution:
Suppose pipe A alone takes x hours to fill the tank.
Then, pipes B and C will take x/2 and x/4 hours respectively to fill the tank.

∴1/x + 2/x + 4/x = 1/6
⇒ 7/x =1/6
⇒ x = 42 hours

৬,২২০.

What is the number of divisor of 1008?

ক
30
খ
25
গ
8
ঘ
None of these

ব্যাখ্যা

1008 = 2⁴ × 3² × 7
the number of divisor of 1008 = (4 + 1)(2 + 1) (1 + 1) = 5 × 3 × 2 = 30

৬,২২১.

The present age of three persons are in the ratio of 4 : 7 : 9. Eight years ago, their total age was 56 years. In five years what will be the age of the youngest person?

ক
20 years
খ
21 years
গ
22 years
ঘ
19 years

ব্যাখ্যা

Question: The present age of three persons are in the ratio of 4 : 7 : 9. Eight years ago, their total age was 56 years. In five years what will be the age of the youngest person?

Solution:
Present age ratio of three persons is 4 : 7 : 9

Let,
Their age is 4X, 7X and 9X respectively.

ATQ,
4X - 8 + 7X - 8 + 9X - 8 = 56
Or, 20X - 24 = 56
Or, 20X = 80
Or, X = 4

∴ The present age of the youngest person is = 4X = 4 x 4 = 16 years.
In five years, his age will be = (16 + 5 ) = 21 years.

৬,২২২.

2 men, working 9 hours a day, can build a dam in 2 days. How many hours a day must 3 men work to build the dam in 1 day?

ক
ক) 6 hours
খ
খ) 10 hours
গ
গ) 12 hours
ঘ
ঘ) 14 hours

ব্যাখ্যা

Question: 2 men, working 9 hours a day, can build a dam in 2 days. How many hours a day must 3 men work to build the dam in 1 day?

Solution:
2 men need 2 days working 9 hours
∴ 1 man need 2 days working (9 × 2) hours
∴ 1 man need 1 day working (9 × 2 × 2) hours
∴ 3 men need 1 day working (9 × 2 × 2)/3 hours
= 12 hours

৬,২২৩.

সুমন একটি কাজ ১৫ দিনে করতে পারে এবং রাজু একই কাজ ২৫ দিনে করতে পারে। সুমন ও রাজু একত্রে ৬ দিন কাজ করার পর সুমন চলে গেল। বাকি কাজ রাজু একা কত দিনে সম্পন্ন করতে পারবে?

ক
৫ দিন
খ
৭ দিন
গ
৯ দিন
ঘ
১২ দিন
ঙ
১৫ দিন

ব্যাখ্যা

প্রশ্ন: সুমন একটি কাজ ১৫ দিনে করতে পারে এবং রাজু একই কাজ ২৫ দিনে করতে পারে। সুমন ও রাজু একত্রে ৬ দিন কাজ করার পর সুমন চলে গেল। বাকি কাজ রাজু একা কত দিনে সম্পন্ন করতে পারবে?

সমাধান:
সুমন ১ দিনে করে = ১/১৫ অংশ কাজ
রাজু ১ দিনে করে = ১/২৫ অংশ কাজ

সুমন ও রাজু একত্রে,
১ দিনে করে = (১/১৫) + (১/২৫) অংশ কাজ
= (৫ + ৩)/৭৫ অংশ কাজ
= ৮/৭৫ অংশ কাজ
∴ ৬ দিনে করে = (৬ × ৮)/৭৫ অংশ কাজ
= ৪৮/৭৫ অংশ কাজ

∴ বাকী কাজ = ১ - (৪৮/৭৫) অংশ কাজ
= ২৭/৭৫ অংশ কাজ

রাজু ১ অংশ কাজ করে = ২৫ দিনে
রাজু ২৭/৭৫ অংশ কাজ করে = (২৫ × ২৭)/৭৫ দিনে
= ৯ দিনে

∴ বাকি কাজ রাজু একা সম্পন্ন করতে পারবে = ৯ দিনে।

৬,২২৪.

The ratio of the angles of a triangle is 2 : 3 : 4. What are the angles?

ক
30°, 45°, 90°
খ
45°, 60°, 120°
গ
90°, 50°, 30°
ঘ
40°, 60°, 80°

ব্যাখ্যা

Question: The ratio of the angles of a triangle is 2 : 3 : 4. What are the angles?

Solution:
The sum of the ratios = 2 + 3 + 4 = 9

We know that,
The sum of the three angles of a triangle = 180°

Now,
First angle = (2/9) × 180° = 40°
Second angle = (3/9) × 180° = 60°
Third angle = (4/9) × 180° = 80°

So, the angles are 40°, 60°, 80°

৬,২২৫.

১৫, ২৫, ৩৩ রাশিগুলোর ৪র্থ সমানুপাতী কত?

ক
ক) ৩৫
খ
খ) ৪৫
গ
গ) ৫৫
ঘ
ঘ) ৬৫

ব্যাখ্যা

এখানে
প্রথম রাশি = ১৫
দ্বিতীয় রাশি = ২৫
তৃতীয় রাশি = ৩৩

আমরা জানি,
১ম রাশি × ৪র্থ রাশি = ২য় রাশি × ৩য় রাশি
১৫ × ৪র্থ রাশি = ২৫ × ৩৩
৪র্থ রাশি = (২৫ × ৩৩)/১৫
= ৫৫
∴ ৪র্থ সমানুপাতি = ৫৫

৬,২২৬.

A clock is started at noon. By 20 minutes past 4, through how many degrees has the hour hand turned?

ক
150°
খ
130 °
গ
120°
ঘ
160°
ঙ
110°

ব্যাখ্যা

Question: A clock is started at noon. By 20 minutes past 4, through how many degrees has the hour hand turned?

From noon (12.00 pm) to 4 pm = 4 hours
and 20 minutes = 20/60 hours = 1/3 hours

Total time = (4 + 1/3) = 13/3 hours

We know,
Angle traced by hour hand in 12 hours = 360°

∴ Angle in 13/3 hours,
= (360/12) × (13/3)
= 130°

৬,২২৭.

If a and b are integers greater than 100 such that a + b = 300, which of the following could be the exact of a to b?

ক
ক) 9 to 1
খ
খ) 5 to 2
গ
গ) 5 to 3
ঘ
ঘ) 3 to 2

ব্যাখ্যা

অপশন a থেকে,
a = {300/(9 + 1)} × 1 = 30 [100 থেকে ছোট, তাই বাদ]

অপশন b থেকে,
a = {300/(5 + 2)} × 5 = 128.7 [ভগ্নাংশ, তাই বাদ]

অপশন c থেকে,
a = {300/(5 + 3)} × 5 = 187.5 [ভগ্নাংশ, তাই বাদ]
b = {300/(5 + 3)} × 3 = 112.5 [ভগ্নাংশ, তাই বাদ]

অপশন d থেকে,
a = {300/(3 + 2)} × 3 = 180
এবং, b = {300/(3 + 2)} × 2 = 120

৬,২২৮.

What is the value of the lesser root of the equation x² - 3x + 2 = 0?

