Bank Math

PrepBank · পাতা ২ / ১৬১ · ১০১–২০০ / ১৬,১২৪

১০১.

The HCF and LCM of the two numbers are 165 and 33 respectively. If the ratio of the two numbers 1 : 5, then what is the largest number?

ক
ক) 105
খ
খ) 115
গ
গ) 135
ঘ
ঘ) 165

সঠিক উত্তর: ঘ

ঘ) 165

উত্তর

সঠিক উত্তর: ঘ

ঘ) 165

ঘ

ব্যাখ্যা

Question: The HCF and LCM of the two numbers are 165 and 33 respectively. If the ratio of the two numbers 1 : 5, then what is the largest number?

Solution:
Let the numbers be x and 5x respectively.

ATQ,
x × 5x = 165 × 33
⇒ x² = (165 × 33)/5
⇒ x² = 33 × 33
⇒ x = 33

The largest number is = 5 × 33 = 165

১০২.

From a poll of 80m high, the angle of depression of a bus is 30°. How far is the bus from the poll?

ক
138.4m
খ
128m
গ
130.4m
ঘ
148.4m
ঙ
198m

সঠিক উত্তর: ক

138.4m

উত্তর

সঠিক উত্তর: ক

138.4m

ক

ব্যাখ্যা

Let AC be the poll.
B be the position of the bus.
∴ BC = the distance of the bus from the foot of the poll.
Given that
height of the poll, AC = 80 m
Angle of depression, ∠DAB = 30°
∠ABC = ∠DAB = 30° (because DA || BC)

tan30 = AC/BC
or, 1/√3 = 80/BC
∴ BC = 80√3 = 138.4 m

১০৩.

A container contains 48 liters of milk. From this container 12 liters of milk was taken out and replaced by water. This process was repeated overall two times. How much milk is now contained by the container?

ক
ক) 25 liter
খ
খ) 24 liter
গ
গ) 26 liter
ঘ
ঘ) 27 liter

সঠিক উত্তর: ঘ

ঘ) 27 liter

উত্তর

সঠিক উত্তর: ঘ

ঘ) 27 liter

ঘ

ব্যাখ্যা

Question: A container contains 48 liters of milk. From this container 12 liters of milk was taken out and replaced by water. This process was repeated overall two times. How much milk is now contained by the container?

Solution:
After first replacement the ratio of milk and water is (48 - 12) : 12 = 36 : 12 = 3 : 1

After second replacement,
remaining milk = 36 - (3/4 of 12) = 36 - 9 = 27

∴ There is 27 liter of milk in the container now.

১০৪.

The price of cooking oil is reduced by 10%. A family buys 4 litres more for Tk. 720 after the reduction. What was the original price per litre?

ক
45 Tk/litre.
খ
40 Tk/litre.
গ
20 Tk/litre.
ঘ
35 Tk/litre.

সঠিক উত্তর: গ

20 Tk/litre.

উত্তর

সঠিক উত্তর: গ

20 Tk/litre.

গ

ব্যাখ্যা

Question: The price of cooking oil is reduced by 10%. A family buys 4 litres more for Tk. 720 after the reduction. What was the original price per litre?

Solution:
Let
Original price of cooking oil = x Tk/litre.
Original quantity = 720/x litre

New price = 0.90x Tk/litre
New quantity = 720/0.90x = 720/(9x/10) = (720 × 10)/9x = 800/x litre

ATQ,
(800/x) - (720/x) = 4
⇒ (800 - 720)/x = 4
⇒ 80/x = 4
⇒ x = 80/4
∴ x = 20

∴ Original price of cooking oil = 20 Tk/litre.

১০৫.

Find averages of the first 97 natural numbers.

ক
ক) 47
খ
খ) 37
গ
গ) 48
ঘ
ঘ) 49.5
ঙ
ঙ) 49

সঠিক উত্তর: ঙ

ঙ) 49

উত্তর

সঠিক উত্তর: ঙ

ঙ) 49

ঙ

ব্যাখ্যা

Shot Cut: to solve this type of problem, sum up the first number and last number of the series and divide by 2.
So, ( 1 + 97)/2 = 49

১০৬.

The ages of Anish and Belal are in the ratio 6 : 5 and the sum of their ages is 44 years. What will be the ratio of their ages after 10 years ?

ক
ক) 17 : 15
খ
খ) 17 : 13
গ
গ) 17 : 12
ঘ
ঘ) 13 : 15

সঠিক উত্তর: ক

ক) 17 : 15

উত্তর

সঠিক উত্তর: ক

ক) 17 : 15

ক

ব্যাখ্যা

Question: The ages of Anish and Belal are in the ratio 6 : 5 and the sum of their ages is 44 years. What will be the ratio of their ages after 10 years ?

Solution:
The ages of Anish and Belal are in the ratio 6 : 5
let, Anish is 6x years old and belal is 5x years old
5x + 6x = 44
⇒ 11x = 44
∴ x = 4 years old

So, Anish is 24 years old and belal is 20 years old

After 10 years, Anish is (24 + 10) = 34 years old and belal is (20 + 10) years or 30 years old
∴ their ratio = 34 : 30
= 17 : 15

১০৭.

An urn contains 6 red, 4 blue, 2 green and 3 yellow marbles. If two marbles are drawn at random from the urn, what is the probability that both are red?

ক
1/6
খ
1/7
গ
2/15
ঘ
2/5

সঠিক উত্তর: খ

1/7

উত্তর

সঠিক উত্তর: খ

1/7

খ

ব্যাখ্যা

Total number of balls = (6 + 4 + 2 + 3)
= 15.
Let,
E be the event of drawing 2 red balls.
Then,
n(E) = ⁶C₂
= (6 × 5)/(2 × 1)
= 15.
Also, n(S) = ¹⁵C₂
= (15 × 14)/(2 × 1)
= 105.
∴ P(E) = n(E)/n(S)
= 15/105
= 1/7.

১০৮.

Find the square of a positive number which when increased by 17 is equal to 60 times the reciprocal of the number.

ক
9
খ
16
গ
25
ঘ
18

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: Find the square of a positive number which when increased by 17 is equal to 60 times the reciprocal of the number.

Solution:
Let the number be x

Then, x + 17 = 60/x
⇒ x² + 17x − 60 = 0
⇒ x² + 20x − 3x − 60 = 0
⇒ (x + 20)(x − 3) = 0
∴ x = - 20, 3

The positive number = 3
Hence, the square of the positive number = 9

১০৯.

A is two years older than B who is twice as old as C. If the total of the ages of A, B and C be 42, then how old is B?

ক
6 years
খ
8 years
গ
12 years
ঘ
16 years

সঠিক উত্তর: ঘ

16 years

উত্তর

সঠিক উত্তর: ঘ

16 years

ঘ

ব্যাখ্যা

Question: A is two years older than B who is twice as old as C. If the total of the ages of A, B and C be 42, then how old is B?

Solution:
Let the age of C be = x years

Then, age of B = 2x years.
age of A = (2x + 2) years.

According to the question,
(2x + 2) + 2x + x = 42
⇒ 5x + 2 = 42
⇒ 5x = 42 - 2
⇒ 5x = 40
⇒ x = 40/5
⇒ x = 8

Hence, age of B = 2x = (2 × 8) years = 16 years.

১১০.

I forgot the last digit of a 7 digit telephone number. If one randomly dial the final three digits after correctly dialling the four, then what is the chance of dialling the correct number?

ক
1/1000
খ
1/1001
গ
1/999
ঘ
1/990

সঠিক উত্তর: ক

1/1000

উত্তর

সঠিক উত্তর: ক

1/1000

ক

ব্যাখ্যা

Question: I forgot the last digit of a 7 digit telephone number. If one randomly dial the final three digits after correctly dialling the four, then what is the chance of dialling the correct number?

Solution:
It is given that last three digits are randomly dialled. Then each of the digit can be selected out of 10 digits in 10 ways.

Hence required probability
= 1/10 × 1/10 × 1/10
= 1/1000

১১১.

A ladder 20 meters long leans against a vertical wall. If the ladder makes an angle of 60 degrees with the ground, what is the distance of the foot of the ladder from the wall?

ক
10 meters
খ
10√3 meters
গ
20√3 meters
ঘ
15 meters

সঠিক উত্তর: ক

10 meters

উত্তর

সঠিক উত্তর: ক

10 meters

ক

ব্যাখ্যা

Question: A ladder 20 meters long leans against a vertical wall. If the ladder makes an angle of 60 degrees with the ground, what is the distance of the foot of the ladder from the wall?

Solution:

মইয়ের দৈর্ঘ্য, AC = 20 m
ধরি, দেয়াল থেকে মইয়ের পাদদেশের দূরত্ব, BC = x
মই ভূমির সাথে যে কোণ তৈরি করে, ∠ACB = 60°

আমরা জানি, cosθ = ভূমি/অতিভুজ
∴ cos 60° = BC/ AC
⇒ 1/2 = x/20
⇒ 2x = 20
∴ x = 10 m

অতএব, দেয়াল থেকে মইয়ের পাদদেশের দূরত্ব = 10 m।

১১২.

What is the volume of a cone if its radius is 3 cm and its height is 14 cm?

ক
132
খ
244
গ
42
ঘ
164

সঠিক উত্তর: ক

132

উত্তর

সঠিক উত্তর: ক

132

ক

ব্যাখ্যা

Question: What is the volume of a cone if its radius is 3 cm and its height is 14 cm?

Solution:
Given that,
radius, r = 3 cm
height, h = 14 cm

Now,
Volume = (1/3) × π × r² × h
= (1/3) × (22/7) × 3² × 14
= 132 cm³

১১৩.

If p(x) = 3x⁴ - 2x² + x - 1, q(x) = 7x⁵ + 2x², then find the value of p(x) + q(x).

ক
- 7x⁵ + 3x⁴ - 4x² - 1
খ
7x⁵ + 3x⁴ + x - 1
গ
7x⁵ + 3x⁴ - 4x² + x - 1
ঘ
None of them

সঠিক উত্তর: খ

7x⁵ + 3x⁴ + x - 1

উত্তর

সঠিক উত্তর: খ

7x⁵ + 3x⁴ + x - 1

খ

ব্যাখ্যা

Question: If p(x) = 3x⁴ - 2x² + x - 1, q(x) = 7x⁵ + 2x², then find the value of p(x) + q(x).

