Bank Math

PrepBank · পাতা ৫২ / ১৬১ · ৫,১০১–৫,২০০ / ১৬,১২৪

৫,১০১.

Two pipes A and B can fill a tank in 40 and 60 minutes respectively. If both the pipes are used together, then how long will it take to fill the tank?

ক
24 minutes
খ
20 minutes
গ
18 minutes
ঘ
15 minutes

ব্যাখ্যা

Question: Two pipes A and B can fill a tank in 40 and 60 minutes respectively. If both the pipes are used together, then how long will it take to fill the tank?

Solution:
Part filled by A in 1 minute = 1/40

Part filled by B in 1 minute = 1/60

Part filled by (A + B) in 1 minute = 1/40 + 1/60 = 5/120 = 1/24

∴ Both pipes can fill the tank in 24 minutes

৫,১০২.

A car starts at A and reaches a destination in 8 hours. If it travels first and second half at the speed of 80 km/hr and 100 km/hr respectively then the distance between A and destination is -

ক
ক) 818.25 km
খ
খ) 979.50 km
গ
গ) 711.11 km
ঘ
ঘ) 850 km

ব্যাখ্যা

Let,
The distance between A and destination be X
The total time taken by the car to cover X = 8 hours
Since X/2 by 80km/hr and remaining X/2 by 100km/hr
Then by Time = distance/speed, we have
(X/2)/80 + (X/2)/100 = 8
⇒ 1/2 {(X/80 + X/100)} = 8
⇒ X/80 + X/100 = 16
⇒ (5X + 4X)/400 = 16
⇒ 5X + 4X = 6400
⇒ 9X = 6400
⇒ X = 6400/9
⇒ X = 711.11
Hence 711.11 km is the required answer.

৫,১০৩.

The average temperature for Wednesday, Thursday and Friday was 40°C. The average for Thursday, Friday and Saturday was 41° C. If temperature on Saturday was 42° C, what was the temperature on Wednesday?

ক
38°C
খ
39°C
গ
41°C
ঘ
43°C

ব্যাখ্যা

Question: The average temperature for Wednesday, Thursday and Friday was 40°C. The average for Thursday, Friday and Saturday was 41°C. If temperature on Saturday was 42°C, what was the temperature on Wednesday?

Solution:
Here,
Total temperature for Wednesday, Thursday and Friday = 3 × 40° = 120°C
Total temperature for Thursday, Friday and Saturday = 3 × 41° = 123°C

Now,
(Thursday + Friday + Saturday) - (Wednesday + Thursday + Friday) = 123° - 120°
⇒ Saturday - Wednesday = 3°
∴ Wednesday = 42° - 3° = 39°C

৫,১০৪.

Among 100 students, the average marks in mathematics is 70. If the 60 girls scored an average of 72, determine the average score of the remaining boys.

ক
65.5
খ
67
গ
68
ঘ
69.5

ব্যাখ্যা

Question: Among 100 students, the average marks in mathematics is 70. If the 60 girls scored an average of 72, determine the average score of the remaining boys.

Solution:
ধরি,
ছাত্রদের গড় নম্বর = ক

100 জন শিক্ষার্থীর মোট নম্বর = 100 × 70 = 7000
এবং 60 জন ছাত্রীর মোট নম্বর = 60 × 72 = 4320

প্রশ্নমতে,
4320 + (100 - 60) × ক = 7000
⇒ 4320 + 40ক = 7000
⇒ 40ক = 7000 - 4320
⇒ 40ক = 2680
⇒ ক = 2680/40
⇒ ক = 67

∴ 40 জন ছাত্রের গড় নম্বর = 67

৫,১০৫.

If rsinθ = 1, rcosθ = √3 then the value of √3tanθ + 1 = ?

ক
ক) √3
খ
খ) 1/√3
গ
গ) 1
ঘ
ঘ) 2

ব্যাখ্যা

Question: If rsinθ = 1, rcosθ = √3 then the value of √3tanθ + 1 = ?

Solution:
দেওয়া আছে,
rsinθ = 1 ......... (1)
rcosθ = √3 .............. (2)

(1) ÷ (2) হতে পাই
⇒ rsinθ/rcosθ =1/√3
⇒ tanθ = 1/√3
⇒ √3tanθ = 1

এখন, √3tanθ + 1 = 1 + 1
∴ √3tanθ + 1 = 2

৫,১০৬.

Find the median of the following data 2, 9, 15, 5, 20, 8, 25, 17, 21, 23, 11.

ক
ক) 11
খ
খ) 15
গ
গ) 17
ঘ
ঘ) 20

ব্যাখ্যা

Question: Find the median of the following data 2, 9, 15, 5, 20, 8, 25, 17, 21, 23, 11.

Solution:
উপাত্তগুলোকে মানের ক্রমানুসারে সাজিয়ে পাই,
2, 5, 8, 9, 11, 15, 17, 20, 21, 23, 25
এখানে মোট পদ 11টি
∴ নির্ণেয় মধ্যক = ৬ষ্ঠ পদ = 15

৫,১০৭.

What is the 8^th term of the series: 2 + 4 + 8 + 16 +..........

ক
128
খ
256
গ
512
ঘ
1024

ব্যাখ্যা

Question: What is the 8^th term of the series: 2 + 4 + 8 + 16 +..........

Solution:
Given,
a = 2
r = 4/2
= 2 > 1
The given series is a geometric series.

We know,
the n^th term of a geometric series = ar^{(n - 1)}

8^th term of the series = 2 × 2^{(8 - 1)} = 2 × 2⁷ = 256

৫,১০৮.

Find the value of cos(7π/6).

ক
- 1/2
খ
1/√2
গ
1
ঘ
- √3/2

ব্যাখ্যা

Question: Find the value of cos(7π/6).

Solution:
cos(7π/6)
= cos(π + π/6) [যেহেতু (π + π/6) তৃতীয় চতুর্ভাগে পড়ে এবং তৃতীয় চতুর্ভাগে cos ঋণাত্মক, তাই cos(π + θ) = - cosθ]
= - cos(π/6)
= - cos(30°)
= - √3/2

৫,১০৯.

Akash took a loan from a bank at the rate of 12 % p.a. simple interest. After three years he had to pay Tk. 5400 interest only for the period. The principal amount borrowed by him was:

ক
Tk. 10000
খ
Tk. 12000
গ
Tk. 15000
ঘ
Tk. 20000

ব্যাখ্যা

Question: Akash took a loan from a bank at the rate of 12 % p.a. simple interest. After three years he had to pay Tk. 5400 interest only for the period. The principal amount borrowed by him was:

Solution:
Let, The principal amount borrowed by him was P

P × (12/100) × 3 = 5400
⇒ P = (5400 × 100)/36
= 15000 taka

৫,১১০.

3cotA = 4 হলে, sinA এর মান কত?

ক
5/3
খ
5/6
গ
3/5
ঘ
1/3

ব্যাখ্যা

প্রশ্ন : 3cotA = 4 হলে, sinA এর মান কত?

সমাধান :
দেওয়া আছে,
3 cotA = 4
⇒ cot = 4/3
⇒ cot²A = 16/9
⇒ cosec²A - 1 = 16/9
⇒ cosec²A = (16/9) + 1
⇒ cosec²A = (16 + 9)/9
⇒ cosec²A = 25/9
⇒ cosecA = 5/3
⇒ 1/sinA = 5/3
⇒ sinA = 3/5

৫,১১১.

What is the angle between the hour and minute hand of a clock when it is 9 : 30 pm?

ক
98.5°
খ
100°
গ
105°
ঘ
110°

ব্যাখ্যা

Question: What is the angle between the hour and minute hand of a clock when it is 9 : 30 pm?

Solution:
9টা 30 মিনিট = 9 + (30/60) ঘন্টা
= 9 + 1/2 = 19/2 ঘন্টা

আমরা জানি,
ঘণ্টার কাঁটা 12 ঘণ্টায় 360° ঘোরে।
∴ 1 ঘণ্টায় ঘোরে = 360°/12 = 30°
∴ 19/2 ঘণ্টায় ঘোরে = (30° × 19)/2
= 15 × 19 = 285°
আবার,

মিনিটের কাঁটা 60 মিনিটে 360° ঘোরে।
∴ 1 মিনিটে ঘোরে = 360°/60 = 6°
∴ 30 মিনিটে ঘোরে = 30 × 6° = 180°

∴ ঘড়ির কাঁটা দুটির মধ্যবর্তী কোণ = |285° - 180°| = 105°

৫,১১২.

Two numbers are in ratio of 3 : 7. If 6 is added in each, the new numbers are in ratio of 5 : 9. Find the ratio of numbers, if 6 subtracted from each number?

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

ব্যাখ্যা

ধরি
সংখ্যা দুটি 3x এবং 7x
⇒ (3x + 6)/(7x + 6) = 5/9
⇒ 27x + 54 = 35x + 30
⇒ 8x = 24
⇒ x = 3
সংখ্যা দুটি 9 এবং 21

নতুন অনুপাত = (9 - 6)/(21- 6)
= 3/15
= 1/5
= 1 : 5

৫,১১৩.