ক
1
খ
2
গ
3
ঘ
- 1

ব্যাখ্যা

Question: What is the value of the lesser root of the equation x² - 3x + 2 = 0?

Solution:
x² - 3x + 2 = 0
⇒ x² - 2x - x + 2 = 0
⇒ x(x - 2) - 1(x - 2)= 0
⇒ (x - 1)(x - 2) = 0
so the solutions to the equation are x₁ = 1, x₂ = 2.
The lesser one is obviously 1.

৬,২২৯.

A question paper has two parts, A and B, each containing 10 questions. If a student has to choose 5 from part A and 8 from part B, in how many ways can he choose the questions?

ক
ক) 10340
খ
খ) 11340
গ
গ) 12340
ঘ
ঘ) 21340

ব্যাখ্যা

Question: A question paper has two parts, A and B, each containing 10 questions. If a student has to choose 5 from part A and 8 from part B, in how many ways can he choose the questions?

Solution:
ways to choose 5 from part A = ¹⁰C₅
ways to choose 8 from part B = ¹⁰C₈

choose 5 from part A and 8 from part B = ¹⁰C₅ × ¹⁰C₈
= {10!/(5! 5!)} × {10!/(2! 8!)}
= 11340

৬,২৩০.

A swimmer covers 3 km against the current in 30 minutes and the same distance with the current in 15 minutes. What is the speed of the swimmer in still water?

ক
6 km/hr
খ
9 km/hr
গ
5 km/hr
ঘ
10 km/hr

ব্যাখ্যা

Question: A swimmer covers 3 km against the current in 30 minutes and the same distance with the current in 15 minutes. What is the speed of the swimmer in still water?

Solution:
Given that,
Distance covered = 3 km
Time consumed = 30 minutes = 1/2 hour.
∴ speed of the swimmer against current (upstream) = 3/(1/2) = 6 km/hr

Again,
With the current,
Distance covered = 3 km
Time consumed = 15 minutes = 15/60 = 1/4 hr
∴ speed of the swimmer with current (downstream) = 3/(1/4) = 12 km/hr

let,
swimmer's speed in still water = x km/hr
speed of current = y km/hr

According to question,
x + y = 12 ------(1)
x - y = 6 -------(2)

(1) + (2) ⇒
2x = 18
⇒ x = 9

∴ swimmer's speed in still water = 9 km/hr

৬,২৩১.

One rabbit saw 6 elephants while going towards River. Every elephant saw 2 monkeys are going towards river. Every monkey holds one tortoise in their hands. How many animals are going towards the river?

ক
14
খ
11
গ
5
ঘ
8

ব্যাখ্যা

Question: One rabbit saw 6 elephants while going towards River. Every elephant saw 2 monkeys are going towards river. Every monkey holds one tortoise in their hands. How many animals are going towards the river?

Solution:

From the given data,

1 rabbit is going towards river not the six elephants. And these 6 elephants saw 2 monkeys are going towards river. Each monkey is holding 1 tortoise.

Hence, number of animals going towards river are 1 rabbit, 2 monkeys and 2 tortoice

= 1 + 2 + 2

= 5.

৬,২৩২.

What must be added to each term of the ratio 7 : 11 so as to make it equal to 5 : 6?

ক
ক) 9
খ
খ) 11
গ
গ) 13
ঘ
ঘ) 17

ব্যাখ্যা

Question: What must be added to each term of the ratio 7 : 11 so as to make it equal to 5 : 6?

Solution:
let, the number be x

(7 + x) : (11 + x) = 5 : 6
⇒ (7 + x) / (11 + x) = 5/6
⇒ 6 (7 + x) = 5 (11 + x)
⇒ 42 + 6x = 55 + 5x
⇒ 6x - 5x = 55 - 42
∴ x = 13

৬,২৩৩.

A train having a length of 270 meter is running at the speed of 120 kmph. It crosses another train running in the opposite direction at the speed of 80 kmph in 9 seconds. What is the length of the other train?

ক
ক) 230 meter
খ
খ) 190 meter
গ
গ) 320 meter
ঘ
ঘ) 210 meter

ব্যাখ্যা

Relative speed = 120 + 80
= 200 km/hr.
= 200 × (5/18)
= 500/9
Time = 9 seconds
Distance covered = (500/9) × 9
= 500 meter.
Length of the train = (500 - 270) meter.
= 230 meter.

৬,২৩৪.

A man can row 40 kmph in still water and the river is running at 10 kmph. If the man takes 1 hr to row to a place and back, how far is the place?

ক
ক) 18.75 km
খ
খ) 16.5 km
গ
গ) 2.25 km
ঘ
ঘ) 12.15 km

ব্যাখ্যা

Let the distance be x
Speed upstream = (40-10) = 30 kmph
Speed downstream = (40+10) = 50 kmph
Total time taken = 1 hr
⇒ x/50 + x/30 = 1
⇒ 8x/150 = 1
⇒ x = 150/8 = 18.75 km

৬,২৩৫.

A man has Tk. 480 in the denominations of one-taka notes, five-taka notes and ten-taka notes. The number of notes of each denomination is equal. What is the total number of notes that he has?

ক
45
খ
60
গ
75
ঘ
90

ব্যাখ্যা

Question: A man has Tk. 480 in the denominations of one-taka notes, five-taka notes and ten-taka notes. The number of notes of each denomination is equal. What is the total number of notes that he has?

Solution:
Let number of notes of each denomination be x.

Then
x + 5x + 10x = 480
⇒ 16x = 480
∴ x = 30.

Hence, total number of notes = 3x = 90.

৬,২৩৬.

If the simple interest on a certain sum of money after 3(1/8) years is (1/4) of the principal, what is the rate of interest per annum?

ক
ক) 4%
খ
খ) 12%
গ
গ) 6%
ঘ
ঘ) 8%

ব্যাখ্যা

Let the sum be x
Then,
Simple interest = x/4
T = 3(1/8)
= 25/8 years
R = {100 × (x/4)}/{x × (25/8)}
= 8
Hence Required interest rate = 8%

৬,২৩৭.

The sum of first five prime numbers is:

ক
11
খ
18
গ
26
ঘ
28

ব্যাখ্যা

Question: The sum of first five prime numbers is -

Solution:
Required sum = (2 + 3 + 5 + 7 + 11)
= 28.

৬,২৩৮.

log(a/b) + log(b/c) + log(c/a) = ?

ক
logabc
খ
abc
গ
1
ঘ
0
ঙ
None of these

ব্যাখ্যা

Question: log(a/b) + log(b/c) + log(c/a) = ?

Solution:
log(a/b) + log(b/c) + log(c/a)
= loga - logb + logb - logc + logc - loga
= 0

৬,২৩৯.

How many coins 3 mm thick and 1.2 cm in diameter should be melted in order to form a right circular cylinder, having base diameter 4 cm and height 27 cm?

ক
850
খ
950
গ
980
ঘ
1000
ঙ
900

ব্যাখ্যা

Question: How many coins 3 mm thick and 1.2 cm in diameter should be melted in order to form a right circular cylinder, having base diameter 4 cm and height 27 cm?

Solution:
Let the number of coins be n.