Solution:
p(x) + q(x)
= 3x⁴ - 2x² + x - 1 + 7x⁵ + 2x²
= 7x⁵ + 3x⁴ + x - 1

১১৪.

12 buckets of water fill a tank when the capacity of each bucket is 13.5 litres. How many buckets will be needed to fill the same tank, if the capacity of each bucket is 9 litres?

ক
14
খ
16
গ
18
ঘ
20
ঙ
None of the above

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: 12 buckets of water fill a tank when the capacity of each bucket is 13.5 litres. How many buckets will be needed to fill the same tank, if the capacity of each bucket is 9 litres?

Solution:
Capacity of the tank
= (12 × 13.5) litres
= 162 litres

Capacity of each bucket = 9 litres

∴ Number of buckets needed
= 162/9
= 18

১১৫.

In how many different ways can a committee of 3 members be selected from 5 people if a particular person must always be included in the committee?

ক
4
খ
5
গ
6
ঘ
7

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: In how many different ways can a committee of 3 members be selected from 5 people if a particular person must always be included in the committee?

Solution:
Since one person must always be in the committee, we need to select the other two members from the remaining 4 people.
∴ Number of ways to choose the other two members = ⁴C₂
= (4 × 3)/(1 × 2)
= 6

১১৬.

If both 11² and 3³ are factors of the number a* 4³ * 6² * 13¹¹ , then what is the smallest possible value of ‘a’?

ক
ক) 363
খ
খ) 121
গ
গ) 3267
ঘ
ঘ) 33

সঠিক উত্তর: ক

ক) 363

উত্তর

সঠিক উত্তর: ক

ক) 363

ক

ব্যাখ্যা

Question: If both 11² and 3³ are factors of the number a* 4³ * 6² * 13¹¹ , then what is the smallest possible value of ‘a’?

Solution:
প্রদত্ত সংখ্যাটির একটি উৎপাদক হলো 11²
a * 4³ * 6² * 13¹¹ সংখ্যাটি 11 এর গুণিতক নয়।
a * 4³ * 6²* 13¹¹ সংখ্যাটি 11 দ্বারা বিভাজ্য হলে a এর মান 11² হবে।

আবার,
3³ হলো a * 4³ * 6² * 13¹¹ এর আরও একটি উৎপাদক
a * 4³ * 6² * 13¹¹ এ 6² হলো সংখ্যাটির একটি অংশ।
এখানে,
6² = 3² * 2²
a * 4³ * 6²* 13¹¹ সংখ্যাটিতে 3² রয়েছে।
a সংখ্যাটিতে আরো একটি 3 লাগবে।

a এর কমপক্ষে মান হবে = 11² × 3 = 363

১১৭.

Find the annual income obtained by investing Tk 3000 in 5% debentures of face value Tk 100 at Tk 125?

ক
Tk 180
খ
Tk 120
গ
Tk 210
ঘ
None of the above

সঠিক উত্তর: খ

Tk 120

উত্তর

সঠিক উত্তর: খ

Tk 120

খ

ব্যাখ্যা

Question: Find the annual income obtained by investing Tk 3000 in 5% debentures of face value Tk 100 at Tk 125?

Solution:
The number of debentures purchased = Tk. 3000/125
One debenture will give an income investment of Tk 100 × 5% = Tk 5.
So, total income = 5 × (3000/125) = Tk 120.

১১৮.

The average age of husband, wife and their child 3 years ago was 27 years and that of wife and the child 5 years ago was 20 years. The sum of the present ages of wife and child is:

ক
40 years
খ
45 years
গ
50 years
ঘ
55 years

সঠিক উত্তর: গ

50 years

উত্তর

সঠিক উত্তর: গ

50 years

গ

ব্যাখ্যা

Question: The average age of husband, wife and their child 3 years ago was 27 years and that of wife and the child 5 years ago was 20 years. The sum of the present ages of wife and child is:

Solution:
Sum of the present ages of husband, wife and child =(27 × 3 + 3 × 3) years
= (81 + 9) years
= 90 years

Sum of the present ages of wife and child =(20 × 2 + 5 × 2) years
= (40 + 10) years
= 50 years

১১৯.

In how may different ways can the letters of the word JUDGE be arranged in such a way that the vowels always come together?

ক
ক) 32
খ
খ) 36
গ
গ) 42
ঘ
ঘ) 48

সঠিক উত্তর: ঘ

ঘ) 48

উত্তর

সঠিক উত্তর: ঘ

ঘ) 48

ঘ

ব্যাখ্যা

The word 'JUDGE' contains 5 different letters.
When the vowels UE are always together, they can be supposed to form one letter.
Then, we have to arrange the letters JDG (UE).
Now, 5 letters can be arranged in 4! ways
= 24 ways.
The vowels (OIA) can be arranged among themselves in 2! = 2 ways.

∴ Required number of ways = (24 x 2) = 48

১২০.

The solution to log₂x = 5 is:

ক
10
খ
25
গ
15
ঘ
32

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: The solution to log₂x = 5 is:

Solution:
log₂x = 5
⇒ x = 2⁵
∴ x = 32

১২১.

Find the number of ways of selecting 4 girls and 4 boys out of 7 girls and 6 boys.

ক
125
খ
145
গ
525
ঘ
575

সঠিক উত্তর: গ

525

উত্তর

সঠিক উত্তর: গ

525

গ

ব্যাখ্যা

Question: Find the number of ways of selecting 4 girls and 4 boys out of 7 girls and 6 boys.

Solution:
The number of ways of selecting 4 girls and 4 boys out of 7 girls and 6 boys is
⁷C₄ × ⁶C₄
= {7!/(4! × 3!)} × {6!/(4! × 2!)}
= (35 × 15)
= 525 ways

১২২.

If m is an integer such that (- 2)^2m = 2^{9 - m}, then m =?

ক
1
খ
2
গ
3
ঘ
4
ঙ
6

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: If m is an integer such that (- 2)^2m = 2^{9 - m}, then m =?

Solution:
First of all, since m is an integer, then 2m = even, and therefore (- 2)^2m = 2^2m

So, we'd have:
2^2m = 2^{9 - m}
⇒ 2m = 9 - m
⇒ 3m = 9
∴ m = 3

১২৩.

If x = 1 + √2 and y = 1 - √2, find the value of x² + y².

ক
ক) 6
খ
খ) 8
গ
গ) 10
ঘ
ঘ) 12

সঠিক উত্তর: ক

ক) 6

উত্তর

সঠিক উত্তর: ক

ক) 6

ক

ব্যাখ্যা

Question: If x = 1 + √2 and y = 1 - √2, find the value of x² + y².

Solution:
Given that,
x = 1 + √2,
y = 1 - √2

∴ x + y = 1 + √2 + 1 - √2
= 2

And,
xy = (1 + √2)(1 - √2)
= 1² - (√2)²
= 1 - 2
= - 1

Now,
x² + y² = (x + y)² - 2xy
= (2)² - 2(- 1)
= 4 + 2
= 6

১২৪.

The length of one side of a square inscribed in a circle is 2. What is the area of the circle?

ক
π/2
খ
π
গ
√2π
ঘ
2π
ঙ
None of these

সঠিক উত্তর: ঘ

2π

উত্তর

সঠিক উত্তর: ঘ

2π

ঘ

ব্যাখ্যা

Question: The length of one side of a square inscribed in a circle is 2. What is the area of the circle?

Solution:
বৃত্তের অন্তর্লিখিত বর্গের বাহুর দৈর্ঘ্য ২ একক
∴ বর্গের কর্ণের দৈর্ঘ্য = ২√২ একক

এখানে বর্গের কর্ণ বৃত্তটির ব্যাসের সমান।
∴ বৃত্তের ব্যাসার্ধ = (২√২)/২ একক = √২ একক

বৃত্তের ক্ষেত্রফল = π(√২)^২ বর্গএকক
= ২π বর্গএকক

১২৫.

Selling price of first article is Tk. 960 and cost price of second article is Tk. 960. If there is a profit of 20% on first article and loss of 20% on second article, then, what will be the total loss?

ক
ক) Tk. 38
খ
খ) Tk. 36
গ
গ) Tk. 34
ঘ
ঘ) Tk. 32

সঠিক উত্তর: ঘ

ঘ) Tk. 32

উত্তর

সঠিক উত্তর: ঘ

ঘ) Tk. 32

ঘ

ব্যাখ্যা

SP of first article = Tk. 960
CP of 2nd article = Tk.960
Profit on first article = 20%
Loss on 2nd article = 20%

After profit of 20%, CP of first article = 960 × (100/120)
= Tk. 800
After loss of 20%, SP of 2nd article = 960 × (80/100)
=Tk. 768
Total CP of both article = 800 + 960 = Tk.1760
Total SP of both article = 960 + 768 = Tk.1728

∴ Total loss = 1760 – 1728 = Tk. 32

১২৬.

Two boats, traveling at 5 and 10 kms per hour, head directly towards each other. They begin at a distance of 20 kms from each other. How far apart are they (in kms) one minute before they collide?

ক
ক) 1/12
খ
খ) 1/6
গ
গ) 1/4
ঘ
ঘ) 1/2

সঠিক উত্তর: গ

গ) 1/4

উত্তর

সঠিক উত্তর: গ

গ) 1/4

গ

ব্যাখ্যা

Notice that the distance between the two boats one minute before they collide is equal to the total distance traveled by the two boats in one minute.

In 1 minute, the boat with speed of 5 km/hr will have traveled 5 x 1/60 = 5/60 km.
Similarly, the boat with speed of 10 km/hr will have traveled 10 x 1/60 = 10/60 km in 1 minute.
Therefore, 1 minute before they collide, they are 5/60 + 10/60 = 15/60 = 1/4 km apart.

১২৭.

There are 8 true-false questions in an examination, these questions can be answered in-

ক
1024
খ
820
গ
256
ঘ
128

সঠিক উত্তর: গ

256

উত্তর

সঠিক উত্তর: গ

256

গ

ব্যাখ্যা

Question: There are 8 true-false questions in an examination, these questions can be answered in-

Solution:
Total number of question = 8
Each question has 2 answer.