A room 8 m long, 6 m high and 22.5 cm thick is made up of bricks, each measuring 25 cm × 11.25 cm × 6 cm. The number of bricks required is.

ক
ক) 7200
খ
খ) 6400
গ
গ) 6000
ঘ
ঘ) 5600

ব্যাখ্যা

Number of bricks = Volume of the wall/Volume of 1 brick
= (800×600×22.5)/(25×11.25×6)
= 6400

৫,১১৪.

6 × 3 (3 - 1) is equal to =?

ক
ক) 19
খ
খ) 20
গ
গ) 53
ঘ
ঘ) 36

ব্যাখ্যা

Question: 6 × 3 (3 - 1) is equal to =?

Solution:
6 × 3 (3 - 1)
= 6 × 3 × 2
= 36

৫,১১৫.

In a lottery, there are 10 prizes and 25 blanks. A lottery is drawn at random. What is the probability of getting a prize?

ক
2/3
খ
3/8
গ
2/7
ঘ
3/5

ব্যাখ্যা

Question: In a lottery, there are 10 prizes and 25 blanks. A lottery is drawn at random. What is the probability of getting a prize?

Solution:
probability of getting a prize = 10/(10 + 25)
= 10/35
= 2/7

৫,১১৬.

A product is sold at a profit of 20%. If the cost price is increased by 10% and sale price by Tk. 26, then the percentage of profit reduce by 5%, cost price is

ক
ক) 400
খ
খ) 500
গ
গ) 600
ঘ
ঘ) 700

ব্যাখ্যা

Let, the cost price be Tk. x
Selling price at 20% profit
= x + 20% of x
= x + 20x/100
= 120x/100
= Tk. 6x/5

The cost price is increased by 10%.
So, new cost price
= (x + 10% of x)
= Tk. 11x/10

Therefore, new selling price = Tk. (6x/5 + 26)

New profit = (6x/5 + 26) - 11x/6
= Tk. (x/10 + 26)

According to the question,
(x/10 + 26) ÷ 11x/10 × 100 = 15
⇒ (x + 260) × 100/11x = 15
⇒ 65x = 26,000
∴ x = 400

৫,১১৭.

If 4sin²(2θ) +1 = 4, then θ = ?

ক
ক) 60°
খ
খ) 45°
গ
গ) 30°
ঘ
ঘ) 0°

ব্যাখ্যা

Question: If 4sin²(2θ) +1 = 4, then θ = ?

Solution:
4sin²(2θ) +1 = 4
বা, 4sin²(2θ) = 4 - 1
বা, sin²(2θ) = 3/4
বা, sin(2θ) = √3/2
বা, sin(2θ) = sin60°
বা, 2θ = 60°
∴ θ = 30°

৫,১১৮.

The present age of three persons are in the ratio 3 : 5 : 6. Four years ago, the sum of their ages was 72 years. The present age of the eldest person is -

ক
32 years
খ
36 years
গ
42 years
ঘ
48 years

ব্যাখ্যা

Question: The present age of three persons are in the ratio 3 : 5 : 6. Four years ago, the sum of their ages was 72 years. The present age of the eldest person is -

Solution:
Let, present ages of three persons 3x, 5x, 6x

Four years ago, their ages were- (3x - 4), (5x - 4), (6x - 4)

ATQ,
⇒ (3x - 4) + (5x - 4) + (6x - 4) = 72
⇒ 3x + 5x + 6x - 12 = 72
⇒ 14x = 72 + 12
⇒ 14x = 84
⇒ x = 84/14
∴ x = 6

So, eldest person’s present age = 6x = 6 × 6 = 36 years.

৫,১১৯.

Jasim was 27 years old when he married Bela. They just celebrated their fifth wedding anniversary, and Bela's age is now 7/8 of Jasim's. How old is Bela?

ক
22 years
খ
25 years
গ
28 years
ঘ
31 years

ব্যাখ্যা

Question: Jasim was 27 years old when he married Bela. They just celebrated their fifth wedding anniversary, and Bela's age is now 7/8 of Jasim's. How old is Bela?

Solution:
jasim's present age = 27 + 5 = 32 years
bela's age = (7/8) × 32 = 28 years

৫,১২০.

The average of first five multiples of 5 is-

ক
ক) 10
খ
খ) 15
গ
গ) 20
ঘ
ঘ) 25

ব্যাখ্যা

Question: The average of first five multiples of 5 is-

Solution:
first five multiples of 5 are
= (5 × 1)
= (5 × 2)
= (5 × 3)
= (5 × 4)
= (5 × 5)

∴ The average of first five multiples of 5 is = {(5 × 1) + (5 × 2) + (5 × 3) + (5 × 4) + (5 × 5)}/5
= 5 (1 + 2 + 3 + 4 + 5)/5
= 1 + 2 + 3 + 4 + 5
= 15

৫,১২১.

What is the probability of getting a sum 9 from two throws of a dice?

ক
1/6
খ
1/8
গ
1/9
ঘ
1/12

ব্যাখ্যা

Question: What is the probability of getting a sum 9 from two throws of a dice?

Solution:

In two throws of a dice, n(S) = (6 × 6) = 36.

Let E = event of getting a sum ={(3, 6), (4, 5), (5, 4), (6, 3)}.

∴ P(E) = n(E)/n(S) = 4/36 = 1/9

৫,১২২.

A rectangular floor that measures 4 meters by 8 meters is to be covered with carpet that each measures 2 meters by 2 meters. If the carpet cost Tk. 14 a piece, what is the total cost to cover the floor?

ক
ক) Tk. 115
খ
খ) Tk. 112
গ
গ) Tk. 240
ঘ
ঘ) Tk. 118

ব্যাখ্যা

Area of the floor = 4 × 8 = 32 m²
Area of each carpet = 2 × 2 = 4 m²
Required number of carpet = 32/4 = 8
one carpet costs Tk. 14
8 carpet costs Tk. 14 × 8 = Tk. 112

৫,১২৩.

How many positive integers less than ten thousand are multiples of both eight and eighteen.

ক
ক) 70
খ
খ) 72
গ
গ) 138
ঘ
ঘ) 139

ব্যাখ্যা

৪ ও 18 এর ল.সা.গু = 72
∴ নির্ণেয় পূর্ণ সংখ্যা =10000/72
= 138.8 ≈ 138 টি

৫,১২৪.

If an exponent or index has base 15 and power zero, then which of the following will be its value?

ক
15
খ
1
গ
0
ঘ
225

ব্যাখ্যা

Question: If an exponent or index has base 15 and power zero, then which of the following will be its value?

Solution:
a⁰= 1 (for any non-zero base a)

Now,
= 15⁰= 1

৫,১২৫.

A piece of wire 156 cm long is bent in the form of an isosceles triangle. If the ratio of one of the equal sides to the base is 3 : 2, then what is the length of the base?

ক
39 cm
খ
43 cm
গ
35 cm
ঘ
41 cm

ব্যাখ্যা

Question: A piece of wire 156 cm long is bent in the form of an isosceles triangle. If the ratio of one of the equal sides to the base is 3 : 2, then what is the length of the base?

Solution:
Given,
Ratio of one of the equal sides to the base is 3 : 2
Therefore, let base be 2x and equal sides be 3x
156 cm piece of wire is bent to form an isosceles triangle.
Thus perimeter of triangle is 156 cm.
Therefore,
2x + 3x + 3x = 156
⇒ 8x = 156
⇒ x = 19.5
Thus the length of the base = 2 × 19.5 = 39 cm.

৫,১২৬.

The age of the father ten years ago was thrice the age of his son. Ten years hence, the father's age will be twice the age of his son. What is the ratio of their present ages?

ক
5 : 2
খ
7 : 3
গ
9 : 2
ঘ
13 : 5

ব্যাখ্যা

Question: The age of the father ten years ago was thrice the age of his son. Ten years hence, the father's age will be twice the age of his son. What is the ratio of their present ages?

Solution:
Let,
son's age 10 years ago be x years.
Then, father’s age 10 years ago = 3x years.

Son's present age = (x + 10) years
Father's present age = (3x + 10) years.

according to the question,
(3x + 10) + 10 = 2 (x + 10 + 10)
⇒ 3x + 20 = 2 (x + 20)
⇒ 3x + 20 = 2x + 40
∴ x = 20

Ratio of present ages of father and the son
= (3x + 10)/(x + 10)
= {(3 × 20) + 10}/(20 + 10)
= 70/30
= 7 : 3

৫,১২৭.

A tradesman marks his goods 30% above the cost price. If he allows a discount of 10% Then his gain percent is?

ক
ক) 17%
খ
খ) 16%
গ
গ) 15%
ঘ
ঘ) 14%

ব্যাখ্যা

Question: A tradesman marks his goods 30% above the cost price. If he allows a discount of 10% Then his gain percent is?