We have
n × π × (1.2/2)² × 0.3 = π × (4/2)² × 27
⇒ n × 0.36 × 0.3 = 4 × 27
⇒ n = (4 × 27 × 100 × 10)/(36 × 3)
⇒ n = 1000

৬,২৪০.

Two boats on the opposite shores of a river start moving toward each other. When they pass each other they are 750 yards from one shoreline. They each continue to the opposite shore, immediately turn around and start back. When they meet again they are 250 yards from the other shoreline. Each boat maintains a constant speed throughout. How wide was the river?

ক
ক) 2400 yards
খ
খ) 3000 yards
গ
গ) 2000 yards
ঘ
ঘ) 4000 yards

ব্যাখ্যা

ধরি,
একটি জাহাজ A speed - এ যাচ্ছে এবং অপরটি B speed এ যাচ্ছে।
নদীর প্রস্থ w
At time = t₁
A (t₁) = 750
B(t₁) =(w - 750)
⇒ B/A = (w - 750)/750
750B = (w - 750)A
At time = t₂
A(t₂) = w + 250
B(t₂) = w +(w - 250)
= 2w - 250
⇒ B/A = (2w - 250)/(w + 250)
(w + 250)B = (2w - 250)A .... Eq.[1]
750B = (w - 750/A
B = [(w - 750)/750]A
Plug into Eq. [1]
(w + 250)[(w - 750)/750]A = (2w - 250) A
(w + 250)(w - 750)/750 =750(2w -250)
w² - 500w - 187,500 = 1500w - 187500
w2 - 2000w = 0
w2 = 2000w
w = 2000 yards.

৬,২৪১.

log_4√3 48 = x², x³ = ?

ক
2
খ
√2
গ
2√2
ঘ
8

ব্যাখ্যা

Question : log_4√3 48 = x², x³ = ?

Solution :
log_4√3 48 = x²
⇒ log_4√3 (4√3)² = x²
⇒ 2log_4√3 4√3 = x² [ log_a a = 1]
⇒ x² = 2
⇒ x = √2
⇒ x³ = (√2)³
∴ x = 2√2

৬,২৪২.

A military camp had enough food supplies to last for a certain number of days. After 10 days, 1/5 of the men desert and it is found that the provisions will now last just as long as before. How long was that?

ক
60
খ
50
গ
40
ঘ
45

ব্যাখ্যা

Question: A military camp had enough food supplies to last for a certain number of days. After 10 days, 1/5 of the men desert and it is found that the provisions will now last just as long as before. How long was that?

Solution:
Let, initially there be 'x' men having foods for y days
After 1/5 of the men left = x - (x/5)
= 4x/5

After 10 days,
x men had foods for (y - 10) days
∴ 1 man had foods for x(y - 10) days
∴ 4x/5 men had foods for = {5x(y - 10)}/4x days
= 5(y - 10)/4 days

ATQ,
5(y - 10)/4 = y
⇒ 5y - 50 = 4y
∴ y = 50

৬,২৪৩.

If 5% is gained by selling an article BDT 350 than selling it for BDT 340, The cost of the article is -

ক
ক) BDT 180
খ
খ) BDT 150
গ
গ) BDT 200
ঘ
ঘ) BDT 250

ব্যাখ্যা

Let, buying price is x Taka
ATQ, 5% of x = 350 - 340 = 10
⇒ x = (100 × 10) / 5
∴ x = 200

৬,২৪৪.

A man can row 6 km/hr in still water. If the speed of the current is 2 km/hr, it takes 3 hrs more in upstream than in the downstream for the same distance. The distance is-

ক
48 km
খ
32 km
গ
24 km
ঘ
12 km

ব্যাখ্যা

Question: A man can row 6 km/hr in still water. If the speed of the current is 2 km/hr, it takes 3 hrs more in upstream than in the downstream for the same distance. The distance is-

Solution:
Let distance = x
Speed of man in still water, = 6 km/h
Speed of current, = 2 km/h

Now
{x/(6 - 2)} - {x/(6 + 2)} = 3
⇒ (x/4) - (x/8) = 3
⇒ (2x - x)/8 = 3
⇒ x/8 = 3
∴ x = 24

∴ Distance = 24 km.

৬,২৪৫.

An observer 1.5 m tall stands 10√3 meters away from a flagpole. The angle of elevation from his eye to the top of the flagpole is 30°. What is the height of the flagpole?

ক
10 m
খ
6 m
গ
12.5 m
ঘ
15 m
ঙ
11.5 m

ব্যাখ্যা

Question: An observer 1.5 m tall stands 10√3 meters away from a flagpole. The angle of elevation from his eye to the top of the flagpole is 30°. What is the height of the flagpole?

Solution:

Here,
Flagpole Height = AB

Now,
tan∠c = AE/CE
⇒ tan30° = AE/10√3
⇒ 1/√3 = AE/10√3
∴ AE = 10

∴ AB = AE + BE
= 10 + 1.5
= 11.5 m

৬,২৪৬.

If 6^th March, 2005 is Sunday, what was the day of the week on 6^th March, 2004?

ক
Friday
খ
Thursday
গ
Sunday
ঘ
Saturday

ব্যাখ্যা

Question: If 6^th March, 2005 is Sunday, what was the day of the week on 6^th March, 2004?

Solution:
Since 6^th March coming after February. Leap year will not count for 2004.
∴ There number of odd day is 1.

So it had to be Saturday, which was one day before Sunday

৬,২৪৭.

The speed of a car increases by 2 kms after every one hour. If the distance travelling in the first one hour was 35 kms. what was the total distance travelled in 12 hours?

ক
456 kms
খ
558 kms
গ
482 kms
ঘ
556 kms
ঙ
None of these

ব্যাখ্যা

Question: The speed of a car increases by 2 kms after every one hour. If the distance travelling in the first one hour was 35 kms. what was the total distance travelled in 12 hours?

Solution:
Total distance travelled in 12 hours = (35 + 37 + 39 + ..... upto 12 terms)
This is an A.P with first term, a = 35,
number of terms, n= 12,
d=2.

Required distance = (12/2)[2 × 35 + {12 - 1) × 2]
= 6(70 + 22)
= 552 kms

৬,২৪৮.

If the radius of a circle is increased by 100%, by what percent is the area of the circle increased?

ক
ক) 500%
খ
খ) 400%
গ
গ) 300%
ঘ
ঘ) 200%

ব্যাখ্যা

মনে করি,
radius = 10
সুতরাং area = πr²
= 100π
radius 100% বাড়ল।
নতুন radius = 20
area = πr²
= 400π
area বাড়ল 300%

৬,২৪৯.

The difference of two number is 20% of the large number, if the smaller number is 20, then the larger number is:

ক
ক) 25
খ
খ) 20
গ
গ) 10
ঘ
ঘ) 30

ব্যাখ্যা

Let the large number be x.
Then,
x - 20 is a 20% of x = 20x/100 = x/5
Or, x - x/5 = 20
Or, 4x = 100
∴ x = 25

৬,২৫০.