These question can be answered in 2⁸ ways = 256 ways

১২৮.

How many 5 will you pass on the way when you count from 1 to 100?

ক
ক) 18
খ
খ) 19
গ
গ) 20
ঘ
ঘ) 21

সঠিক উত্তর: গ

গ) 20

উত্তর

সঠিক উত্তর: গ

গ) 20

গ

ব্যাখ্যা

Numbers in which digit 5 is used between 1 & 100 are the following = 5 , 15 , 25 , 35 , 45 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 65 , 75 , 85 , 95 = 20 times

১২৯.

HCF and LCM of two fractions is 1/35 and 15/1, if one fraction is 3/5, then the second fraction is:

ক
3/7
খ
7/3
গ
5/7
ঘ
7/5

সঠিক উত্তর: গ

5/7

উত্তর

সঠিক উত্তর: গ

5/7

গ

ব্যাখ্যা

Question: HCF and LCM of two fractions is 1/35 and 15/1, if one fraction is 3/5, then the second fraction is:

Solution:
Given,
HCF of two fractions = 1/35
LCM of two fractions = 15/1
One fraction = 3/5

HCF of fractions = HCF of numerators/LCM of denominators
LCM of fractions = LCM of numerators/HCF of denominators
LCM × HCF = Product of two fractions

Let second fraction be a/b,
then
a/b × 3/5 = 15/1 × 1/35
⇒ a/b = 15/1 × 1/35 × 5/3
∴ a/b = 5/7

১৩০.

Abir and Babul invest in a business in the ratio 5 : 4. If 10% of the total profit goes to charity and abir’s share is Tk 945, the total profit is -

ক
1500 tk
খ
2100 tk
গ
1890 tk
ঘ
2400 tk

সঠিক উত্তর: গ

1890 tk

উত্তর

সঠিক উত্তর: গ

1890 tk

গ

ব্যাখ্যা

Question: Abir and Babul invest in a business in the ratio 5 : 4. If 10% of the total profit goes to charity and abir’s share is Tk 945, the total profit is -

Solution:
Let,
The total profit = 100 tk

After paying 10% to charity,
Abir’s share = 90 × (5/9)
= 50 tk

Now,
If abir’s share is tk 50, total profit = 100 tk
If abir’s share is tk 1, total profit = 100/50 tk
If abir’s share is tk 945, total profit = (100/50) × 945
= 1890 tk

So, the total profit is 1890 tk.

১৩১.

In how many ways can 10 examination papers be arranged so that the best and the worst papers never come together?

ক
2 × 9!
খ
2 × 7!
গ
7!
ঘ
8 × 9!

সঠিক উত্তর: ঘ

8 × 9!

উত্তর

সঠিক উত্তর: ঘ

8 × 9!

ঘ

ব্যাখ্যা

Question: In how many ways can 10 examination papers be arranged so that the best and the worst papers never come together?

Solution:
No. of ways in which 10 paper can arranged is 10! Ways.
When the best and the worst papers come together, regarding the two as one paper, we have only 9 papers.
These 9 papers can be arranged in 9! Ways.
And two papers can be arranged themselves in 2! Ways.
No. of arrangement when best and worst paper do not come together,
= 10! - 9! × 2!
= 9!(10 - 2)
= 8 × 9!

১৩২.

The sum of the present ages of a son and his father is 70 years. After 5 years, the age of the father will be three times that of the son. At present their ages are?

ক
10 years, 60 years
খ
12 years, 58 years
গ
14 years, 56 years
ঘ
15 years, 55 years

সঠিক উত্তর: ঘ

15 years, 55 years

উত্তর

সঠিক উত্তর: ঘ

15 years, 55 years

ঘ

ব্যাখ্যা

Question: The sum of the present ages of a son and his father is 70 years. After 5 years, the age of the father will be three times that of the son. At present their ages are?

Solution:
Let,
the son's age be x years.
Then, father's age = (70 - x) years.

ATQ,
(70 - x) + 5 = 3(x + 5)
⇒ 75 - x = 3x + 15
⇒ 4x = 60
∴ x = 15

∴ Son's age = 15 years
And the father's age = (70 - 15) = 55 years.

১৩৩.

A plane traveling at 600 miles per hour is heading for Chittagong airport. At 3.58 pm it was 30 miles from the airport. At what time it will arrive at the airport?

ক
4.01 pm
খ
4.02 pm
গ
4.18 pm
ঘ
4.20 pm

সঠিক উত্তর: ক

4.01 pm

উত্তর

সঠিক উত্তর: ক

4.01 pm

ক

ব্যাখ্যা

Question: A plane traveling at 600 miles per hour is heading for Chittagong airport. At 3.58 pm it was 30 miles from the airport. At what time it will arrive at the airport?

Solution:
প্লেনটি ৬০০ মাইল অতিক্রম করে ৬০ মিনিটে
প্লেনটি ১ মাইল অতিক্রম করে ৬০/৬০০ মিনিটে
প্লেনটি ৩০ মাইল অতিক্রম করে (৬০ × ৩০)/৬০০ মিনিটে
= ৩ মিনিটে

প্লেনটি বিমানবন্দরে পৌঁছাতে ৩ মিনিট সময় নিবে।
যদি বিকাল ৩ : ৫৮ এ প্লেনটি ৩০ মাইল দূরে থাকে, তাহলে ৩ মিনিট যোগ করলে পৌঁছানোর সময় হবে:
৩ : ৫৮ বিকাল + ৩ মিনিট = ৪ : ০১ বিকাল

১৩৪.

Excluding stoppages, the speed of a train is 80 kmph, and including stoppages, it is 64 kmph. For how many minutes does the train stop per hour?

ক
12 minutes
খ
15 minutes
গ
22 minutes
ঘ
16 minutes

সঠিক উত্তর: ক

12 minutes

উত্তর

সঠিক উত্তর: ক

12 minutes

ক

ব্যাখ্যা

Question: Excluding stoppages, the speed of a train is 80 kmph, and including stoppages, it is 64 kmph. For how many minutes does the train stop per hour?

Solution:
Excluding stoppages speed = 80 kmph
Including stoppages speed = 64 kmph

Loss in distance per hour due to stoppages
= (80 - 64) km
= 16 km

Time taken to cover 16 km at 80 kmph
= (16/80) × 60 minutes
= 12 minutes

১৩৫.

A train running at the speed of 84 km/hr passes a man walking in opposite direction at the speed of 6 km/hr in 4 seconds. What is the length of train (in meter)?

ক
130 meters
খ
150 meters
গ
90 meters
ঘ
100 meters

সঠিক উত্তর: ঘ

100 meters

উত্তর

সঠিক উত্তর: ঘ

100 meters

ঘ

ব্যাখ্যা

Question: A train running at the speed of 84 km/hr passes a man walking in opposite direction at the speed of 6 km/hr in 4 seconds. What is the length of train (in meter)?

Solution:
Given that,
Train speed = 84 km/hr
Man speed = 6 km/hr (opposite direction)
∴ Relative speed = 84 + 6 = 90 km/hr
= 90 × (5/18) = 25 m/s

And,
Time taken to pass the man,
Given = 4 seconds

∴ Length = Relative speed × Time = 25 × 4 = 100 m

So the length of the train is 100 meters.

১৩৬.

If x books cost Tk. 5 each and y books cost Tk. 8 each, then the average (arithmetic mean) cost per book is equal to-

ক
(5x + 8y)/(x + y)
খ
(5x + 8y)/(xy)
গ
(5x + 8y)/13
ঘ
(40xy)/(x + y)
ঙ
(40xy)/13

সঠিক উত্তর: ক

(5x + 8y)/(x + y)

উত্তর

সঠিক উত্তর: ক

(5x + 8y)/(x + y)

ক

ব্যাখ্যা

Question: If x books cost Tk. 5 each and y books cost Tk. 8 each, then the average (arithmetic mean) cost per book is equal to-

Solution:
x books cost Tk. 5 each and y books cost Tk. 8 each
Cost of x books at Tk. 5 apiece = 5x
Cost of y books at Tk. 8 apiece = 8y

TOTAL cost of all books = 5x + 8y
TOTAL number of books = x + y

∴ Average cost per book = (5x + 8y)/(x + y)

১৩৭.

(5³ + 5³ + 5³ + 5³) = ?

ক
5⁴
খ
5⁶
গ
5⁸
ঘ
5¹²
ঙ
None of these

সঠিক উত্তর: ঙ

None of these

উত্তর

সঠিক উত্তর: ঙ

None of these

ঙ

ব্যাখ্যা

Question: (5³ + 5³ + 5³ + 5³) = ?

Solution:
5³ + 5³ + 5³ + 5³
= 5³ (1 + 1 + 1 + 1)
= 5³ · 4
= 125 · 4
= 500

১৩৮.

A bond gets matured in 10 years. Ram invested Tk. 42,000 in this bond and received Tk.1,05,000 once the bond matured. If the bond was under simple interest, then what was the rate of interest per annum?

ক
ক) 10%
খ
খ) 12.5%
গ
গ) 15%
ঘ
ঘ) 7.5%

সঠিক উত্তর: গ

গ) 15%

উত্তর

সঠিক উত্তর: গ

গ) 15%

গ

ব্যাখ্যা

এখানে,
সরল সুদের পরিমাণ =( ১,০৫,০০০ - ৪২,০০০) টাকা
= ৬৩,০০০ টাকা
আমরা জানি
সুদের পরিমাণ = আসল × হার×সময় / ১০০
৬৩০০০ = ৪২০০০ × হার × ১০ / ১০০
হার = ৬৩০০০× ১০০/৪২০০০× ১০
= ১৫%

১৩৯.