Solution:
At 30% above,
Market price of goods = 100 + 30 = Tk. 130

At 10% discount,
Selling price = 130 - 10% of 130
= 130 - 13
= Tk. 117

∴ Gain = 117 - 100 = Tk. 17
Gain % = (17/100) × 100
= 17%

৫,১২৮.

How many distinct arrangements can be made using all the letters of the word "BANANA" such that no two N’s appear together?

ক
60
খ
40
গ
35
ঘ
30
ঙ
None

ব্যাখ্যা

Question: How many distinct arrangements can be made using all the letters of the word "BANANA" such that no two N’s appear together?

Solution:
BANANA has 6 letters, where A = 3 times, N = 2 times and B = 1 time

∴ Total arrangements = 6!/(3! × 2!) = 720/(6 × 2)) = 60

Arrangements where the two N's are together = 5!/3! = (5 × 4 × 3!)/3! = 20

∴ N’s not together = Total arrangements − N’s together
= 60 - 20
= 40

৫,১২৯.

Two pipes A and B can fill a tank in 20 hours and 30 hours respectively. If both the pipes are opened simultaneously, find after how much time should pipe B be closed so that the tank is full in 18 hours?

ক
1 hour
খ
2 hours
গ
3 hours
ঘ
4 hours
ঙ
None of these

ব্যাখ্যা

Question: Two pipes A and B can fill a tank in 20 hours and 30 hours respectively. If both the pipes are opened simultaneously, find after how much time should pipe B be closed so that the tank is full in 18 hours?

Solution:
Let the capacity of the tank be LCM (20, 30) = 60 units
Efficiency of pipe A = 60/20 = 3 units/hour
Efficiency of pipe B = 60/30 = 2 units/hour
Combined efficiency of pipes A and B = 5 units/hour

Let both A and B be opened for ‘n’ hours and then, B be closed and only A be opened for the remaining ’18 - n’ hours.
5n + 3 × (18 - n) = 60
⇒ 2n + 54 = 60
⇒ 2n = 6
∴ n = 3
Therefore, B should be closed after 3 hours.

৫,১৩০.

If x + 2y = 4 and x/y = 2, then determine the value of x and y.

ক
1,2
খ
2,3
গ
2,1
ঘ
3,4

ব্যাখ্যা

দেয়া আছে
x + 2y = 4..............(1)
এবং
x/y = 2
x = 2y...............(2)

x এর মান (1) নং সমীকরণে বসিয়ে পাই
x + 2y = 4
2y + 2y = 4
4y = 4
y = 1

y এর মান (2) নং সমীকরণে বসিয়ে পাই
x = 2y
x = 2 × 1 = 2

নির্ণেয় সমধান (x, y) = (2,1)

৫,১৩১.

Father is 3 times more aged than his daughter. If after 5 years, he would be 3 times of daughter’s age, then further after 5 years, how many times would he be of his daughter’s age?

ক
ক) 1 (1/2) times
খ
খ) 2 times
গ
গ) 2.5 times
ঘ
ঘ) 3 times

ব্যাখ্যা

Let daughter’s age be x and father’s age be 3x.
Father's age is 3 times more aged than his daughter,
therefore father’s present age = x + 3x = 4x
After 5 years,
The father’s age is 3 times more than his daughter's age.
(4x + 5) = 3 (x + 5)
(4x+5) = 3 (x+5)
(4x + 5) = 3 (x + 5)
x = 10

After 5 years it was (4x + 5),
then after further 5 years, father’s age = (4x +10) and daughter’s age = (x + 10)
(4x + 10)/(x + 10)
= [{(4 × 10) + 10}/(10 + 10)]
= 50/20
= 5/2
= 2.5
After further 5 years, the father will be 2.5 times of daughter’s age.

৫,১৩২.

If a and b are non-negative real numbers such that a + 2b = 6, then the average of the maximum and minimum possible values of (a + b) is

ক
3
খ
3.5
গ
4
ঘ
4.5

ব্যাখ্যা

Question: If a and b are non-negative real numbers such that a + 2b = 6, then the average of the maximum and minimum possible values of (a + b) is

Solution:
a + 2b = 6; the maximum value of b can be 3

⇒ a + b = 6 - b

maximum value of a + b = 6 - 0 = 6
minimum value of a + b = 6 - 3 = 3

average = (6 + 3)/2 = 9/2 = 4.5

৫,১৩৩.

Which of the following fractions is the largest ?

ক
7/4
খ
3/2
গ
5/3
ঘ
6/5

ব্যাখ্যা

Question: Which of the following fractions is the largest?

Solution:
3/2 = 1.5
7/4 = 1.75
5/3 = 1.66
6/5 = 1.2
∴ the largest fraction is = 7/4

৫,১৩৪.

A forester wants to plant 88 coconut trees, 110 date trees and 132 plum trees in equal sized rows (in terms of number of tress). Also he wants to make distinct rows of trees (i.e only one type of trees in one row). What is the minimum number of rows required?

ক
ক) 12
খ
খ) 13
গ
গ) 15
ঘ
ঘ) 18

ব্যাখ্যা

Question: A forester wants to plant 88 coconut trees, 110 date trees and 132 plum trees in equal sized rows (in terms of number of tress). Also he wants to make distinct rows of trees (i.e only one type of trees in one row). What is the minimum number of rows required?

Solution:
এখানে
88, 110 এবং 132 এর গ.সা.গু = 22

নারিকেল গাছের সারি লাগবে = 88/22 = 4টি
খেজুর গাছের সারি লাগবে = 110/22 = 5টি
পাম গাছের সারি লাগবে = 132/22 = 6টি

মোট সারি লাগবে = 4 + 5 + 6 = 15 টি

৫,১৩৫.

If a man were to sell his chair for 930 taka, he would lose 25%. To gain 25%, what amount should he sell it?

ক
1400
খ
1600
গ
1280
ঘ
930
ঙ
None

ব্যাখ্যা

Let the Cost price of the Chair be y taka.
Selling price = y - 25% of y
= y - 25y/100
= 75y/100
= 3y/4
Therefore 3y/4 = 930
or, y = 1240
So, To gain 25%, Selling Price would be
= 1240 + 25% of 1280
= 1240 + 310
= 1550
= 1600 taka

৫,১৩৬.

If a² + b² = 5ab then find the value of (a²/b² + b²/a²)

ক
ক) 32
খ
খ) 16
গ
গ) 23
ঘ
ঘ) -23

ব্যাখ্যা

a² + b² = 5ab
a²/ab + b²/ab = 5
a/b + b/a = 5
Squaring the both sides
(a/b)² + (b/a)² = (5)²
a²/b² + b²/a² + 2 × (a/b) × (b/a)= 25
a²/b² + b²/a² + 2 = 25
a²/b² + b²/a² = 25 -2
a²/b² + b²/a² = 23.

৫,১৩৭.

Three years ago, the average of A, B, and C was 27 years and that of A and C, 5 years ago was 20 years. B’s present age is-

ক
ক) 20 years
খ
খ) 30 years
গ
গ) 40 years
ঘ
ঘ) 50 years

ব্যাখ্যা

Question: Three years ago, the average of A, B, and C was 27 years and that of A and C, 5 years ago was 20 years. B’s present age is-

Solution:
Three years ago, the average of A, B, and C was 27 years old
total age 3 years ago = (27 × 3) = 81 years
So, at present their total age is = 81 + 3 + 3 + 3 = 90 years

average of A and C, 5 years ago was 20 years
total age 5 years ago = (20 × 2) = 40 years
At present, total age of A and C is = 40 + 5 + 5 = 50 years

∴ B's present age is = 90 - 50 years = 40 years

৫,১৩৮.

The difference between compound interest and simple interest on a sum of Taka 10,000 for 2 years at 10% per annum compounded annually is -

ক
100 Taka
খ
120 Taka
গ
150 Taka
ঘ
200 Taka

ব্যাখ্যা

Question: The difference between compound interest and simple interest on a sum of Taka 10,000 for 2 years at 10% per annum compounded annually is -

Solution:
Simple interest = 10000 × 2 × (10/100)
= 2000 Taka

Compound Interest,

So, difference = 2100 - 2000 = 100 Taka

৫,১৩৯.

A boat travels 24 km downstream in 40 minutes. If the speed of the stream is 6 km/h, what is the speed of the boat in still water?

ক
18 km/h
খ
24 km/h
গ
30 km/h
ঘ
36 km/h

ব্যাখ্যা

Question: A boat travels 24 km downstream in 40 minutes. If the speed of the stream is 6 km/h, what is the speed of the boat in still water?

Solution:
স্রোতের অনুকূলে 40 মিনিটে যায় 24 কিমি
স্রোতের অনুকূলে 1 মিনিটে যায় 24/40 কিমি
স্রোতের অনুকূলে 1 ঘণ্টা বা 60 মিনিটে যায় (24 × 60)/40 কিমি
= 36 কিমি

∴ স্রোতের অনুকূলে বেগ = 36 কিমি/ঘণ্টা

দেওয়া আছে,
স্রোতের বেগ = 6 কিমি/ঘণ্টা।

∴ স্থির পানিতে নৌকার বেগ = স্রোতের অনুকূলে বেগ - স্রোতের বেগ
= 36 - 6 = 30 কিমি/ঘণ্টা।

৫,১৪০.