If one-third of one-fourth of a number is 15, then three-tenth of that number is-

ক
180
খ
155
গ
54
ঘ
144
ঙ
66

ব্যাখ্যা

Question: If one-third of one-fourth of a number is 15, then three-tenth of that number is -

Solution:
Let the number be x.
The equation:
⇒ 1/3 × 1/4 × x = 15
⇒ 1/12 × x = 15
⇒ x = 12 × 15
⇒ x = 180

three-tenth of the number:
= 3/10 × 180
= 54

∴ That number is 54

৬,২৫১.

If a person walks at 14 km/hr instead of 10 km/hr, he would have walked 20 km more. The actual distance travelled by him is-

ক
50 km
খ
56 km
গ
70 km
ঘ
80 km
ঙ
None of these

ব্যাখ্যা

Question: If a person walks at 14 km/hr instead of 10 km/hr, he would have walked 20 km more. The actual distance travelled by him is-

Solution:
Let the actual distance travelled be x km.
Then,
x/10 = (x + 20)/14
⇒ 14x = 10x + 200
⇒ 4x = 200
∴ x = 50 km.

৬,২৫২.

A candidate scoring 30% marks in an examination fails by 24 marks while another candidate who score 60% marks gets 36 marks more than the minimum pass marks. What is the minimum pass mark?

ক
75
খ
80
গ
84
ঘ
86

ব্যাখ্যা

Question: A candidate scoring 30% marks in an examination fails by 24 marks while another candidate who score 60% marks gets 36 marks more than the minimum pass marks. What is the minimum pass mark?

Solution:
Let,
the total marks of the exam be x.

∴ 30% of x = 3x/10
60% of x = 3x/5

ATQ,
3x/10 + 24 = 3x/5 - 36
⇒ (3x + 240)/10 = (3x - 180)/5
⇒ 5(3x + 240) = 10(3x - 180)
⇒ 15x + 1200 = 30x - 1800
⇒ 1200 + 1800 = 30x - 15x
⇒ 15x = 3000
∴ x = 200

∴ the minimum pass mark = (3 × 200)/10 + 24
= 600/10 + 24
= 84

৬,২৫৩.

A and B are centers of two circles that touch each other externally, as shown in the figure. What is the area of the circle whose diameter is AB?

ক
49π/4 square cm
খ
49π square cm
গ
25π/4 square cm
ঘ
36π square cm

ব্যাখ্যা

Question: A and B are centers of two circles that touch each other externally, as shown in the figure. What is the area of the circle whose diameter is AB?

Solution:
যেহেতু বৃত্ত দুটি পরস্পরকে বহিস্থভাবে স্পর্শ করে, তাই তাদের কেন্দ্রবিন্দুদ্বয়ের মধ্যবর্তী দূরত্ব (AB) হবে তাদের ব্যাসার্ধের যোগফলের সমান।

এখন, নতুন বৃত্তের ব্যাস, AB = (4 + 3) সেমি = 7 সেমি।
সুতরাং, নতুন বৃত্তের ব্যাসার্ধ, r = 7/2 সেমি।

∴নতুন বৃত্তের ক্ষেত্রফল = πr²
= π(7/2)²
= π(49/4)
= 49π/4 বর্গ সেমি।

সুতরাং, নতুন বৃত্তের ক্ষেত্রফল হবে 49π/4 বর্গ সেমি।

৬,২৫৪.

Alam sold an item for Tk. 6,384 and incurred a loss of 30%. At what price should he have sold the item to have gained a profit of 30% ?

ক
ক) Tk. 14,656
খ
খ) Tk. 11,856
গ
গ) Tk. 13,544
ঘ
ঘ) None of these

ব্যাখ্যা

30% ক্ষতিতে দাম, 70% = 6384
30% লাভে দাম, 130% = (6384×130) / 70 = 11856

৬,২৫৫.

The area of a square is 1024 sq.cm. What is the ratio of the length to the breadth of a rectangle whose length is twice the side of the square and breadth is 12 cm less than the side of this square?

ক
5 : 18
খ
14 : 5
গ
16 : 5
ঘ
32 : 5

ব্যাখ্যা

Question: The area of a square is 1024 sq.cm. What is the ratio of the length to the breadth of a rectangle whose length is twice the side of the square and breadth is 12 cm less than the side of this square?

Solution:
Let,
Arm of the square be a cm.
Area of square = 1024 sq.cm.
∴ a² = 1024
⇒ a = √1024
∴ a = 32

Length of rectangle = 2a = (2 × 32) cm = 64 cm.
Breadth of rectangle = (32 - 12) cm = 20 cm.

∴ Required ratio = 64 : 20 = 16 : 5.

৬,২৫৬.

Jabed spends 60% of her salary and donates 20% of her salary to charity. If he is left with Tk. 2000, what is her monthly salary?

ক
10000 tk
খ
11000 tk
গ
12000 tk
ঘ
12600 tk

ব্যাখ্যা

Question: Jabed spends 60% of her salary and donates 20% of her salary to charity. If he is left with Tk. 2000, what is her monthly salary?

Solution:
Let Jabed's salary be x

ATQ,
x - (60% of x + 20% of x) = 2000
⇒ x - 80% of x = 2000
⇒ 20% of x = 2000
⇒ 20x/100 = 2000
⇒ 20x = 200000
⇒ x = 200000/20
⇒ x = 10000

Therefore, Jabed's monthly salary is Tk. 10,000.

৬,২৫৭.

If θ be an acute angle and 7sin²θ + 3cos²θ = 4, then the value of tanθ is?

ক
1
খ
1/√2
গ
0
ঘ
1/√3

ব্যাখ্যা

Question: If θ be an acute angle and 7sin²θ + 3cos²θ = 4, then the value of tanθ is?

solution:
7sin²θ + 3cos²θ = 4
⇒7sin²θ + 3(1 - sin²θ) = 4 [cos²θ = 1 - sin²θ]
⇒7sin²θ + 3 - 3sin²θ = 4
⇒ 4sin²θ = 1
⇒ sin²θ = 1/4
⇒ sinθ = 1/2
⇒ sinθ = sin30°
∴ θ = 30°

Now,
tanθ
= tan30^∘
= 1/√3

৬,২৫৮.

Three numbers are in the ratio 1 : 2 : 3 and their HCF 12. The average of three numbers is -

ক
ক) 24
খ
খ) 36
গ
গ) 48
ঘ
ঘ) 72

ব্যাখ্যা

Question: Three numbers are in the ratio 1 : 2 : 3 and their HCF 12. The sum of three numbers is -

Solution:
Let the three numbers are x, 2x, 3x respectively
Their HCF = x

ATQ,
x = 12

So, the three numbers are 12, 24, and 36 respectively
The average of the three number = (12 + 24 + 36)/3 = 24

৬,২৫৯.

The difference of two positive numbers is 7 and the difference of their squares is 112. What is the sum of two numbers?

ক
ক) 13
খ
খ) 14
গ
গ) 16
ঘ
ঘ) 18

ব্যাখ্যা

Let
The two numbers are x and y.
According to the question
x - y = 7
x² - y² = 112

We know
x² - y² = (x + y)(x - y)
112 = 7(x + y)
(x + y) = 112/7
x + y = 16

৬,২৬০.