If x² - 2x + 1 = 0 then the value of (x⁴ + 2x² + 1)/x² is -

ক
ক) 1
খ
খ) 2
গ
গ) 3
ঘ
ঘ) 4

সঠিক উত্তর: ঘ

ঘ) 4

উত্তর

সঠিক উত্তর: ঘ

ঘ) 4

ঘ

ব্যাখ্যা

Question: If x² - 2x + 1 = 0 then the value of (x⁴ + 2x² + 1)/x² is -

Solution:
Given,
x² - 2x + 1 = 0
⇒ (x - 1)² = 0
⇒ x - 1 = 0
∴ x = 1

(x⁴ + 2x² + 1)/x² = (1⁴+ 2 . 1² + 1)/1²
= 4/1
= 4

১৪০.

A company issued 20000 shares of par value Tk. 10 each. If the total dividend declared by the company is Tk. 24000, find the rate of dividend paid by the company.

ক
ক) 12%
খ
খ) 10%
গ
গ) 9%
ঘ
ঘ) 15%

সঠিক উত্তর: ক

ক) 12%

উত্তর

সঠিক উত্তর: ক

ক) 12%

ক

ব্যাখ্যা

Number of shares = 20000
Face value of each share = Tk. 10
dividend per share = (10 × R/100) where R is the Rate of interest.
Total dividend = 20000 ×10 ×R/100
20000 ×10 ×R/100 = 24000
R = 24000/2000
= 12
Hence the dividend is 12%.

১৪১.

Find the equation of the line with x-intercept = 6 and y-intercept = 5.

ক
6x + 5y - 30 = 0
খ
6x + 5y + 30 = 0
গ
5x - 6y - 30 = 0
ঘ
5x + 6y - 30 = 0

সঠিক উত্তর: ঘ

5x + 6y - 30 = 0

উত্তর

সঠিক উত্তর: ঘ

5x + 6y - 30 = 0

ঘ

ব্যাখ্যা

Question: Find the equation of the line with x-intercept = 6 and y-intercept = 5.

Solution:
Given,
x-intercept = 6, So, the line passes through (6, 0).
y-intercept = 5, So, the line passes through (0, 5).

We know, The intercept form of a line is:
(x/a) + (y/b) = 1, where a = x-intercept, b = y-intercept.
⇒ (x/6) + (y/5) = 1
⇒ (5x + 6y)/30 = 1
⇒ 5x + 6y = 30
⇒ 5x + 6y - 30 = 0

∴ The equation of the line is 5x + 6y - 30 = 0

১৪২.

The average mark obtained by 22 candidates in an examination is 43. The average marks of the first ten are 45 and the last eleven are 40. The number of marks obtained by the 11^th candidate is-

ক
56
খ
54
গ
52
ঘ
48

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: The average mark obtained by 22 candidates in an examination is 43. The average marks of the first ten are 45 and the last eleven are 40. The number of marks obtained by the 11^th candidate is-

Solution:
Total marks scored by 22 candidates = 22 × 43
= 946
Total marks scored by first 10 candidates =10 × 45
= 450
Total marks scored by last 11 candidates = 11 × 40
= 440

∴ Marks scored by 11th candidate = 946 - (450 + 440)
= 56

১৪৩.

The average monthly income of P and Q is TK. 5050. The average monthly income of Q and R is TK. 6250 and the average monthly income of P and R is TK. 5200. The monthly income of P is:

ক
ক) 3500
খ
খ) 4050
গ
গ) 5000
ঘ
ঘ) 5050
ঙ
ঙ) 4000

সঠিক উত্তর: ঙ

ঙ) 4000

উত্তর

সঠিক উত্তর: ঙ

ঙ) 4000

ঙ

ব্যাখ্যা

Let P, Q and R represent their respective monthly incomes. Then, we have:

P + Q = (5050 x 2) = 10100 .... (i)
Q + R = (6250 x 2) = 12500 .... (ii)
P + R = (5200 x 2) = 10400 .... (iii)

Adding (i), (ii) and (iii), we get:
2(P + Q + R) = 33000 or P + Q + R = 16500 .... (iv)

Subtracting (ii) from (iv), we get,
P = 4000.
∴ P's monthly income = TK. 4000.

১৪৪.

A clock seen through a mirror shows quarter past five. What is the correct time shown by the clock?

ক
11 : 45
খ
6 : 45
গ
11 : 15
ঘ
6 : 15

সঠিক উত্তর: খ

6 : 45

উত্তর

সঠিক উত্তর: খ

6 : 45

খ

ব্যাখ্যা

Question: A clock seen through a mirror shows quarter past five. What is the correct time shown by the clock?

Solution:
The time quarter past 5 is 5 : 15

প্রকৃতপক্ষে সময়
= 11 : 60 - আয়নায় দেখা সময়
= 11 : 60 - 5 : 15
= 6 : 45

১৪৫.

Mina has 3 Taka more than Babu has, but 5 Taka less than Shelly has. If Mina has 450 taka, how much money Shelly and Babu have altogether?

ক
902 taka
খ
900 taka
গ
850 taka
ঘ
952 taka

সঠিক উত্তর: ক

902 taka

উত্তর

সঠিক উত্তর: ক

902 taka

ক

ব্যাখ্যা

Question: Mina has 3 Taka more than Babu has, but 5 Taka less than Shelly has. If Mina has 450 taka, how much money Shelly and Babu have altogether?

Solution:
মিনার কাছে আছে = 450 টাকা
বাবুর কাছে আছে = 450 - 3 টাকা = 447 টাকা
শেলির কাছে আছে = 450 + 5 টাকা = 455 টাকা

বাবু ও শেলির কাছে আছে = 447 + 455 টাকা
= 902 টাকা

১৪৬.

X is 50% more efficient than Y. How much time will they, working together, take to complete a job which Y alone could have done in 25 days?

ক
7 days
খ
10 days
গ
14 days
ঘ
18 days

সঠিক উত্তর: খ

10 days

উত্তর

সঠিক উত্তর: খ

10 days

খ

ব্যাখ্যা

Question: X is 50% more efficient than Y. How much time will they, working together, take to complete a job which Y alone could have done in 25 days?

Solution:
X, Y এর থেকে 50% বেশি দক্ষ।
⇒ X : Y = 150 : 100 = 3 : 2

Y এক দিনে কাজ করে = 2 ইউনিট
X এক দিনে কাজ করে = 3 ইউনিট

মোট কাজ = Y এর দৈনিক কাজ × Y এর দিন = 2 × 25 = 50 ইউনিট

একসাথে এক দিনে কাজ করে = 3 + 2 = 5 ইউনিট
তাহলে কাজ শেষ করতে সময় লাগবে = 50 ÷ 5 দিন = 10 দিন

∴ সুতরাং, X এবং Y একত্রে কাজটি শেষ করতে 10 দিন সময় নেবে।

১৪৭.

If 5 persons working 7 hours a day earn Tk. 7000 per week, then 8 persons working 5 hours a day will earn per week?

ক
ক) Tk. 7500
খ
খ) Tk. 8000
গ
গ) Tk. 8250
ঘ
ঘ) Tk. 9100

সঠিক উত্তর: খ

খ) Tk. 8000

উত্তর

সঠিক উত্তর: খ

খ) Tk. 8000

খ

ব্যাখ্যা

Question: If 5 persons working 7 hours a day earn Tk. 7000 per week, then 8 persons working 5 hours a day will earn per week?

Solution:
Earning of 5 × 7 = 35 hours = 7000 Tk.
Earning of 1 hour is = 7000/35 Tk.
Earning of 8 × 5 = 40 hours is = (7000 × 40)/35 = Tk. 8000

১৪৮.

In a bag, there are 5 red, 3 green, and 2 blue marbles. If two marbles are drawn one after the other without replacement, what is the probability that the first one is red and the second one is green?

ক
1/5
খ
1/3
গ
3/10
ঘ
1/6
ঙ
3/8

সঠিক উত্তর: ঘ

1/6

উত্তর

সঠিক উত্তর: ঘ

1/6

ঘ

ব্যাখ্যা

Question: In a bag, there are 5 red, 3 green, and 2 blue marbles. If two marbles are drawn one after the other without replacement, what is the probability that the first one is red and the second one is green?

Solution:
মোট মার্বেলের সংখ্যা = 5 (লাল) + 3 (সবুজ) + 2 (নীল) = 10টি
প্রথম মার্বেলটি লাল হওয়ার সম্ভাবনা = 5/10 = 1/2
প্রথম মার্বেলটি তোলার পর থলেতে মোট মার্বেলের সংখ্যা = 10 - 1 = 9টি
দ্বিতীয় মার্বেলটি সবুজ হওয়ার সম্ভাবনা = 3/9 = 1/3

∴ প্রথমটি লাল এবং দ্বিতীয়টি সবুজ হওয়ার সম্ভাবনা = (প্রথমটি লাল হওয়ার সম্ভাবনা) × (দ্বিতীয়টি সবুজ হওয়ার সম্ভাবনা)
= 1/2 × 1/3
= 1/6

১৪৯.

The average of X and Y is 50, and the average of Y and Z is 34. Find the value of (X - Z)/2

ক
32
খ
24
গ
16
ঘ
28

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: The average of X and Y is 50, and the average of Y and Z is 34. Find the value of (X - Z)/2

Solution:
Given,
(X + Y)/2 = 50
⇒ X + Y = 100 ...... (1)

and,
(Y + Z)/2 = 34
⇒ Y + Z = 68 ...... (2)

from (1) - (2) we get,
X + Y - Y - Z = 100 - 68
⇒ X - Z = 32
⇒ (X - Z)/2 = 32/2
∴ (X - Z)/2 = 16

১৫০.

A rectangular prism has dimensions 12 cm, 8 cm, and 5 cm. Calculate the volume of the prism.

ক
ক) 420 cm³
খ
খ) 440 cm³
গ
গ) 450 cm³
ঘ
ঘ) 480 cm³

সঠিক উত্তর: ঘ

ঘ) 480 cm³

উত্তর

সঠিক উত্তর: ঘ

ঘ) 480 cm³

ঘ

ব্যাখ্যা

Question: A rectangular prism has dimensions 12 cm, 8 cm, and 5 cm. Calculate the volume of the prism.

Solution:
The volume of a rectangular prism can be found using the formula:
Volume = length × width × height
= 12 × 8 ×5 cm³
= 480 cm³

১৫১.