If log₄[log₃(log₂x)] = 0, then find the value of x.

ক
2
খ
3
গ
5
ঘ
8
ঙ
125

ব্যাখ্যা

Question: If log₄[log₃(log₂x)] = 0, then find the value of x.

Solution:
log₄[log₃(log₂x)] = 0
⇒ log₃(log₂x) = 4⁰ [log_bM = c ⇒ M = b^c]
⇒ log₃(log₂x) = 1
⇒ log₂x = 3¹ [log_bM = c ⇒ M = b^c]
⇒ log₂x = 3
⇒ x = 2³ [log_bM = c ⇒ M = b^c]
∴ x = 8

৫,১৪১.

When the positive integer n is divided by 5, the remainder is 2. Which of the following must be true?

ক
n is odd
খ
n is even
গ
n - 1 is divisible by 3
ঘ
None

ব্যাখ্যা

Question: When the positive integer n is divided by 5, the remainder is 2. Which of the following must be true?

Solution:
Given that when the positive integer n is divided by 5, the remainder is 2, we can express n as:
n = 5k + 2 where k is an integer.

Option ক)
n is odd
Since n = 5k + 2 the value of n depends on k. If k is even, n is even; if k is odd, n is odd.
Therefore, n is not necessarily odd.

Option খ)
n is even
Similarly, as discussed above, n = 5k + 2 could be odd or even depending on the value of k. If k is even, n is even, but if k is odd, n is odd.
Therefore, n is not necessarily even.

Option গ)
n - 1 is divisible by 3
Let's calculate n - 1:
n - 1 = 5k + 2 - 1 = 5k + 1
We need to check if 5k + 1 is divisible by 3.
We know that any integer 5k leaves a remainder of 2 when divided by 3
∴ For n - 1 to be divisible by 3, 5k + 1 mod 3 must equal 0. This can occur depending on k, but it's not necessarily true for all k.

None of the provided options must be true in all cases. Therefore, none of the options is necessarily true based on the information given.

৫,১৪২.

The market value of a certain machine decreased by 30 percent of its purchase price each year. If the machine was purchased in 1982 for its market value of Tk. 8000, what was its market value two years later?

ক
Tk. 8000
খ
Tk. 3200
গ
Tk. 5600
ঘ
Tk. 2400

ব্যাখ্যা

Question: The market value of a certain machine decreased by 30 percent of its purchase price each year. If the machine was purchased in 1982 for its market value of Tk. 8000, what was its market value two years later?

Solution:

Total price reduce after 2 years = 60% of purchase price = 60% of 8000

= 4800

Market value after 2 years = 8000 - 4800

= 3200

৫,১৪৩.

What is the value of (log√64) / (3log√4)?

ক
ক) 4
খ
খ) √2
গ
গ) 1
ঘ
ঘ) 0

ব্যাখ্যা

Question: What is the value of (log√64) / (3log√4)?

Solution:
(log√64) / (3log√4)
= log8 / 3log2
= log8 / log2³
= log8 / log8
= 1

৫,১৪৪.

A tap can fill a tank in 48 minutes where as another tap can empty it in 2 hours. If both the taps are opened at 11 : 40 A. M, then the tank will be filled at-

ক
1 : 20 P.M
খ
12 : 40 P.M
গ
1 : 00 P.M
ঘ
1 : 30 P.M

ব্যাখ্যা

Question: A tap can fill a tank in 48 minutes where as another tap can empty it in 2 hours. If both the taps are opened at 11 : 40 A. M, then the tank will be filled at-

Solution:
1 মিনিটে পূর্ণ হয় = 1/48 অংশ

আবার,
1 মিনিটে খালি হয় = 1/(2 × 60) = 1/120 অংশ

∴ 1 মিনিটে পূর্ণ হয় = {(1/48) - (1/120)} অংশ
= (5 - 2)/240
= 3/240
= 1/80 অংশ

∴ 1/80 অংশ পূর্ণ হয় = 1 মিনিটে
1 বা সম্পূর্ণ অংশ পূর্ণ হয় = 1/(1/80) = 80 মিনিটে

সুতরাং ট্যাংকটি পূর্ণ হবে = 11 : 40 A. M + 80 মিনিটে = 1 : 00 P.M

৫,১৪৫.

What is the next term in the sequence - 4, 9, 6, 11, 8, 13, …… ?

ক
ক) 18
খ
খ) 10
গ
গ) 16
ঘ
ঘ) 9

ব্যাখ্যা

এখানে দুটি ধারা বিদ্যমান।
বিজোড় অবস্থানের ধারা = 4, 6, 8, 10
জোড় অবস্থানের ধারা = 9, 11, 13, 15

৫,১৪৬.

A team of 6 members is to be selected from 9 people, with Mr. Z as the team leader (so he is always included). In how many ways can the team be formed?

ক
56 ways
খ
72 ways
গ
126 ways
ঘ
84 ways

ব্যাখ্যা

Question: A team of 6 members is to be selected from 9 people, with Mr. Z as the team leader (so he is always included). In how many ways can the team be formed?

solution:
Total people = 9
Mr. Z must be included, so we are selecting the remaining 5 members from the remaining 8 people.

So, the number of ways to choose the remaining 5 members from 8,
⁸C₅ = 8!/5!(8 - 5)!
= (8 × 7 × 6)/(3 × 2 × 1)
= 56 ways

৫,১৪৭.

A working partner receives a commission equal to 30% of the profit remaining after paying his commission. If his commission is Tk 12,000, find the total profit.

ক
50,000 TK
খ
62,000 TK
গ
52,000 TK
ঘ
60,000 TK

ব্যাখ্যা

Question: A working partner receives a commission equal to 30% of the profit remaining after paying his commission. If his commission is Tk 12,000, find the total profit.

Solution:
Let,
the total profit be K taka
According to the question,
remaining profit after paying 30% working partners commission = (K - 12000)

∴ (K - 12000) × (30/100) = 12000
⇒ (K - 12000) = 12000 × (100/30)
⇒ (K - 12000) = 40,000
⇒ K = 40,000 + 12,000
∴ K = 52,000

so, Total profit = 52,000 TK

৫,১৪৮.

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

ক
200
খ
220
গ
250
ঘ
270

ব্যাখ্যা

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

Solution:
let, larger number be x
smaller number be y

x - y = 1365
x = y + 1365

x = 6y + 15
⇒ y + 1365 = 6y + 15
⇒ 6y - y = 1365 - 15 = 1350
⇒ y = 1350/5 = 270

৫,১৪৯.

A train passes a station platform in 36 seconds and a man standing on the platform in 20 seconds. If the speed of the train is 54 km/hr, what is the length of the platform?

ক
200m
খ
220m
গ
236m
ঘ
240m

ব্যাখ্যা

Question: A train passes a station platform in 36 seconds and a man standing on the platform in 20 seconds. If the speed of the train is 54 km/hr, what is the length of the platform?

Solution:
Speed = {54 × (5/18)} m/sec
=15 m/sec
∴ Length of the train =(15 × 20)m
=300m

Let,
the length of the platform be = a metres
Then,
(a + 300)/36 = 15
⇒ a + 300 = 540
⇒ a = 240m

৫,১৫০.

Hasibul bought an AC for Tk.12160 and paid Tk.340 for transportation. Then he sold it for Tk.12875. Find his profit %.

ক
ক) 3%
খ
খ) 4%
গ
গ) 5%
ঘ
ঘ) 6%

ব্যাখ্যা

প্রশ্ন: Hasibul bought an AC for Tk.12160 and paid Tk.340 for transportation. Then he sold it for Tk.12875. Find his profit %.

সমাধান:
হাসিবুলের AC বাবদ মোট খরচ ১২১৬০ + ৩৪০ টাকা
= ১২৫০০ টাকা

লাভ হয় = (১২৮৭৫ - ১২৫০০) টাকা
= ৩৭৫ টাকা

১২৫০০ টাকায় লাভ হয় ৩৭৫ টাকা
∴ ১০০ টাকায় লাভ হয় (৩৭৫ × ১০০)/১২৫০০ টাকা
= ৩%

৫,১৫১.

The average pocket money of 3 friends Sharifullah, Nawshad, Tuhin is Tk. 80 in a particular month. If Nawshad spends double and Tuhin spends triple of what Sharifullah spends during that month and if the average of their unspent pocket money is Tk. 60, then Tuhin spends -

ক
Tk. 10
খ
Tk. 12
গ
Tk. 30
ঘ
Tk. 36

ব্যাখ্যা

Question: The average pocket money of 3 friends Sharifullah, Nawshad, Tuhin is Tk. 80 in a particular month. If Nawshad spends double and Tuhin spends triple of what Sharifullah spends during that month and if the average of their unspent pocket money is Tk. 60, then Tuhin spends -

Solution:
Total pocket money = 3 × 80
= 240 tk

total unspent money = 3 × 60
= 180 tk

Total spent money = 240 - 180 tk
= 60 tk

let, sharifullah spends x tk
nawshad spends 2x tk
tuhin spends 3x tk

x + 2x + 3x = 60
⇒ 6x = 60
∴ x = 10 taka

Tuhin spends = 3 × 10 taka
= 30 taka

৫,১৫২.