The cost of variety A wheat flour is Tk. 42 per kg and variety B wheat flour is Tk. 35 per kg. If both variety A and variety B are mixed in the ratio of 3 : 2, then the price per kg of the mixed variety of wheat flour is:

ক
Tk. 37.40
খ
Tk. 39.20
গ
Tk. 38.50
ঘ
Tk. 41.50

ব্যাখ্যা

Question: The cost of variety A wheat flour is Tk. 42 per kg and variety B wheat flour is Tk. 35 per kg. If both variety A and variety B are mixed in the ratio of 3 : 2, then the price per kg of the mixed variety of wheat flour is:

Solution:
Let,
Quantity of variety A flour is 3x kg.
Quantity of variety B flour is 2x kg.
The price per kg of the mixed variety of flour is y taka
∴ Total price of variety A flour is 42 × 3x = 126x Taka
∴ Total price of variety B flour is 35 × 2x = 70x Taka

ATQ,
126x + 70x = y(3x + 2x)
⇒ 196x = y × 5x
⇒ y = (196x)/(5x)
∴ y = 39.2
Therefore, the price per kg of the mixed variety of wheat flour is Tk. 39.20

৬,২৬১.

If 8 machines can produce 240 units in 6 days, how many units can 12 machines produce in 4 days?

ক
240 units
খ
360 units
গ
480 units
ঘ
720 units

ব্যাখ্যা

Question: If 8 machines can produce 240 units in 6 days, how many units can 12 machines produce in 4 days?

Solution:
In 6 days 8 machines can produce 240 units
In 4 days 12 machines can produce (240 × 4 × 12)/(6 × 8) units
= 240 units

৬,২৬২.

A sum of money at compound interest doubles itself in 15 years. It will become eight times of itself in-

ক
45 years
খ
42 years
গ
35 years
ঘ
48 years

ব্যাখ্যা

Question: A sum of money at compound interest doubles itself in 15 years. It will become eight times of itself in-

Solution:
let the sum P

2P = P (1 + r)¹⁵
⇒ (1 + r)¹⁵ = 2

let, sum will 8 times in n years
8P = P(1 + r)ⁿ
⇒ 8 = (1 + r)ⁿ
⇒ 2³ = (1 + r)ⁿ
⇒ {(1 + r)¹⁵}³ = (1 + r)ⁿ
⇒ (1 + r)⁴⁵ = (1 + r)ⁿ
∴ n = 45 years

৬,২৬৩.

If the diagonal of a square measures 16√2 cm, what is the area of the square in sq. cm?

ক
ক) 32√2
খ
খ) 64√2
গ
গ) 128
ঘ
ঘ) 256

ব্যাখ্যা

Given, the diagonal of a square measures, a√2 = 16√2 cm
So, side of the square is = 16 cm
Area of the square = 16² = 256 cm²

৬,২৬৪.

The length of a rope, to which a cow is tied assume that the cow is able to move on all sides with equal ease, is increased from 21 m to 28 m. How much additional ground will it be able to graze?

ক
995 sq m.
খ
1055 sq m.
গ
1078 sq m.
ঘ
1135 sq m.
ঙ
None

ব্যাখ্যা

Question: The length of a rope, to which a cow is tied assume that the cow is able to move on all sides with equal ease, is increased from 21 m to 28 m. How much additional ground will it be able to graze?

Solution:
We know,
Area of a circle = π × (radius)²

Given,
The cow can graze the area covered by the circle of radius 21 m initially, because the length of the rope is 21 m.
Therefore, the initial area that the cow can graze = (22/7) × 21² sq m.
= 1386 sq m.

When the length of the rope is increased to 28 m, grazing area becomes = (22/7) × 28² sq m.
= 2464 sq m.

The additional area it could graze when length is increased from 21 m to 28 m = (2464 - 1386) sq m.
= 1078 sq m.

৬,২৬৫.

Which line is parallel to y = x – 2?

ক
ক) y = 2x+1
খ
খ) 2y = 2x – 6
গ
গ) 2y = x+7
ঘ
ঘ) y= 3x+1

ব্যাখ্যা

দেয়া আছে, y = x - 2
যেহেতু, y = mx + c
সুতরাং, ঢাল = 1

a, c, d এই তিনটি অপশনের সমীকরণ দেখলে বুঝা যায় যে এগুলোর সমাধান করলে x এর সহগ 1 হবে না

সমান্তরাল হতে হলে দুটির ঢাল ই সমান হতে হবে। অপশনের মধ্য থেকে
2y = 2x - 6
⇒ y = x - 3

সুতরাং, ঢাল = 1

৬,২৬৬.

If a + b = 3 and ab = 2, then a³ + b³ = ?

ক
ক) 6
খ
খ) 7
গ
গ) 8
ঘ
ঘ) 9

ব্যাখ্যা

Question: If a + b = 3 and ab = 2, then a³ + b³ = ?

Solution:
a + b = 3
ab = 2

a³ + b³ = (a + b)³ - 3ab(a + b)
= 3³ - 3 × 3 × 2
= 27 - 18
= 9

৬,২৬৭.

Two trains running in opposite directions cross a man standing on the platform in 29 seconds and 17 seconds respectively and they cross each other in 23 seconds. The ratio of their speeds is -

ক
ক) 1:3
খ
খ) 3:2
গ
গ) 3:4
ঘ
ঘ) 1:1

ব্যাখ্যা

Let the speeds of the two trains be x m/sec and y m/sec respectively.
Then, length of the first train = 29x metres
and length of the second train = 17y metres
∴ (29x+17y)/(x+y) = 23
⇒ 29x+17y = 23x+23y
⇒ 6x=6y
⇒ x/y= 1/1

৬,২৬৮.

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

ক
250
খ
150
গ
135
ঘ
125

ব্যাখ্যা

Question: The average of a natural number and Its cube Is 13 times the number. The cube of the number is:
(Janata RC 2022 অনুযায়ী)

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.
Hence, x³ = 5³ = 125

৬,২৬৯.

At what rate percent per annum will the simple interest on a sum of money be 2/5 of the amount in 10 years?

ক
ক) 5%
খ
খ) 4%
গ
গ) 6%
ঘ
ঘ) 7%

ব্যাখ্যা

Principal P = x
S.I.= (2/5) of x = 2x/5
Time n = 10 years

S.I. = (P × r × n)/100
⇒ (2x/5)= (x × r × 10)/100
⇒ (2/5) = (r × 10)/100
⇒ (2/5) = r/10
⇒ 2 = r/2
⇒ r = 4%

∴ Rate is 4% per annum

৬,২৭০.

David got two and a half time as many marks in English as in History. If his total marks in the two subjects are 140, the marks obtained by him in English are:

ক
40
খ
75
গ
90
ঘ
100

ব্যাখ্যা

Question: David got two and a half time as many marks in English as in History. If his total marks in the two subjects are 140, the marks obtained by him in English are:

Solution:
Let,
The marks of History = x
∴ The marks of English = 2.5x

ATQ,
x + 2.5x = 140
⇒ 3.5x = 140
⇒ x = 140/3.5
∴ x = 40

∴ The marks of English = 2.5x = 2.5 × 40 = 100

৬,২৭১.