4 kg of metal contains 1/5 copper and rest in Zinc. Another 5 kg of metal contains 1/6 copper and rest in Zinc.The ratio of Copper and Zinc into the mixture of these two metals:

ক
ক) 39 : 231
খ
খ) 49 : 221
গ
গ) 94 : 181
ঘ
ঘ) 36 : 272

সঠিক উত্তর: খ

খ) 49 : 221

উত্তর

সঠিক উত্তর: খ

খ) 49 : 221

খ

ব্যাখ্যা

Copper in 4 kg = 4/5 kg and
Zinc in 4 kg = 4 x (4/5) kg

Copper in 5 kg = 5/6 kg and
Zinc in 5 kg = 5 x (5/6) kg

Therefore, Copper in mixture = (4/5) + (5/6) = 49/30 kg
and Zinc in the mixture = (16/5) + (25/6) = 221/30 kg

Therefore the required ratio = 49/30 : 221/30
= 49 : 221.

১৫২.

There is a shopkeeper whose claim is that he sells a certain product at a cost of Tk 23/kg, which actually costs him Tk 25/kg. The shopkeeper says that he is taking the loss to let his customers get a better deal. When examined thoroughly, the constabulary find that the shopkeeper is actually using an 800 gms weight in place of a 1 kg weight. How much does he gain or lose?

ক
10% loss
খ
15% profit
গ
20% loss
ঘ
12.5% profit

সঠিক উত্তর: খ

15% profit

উত্তর

সঠিক উত্তর: খ

15% profit

খ

ব্যাখ্যা

Question: There is a shopkeeper whose claim is that he sells a certain product at a cost of Tk 23/kg, which actually costs him Tk 25/kg. The shopkeeper says that he is taking the loss to let his customers get a better deal. When examined thoroughly, the constabulary find that the shopkeeper is actually using an 800 gms weight in place of a 1 kg weight. How much does he gain or lose?

Solution:
Selling price of 0.8 kg = 23 taka

Now,
Selling price of 1 kg = 23/0.8 taka
= 115/4 taka
= 28.75 taka

∴ Percentage profit = {(28.75 - 25)/25} × 100%
= (3.75/25) × 100%
= 15%

১৫৩.

A farmer grows paddy on one acre and gets 400 kg of it. If each kilogram of paddy yields 700 grams of rice, how much rice does he get in total?

ক
250 kg
খ
260 kg
গ
280 kg
ঘ
350 kg

সঠিক উত্তর: গ

280 kg

উত্তর

সঠিক উত্তর: গ

280 kg

গ

ব্যাখ্যা

Question: A farmer grows paddy on one acre and gets 400 kg of it. If each kilogram of paddy yields 700 grams of rice, how much rice does he get in total?

Solution:
1 কেজি ধানে চাল হয় = 700 গ্রাম
= 700/1000 কেজি
= 0.7 কেজি

∴ 400 কেজি ধানে চাল হয় = (400 × 0.7) কেজি = 280 কেজি

১৫৪.

If tanA = 3/4 then, cosA = ?

ক
3/5
খ
5/4
গ
2/3
ঘ
4/3
ঙ
4/5

সঠিক উত্তর: ঙ

4/5

উত্তর

সঠিক উত্তর: ঙ

4/5

ঙ

ব্যাখ্যা

Question: If tanA = 3/4 then, cosA = ?

Solution:
We know,
sec²A = 1 + tan2A
⇒ sec²A = 1 + (3/4)²
⇒ sec²A = 1 + (9/16)
⇒ sec²A = (16 + 9)/16
⇒ sec²A = 25/16
⇒ secA = √(25/16)
⇒ 1/cosA = 5/4
∴ cosA = 4/5

১৫৫.

A bicycle wheel has a diameter of 70 cm and is making 200 revolutions per minute. What is the speed of the bicycle in km/h?

ক
25.5 km/h
খ
28.2 km/h
গ
27.0 km/h
ঘ
26.4 km/h

সঠিক উত্তর: ঘ

26.4 km/h

উত্তর

সঠিক উত্তর: ঘ

26.4 km/h

ঘ

ব্যাখ্যা

Question: A bicycle wheel has a diameter of 70 cm and is making 200 revolutions per minute. What is the speed of the bicycle in km/h?

Solution:
Given that,
Diameter of wheel, D = 70 cm
Radius, r = 70/2 = 35 cm
And revolutions per minute (rpm) = 200

We know,
Circumference C = 2πr = 2 × (22/7) × 35
= 44 × 5 = 220 cm

∴ Distance per minute = 220 × 200 = 44000 cm/min
= (44000 × 60)/100000 km/h ; [1 km = 100000 cm and 1 hour = 60 min]
= (44 × 6)/10 km/h
= 26.4 km/h

১৫৬.

Last year, the ratio between the salaries of A and B was 3 : 4. But the ratios of their individual salaries between last year and this year were 4 : 5 and 2 : 3 respectively. If the sum of their present salaries is Tk. 4160, then how much is the salary of A now?

ক
Tk.1200
খ
Tk.1400
গ
Tk.1600
ঘ
Tk.1800

সঠিক উত্তর: গ

Tk.1600

উত্তর

সঠিক উত্তর: গ

Tk.1600

গ

ব্যাখ্যা

Question: Last year, the ratio between the salaries of A and B was 3 : 4. But the ratios of their individual salaries between last year and this year were 4 : 5 and 2 : 3 respectively. If the sum of their present salaries is Tk. 4160, then how much is the salary of A now?

Solution:
Let,
the salaries of A and B last year be Tk. 3x and Tk. 4x respectively.
Then,
A's present salary = Tk. (5/4) × 3x
= Tk. 15x/4

B's present salary = Tk.(3/2) × 4x
= Tk. 6x.

According to the question,
(15x/4) + 6x = 4160
⇒ 39x = 4160 × 4
⇒ x = (4160 × 4)/39

So, A's present salary = Tk. (15/4) × (4160 × 4)/39
= Tk.1600

১৫৭.

A chord of length 18 cm is drawn in a circle of radius 12 cm. The distance of the chord from the center of the circle is -

ক
√65 cm
খ
√60 cm
গ
√66 cm
ঘ
√63 cm

সঠিক উত্তর: ঘ

√63 cm

উত্তর

সঠিক উত্তর: ঘ

√63 cm

ঘ

ব্যাখ্যা

Question: A chord of length 18 cm is drawn in a circle of radius 12 cm. The distance of the chord from the center of the circle is -

Solution:
Given,
r = 12 cm
and c = 18 cm.
Half of the chord length = 18/2 = 9 cm

∴ Distance = √(12² - 9²)
= √(144 - 81)
= √63 cm

The chord's distance from the circle's center is √63 cm.

১৫৮.

Find the value of i^-42

ক
ক) 1
খ
খ) -1
গ
গ) i
ঘ
ঘ) -i

সঠিক উত্তর: খ

খ) -1

উত্তর

সঠিক উত্তর: খ

খ) -1

খ

ব্যাখ্যা

Question: Find the value of i^-42

Solution:
i^-42
= 1 / i⁴²
= 1 / {(i²)²¹}
= 1 / {(-1)²¹} [i² = -1]
= 1 / (-1)
= - 1

১৫৯.

How many numbers from 11 to 50 are there which are exactly divisible by 7 not by 3?

ক
ক) 6
খ
খ) 5
গ
গ) 4
ঘ
ঘ) 2

সঠিক উত্তর: গ

গ) 4

উত্তর

সঠিক উত্তর: গ

গ) 4

গ

ব্যাখ্যা

Question: How many numbers from 11 to 50 are there which are exactly divisible by 7 not by 3?

Solution:
11 থেকে 50 পর্যন্ত 7 দ্বারা বিভাজ্য সংখ্যাগুলো হলো 14, 21, 28, 35, 42, এবং 49
14, 28, 35, এবং 49 সংখ্যাগুলো 3 দ্বারা বিভাজ্য নয়।
7 দ্বারা বিভাজ্য এবং 3 দ্বারা বিভাজ্য নয় এমন সংখ্যা = 4

১৬০.

At present, the sum of the ages of A, B and C is 150 years, whereas 10 years ago, the ratio of their age was 5 : 4 : 3. Find the ratio of their ages 10 years hence.

ক
7 : 5 : 3
খ
7 : 6 : 5
গ
4 : 3 : 5
ঘ
6 : 9 : 5
ঙ
None of these

সঠিক উত্তর: খ

7 : 6 : 5

উত্তর

সঠিক উত্তর: খ

7 : 6 : 5

খ

ব্যাখ্যা

Question: At present, the sum of the ages of A, B and C is 150 years, whereas 10 years ago, the ratio of their age was 5 : 4 : 3. Find the ratio of their ages 10 years hence.

Solution:
A, B, and C together have ages that are equal to 150 years.
10 years ago, their ages were 5 : 4 : 3, respectively.

The sum of the ages of A, B, and C before 10 years ago: ⇒ 150 - 3 × 10 ⇒ 150 - 30 = 120 years
10 years ago, their ages were 5 : 4 : 3,
which ⇒ 5x + 4x + 3x = 120
⇒ 12x = 120
⇒ x = 120/12 = 10 years

Hence, their age ratio after 10 years. ⇒ (5x + 20) : (4x + 20) : (3x + 20)
⇒ (5 × 10 + 20) : (4 × 10 + 20): (3 × 10 + 20)
⇒ 70 : 60 : 50 = 7 : 6 : 5

∴ Their ages are 7 : 6 : 5 ten years from now.

১৬১.

If the current month is January, what month will it be after 50 months?

ক
February
খ
March
গ
April
ঘ
None of the above

সঠিক উত্তর: খ

March

উত্তর

সঠিক উত্তর: খ

March

খ

ব্যাখ্যা

Question: If the current month is January, what month will it be after 50 months?

Solution:
We know that there are 12 months in a year, and the months repeat after every 12 months.

প্রথমে 50 কে 12 দিয়ে ভাগ দেই,
50 ÷ 12 = 4 remainder 2,

January মাসের এক মাস পরে হবে February.
January মাসের দুই মাস পরে আসবে March.

১৬২.

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

ক
log 7.2
খ
log 12
গ
log 9
ঘ
log 16

সঠিক উত্তর: ঘ

log 16

উত্তর

সঠিক উত্তর: ঘ

log 16

ঘ

ব্যাখ্যা

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

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

১৬৩.