If 'x' is an odd number and 'y' is an even number, which one of the following must be an even number?

ক
ক) xy + 1
খ
খ) x² + y²
গ
গ) (x + y)² + 1
ঘ
ঘ) x + y

ব্যাখ্যা

Question: If 'x' is an odd number and 'y' is an even number, which one of the following must be an even number?

Solution:
Suppose, x = 1 and y = 2

• xy + 1 = 1 × 2 + 1 = 3
• x² + y²= 1² + 2² = 5
• (x + y)² + 1 = (1 + 2)² + 1 = 10
• x + y = 1 + 2 = 3

৫,১৫৩.

Two trains 180 m and 120 m long respectively pass each other in 54 seconds when they run in the same direction and in 18 seconds when run in opposite directions. Find the speed of first train.

ক
40 km/hr
খ
45 km/hr
গ
35 km/hr
ঘ
20 km/hr

ব্যাখ্যা

Question: Two trains 180 m and 120 m long respectively pass each other in 54 seconds when they run in the same direction and in 18 seconds when run in opposite directions. Find the speed of first train.

Solution:
Let the speed of 1st train is S1 and speed of 2nd train is S2
Time = total distance/ relative speed

1) In same direction
54 = (180 + 120)/(S1 - S2) × (5/18)
⇒ (S1 - S2)54 = (300 × 18)/5
⇒ (S1 - S2) = 20

2) In opposite direction
18 = (180 + 120) / (S1 + S2) × (5/18)
⇒ (S1 + S2)18 = (300 × 18)/5
⇒ (S1 + S2) = 60

from 1 and 2
S1 = 40 km/hr

৫,১৫৪.

By selling an umbrella for Tk. 300, a shopkeeper gains 20%. During a clearance sale, the shopkeeper allows a discount of 10% on the marked price. His gain percent during the sale is-

ক
ক) 4%
খ
খ) 8%
গ
গ) 6%
ঘ
ঘ) 10%

ব্যাখ্যা

Marked price = Tk. 300
C.P = (100/120) × 300 = Tk. 250

Sale price = 90% of Tk. 300
= 300 × (90/100)
= Tk. 270

Required gain% = {(20/250) × 100}% = 8%

৫,১৫৫.

An agent sells goods worth Tk. 22000. If his commission rate was 15.5%, what was the amount of his commission?

ক
3410
খ
2820
গ
3550
ঘ
3230

ব্যাখ্যা

Question: An agent sells goods worth Tk. 22000. If his commission rate was 15.5%, what was the amount of his commission?

Solution:
Commission = 15.5% of 22000
= (155/10) × (1/100) × 22000
= (155/1000) × 22000
= 3410

৫,১৫৬.

The ages of Saha and Akash are in the ratio 9 : 8 respectively. After 5 years, the ratio of their ages will be 10 : 9. What is the difference in their ages?

ক
ক) 4 years
খ
খ) 5 years
গ
গ) 6 years
ঘ
ঘ) 7 years

ব্যাখ্যা

Question: The ages of Saha and Akash are in the ratio 9 : 8 respectively. After 5 years, the ratio of their ages will be 10 : 9. What is the difference in their ages?

Solution:
The ages of Saha and Akash are 9x and 8x
After 5 years, the ratio of their ages will be (9x + 5) and (8x + 5)

(9x + 5) /(8x + 5) = 10/9
⇒ 81x + 45 = 80x + 50
⇒ x = 5

∴ the difference in their ages is = 9x - 8x
= x
= 5 years

৫,১৫৭.

Which of the following is the polynomial equation 7x⁴ + 3x² - 2x + 1 = 0?

ক
Cubic equation
খ
Linear equation
গ
Quadratic equation
ঘ
Biquadratic equation

ব্যাখ্যা

Question: Which of the following is the polynomial equation 7x⁴ + 3x² - 2x + 1 = 0?

Solution:
প্রদত্ত বহুপদী সমীকরণটি x এর সাপেক্ষে।
x এর সর্বোচ্চ ঘাত হলো 4 এবং তাই সমীকরণটির ঘাত 4
∴ এই সমীকরণটির মাত্রা (degree) হলো 4
সুতরাং, এটি একটি চতুর্ঘাত বা দ্বি-দ্বিঘাত সমীকরণ (Biquadratic equation)।

• বহুপদীর সর্বোচ্চ ঘাত 1 হলে তা রৈখিক (linear) সমীকরণ।
• বহুপদীর সর্বোচ্চ ঘাত 2 হলে তা দ্বিঘাত (quadratic) সমীকরণ।
• বহুপদীর সর্বোচ্চ ঘাত 3 হলে তা ত্রিঘাত (cubic) সমীকরণ।
• বহুপদীর সর্বোচ্চ ঘাত 4 হলে তা চতুর্ঘাত বা দ্বি-দ্বিঘাত (biquadratic) সমীকরণ।

৫,১৫৮.

The bankers discount and true discount on a sum of money due 8 months hence are Tk.120 & Tk.110 resp. Find the sum.

ক
ক) 1457
খ
খ) 1320
গ
গ) 1140
ঘ
ঘ) 1260

ব্যাখ্যা

Sum = (B.D.×T.D.)/(B.D.−T.D.)
= (120×110) / (120−110)
= 1320

৫,১৫৯.

Three pipes A, B and C can fill a tank in 6 hours. After working at it together for 2 hours, C is closed and A and B can fill the remaining part in 6 hours. The number of hours taken by C alone to fill the tank is

ক
18 hours
খ
14 hours
গ
10 hours
ঘ
9 hours

ব্যাখ্যা

Question: Three pipes A, B and C can fill a tank in 6 hours. After working at it together for 2 hours, C is closed and A and B can fill the remaining part in 6 hours. The number of hours taken by C alone to fill the tank is

Solution:
A, B and C together can fill in one hour = 1/6
in two hours = 2/6 = 1/3

the remaining part is = 1 - 1/3 = 2/3

in 6 hours, A and B can do = 2/3
so, in one hour A and B can do = 2/(3 × 6) = 2/18

so, C can do in one hour = (A, B and C in one hour) - (A and B in one hour)
= 1/6 - 2/18
= 1/18

hence, C take 18 hours to fill the tank

৫,১৬০.

A cricket team has won 40 games out of 60 played. It has 30 more games to play. How many of these must the team win to make a record 70% win for them?

ক
ক) 20
খ
খ) 25
গ
গ) 23
ঘ
ঘ) 32

ব্যাখ্যা

Team won 40 games out of 60 and the remaining games were 30.
Total games = 60 + 30
= 90
70% of 90 = 63
Team has to win 63 games in total.
Team has already won 40.
∴ Games to win = 63 - 40
= 23

৫,১৬১.

A train reaches from A to B in 5 hours travelling at a speed of 63 km/h. If its speed is reduced by 18 km/h, then the time of journey is increased by-

ক
1 hours
খ
1.5 hours
গ
3 hours
ঘ
2 hours

ব্যাখ্যা

Question: A train reaches from A to B in 5 hours travelling at a speed of 63 km/h. If its speed is reduced by 18 km/h, then the time of journey is increased by-

Solution:
Here,
Total distance = speed × time
= 63 × 5 = 315 km

If speed reduced then new speed = (63 - 18) = 45 km/hr

New time = Total distance/speed
= 315/45
= 7 hours

Time increased by (7 - 5) = 2 hours

∴ The time of journey is increased by 2 hours.

৫,১৬২.

If tanθ = a/b, find the value of asinθ + bcosθ.

ক
1/(a² + b²)
খ
(a² + b²)
গ
1/√(a² + b²)
ঘ
√(a² + b²)

ব্যাখ্যা

Question: If tanθ = a/b, find the value of asinθ + bcosθ.

Solution:
tanθ = লম্ব/ভূমি = a/b

অতিভুজ = √(a² + b²)

asinθ + bcosθ
= a{a/√(a² + b²)} + b{b/√(a² + b²)}
= a²/√(a² + b²) + b²/√(a² + b²)
= (a² + b²)/√(a² + b²)
= √(a² + b²)

৫,১৬৩.

In a right triangle, the length of one of the legs is 12 and the length of the hypotenuse is 13. What is the length of the other leg?

ক
17
খ
9
গ
8
ঘ
5

ব্যাখ্যা

Question: In a right triangle, the length of one of the legs is 12 and the length of the hypotenuse is 13. What is the length of the other leg?