The square root of (6 + 5√2)(6 - 5√2) is:

ক
14i
খ
- 2
গ
2i
ঘ
i√14

ব্যাখ্যা

Question: The square root of (6 + 5√2)(6 - 5√2) is:

Solution:
√{(6 + 5√2)(6 - 5√2)}
= √{6² - (5√2)²}
= √{36 - (25 × 2)}
= √(36 - 50)
= √(- 14)
= √{14 × (- 1)}
= √(14 × i²) [∵ i² = - 1]
= i√14

৬,২৭২.

Six consecutive whole numbers are given. The sum of the first three numbers is 27. What is the product of the last three numbers?

ক
1832
খ
1542
গ
1852
ঘ
1716

ব্যাখ্যা

Question: Six consecutive whole numbers are given. The sum of the first three numbers is 27. What is the product of the last three numbers?

Solution:
let the numbers be n - 2, n - 1, n, n + 1, n + 2, n + 3

n - 2 + n - 1 + n = 27
⇒ 3n - 3 = 27
⇒ 3n = 27 + 3
⇒ n = 30/3
∴ n = 10

the sum of the last three numbers is = (n + 1) × (n + 2) × (n + 3)
= (10 + 1) × (10 + 2) × (10 + 3)
= 1716

৬,২৭৩.

If x and y are positive integers satisfying x + y = 7, what is the probability that x < y?

ক
1/2
খ
3/5
গ
2/5
ঘ
4/6

ব্যাখ্যা

Question: If x and y are positive integers satisfying x + y = 7, what is the probability that x < y?

Solution:
Since both x and y must be positive integers,
total possible ways = (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1) = 6 pairs.
Next, we identify the pairs where x < y,
(1, 6), (2, 5), (3, 4) ; There are 3 pairs satisfying x < y.

∴ Probability = Number of pairs where x < y/Total number of pairs = 3/6 = 1/2

So the probability that x < y is 1/2

৬,২৭৪.

If a + b + c = 9, a² + b² + c² = 29 then what is the value of (ab + bc + ca) = ?

ক
110
খ
52
গ
26
ঘ
25

ব্যাখ্যা

Question: If a + b + c = 9, a² + b² + c² = 29 then what is the value of (ab + bc + ca) = ?

Solution:
Given,
a + b + c = 9,
a² + b² + c² = 29

We know,
(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
⇒ 2(ab + bc + ca) = (a + b + c)² - (a² + b² + c²)
⇒ 2(ab + bc + ca) = (9)² - 29
⇒ 2(ab + bc + ca) = 81 - 29
⇒ 2(ab + bc + ca) = 52
∴ (ab + bc + ca) = 26

৬,২৭৫.

If the length of each side of an equilateral triangle is decreased by 1 units, the area is found to be decreased by 2√3 square unit. The length of each side of the triangle is-

ক
6 units
খ
5.5 units
গ
3.75 units
ঘ
9 units
ঙ
4.5 units

ব্যাখ্যা

Question: If the length of each side of an equilateral triangle is decreased by 1 units, the area is found to be decreased by 2√3 square unit. The length of each side of the triangle is-

Solution:
Let,
Original side length = x
New side length = x - 1
Decrease in area = 2√3

Now,
⇒ (√3/4)x² - (√3/4)(x - 1)² = 2√3
⇒ x² - (x - 1)² = 8
⇒ x² - (x² - 2x + 1) = 8
⇒ x² - x² + 2x - 1 = 8
⇒ 2x - 1 = 8
⇒ 2x = 9
⇒ x = 9/2
∴ x = 4.5 units

So the original length of each side of the equilateral triangle is 4.5 units.

৬,২৭৬.

At a party, everyone shakes hands with everybody else. If there were 66 handshakes, how many people were at the party?

ক
ক) 33
খ
খ) 34
গ
গ) 12
ঘ
ঘ) 31

ব্যাখ্যা

প্রশ্ন: At a party, everyone shakes hands with everybody else. If there were 66 handshakes, how many people were at the party?

সমাধান:
ধরি, x সংখ্যক লোক ছিল।

প্রশ্নমতে,
^xC₂ = 66
⇒ x!/{2!(x - 2)!} = 66
⇒ x (x - 1) (x - 2)!/2(x - 2)! = 66
⇒ x (x - 1)/2 = 66
⇒ x (x - 1) = 132
⇒ x² - x - 132 = 0
⇒ x² - 12x + 11x - 132 = 0
⇒ x (x - 12) + 11(x - 12) = 0
⇒ (x + 11) (x - 12) = 0
∴ x + 11 = 0 বা, x - 12 = 0

x = -11; লোকসংখ্যা ঋণাত্মক হতে পারে না।

∴ x = 12
অতএব, পার্টিতে ১২ জন ব্যক্তি ছিল।

৬,২৭৭.

A car reaches from City A to City B in 9 hours travelling at a speed of 40 km/hr. If its speed is increased by 20 km/hr, then the time of journey is reduced by-

ক
4.5 hours
খ
3 hours
গ
5.5 hours
ঘ
4 hours

ব্যাখ্যা

Question: A car reaches from City A to City B in 9 hours travelling at a speed of 40 km/hr. If its speed is increased by 20 km/hr, then the time of journey is reduced by-

Solution:
Original speed = 40 km/hr
Time taken = 9 hours

∴ Distance between City A and City B = speed × time
= 40 × 9 = 360 km

∴ New speed = 40 + 20 = 60 km/hr

∴ New time taken = distance/new speed
= 360/60
= 6 hours

∴ Reduction in time = original time - new time
= 9 - 6
= 3 hours

So the time of journey is reduced by 3 hours.

৬,২৭৮.

An outlet pipe can empty a cistern in 3 hours. In what time will empty 2/9 of the cistern?

ক
20 minutes
খ
30 minutes
গ
40 minutes
ঘ
50 minutes
ঙ
70 minutes

ব্যাখ্যা

The outlet pipe empties the one complete cistern in 3 hours
Time taken to empty 2/9 part of the cistern
= (2/9) × 3 hour
= (2/9) × 3 × 60 minutes
= (2/9) × 3 × 60 minutes
= 40 minutes

৬,২৭৯.

The radius of a circular plate is 20 cm. If the radius is decreased by 20%, what is the percentage decrease in its area?

ক
32%
খ
44%
গ
40%
ঘ
36%

ব্যাখ্যা

Question: The radius of a circular plate is 20 cm. If the radius is decreased by 20%, what is the percentage decrease in its area?

Solution:
Given that,
Original radius = 20 cm
And decreased by 20%
∴ Decrease = 20% of 20 = 4 cm
∴ New radius = 20 - 4 = 16 cm

We know,
Area of circle = πr²
∴ Original area = π × (20)² = 400π cm²
∴ New area = π × (16)² = 256π cm²

∴ Decrease in area = original area - new area = 400π - 256π = 144π cm²

∴ Percentage decrease in area,
= (decrease/original area) × 100%
= (144π/400π) × 100%
= (144/400) × 100%
= 0.36 × 100%
= 36%

So the area decreases by 36%.

৬,২৮০.

A clock loses (falls behind) 10 minutes each day. How many days will take to reach a point where the clock will indicate the correct time?