A certain sum of money becomes three times of itself in 20 years at simple interest. In how many years does it become double of itself at the same rate of simple interest?

ক
ক) 6 years
খ
খ) 9 years
গ
গ) 10 years
ঘ
ঘ) 8 years

সঠিক উত্তর: গ

গ) 10 years

উত্তর

সঠিক উত্তর: গ

গ) 10 years

গ

ব্যাখ্যা

Let the principal be P.
After 20 years, Amount = 3P
SI = 3P - P = 2P

We know that, SI = PRT/100
⇒ 2P = (P × R × 20)/100
⇒ R = 10%

Now,
Amount = 2P
SI = Amount - Principal = 2P - P = P
⇒ SI = PRT/100
⇒ P = (P × 10 × T)/100
⇒ T = 10 years

১৬৪.

If 5n + 4 = 11, what is the value of 10n - 2?

ক
68
খ
14
গ
12
ঘ
7
ঙ
- 1

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: If 5n + 4 = 11, what is the value of 10n - 2?

Solution:
5n + 4 = 11
⇒ 5n = 7
⇒ 5n × 2 = 7 × 2
∴ 10n = 14

∴ 10n - 2 = 14 - 2 = 12

১৬৫.

Find the LCM of 2.5, 0.5 and 0.175.

ক
ক) 0.5
খ
খ) 0.05
গ
গ) 17.5
ঘ
ঘ) 1.75

সঠিক উত্তর: গ

গ) 17.5

উত্তর

সঠিক উত্তর: গ

গ) 17.5

গ

ব্যাখ্যা

Question: Find the LCM of 2.5, 0.5 and 0.175.

Solution:
2.5 = 25/10
0.5 = 5/10
0.175 = 175/1000

দুই বা ততোধিক ভগ্নাংশের ল.সা.গু = (লব গুলোর ল.সা.গু)/(হর গুলোর গ.সা.গু)
= (২৫, ৫, ১৭৫ এর ল.সা.গু)/(১০, ১০, ১০০০ এর গ.সা.গু)
= ১৭৫/১০
= ১৭.৫

১৬৬.

4200 Taka is lent at compound interest of 10% per annum for 2 years. Find the amount after two years.

ক
Tk. 6068
খ
Tk. 5028
গ
Tk. 4082
ঘ
Tk. 5082

সঠিক উত্তর: ঘ

Tk. 5082

উত্তর

সঠিক উত্তর: ঘ

Tk. 5082

ঘ

ব্যাখ্যা

Question: 4200 Taka is lent at compound interest of 10% per annum for 2 years. Find the amount after two years.

Solution:
Compound amount = P (1 + r)ⁿ
= 4200 {1 + (10/100)}²
= 4200 × (110/100) × (110/100)
= 42 × 11 × 11
= 5082

১৬৭.

Salaries of Ravi and Sumit are in the ratio 2 : 3. If the salary of each is increased by Tk. 4000, the new ratio becomes 40 : 57. What is Sumit's salary?

ক
Tk. 17,000
খ
Tk. 20,000
গ
Tk. 25,500
ঘ
Tk. 38,000

সঠিক উত্তর: ঘ

Tk. 38,000

উত্তর

সঠিক উত্তর: ঘ

Tk. 38,000

ঘ

ব্যাখ্যা

Question: Salaries of Ravi and Sumit are in the ratio 2 : 3. If the salary of each is increased by Tk. 4000, the new ratio becomes 40 : 57. What is Sumit's salary?

Solution:
Let the original salaries of Ravi and Sumit be Tk. 2x and Tk. 3x respectively.
Then,
(2x + 4000)/(3x + 4000) = 40/57
⇒ 57(2x + 4000) = 40(3x + 4000)
⇒ 6x = 68,000
⇒ 3x = 34,000

∴ Sumit's present salary = (3x + 4000) = Tk.(34000 + 4000) = Tk. 38,000.

১৬৮.

When x = (y + 3)² , which of the following matches (- 2y - 6)²?

ক
4x
খ
2x
গ
3x
ঘ
- 2x
ঙ
- 4x

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: When x = (y + 3)² , which of the following matches (- 2y - 6)²?

Solution:
Here,
x = (y + 3)²

∴ (- 2y - 6)² ={- 2 (y + 3)}²
= 4 × (y + 3)²
= 4x

১৬৯.

What is the angle between the hour and minute hand of a clock when it is 12 : 20 pm by the clock?

ক
100°
খ
90°
গ
110°
ঘ
80°
ঙ
None of these

সঠিক উত্তর: গ

110°

উত্তর

সঠিক উত্তর: গ

110°

গ

ব্যাখ্যা

Question: What is the angle between the hour and minute hand of a clock when it is 12 : 20 pm by the clock?

Solution:
Angle = |(11M - 60H)/2|°
= |{(11 × 20) - (60 × 0)}/2|° [Hour consider 0 at 12'o clock]
= |220/2|°
= 110°

১৭০.

If the total surface area of a cube is 150 cm², then what is the volume of the cube?

ক
125 cm³
খ
216 cm³
গ
512 cm³
ঘ
343 cm³

সঠিক উত্তর: ক

125 cm³

উত্তর

সঠিক উত্তর: ক

125 cm³

ক

ব্যাখ্যা

Question: If the total surface area of a cube is 150 cm², then what is the volume of the cube?

Solution:
ধরি,
ঘনকের এক বাহুর দৈর্ঘ্য = a সেমি

আমরা জানি, একটি ঘনকের মোট ৬টি সমান বর্গাকার তল থাকে।
সুতরাং, সমগ্র তলের ক্ষেত্রফল = 6a²

প্রশ্নমতে,
6a² = 150
⇒ a² = 150/6
⇒ a² = 25
⇒ a = √25 = 5 সেমি

আবার,
ঘনকের আয়তন = a³∴ আয়তন = 5³ = 125 ঘন সেমি

সুতরাং, ঘনকটির আয়তন = 125 ঘন সেমি

১৭১.

If a - b = 3 and ab = 2, then the value of a³ - b³ - 3ab will be?

ক
20
খ
27
গ
29
ঘ
39

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: If a - b = 3 and ab = 2, then the value of a³ - b³ - 3ab will be?

Solution:
We know,
a³ - b³ = (a - b)(a² + ab + b²)

So,
a³ - b³ - 3ab = (a - b)(a² + ab + b²) - 3ab
= 3 × (a² + b² + ab) - 3ab
= 3(a² + b²) + 3ab - 3ab
= 3(a² + b²)

Also, a² + b² = (a - b)²+ 2ab = 3² + 2×2 = 9 + 4 = 13

Therefore,
∴ a³- b³ - 3ab = 3 × 13 = 39

১৭২.

The sum of the present ages of a mother and her daughter is 50 years. Ten years ago, the mother’s age was four times the daughter’s age. Find the daughter’s present age.

ক
12 years
খ
16 years
গ
8 years
ঘ
14 years

সঠিক উত্তর: খ

16 years

উত্তর

সঠিক উত্তর: খ

16 years

খ

ব্যাখ্যা

Question: The sum of the present ages of a mother and her daughter is 50 years. Ten years ago, the mother’s age was four times the daughter’s age. Find the daughter’s present age.

Solution:
Let the present ages of the daughter and mother be x and (50 -x) years respectively.
Then,
(50 - x) - 10 = 4(x - 10)
⇒ 40 - x = 4x - 40
⇒ 5x = 80
∴ x = 16

∴ daughter's age is 16 years.

১৭৩.

If √2ⁿ = 64 , what will be the value of n?

ক
ক) 12
খ
খ) 8
গ
গ) 4
ঘ
ঘ) 16

সঠিক উত্তর: ক

ক) 12

উত্তর

সঠিক উত্তর: ক

ক) 12

ক

ব্যাখ্যা

√(2ⁿ) = 64
⇒ √(2ⁿ) = 2⁶
⇒ 2ⁿ = (2⁶)²
⇒ 2ⁿ= 2¹²
⇒ n = 12

১৭৪.

Three years back, a father was 24 years older than his son. At present the father is 5 times as old as the son. How old will the son be three years from now?

ক
ক) 12 years
খ
খ) 6 years
গ
গ) 9 years
ঘ
ঘ) 27 years

সঠিক উত্তর: গ

গ) 9 years

উত্তর

সঠিক উত্তর: গ

গ) 9 years

গ

ব্যাখ্যা

Question: Three years back, a father was 24 years older than his son. At present the father is 5 times as old as the son. How old will the son be three years from now?

Solution:
Let
Y is the age of the father and X the age of the son nowadays
According to the statement,
(Y - 3) - (X - 3) = 24............(1)
Again
5X = Y
From(1)
(Y-3) - ( X - 3) = 24
5X - 3 - X + 3 = 24
4X = 24
X = 6

The age of the son 3 years from now is = X + 3 = 9 years

১৭৫.

The average run of a cricket player of 10 innings was 35. How many runs must be made in his next innings so as to increase his average of runs by 5?

ক
ক) 80
খ
খ) 90
গ
গ) 85
ঘ
ঘ) 82

সঠিক উত্তর: খ

খ) 90

উত্তর

সঠিক উত্তর: খ

খ) 90

খ

ব্যাখ্যা

Question: The average run of a cricket player of 10 innings was 35. How many runs must be made in his next innings so as to increase his average of runs by 5?

Solution:
Average after 11 innings = 35 + 5 = 40
Required number of runs,
= (40 × 11) - (35 × 10)
= 440 - 350
= 90

১৭৬.

Today is Aziz's 12th Birthday and his father's 40th Birthday. How many years from today will Aziz's father be twice as old as Aziz's at that time?

ক
ক) 12
খ
খ) 24
গ
গ) 18
ঘ
ঘ) 16

সঠিক উত্তর: ঘ

ঘ) 16

উত্তর

সঠিক উত্তর: ঘ

ঘ) 16

ঘ

ব্যাখ্যা

Let, after x years father's age will be twice
ATQ, 2(12 + x) = 40 + x
Or, 24 + 2x = 40 + x
Or, x = 16

১৭৭.