Solution:
এখানে, সমকোণী ত্রিভুজের (right triangle) অতিভুজ (hypotenuse) = 13 একক সমকোণ সংলগ্ন এক বাহু = 12 একক
সমকোণ সংলগ্ন অপর বাহু = a একক

প্রশ্নমতে,
a² + 12² = 13²
⇒ a² + 144 = 169
⇒ a² = 169 - 144
⇒ a² = 25
⇒ a = √25
⇒ a = 5

∴ সমকোণ সংলগ্ন অপর বাহুর দৈর্ঘ্য = 5 একক

৫,১৬৪.

Speed of motorboat in still water is 35 kmph. If the motorboat travels 100 km along the stream in 2 hour 30 min, then the time taken by it to cover the same distance against the stream is-

ক
2 hour 20 min
খ
4 hour 20 min
গ
3 hour 20 min
ঘ
5 hour 20 min

ব্যাখ্যা

Question: Speed of motorboat in still water is 35 kmph. If the motorboat travels 100 km along the stream in 2 hour 30 min, then the time taken by it to cover the same distance against the stream is-

Solution:
The speed of the motorboat in still water is 35 km/hr.
let the speed of the stream = x km/hr
Downstream speed = Distance/time
= 100/2.5
= 40 km/hr

Speed of stream = 35 + x = 40
∴ x = 5 km/hr

Upstream speed = 35 - 5 = 30 km/hr
Time taken in upstream = 100/30 = 3 hour 20 min

৫,১৬৫.

Present ages of Abir and Rubel are in the ratio of 5 : 4 respectively. Three years hence, the ratio of their ages will become 11 : 9 respectively. What is Rubel's present age in years?

ক
22 years
খ
24 years
গ
26 years
ঘ
28 years

ব্যাখ্যা

Question: Present ages of Abir and Rubel are in the ratio of 5 : 4 respectively. Three years hence, the ratio of their ages will become 11 : 9 respectively. What is Rubel's present age in years?

Solution:
Let,
the present ages of Abir and Rubel be 5a years and 4a years respectively

Then,
(5a + 3)/(4a +3) = 11/9
⇒ 9(5a + 3) = 11(4a + 3)
⇒ 45a + 27 = 44a + 33
⇒ 45a - 44a = 33 - 27
⇒ a = 6

∴ Rubal's present age = 4a = 4 × 6 = 24 years

৫,১৬৬.

A train overtakes two persons who are walking in the same direction to that of the train at 2 kmph and 4 kmph and passes them completely in 9 and 10 seconds respectively. What is the length of the train?

ক
ক) 45
খ
খ) 50
গ
গ) 55
ঘ
ঘ) 65
ঙ
ঙ) 60

ব্যাখ্যা

Let length and speed of the train be x metre and v kmph respectively.
x/9 = (v − 2) × 5/18 ⋯ ( 1 )
x/10 = (v − 4) × 5/18 ⋯ ( 2 )
Dividing (1) by (2) gives,
10/9 = (v − 2)/(v − 4)
⇒ 10v − 40 = 9v − 18
⇒ v = 22
Substituting the value of v in (1)
x/9 = 100/18
⇒ x = 50

৫,১৬৭.

Find the smallest number of 6 digits which is exactly divisible by 349.

ক
101063
খ
100163
গ
160063
ঘ
None of the above

ব্যাখ্যা

Question: Find the smallest number of 6 digits which is exactly divisible by 349.

Solution:
The smallest 6 digit number is 100000.
When divided by 349, the remainder is 186.
So, Required Number = 100000 + (349 - 186) = 100163

৫,১৬৮.

(12)³ × 6⁴ ÷ 432 = ?

ক
5184
খ
5060
গ
5148
ঘ
5084

ব্যাখ্যা

Question: (12)³ × 6⁴ ÷ 432 =?

Solution:
(12)³ × 6⁴ ÷ 432
= {(12)³ × 6⁴}/432
= {(12)³ × 6⁴}/(12 × 6²)
= (12)² × 6²
= (72)²
= 5184

৫,১৬৯.

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

ক
ক) 630
খ
খ) 1260
গ
গ) 1920
ঘ
ঘ) 320

ব্যাখ্যা

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

Solution:
12 = 2 × 2 × 3
18 = 2 × 3 × 3
21 = 3 × 7
30 = 2 × 3 × 5

L.C.M. of 12, 18, 21 30 = 2 × 3 × 2 × 3 × 7 × 5 = 1260
Required number = 1260/2 = 630

৫,১৭০.

What is the rate of discount is a car that had a regular price of Tk. 30,00,000 is sold for Tk.27,90,000?

ক
ক) 7%
খ
খ) 8%
গ
গ) 9%
ঘ
ঘ) 10%

ব্যাখ্যা

লিখিতমূল্য = 30,00,000 টাকা
বিক্রয় মূল্য = 27,90,000 টাকা

ছাড় = (30,00,000 - 27,90,000) টাকা
= 2,10,000 টাকা
ছাড়ের হার = {(2,10,000/30,00,000) × 100}%
=7%

৫,১৭১.

A train 165 m long is running at a uniform speed of 54 km/hr. How much time will it take to cross a pole?

ক
ক) 10 seconds
খ
খ) 11 seconds
গ
গ) 12 seconds
ঘ
ঘ) 13 seconds

ব্যাখ্যা

Question: A train 165 m long is running at a uniform speed of 54 km/hr. How much time will it take to cross a pole?

Solution:
Speed of train = 54 km/hr
= 54 × 5/18 m/sec
= 15 m/sec

Length of the train = 165 meters

Therefore, time taken by the train to cross a pole = length of train/speed of train
= 165/15 sec
= 11 sec

Thus, train takes 11 seconds to cross the pole.

৫,১৭২.

A starts business with tk. 3500 and after 5 months, B joins with A as his partner. After a year, the profit is divided in the ratio 2:3. What is B's contribution in the capital?

ক
ক) 7500
খ
খ) 8000
গ
গ) 8500
ঘ
ঘ) 9000

ব্যাখ্যা

Let B's capital be tk. x.
Then, (3500 x 12)/7x = 2/3
or, 14x = 126000
so, x = 9000 tk

৫,১৭৩.

The lengths of the sides of a triangle are in the ratio 2 : 4 : 5. If the perimeter of the triangle is 66 cm, what is the length of the longest side?

ক
12 cm
খ
24 cm
গ
30 cm
ঘ
36 cm

ব্যাখ্যা

Question: The lengths of the sides of a triangle are in the ratio 2 : 4 : 5. If the perimeter of the triangle is 66 cm, what is the length of the longest side?

Solution:
মনে করি,
ত্রিভুজের তিন বাহুর দৈর্ঘ্য যথাক্রমে 2x, 4x এবং 5x
∴ ত্রিভুজের পরিসীমা = (2x + 4x + 5x)
= 11x

প্রশ্নমতে,
⇒ 11x = 66
⇒ x = 66/11
⇒ x = 6

∴ দীর্ঘতম বাহুর দৈর্ঘ্য = (6 × 5) cm
= 30 cm

৫,১৭৪.

If the average of four consecutive odd integers is x, then in terms of x, which is the smallest among the following options?

ক
x - 3
খ
0.25x - 2
গ
x - 4
ঘ
x - 2.5
ঙ
None

ব্যাখ্যা

Question: If the average of four consecutive odd integers is x, then in terms of x, which is the smallest among the following options?

Solution:
Let the four consecutive odd integers be represented by n, n + 2, n + 4, n + 6.
Their average is calculated by summing them and dividing by 4, which equals x.

Their average is,
x = {(n + (n + 2) + (n + 4) + (n + 6)}/4
⇒ x = (4n + 12)/4 = 4(n + 3)/4
⇒ x = n + 3
∴ n = x - 3

So the smallest integer in terms of (x - 3).

৫,১৭৫.

Amir is 40 years old and Rezwan is 50 years old. How many years ago was the ratio of their ages 3 : 5?

ক
ক) 10 years
খ
খ) 18 years
গ
গ) 25 years
ঘ
ঘ) 30 years

ব্যাখ্যা

Question: Amir is 40 years old and Rezwan is 50 years old. How many years ago was the ratio of their ages 3 : 5?

Solution:
Let x years ago the ratio of their ages were 3 : 5

ATQ,
(40 - x)/(50 - x) = 3/5
⇒ 200 - 5x = 150 - 3x
⇒ 5x - 3x = 200 - 150
⇒ 2x = 50
∴ x = 25

৫,১৭৬.

If the 5^th number in a series of a 5 consecutive integers has the value n + 3, find the 1^st number in the series expressed in terms of n?

ক
0
খ
n - 3
গ
- 3n
ঘ
n - 1
ঙ
1

ব্যাখ্যা

Question: If the 5^th number in a series of a 5 consecutive integers has the value n + 3, find the 1^st number in the series expressed in terms of n?

Solution:
Given that,
A series of 5 consecutive integers.
Let the first number be x
Then the series is,
x, (x + 1), (x + 2), (x + 3), (x + 4)
And
The 5^th number in the series is given as n + 3
So,
⇒ x + 4 = n + 3
⇒ x = n + 3 - 4
∴ x = n - 1

So the first number in terms of nnn is n - 1.