ক
ক) 36
খ
খ) 72
গ
গ) 120
ঘ
ঘ) None

ব্যাখ্যা

Question: A clock loses (falls behind) 10 minutes each day. How many days will take to reach a point where the clock will indicate the correct time?

Solution:
[ঘড়ি সময় হারালে অর্থাৎ ঘড়িটি প্রতি ঘণ্টায় স্বাভাবিক সময় থেকে পিছিয়ে গেলে পুনরায় সঠিক সময় দেওয়ার জন্য ১২ ঘণ্টা হয়ে যেতে হবে। কারণ একটি ঘড়িতে একবার সম্পূর্ণ ঘুরে আসার জন্য মোট ১২ ঘণ্টা ঘুরতে হবে।]

১২ ঘন্টা = ১২ × ৬০ মিনিট
= ৭২০ মিনিট

১০ মিনিট পিছিয়ে পড়ে ১ দিনে
১ মিনিট পিছিয়ে পড়ে ১/১০ দিনে
৭২০ মিনিট পিছিয়ে পড়ে = ৭২০/১০ দিনে
= ৭২ দিনে

৬,২৮১.

If all Bloops are Razzies and all Razzies are Lazzies, then ___.

ক
All Bloops are definitely Lazzies.
খ
Some Bloops are Lazzies.
গ
No Lazzies are Bloops.
ঘ
Some Razzies are not Lazzies.

ব্যাখ্যা

Question: If all Bloops are Razzies and all Razzies are Lazzies, then ___.

Solution:
From the information we get,

∴ All Bloops are definitely Lazzies.

৬,২৮২.

x² + y² = ১৪ এবং xy = ৩ হলে (x - y)² = কত?

ক
ক) ৮
খ
খ) ১১
গ
গ) ১৪
ঘ
ঘ) ১৭

ব্যাখ্যা

প্রশ্ন: x² + y² = ১৪ এবং xy = ৩ হলে (x - y)² = কত?

সমাধান:
আমরা জানি,
(x + y)²
= x² + y² + ২xy
= ১৪ + ২ × ৩
= ১৪ + ৬
= ২০

আবার,
(x - y)²
= (x + y)² - ৪xy
= ২০ - ৪ × ৩
= ২০ - ১২
= ৮

৬,২৮৩.

If the simple interest on a sum of money is 40% of the principal over 8 years, what is the annual interest rate?

ক
5%
খ
6%
গ
7%
ঘ
7.5%

ব্যাখ্যা

Question: If the simple interest on a sum of money is 40% of the principal over 8 years, what is the annual interest rate?

Solution:
We know, Simple Interest, I = Pnr

Given,
Simple Interest = 0.4 × Principal (since 40% of the principal)
= 0.4 × P

Where, Principal = P
Rate = r
Time, n = 8 years

So, 0.4 × P = P × 8 × r
⇒ r = 0.4/8
⇒ r = 0.05 × 100%
∴ r = 5%

৬,২৮৪.

In how many ways can a group of 4 men and 3 women be made out of a total of 8 men and 4 women?

ক
120 ways
খ
210 ways
গ
280 ways
ঘ
350 ways

ব্যাখ্যা

Question: In how many ways can a group of 4 men and 3 women be made out of a total of 8 men and 4 women?

Solution:
There are 8 men and 4 women.
We have to select 5 men out of 7 and 2 women out of 3.

∴ The number of ways of making the selection = ⁸C₄ × ⁴C₃
= 70 × 4
= 280 ways

৬,২৮৫.

If a bucket is 80% full, then it contains 4 liters more water than when it is 200/3% full, What is the capacity of the bucket?

ক
ক) 15 liters
খ
খ) 30 liters
গ
গ) 25 liters
ঘ
ঘ) 7.5 liters

ব্যাখ্যা

Question : If a bucket is 80% full, then it contains 4 liters more water than when it is 200/3% full, What is the capacity of the bucke?

Solution:
Let the capacity of the bucket be x.
(80 - 200/3)% of x =4 litres.
⇒ 40/3 % of x = 4 litres
⇒ 40x/(3 × 100)= 4
⇒40x = 3 × 4 × 100
x = (3 × 4 × 100)/40
x = 30

৬,২৮৬.

Rajib does 20% less work than Pavel. If Rajib can complete a piece of work in 15/2 hours, then Pavel can do it in

ক
9 hours
খ
6 hours
গ
7 hours
ঘ
8 hours

ব্যাখ্যা

Question: Rajib does 20% less work than Pavel. If Rajib can complete a piece of work in 15/2 hours, then Pavel can do it in

Solution:
Ratio of times taken by Rajib and Pavel = 100 : 80 = 5 : 4
Let
Pavel takes x days to do the work

Here
5 : 4 = (15/2) : x
⇒ 5/4 = (15/2)/x
⇒ 5x = (15 × 4/2)
⇒ 5x = 30
∴ x = 6 hours

৬,২৮৭.

The ratio of two positive numbers is 3 : 4. The sum of their squares is 400. What is the sum of the numbers?

ক
28
খ
32
গ
26
ঘ
30

ব্যাখ্যা

Question: The ratio of two positive numbers is 3 : 4. The sum of their squares is 400. What is the sum of the numbers?

Solution:
Let two positive numbers be 3x and 4x.
ATQ,
(3x)² + (4x)² = 400
⇒ 9x² + 16x² = 400
⇒ 25x² = 400
⇒ x² = 400/25
⇒ x²= 16
∴ x = 4

Sum of numbers = (3 × 4) + (4 × 4) = 28

৬,২৮৮.

A college has 10 basketball players. A 5-member team and a captain will be selected out of these 10 players. How many different selections can be made?

ক
1260
খ
1400
গ
1250
ঘ
1600
ঙ
1700

ব্যাখ্যা

A team of 6 members has to be selected from the 10 players.
This can be done in 10C6 or 210 ways.
Now, the captain can be selected from these 6 players in 6 ways.
Therefore, total ways the selection can be made is = 210 × 6
= 1260

৬,২৮৯.

A sum of Tk. 25000 amounts to Tk. 31000 in 4 years at the rate of simple interest, what is the rate of interest?

ক
3%
খ
4%
গ
5%
ঘ
6%

ব্যাখ্যা

Question: A sum of Tk. 25000 amounts to Tk. 31000 in 4 years at the rate of simple interest, what is the rate of interest?

Solution:
সুদ = ৩১০০০ - ২৫০০০ টাকা
= ৬০০০ টাকা

ধরি, সুদের হার r%

I = pnr
⇒ ৬০০০ = ২৫০০০ × ৪ × r/১০০
⇒ r = (৬০০০ × ১০০)/(২৫০০০ × ৪)
⇒ r = ৬

সুদের হার ৬%

৬,২৯০.

Today is Monday. After 59 days, it will be-

ক
Monday
খ
Friday
গ
Wednesday
ঘ
Thursday

ব্যাখ্যা

Question: Today is Monday. After 59 days, it will be-

Solution:
We know that each day of the week is repeated after 7 days.

59 ÷ 7 = 8 (remainder 3)

So, after (7 × 8) = 56 days, it will be Monday.
∴ After 59 days, it will be (Monday + 3 days) = Thursday

৬,২৯১.