In a trapezoid, the lengths of the two parallel bases are 12 and 20. If the height of the trapezoid is 5, find the area of the trapezoid.

ক
80 square units
খ
40 square units
গ
68 square units
ঘ
72 square units

সঠিক উত্তর: ক

80 square units

উত্তর

সঠিক উত্তর: ক

80 square units

ক

ব্যাখ্যা

Question: In a trapezoid, the lengths of the two parallel bases are 12 and 20. If the height of the trapezoid is 5, find the area of the trapezoid.

Solution:
Given,
Trapezoid with bases a = 12 and b = 20
Height, h = 5

We know,
Area of trapezoid = (1/2) × (sum of bases) × height
= (1/2) × (a + b) × h
= (1/2) × (12 + 20) × 5
= (1/2) × 32 × 5
= 16 × 5
= 80

So, the area of the trapezoid is 80 square units.

১৭৮.

In how many ways can the letters of the word ''LEADER'' be arranged?

ক
360
খ
500
গ
720
ঘ
800

সঠিক উত্তর: ক

360

উত্তর

সঠিক উত্তর: ক

360

ক

ব্যাখ্যা

Question: In how many ways can the letters of the word ''LEADER'' be arranged?

Solution:
The word,'LEADER'' contains 6 letters, 1L, 2E, 1A, 1D and 1R.
∴ Required number of ways = 6!/2!
= 360

১৭৯.

A mixture of 200 liters of milk and water contains 15% water. How much more water should be added so that water becomes 20% of the new mixture?

ক
10 liters
খ
12.5 liters
গ
16 liters
ঘ
24.5 liters

সঠিক উত্তর: খ

12.5 liters

উত্তর

সঠিক উত্তর: খ

12.5 liters

খ

ব্যাখ্যা

Question: A mixture of 200 liters of milk and water contains 15% water. How much more water should be added so that water becomes 20% of the new mixture?

Solution:
Amount of water in the 200-liter mixture = 15% of 200
= (15/100) × 200
= 30 liters

Let,
P liters of water are added.
So, new amount of water = (30 + P)
and new total mixture = (200 + P)

ATQ,
(30 + P) = 20% of (200 + P)
⇒ 30 + P = (20/100) × (200 + P)
⇒ 30 + P = (1/5) × (200 + P)
⇒ 150 + 5P = 200 + P
⇒ 5P - P = 200 - 150
⇒ 4P = 50
∴ P = 12.5

∴ 12.5 liters more water should be added.

১৮০.

If the sum of 3 consecutive integers is 210, then the sum of the two smaller integer is-

ক
140
খ
150
গ
139
ঘ
145

সঠিক উত্তর: গ

139

উত্তর

সঠিক উত্তর: গ

139

গ

ব্যাখ্যা

Question: If the sum of 3 consecutive integers is 210, then the sum of the two smaller integer is-

Solution:
Let,
Three consecutive integers is, x - 1 , x, x + 1

ATQ,
x - 1 + x + x + 1 = 210
⇒ 3x = 210
∴ x = 70

The sum of the two smaller integer is = x - 1 + x
= 70 - 1 + 70
= 140 - 1
= 139

১৮১.

A dealer has 1000 kg sugar and he sells a part of it at 8% profit and the rest of it at 18% profit. The overall profit he earns is 14%. What is the quantity which is sold at 18% profit?

ক
ক) 250 kg
খ
খ) 600 kg
গ
গ) 620 kg
ঘ
ঘ) 400 kg

সঠিক উত্তর: খ

খ) 600 kg

উত্তর

সঠিক উত্তর: খ

খ) 600 kg

খ

ব্যাখ্যা

The basic formula which is used to find the ratio in which the ingredients are mixed is :

As per the rule of Allegation,

Quantity of Cheaper : Quantity of Dearer = (18-14) : (14-8) = 4:6 = 2:3
Quantity of sugar sold at 18% profit = 3/5 × 1000 = 600kg

১৮২.

Given that a shop opens at 10 a.m. and closes at 6:45 p.m., with a 20-minute tea break, what is the proportion of the break to the total working hours?

ক
1/15
খ
4/10
গ
2/105
ঘ
4/105
ঙ
None

সঠিক উত্তর: ঘ

4/105

উত্তর

সঠিক উত্তর: ঘ

4/105

ঘ

ব্যাখ্যা

Question: Given that a shop opens at 10 a.m. and closes at 6:45 p.m., with a 20-minute tea break, what is the proportion of the break to the total working hours?

Solution:
Total working time = 6:45 - 10:00
= 8 hours 45 minutes
= (8 × 60) + 45
= 525 minutes

The ratio of the break to the total working period
= 20/525
= 4/105

১৮৩.

If TALE = LATE, then CAFE = ?

ক
FAFC
খ
EAFC
গ
AFCE
ঘ
FACE

সঠিক উত্তর: ঘ

FACE

উত্তর

সঠিক উত্তর: ঘ

FACE

ঘ

ব্যাখ্যা

Question: If TALE = LATE, then CAFE = ?

Solution:
এখানে, TALE = LATE এর T আর L পরস্পরের স্থান পরিবর্তন করেছে, A আর E এর কোনো পরিবর্তন হয়নি।
তাই CAFE শব্দের A আর E এর কোনো পরিবর্তন হবে না, C আর F পরস্পরের স্থান পরিবর্তন করবে।
∴ CAFE = FACE

১৮৪.

An uneducated retailer marks all his goods at 50% above the cost price and thinking that he will still make 25% profit, offers a discount of 25% on the marked price. What is his actual profit on the sales?

ক
10%
খ
12.5%
গ
15.8%
ঘ
20%

সঠিক উত্তর: খ

12.5%

উত্তর

সঠিক উত্তর: খ

12.5%

খ

ব্যাখ্যা

Question: An uneducated retailer marks all his goods at 50% above the cost price and thinking that he will still make 25% profit, offers a discount of 25% on the marked price. What is his actual profit on the sales?

Solution:
Let, cost price is 100 taka

After 50% increase = 100 + 100 × .5
= 150 taka

25% discounted price = 150 - 150 × .25
= 112.5 taka

Actual profit percentage = (112.5 - 100) × 100%/100
= 12.5%

১৮৫.

ক
2.11
খ
2.13
গ
2.58
ঘ
2.03

সঠিক উত্তর: খ

2.13

উত্তর

সঠিক উত্তর: খ

2.13

খ

ব্যাখ্যা

Question:

Solution:

১৮৬.

If px² + 24x + 16 is a perfect square number, then p = ?

ক
6
খ
3
গ
4
ঘ
9

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: If px² + 24x + 16 is a perfect square number, then p = ?

Solution:
দেওয়া আছে, রাশিটি একটি পূর্ণবর্গ সংখ্যা।
px² + 24x + 16
= (3x)² + 2(3x)(4) + (4)²

একটি রাশি পূর্ণবর্গ হওয়ার জন্য, এটি (a² + 2ab + b²) অথবা (a² - 2ab + b²) আকারের হতে হবে। এখানে,
a = 3x এবং b = 4

যেহেতু প্রথম পদটি a² এর সমান,
∴ px² = a²
⇒ px² = (3x)²
⇒ px² = 9x²
⇒ p = 9

অতএব, p এর মান হলো 9।

১৮৭.

A 280 metres long train running at the speed of 120 kmph crosses another train running in opposite direction at the speed of 80 kmph in 9 seconds. What is the length of the other train?

ক
ক) 220 m
খ
খ) 240 m
গ
গ) 260 m
ঘ
ঘ) 210 m

সঠিক উত্তর: ক

ক) 220 m

উত্তর

সঠিক উত্তর: ক

ক) 220 m

ক

ব্যাখ্যা

Relative speed = (120 + 80) km/hr
= 200 km/hr
= 200 ×(5/18) m/sec
= 500/9 m/sec

Let the length of the other train be x metres.
Now
(x + 280)/9 = 500/9
x + 280 = 500
x = 500 - 280
x = 220

১৮৮.

Two trains of equal length are running on parallel lines in the same direction at 46 km/hr and 36 km/hr. The faster train passes the slower train in 36 seconds. The length of each train is:

ক
40 m
খ
50 m
গ
45 m
ঘ
55 m

সঠিক উত্তর: খ

50 m

উত্তর

সঠিক উত্তর: খ

50 m

খ

ব্যাখ্যা

Question: Two trains of equal length are running on parallel lines in the same direction at 46 km/hr and 36 km/hr. The faster train passes the slower train in 36 seconds. The length of each train is:

Solution:
Let the length of each train be x metres.
Then, distance covered = 2x metres.
Relative speed = (46 - 36) km/hr
= 10 × (5/18) m/sec
= 25/9 m/sec

ATQ,
2x/36 = 25/9
⇒ 18x = 25 × 36
⇒ x = (25 × 36)/18
= 50

১৮৯.

A gardener planted trees in rows and columns such that the number of rows is seven more than the number of columns. If the total number of rows and columns is 87, find the number of trees.

ক
1880
খ
2160
গ
1680
ঘ
2450

সঠিক উত্তর: ক

1880

উত্তর

সঠিক উত্তর: ক

1880

ক

ব্যাখ্যা

Question: A gardener planted trees in rows and columns such that the number of rows is seven more than the number of columns. If the total number of rows and columns is 87, find the number of trees.

Solution:
Let the number of columns = x.
Then, number of rows = x + 7

According to the question,
x + (x + 7) = 87
⇒ 2x + 7 = 87
⇒ 2x = 80
⇒ x = 40

Number of rows = x + 7 = 40 + 7 = 47

Total number of trees = rows × columns
= 47 × 40 = 1880

১৯০.

6 years ago, the ratio of the ages of Kamrul and Sagar was 6 : 5. Four years hence, the ratio of their ages will be 11 : 10. What is Sagar's age at present ?

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

সঠিক উত্তর: গ

16 years

উত্তর

সঠিক উত্তর: গ

16 years

গ

ব্যাখ্যা

Question: 6 years ago, the ratio of the ages of Kamrul and Sagar was 6 : 5. Four years hence, the ratio of their ages will be 11 : 10. What is Sagar's age at present ?