৫,১৭৭.

যদি log₁₀125 + log₁₀8 = x হয়, তবে x এর মান কত?

ক
ক) 2
খ
খ) 1
গ
গ) 0
ঘ
ঘ) 3

ব্যাখ্যা

প্রশ্ন: যদি log₁₀125 + log₁₀8 = x হয়, তবে x এর মান কত?

সমাধান:
log₁₀125 + log₁₀8 = x
⇒ log₁₀(125 × 8) = x
⇒ log₁₀(1000) = x
⇒ x = log₁₀(1000)
⇒ x = log₁₀(10)³
⇒ x = 3log₁₀10
⇒ x = 3

৫,১৭৮.

একটি পণ্য ২০% লাভে বিক্রি করা হয়েছে। যদি ক্রয়মূল্য ১০% বৃদ্ধি করা হয় এবং বিক্রয়মূল্য ২৬ টাকা বাড়ানো হয়, তাবে লাভের হার ৫% কমে যায়। পণ্যটির ক্রয়মূল্য কত?

ক
৪০০ টাকা
খ
৪৬৬ টাকা
গ
৪৮০ টাকা
ঘ
৫০০ টাকা
ঙ
কোনোটিই নয়

ব্যাখ্যা

প্রশ্ন: একটি পণ্য ২০% লাভে বিক্রি করা হয়েছে। যদি ক্রয়মূল্য ১০% বৃদ্ধি করা হয় এবং বিক্রয়মূল্য ২৬ টাকা বাড়ানো হয়, তাবে লাভের হার ৫% কমে যায়। পণ্যটির ক্রয়মূল্য কত?

সমাধান:
মনে করি,
ক্রয়মূল্য = ক টাকা

২০% লাভে বিক্রয়মূল্য = (ক × ১২০)/১০০ = (১২০ক/১০০) = ৬ক/৫ টাকা

দেওয়া আছে,
ক্রয়মূল্য ১০% বৃদ্ধি করা হয় এবং বিক্রয়মূল্য ২৬ টাকা বাড়ানো হয়
∴ নতুন ক্রয়মূল্য = (ক × ১১০)/১০০ = (১১০ক/১০০) = ১১ক/১০ টাকা
নতুন বিক্রয়মূল্য = {(৬ক/৫) + ২৬} টাকা
= (৬ক + ১৩০)/৫ টাকা

∴ লাভ = [{(৬ক + ১৩০)/৫} - (১১ক/১০)] টাকা
= (১২ক + ২৬০ - ১১ক)/১০ টাকা
= (ক + ২৬০)/১০ টাকা
= {(ক/১০) + ২৬} টাকা

প্রশ্নমতে,
{(ক/১০) + ২৬} × (১০/১১ক) × ১০০ = ১৫
⇒ ১০০(ক + ২৬০) = ১৬৫ক
⇒ ১০০ক + ২৬০০ = ১৬৫ক
⇒ ৬৫ক = ২৬০০
⇒ ক = ৪০০

সুতরাং, ক্রয়মূল্য ৪০০ টাকা

৫,১৭৯.

If 0.06 is x% of 0.6, then the value of x is

ক
ক) 6
খ
খ) 60
গ
গ) 10
ঘ
ঘ) 5

ব্যাখ্যা

x% of 0.6 = 0.06
⇒ (x/100) × 0.6 = 0.06
⇒ x = 0.06 × 100/0.60
⇒ x = 10

৫,১৮০.

Income tax is raised from 4% to 5% but the revenue is increased by 10% only. Find the decreased percentage in the amount taxed.

ক
ক) 8
খ
খ) 10
গ
গ) 12
ঘ
ঘ) 15

ব্যাখ্যা

ধরি,
প্রথম আয়কর ছিল x টাকা
নতুন আয়কর ছিল y টাকা

ধরি,
প্রথম রাজস্ব ছিলো = 100 টাকা

x এর 4% = 100
4x/100= 100
x/25 = 100
x = 2500

আবার ,
রাজস্ব 10% বৃদ্ধিতে নতুন রাজস্ব = 100 + 10 = 110 টাকা
y এর 5% = 110
5y/100 =110
y/20 = 110
y = 2200

আয়কর কমেছে = (2500 - 2200) টাকা = 300 টাকা

আয়কর শতকরা কমেছে = {(300/2500) × 100}% = 12%

৫,১৮১.

If A : B : C = 2 : 3 : 5 and A = x% of (B + C), then x equal to:

ক
ক) 35
খ
খ) 32
গ
গ) 30
ঘ
ঘ) 25

ব্যাখ্যা

Let A = 2k, B = 3k, C = 5k
A = x% of (B + C)
⇒ 2k = x% of (3k + 5k)
⇒2k = x% of 8k
⇒ 2k = 8kx/100
⇒ 1 = 4x/100
⇒ 4x = 100
x = 25

৫,১৮২.

In a certain store, the profit is 320% of the cost, If the cost increase by 25% but the selling price remains constant, approximately what percentage of the selling price is the profit?

ক
ক) 60%
খ
খ) 70%
গ
গ) 50%
ঘ
ঘ) 30%

ব্যাখ্যা

Question: In a certain store, the profit is 320% of the cost, If the cost increase by 25% but the selling price remains constant, approximately what percentage of the selling price is the profit?

Solution:

Let the cost price be 100x
so, Profit = 320% of 100x = 320x
then, selling price = 100x + 320x = 420x

Now, cost price increased by 25%
New, Cost price = 125x
New Profit = 420x - 125x = 295x

Profit percent on selling price = (295x/420x) × 100 = 70%

∴ The required percentage is 70%

৫,১৮৩.

If 4 times of a number is subtracted from its 13 times, the result is 171. What is the number?

ক
ক) 17
খ
খ) 19
গ
গ) 29
ঘ
ঘ) 21

ব্যাখ্যা

Question: If 4 times of a number is subtracted from its 13 times, the result is 171. What is the number?

Solution:
Let,
The number is x

ATQ,
13x - 4x = 171
Or, 9x = 171
Or, x = 171/9
∴ x = 19

৫,১৮৪.

A does a work in 12 days, B in 15 days, and C can do half the work in 10 days. How long will it take them to complete the whole work if they work together?

ক
3 days
খ
5 days
গ
8 days
ঘ
12 days

ব্যাখ্যা

Question: A does a work in 12 days, B in 15 days, and C can do half the work in 10 days. How long will it take them to complete the whole work if they work together?

Solution:
A,
12 দিনে করে কাজটির = 1 অংশ
∴ 1 দিনে করে কাজটির = 1/12 অংশ

B,
15 দিনে করে কাজটির = 1 অংশ
∴ 1 দিনে করে কাজটির = 1/15 অংশ

C,
10 দিনে করে কাজটির = 1/2 অংশ
∴ 1 দিনে করে কাজটির = 1/20 অংশ

A, B ও C একত্রে করে = (1/12) + (1/15) + (1/20)
= (5 + 4 + 3)/60
= 12/60
= 1/5 অংশ

A, B ও C একত্রে 1/5 অংশ করে = 1 দিনে
∴ 1 বা সম্পূর্ণ অংশ করে = (1 × 5) দিনে
= 5 দিনে

৫,১৮৫.

Sabit, Sabbir, and Salman started a business. Sabit 1/2 part, Sabbir 1/3 part, and the rest of the capital were invested by Salman. The ratio of their profits will be?

ক
ক) 1 : 2 : 4
খ
খ) 3 : 2 : 1
গ
গ) 2 : 3 : 4
ঘ
ঘ) 3 : 4 : 5

ব্যাখ্যা

Question: Sabit, Sabbir, and Salman started a business. Sabit 1/2 part, Sabbir 1/3 part, and the rest of the capital were invested by Salman. The ratio of their profits will be?

Solution:
Let the total capital be 6x
Then, Sabit's share = 6x × (1/2) = 3x
Sabbir's share = 6x × (1/3) = 2x
Salman's share = 6x - (3x + 2x) = x

So, required ratio = 3x : 2x : x = 3 : 2 : 1

৫,১৮৬.

The B.D. and T.D. on a certain sum is Tk. 200 and Tk. 100 respectively. Find out the sum.

ক
ক) 400
খ
খ) 300
গ
গ) 100
ঘ
ঘ) 200

ব্যাখ্যা

F = (BD×TD)/(BD−TD)
= (200×100)/(200−100)
= 200−100/ 100
= Tk. 200

৫,১৮৭.

In a school, as many students as there are, each gives 10 taka and thus Tk.25000 was collected. What is the number of students?

ক
45
খ
50
গ
55
ঘ
60

ব্যাখ্যা

Question: In a school, as many students as there are, each gives 10 taka and thus Tk.25000 was collected. What is the number of students?