The city library donated some books to a class. If each student takes 4 books, there will be 20 books left. If 3 students do not take a book and the rest of the students take 5 books each, there will be no books left. How many books were donated to the class?

ক
ক) 120
খ
খ) 140
গ
গ) 160
ঘ
ঘ) 175

ব্যাখ্যা

Let be the number of students n
ATQ, 4n +20 = (n-3)5
⇒ n = 35
So, No. of books = 4×35 + 20 = 160

৬,২৯২.

Tanveer's investment doubles in every 5 years. If he invested Tk. 50000 in each of the years 2010, 2015, 2020 then what will be the amount received by him in 2025?

ক
Tk. 300000
খ
Tk. 700000
গ
Tk. 800000
ঘ
Tk. 1500000

ব্যাখ্যা

Question: Tanveer's investment doubles in every 5 years. If he invested Tk. 50000 in each of the years 2010, 2015, 2020 then what will be the amount received by him in 2025?

Solution:
Given,
Investment doubles every 5 years.

Tanveer invested Tk. 50000 in each of the years 2010, 2015, 2020

Now, he invested 50000 in 2010 so the amount for 2010 is 50000

In 2015 the amount gets double the amount become 100000 and then he again invest 50000 so the total amount of the year 2015 is 150000.

In 2020 the amount gets double the amount become 300000 and then he again invest 50000 so the total amount of the year 2020 is 350000.

In the year 2025 the amount gets double the amount become 700000 finally, the amount in the year 2025 is 700000.

Hence, '700000' is the correct answer.

৬,২৯৩.

The inverse of f(x) = 2x - 1 is -

ক
ক) (x - 1)/2
খ
খ) (x+1)/2
গ
গ) 2x - 1
ঘ
ঘ) 2x + 1

ব্যাখ্যা

Let, y = f(x) = 2x - 1
or, y = 2x - 1
or, 2x = y + 1
or, x = (y + 1)/2
∴ y = f(x)
Or, f^-1(y) = x
or, f^-1(y) = (y + 1)/2
∴ f^-1(x) = (x + 1)/2

৬,২৯৪.

A fair six sided dice is rolled. Find the probability of getting an odd number or a number less than 4.

ক
ক) 2:3
খ
খ) 3:4
গ
গ) 5:6
ঘ
ঘ) 1:6

ব্যাখ্যা

Given that a single dice is rolled.
Sample space = {1, 2, 3, 4, 5, 6}
Let P(A) be the probability of getting an odd number, where A = {1, 3, 5}
Let P(B) be the probability of getting a number less than 4, where B = {1,2,3}
A ⋂ B ={1, 3}
So, P(A) = 3/6 = 1/2
P(B) = 3/6 = 1/2
P(A ⋂ B) = 2/6 = 1/3

Let P be the required probability of getting an odd number or a number less than 4
P = P(A ⋃ B) = P(A) + P(B) - P(A ⋂ B)
Or, P(A ⋃ B) = 1/2 + 1/2 - 1/3
Or, P(A ⋃ B) = 1 - 1/3
Or, P(A ⋃ B) = 2/3

Therefore, the probability of rolling an odd number or a number less than 4 is 2/3.

৬,২৯৫.

Pipe A is 4 times as fast as B in filling a tank. If A takes 20 minutes to fill a tank, then what is the time taken by both the pipe A and B to fill the tank?

ক
ক) 8
খ
খ) 12
গ
গ) 16
ঘ
ঘ) 20

ব্যাখ্যা

A takes 20 minutes and it is 4 times faster than B,
it means B will take 80 minutes to fill the tank.
(1/20 + 1/80) × t = 1.
So, We get t = 16.

৬,২৯৬.

What will come at the place of the question mark?
4, 8, 10, 14, 16, 20,?

ক
22
খ
24
গ
26
ঘ
28

ব্যাখ্যা

Question: What will come at the place of the question mark?
4, 8, 10, 14, 16, 20,?

Solution:
The above series contains two sub-series in it.
4, 10, 16, 22
8, 14, 20
For each of both terms, there is a difference of 6. The next term is 6 greater than the previous term.

৬,২৯৭.

One card is drawn at random from a pack of 52 cards. What is the probability that the card drawn is a face card (Jack, Queen and King) only?

ক
ক) 1/13
খ
খ) 3/13
গ
গ) 3/52
ঘ
ঘ) 9/52

ব্যাখ্যা

There are 52 cards, out of which there are 12 face (Jack, King and Queen)) cards.
∴ Probability = 12/52 = 3/13

৬,২৯৮.

Abu and Salim started a partnership business investing some amount of money in the ratio of 4 : 6. Shakeel joined them after six months with an amount equal to that of Salim. In what proportion should the profit at the end of one year be distributed among Abu, Salim and Shakeel?

ক
5 : 3 : 4
খ
4 : 6 : 2
গ
5 : 3 : 2
ঘ
4 : 6 : 3

ব্যাখ্যা

Question: Abu and Salim started a partnership business investing some amount of money in the ratio of 4 : 6. Shakeel joined them after six months with an amount equal to that of Salim. In what proportion should the profit at the end of one year be distributed among Abu, Salim and Shakeel?

Solution:
As (4 : 6 is equivalent to 2 : 3)
So let the initial investment of money ratio of Abu and Salim is 2x and 3x.
So Abu , Salim and Shakeel ratio of investment will be ( Abu : Salim : Shakeel ) = (2x × 12) : (3x × 12) : (3x × 6) = 24 : 36 : 18 = 4 : 6 : 3

৬,২৯৯.

The sum of the three consecutive even numbers is 44 more than the average of these numbers. Which of the following is the smallest of these numbers?

ক
ক) 18
খ
খ) 20
গ
গ) 22
ঘ
ঘ) 24

ব্যাখ্যা

Question: The sum of the three consecutive even numbers is 44 more than the average of these numbers. Which of the following is the smallest of these numbers?

Solution:
Let the number be x, x + 2, and x + 4

ATQ,
(x + x + 2 + x + 4) - (x + x + 2 + x + 4)/3 = 44
⇒ (3x + 6) - (3x + 6)/3 = 44
⇒ 9x + 18 - 3x - 6 = 132
⇒ 6x = 120
⇒ x = 20

৬,৩০০.

A ladder is leaning against a wall. It makes a 60° angle with the ground. If the distance between the foot of the ladder and the wall is 8 meters, what is the length of the ladder?

ক
12 meters
খ
8√2 meters
গ
16 meters
ঘ
20 meters

ব্যাখ্যা

Question: A ladder is leaning against a wall. It makes a 60° angle with the ground. If the distance between the foot of the ladder and the wall is 8 meters, what is the length of the ladder?

Solution:

ধরি, দেয়ালটি হলো AB এবং মইটি হলো AC
মইটি ভূমির সাথে ∠ACB = 60° কোণ তৈরি করে।
মইয়ের গোড়া থেকে দেয়ালের দূরত্ব, BC = 8 মিটার।
মইয়ের দৈর্ঘ্য হলো AC

এখন, ΔABC -এ
cos 60° = BC/AC
⇒ 1/2 = 8/AC
⇒ AC = 8 × 2
⇒ AC = 16
∴ মইয়ের দৈর্ঘ্য 16 মিটার।