Solution:
Let the ages of Kamrul and Sagar 6 years ago be 6x and 5x years
Then,
{(6x + 6) + 4}/{(5x + 6) + 4} = 11/10
⇒ (6x + 10)/(5x + 10) = 11/10
⇒ 10(6x + 10) = 11(5x + 10)
⇒ 60x + 100 = 55x + 110
⇒ 5x = 10
∴ x = 2

∴ Sagar's present age,
= (5x + 6) years
= (5 × 2 + 6) years
= 16 years

∴ Sagar's present age is 16 years.

১৯১.

What is the probability of getting a sum 7 from two throws of a dice?

ক
1/6
খ
1/12
গ
1/9
ঘ
1/8

সঠিক উত্তর: ক

1/6

উত্তর

সঠিক উত্তর: ক

1/6

ক

ব্যাখ্যা

Question: What is the probability of getting a sum 7 from two throws of a dice?

Solution:
দুটি ছক্কা নিক্ষেপ করলে মোট সম্ভাব্য ফলাফল (Sample Space), n(S) = 6 × 6 = 36

ধরি, E হলো প্রাপ্ত সংখ্যাদ্বয়ের যোগফল 7 হওয়ার ঘটনা।
∴ অনুকূল ফলাফলগুলো হলো, E = {(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)}

এখানে, n(E) = 6

আমরা জানি,সম্ভাবনা P(E) = n(E)/n(S)
= 6/36
= 1/6

১৯২.

Which number is 60% less than 80?

ক
30
খ
32
গ
35
ঘ
40

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: Which number is 60% less than 80?

Solution:
let, the number is x

x = 80 - 80 × . 6
= 80 - 48
= 32

১৯৩.

Tk 7,500 is divided in the ratio of 1 : 2 : 3 : 4 : 5. Find the difference between the greatest and smallest shares?

ক
ক) 1500
খ
খ) 2000
গ
গ) 2600
ঘ
ঘ) 2700

সঠিক উত্তর: খ

খ) 2000

উত্তর

সঠিক উত্তর: খ

খ) 2000

খ

ব্যাখ্যা

Question: Tk 7,500 is divided in the ratio of 1 : 2 : 3 : 4 : 5. Find the difference between the greatest and smallest shares?

Solution:
প্রদত্ত অনুপাত 1 : 2 : 3 : 4 : 5
প্রদত্ত অনুপাতের রাশিগুলোর যোগফল = 1 + 2 + 3 + 4 + 5 = 15

সবচেয়ে বেশি শেয়ারের পরিমাণ = 7500 এর 5/15 = 2500 টাকা
সবচেয়ে কম শেয়ারের পরিমাণ = 7500 এর 1/15 = 500 টাকা
পার্থক্য = (2500 - 500) = 2000 টাকা

১৯৪.

এক ব্যক্তি ৩০ ঘণ্টায় একটি নির্দিষ্ট দূরত্ব অতিক্রম করেন। যাত্রাপথের অর্ধেক দূরত্ব তিনি ২১ কি.মি./ঘণ্টা এবং বাকী অর্ধেক দূরত্ব ২৪ কি.মি./ঘণ্টা গতিতে চলেন। মোট দূরত্ব কত ছিল?

ক
৬৯৪ কি.মি.
খ
৬৭৫ কি.মি.
গ
৬৮৮ কি.মি.
ঘ
৬৭২ কি.মি.
ঙ
কোনটিই নয়

সঠিক উত্তর: ঘ

৬৭২ কি.মি.

উত্তর

সঠিক উত্তর: ঘ

৬৭২ কি.মি.

ঘ

ব্যাখ্যা

প্রশ্ন: এক ব্যক্তি ৩০ ঘণ্টায় একটি নির্দিষ্ট দূরত্ব অতিক্রম করেন। যাত্রাপথের অর্ধেক দূরত্ব তিনি ২১ কি.মি./ঘণ্টা এবং বাকী অর্ধেক দূরত্ব ২৪ কি.মি./ঘণ্টা গতিতে চলেন। মোট দূরত্ব কত ছিল?

সমাধান:
ধরি,
দূরত্ব = ক কি.মি.
ব্যক্তিটি ৩০ ঘণ্টায় ক দূরত্ব অতিক্রম করেন।
যাত্রাপথের অর্ধেক দূরত্ব অতিক্রম করতে সময় লাগে = অর্ধেক দূরত্ব/যাত্রাপথের অর্ধেক দূরত্ব যাওয়ার গতিবেগ
= (ক/২)/২১
= ক/৪২ ঘণ্টা

যাত্রাপথের বাকি অর্ধেক দূরত্ব অতিক্রম করতে সময় লাগে = বাকি অর্ধেক দূরত্ব/যাত্রাপথের বাকি অর্ধেক দূরত্ব যাওয়ার গতিবেগ
= (ক/২)/২৪
= ক/৪৮ ঘণ্টা

প্রশ্নমতে,
ক/৪২ + ক/৪৮ = ৩০
⇒ (৮ক + ৭ক)/৩৩৬ = ৩০
⇒ ১৫ক = ৩০ × ৩৩৬
∴ ক = ৬৭২ কি.মি.

১৯৫.

Determine the value of the 3^rd term of the sequence: sin⁡(nπ/6)

ক
√3/2
খ
1
গ
1/2
ঘ
0

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: Determine the value of the 3^rd term of the sequence: sin⁡(nπ/6)

Solution:
The sequence is defined as sin(nπ/6), where n = 1, 2, 3, 4, ...
We need the 3^rd term, so substitute n = 3
3^rd term = sin(3π/6)
= sin(π/2)
= 1

So the 3^rd term of the sequence is 1.

১৯৬.

If Tk. 782 is allocated into three portions according to the ratio 1/2 : 2/3 : 3/4, what is the amount of the first portion?

ক
Tk. 204
খ
Tk. 202
গ
Tk. 282
ঘ
Tk. 180
ঙ
None of the above

সঠিক উত্তর: ক

Tk. 204

উত্তর

সঠিক উত্তর: ক

Tk. 204

ক

ব্যাখ্যা

Question: If Tk. 782 is allocated into three portions according to the ratio 1/2 : 2/3 : 3/4, what is the amount of the first portion?

Solution:
The given ratio = 1/2 : 2/3 : 3/4
= 6 : 8 : 9

∴ The first portion = 782 × (6/23)
= 204 Tk.

১৯৭.

In a certain time, a sum becomes 4 times of itself on simple interest at the rate of 10% per annum. What is the rate of interest if the same sum becomes 7 times of itself in the same duration?

ক
20%
খ
16%
গ
21%
ঘ
9%

সঠিক উত্তর: ক

20%

উত্তর

সঠিক উত্তর: ক

20%

ক

ব্যাখ্যা

Question: In a certain time, a sum becomes 4 times of itself on simple interest at the rate of 10% per annum. What is the rate of interest if the same sum becomes 7 times of itself in the same duration?

Solution:
Principal = P
Time = n (same in both cases)
Rate, r = 10%
Simple Interest, SI = 4P - P = 3P

We know,
SI = Prn
⇒ 3P = (P × 10 × n)/100
⇒ 3P = (P × n)/10
∴ n = 30 years

Again,
In the second scenario, the same sum becomes 7 times itself in the same duration (30 years).
Final Amount, A = 7P
Simple Interest, SI = A - P = 7P - P = 6P
Time, n = 30 years

SI = Prn
⇒ 6P = (P × r × 30)/100
⇒ 6 = (r × 3)/10
⇒ r = 60/3
∴ r = 20%

So the new rate of interest is 20% per annum.

১৯৮.

How many years will it take for an investment of Tk. 15000 to earn Tk. 3600 in simple interest rate of 4%?

ক
5 years
খ
6 years
গ
8 years
ঘ
4 years

সঠিক উত্তর: খ

6 years

উত্তর

সঠিক উত্তর: খ

6 years

খ

ব্যাখ্যা

Question: How many years will it take for an investment of Tk. 15000 to earn Tk. 3600 in simple interest rate of 4%?

Solution:
Given that,
Principal, P = 15000
Simple Interest, I = 3600
Rate of interest, r = 4%
Time, n = ?

We know,
I = Pnr/100
⇒ n = (I × 100)/ (P × r)
= (3600 × 100)/(15000 × 4)
= 6

So, it will take 6 years for the investment to earn Tk. 3600 at 4% simple interest.

১৯৯.

The present ratio of students to teachers at a certain school is 30 to 1. If the number of students were to increase by 50 and the number of teachers were to increase by 5, the ratio of students to teachers would then be 25 to 1. What is the present number of students?

ক
290
খ
380
গ
450
ঘ
None of these

সঠিক উত্তর: গ

450

উত্তর

সঠিক উত্তর: গ

450

গ

ব্যাখ্যা

Question: The present ratio of students to teachers at a certain school is 30 to 1. If the number of students were to increase by 50 and the number of teachers were to increase by 5, the ratio of students to teachers would then be 25 to 1. What is the present number of students?

Solution:
Let the number of student and teacher be represented by x and y, respectively.
x : y = 30 : 1
⇒ x/y = 30/1
⇒ x = 30y....................(1)

If the number of students increases by 50 and the number of teachers increases by 5
(50 + x) : (y + 5) = 25 : 1
⇒ (50 + x) / (y + 5) = 25/1
⇒ 50 + x = 25y + 125
⇒ 50 + 30y = 25y + 125
⇒ 30y - 25y = 125 - 50
⇒ 5y = 75
⇒ y = 75/5
⇒ y = 15

∴ Number of teachers = 15
And the number of students = 30 × 15 = 450

২০০.

A truck travels at 96 km/h. How much distance will it cover in 75 minutes?

ক
118 km
খ
120 km
গ
124 km
ঘ
122 km

সঠিক উত্তর: খ

120 km

উত্তর

সঠিক উত্তর: খ

120 km

খ

ব্যাখ্যা

Question: A truck travels at 96 km/h. How much distance will it cover in 75 minutes?

Solution:
Given that,
Speed = 96 km/h
Time = 75 minutes = 75/60 hours = 5/4 hours

We know,
Distance = Speed × Time
= 96 × (5/4)
= 96 × 1.25
= 120 km

So the truck will cover 120 km in 75 minutes.