Solution:
ধরি,
শিক্ষার্থী সংখ্যা = ক জন
প্রত্যেকে চাঁদা দেয় = ১০ক টাকা

প্রশ্নমতে,
ক × ১০ক = ২৫০০০
⇒ ১০ক^২ = ২৫০০০
⇒ ক^২ = ২৫০০০/১০
⇒ ক^২ = ২৫০০
⇒ ক = √২৫০০
∴ ক = ৫০

∴ উক্ত শ্রেণিতে শিক্ষার্থী সংখ্যা ৫০ জন।

৫,১৮৮.

The sum of the digits of two-digit number is 10, while when the digits are reversed, the number decrease by 54. Find the changed number.

ক
28
খ
46
গ
19
ঘ
37
ঙ
None

ব্যাখ্যা

Question: The sum of the digits of two-digit number is 10, while when the digits are reversed, the number decrease by 54. Find the changed number.

Solution:
Let number be (10x + y)

According to question,
(10x + y) - (10y + x) = 54
10x - 10y + y - x = 54
⇒ 9x - 9y = 54
⇒ x - y = 6 ................(i)

Sum of digits,
(x + y) = 10 ................. (ii)

(i) - (ii)
x - y - x - y = 6 - 10
⇒ - 2y = - 4
⇒ y = 2 and, x = 8

The required number is
= (10y + x)
= 10 × 2 + 8
= 28

৫,১৮৯.

The line perpendicular to the tangent line is called -

ক
ক) Normal line
খ
খ) Secant line
গ
গ) Limit
ঘ
ঘ) Derivative

ব্যাখ্যা

The line perpendicular to the tangent line to a curve at the point of tangency is called the normal line to the curve at that point.

৫,১৯০.

Working alone, Protik takes complete April to build a pavement. His friend Mahmud is 25% faster than him at the same work. Working alone, how many days will Mahmud take to build the same pavement?

ক
ক) 20 days
খ
খ) 24 days
গ
গ) 22.5 days
ঘ
ঘ) 37.5 days

ব্যাখ্যা

April has 30 days. So Protik takes 30 days to build the pavement.
Mahmud is 25% faster than Protik

25% = 25/100 = .25
This means, if Protik is 1, then Mahmud is (1 + 0.25) = 1.25
Protik takes 30 days to do the work.
Mahmud will take = 30/1.24 = 24 days to get the work done.

৫,১৯১.

Two coins are tossed. What is the probability of getting at most one head?

ক
1/2
খ
1/4
গ
3/4
ঘ
1/6

ব্যাখ্যা

Question: Two coins are tossed. What is the probability of getting at most one head?

Solution:
Total cases = {HH, HT, TH, TT} = 4
Favorable cases = {TT, HT, TH} = 3

∴ Required Probability = 3/4

৫,১৯২.

Arif bought a ticket of a cricket match for Tk. 25 and later sold the ticket to Rafi for Tk. 75. What was the percent increase in the price of the ticket?

ক
180%
খ
200%
গ
250%
ঘ
280%

ব্যাখ্যা

Question: Arif bought a ticket of a cricket match for Tk. 25 and later sold the ticket to Rafi for Tk. 75. What was the percent increase in the price of the ticket?

Solution:
ক্রয়মূল্য = 25 টাকা
বিক্রয়মূল্য = 75 টাকা
লাভ = 75 - 25 = 50 টাকা

25 টাকায় লাভ হয় = 50 টাকা
1 টাকায় লাভ হয় = 50 / 25 টাকা
100 টাকায় লাভ হয় = (50 × 100)/25 টাকা = 200 টাকা

শতকরা লাভ 200%

৫,১৯৩.

If the median of 21 observations is 40 and if the observations greater than the median are increased by 5 then the median of the new data will be-

ক
45 - (50/21)
খ
45
গ
40
ঘ
40 + (50/21)

ব্যাখ্যা

Question: If the median of 21 observations is 40 and if the observations greater than the median are increased by 5 then the median of the new data will be-

Solution:
• The median of a set of data is the middle value when the data is arranged in ascending order.

For 21 observations, the median is the 11th value.
According to the question, the original median is 40. Now, observations greater than the median are increased by 5. Since only the values greater than the median are increased, the position of the median (the 11th observation) remains unchanged, and the value of the median (40) itself does not change.

⇒ The median of the new data will still be 40.
∴ The median of the new data remains 40.

৫,১৯৪.

Two pipes can fill a tank in 15 and 20 minutes respectively and a waste pipe can empty 2 gallons per minute. All the three pipes working together can fill the tank in 9 minutes. The capacity of the tank is-

ক
360 gallons
খ
260 gallons
গ
200 gallons
ঘ
350 gallons

ব্যাখ্যা

Question: Two pipes can fill a tank in 15 and 20 minutes respectively and a waste pipe can empty 2 gallons per minute. All the three pipes working together can fill the tank in 9 minutes. The capacity of the tank is-

Solution:
Let, the waste pipe empty the tank in x minutes.

According to the question,
⇒ (1/15) + (1/20) - (1/x) = (1/9)
⇒ 1/x = (1/15) + (1/20) - (1/9)
⇒ 1/x = (12 + 9 - 20)/180
⇒ 1/x = 1/180
∴ x = 180

A waste pipe can empty 2 gallons per minute In 180 minutes it can empty = 2 × 180 = 360 gallons.
∴ Capacity of the tank = 360 gallons.

৫,১৯৫.

Five siblings, born three years apart, have a combined age of 100 years. What is the age of the youngest one?

ক
24 years
খ
10 years
গ
12 years
ঘ
14 years

ব্যাখ্যা

Question: Five siblings, born three years apart, have a combined age of 100 years. What is the age of the youngest one?

Solution:
Let the ages of siblings be x, (x + 3), (x + 6), (x + 9) and (x + 12) years.

Then,
x + (x + 3) + (x + 6) + (x + 9) + (x + 12) = 100
⇒ 5x = 100 - 30
⇒ 5x = 70
⇒ x = 14

∴ Age of the youngest sibling = x = 14 years.

৫,১৯৬.

A number is mistakenly divided by 2 instead of being multiplied by 2. Find the percentage change in the result.

ক
25%
খ
40%
গ
55%
ঘ
75%

ব্যাখ্যা

Question: A number is mistakenly divided by 2 instead of being multiplied by 2. Find the percentage change in the result.

Solution:
Let the number be N.
2N is the correct outcome.
0.5N is the mistakenly calculated outcome.

Change in the number= 2N - 0.5N = 1.5N

Percentage change= (1.5N/2N) × 100 = 75%

৫,১৯৭.

How many shares of market value Tk. 25 can be purchased for Tk. 15775?

ক
621
খ
631
গ
641
ঘ
651

ব্যাখ্যা

Question: How many shares of market value Tk. 25 can be purchased for Tk. 15775?

Solution:
Number of shares that can be purchased = 15775/25 = 631.

৫,১৯৮.

Rafi and Nabil started a business. Rafi invested Tk. 15,000 for 12 months, while Nabil invested Tk. 10,000 for 6 months. After 6 months, Nabil added another Tk. 5,000. If the total profit is Tk. 11,000, what is Nabil’s share of the profit?

ক
Tk 2,000
খ
Tk 3,000
গ
Tk 4,000
ঘ
Tk 5,000

ব্যাখ্যা

Question: Rafi and Nabil started a business. Rafi invested Tk. 15,000 for 12 months, while Nabil invested Tk. 10,000 for 6 months. After 6 months, Nabil added another Tk. 5,000. If the total profit is Tk. 11,000, what is Nabil’s share of the profit?

Solution:
Rafi's share 15,000 × 12 = 180,000
Nabil's share (10,000 × 6) + (15,000 × 6)
= 60,000 + 90,000
= 150,000

Ratio of Rafi : Nabil = 180,000 : 150,000 = 6 : 5

∴ Nabil's profit = (5/11) × 11,000
= Tk 5,000

৫,১৯৯.

What is the minimum percentage increase in the mean of set X {-4, -1, 0, 6, 9} if its two smallest elements are replaced with two different primes?

ক
50%
খ
80%
গ
100%
ঘ
150%

ব্যাখ্যা

Question: What is the minimum percentage increase in the mean of set X {-4, -1, 0, 6, 9} if its two smallest elements are replaced with two different primes?

Solution:
old mean = - 4 - 1 + 6 + 9 /5 = 2

-4, -1 will be replaced by 2, 3

new mean = 2 + 3 + 6 + 9 /5 = 4

%increase = {(4 - 2)/2} × 100%
= 100%

৫,২০০.

If a * b = 2a - 3b + ab, then 3 * 5 + 5 * 3 is equal to?

ক
22
খ
24
গ
26
ঘ
28

ব্যাখ্যা

Question: If a * b = 2a - 3b + ab, then 3 * 5 + 5 * 3 is equal to?

Solution:
3 ∗ 5 + 5 ∗ 3

3 ∗ 5 = 2 × 3 - 3 × 5 + 3 × 5
= 6 - 15 + 15
= 6

5 ∗ 3 = 2 × 5 - 3 × 3 + 3 × 5
= 10 - 9 +15
= 16

∴ 3 ∗ 5 + 5 ∗ 3
= 6 + 16
= 22