Bank Math

PrepBank · পাতা ১০৮ / ১৬১ · ১০,৭০১–১০,৮০০ / ১৬,১২৪

১০,৭০১.

The figure above represents a rectangular desk blotter in a holder with dimensions shown. If x = 8 centimeters, what is the area, in square centimeters, of the shaded portion of the blotter?

ক
4200
খ
4184
গ
4124
ঘ
4072
ঙ
3944

ব্যাখ্যা

Question:

The figure above represents a rectangular desk blotter in a holder with dimensions shown. If x = 8 centimeters, what is the area, in square centimeters, of the shaded portion of the blotter?

Solution:
total area = 84 × 50 = 4200
And we have 4 isoscles triangle so 4 × (0.5 × 8 × 8) = 128
∴ shaded area = 4200 - 128 = 4072

১০,৭০২.

In a certain code FIRE is coded as DGPC. What will be the last letter of the coded word for SHOT.

ক
P
খ
R
গ
S
ঘ
Q

ব্যাখ্যা

Question: In a certain code FIRE is coded as DGPC. What will be the last letter of the coded word for SHOT.

Solution:
F এর ১ বর্ণ আগের বর্ণ D
I এর ১ বর্ণ আগের বর্ণ G
R এর ১ বর্ণ আগের বর্ণ P
E এর ১ বর্ণ আগের বর্ণ C

অতএব, SHOT কোডিং করলে শেষ বর্ণ T এর এক বর্ণ আগের বর্ণ R

১০,৭০৩.

When a 60 kg member exits a group of 50, the average weight of the remaining 49 rises by 0.3 kg. Determine the new average weight of those left.

ক
72 kg
খ
75 kg
গ
65 kg
ঘ
60 kg

ব্যাখ্যা

Question: When a 60 kg member exits a group of 50, the average weight of the remaining 49 rises by 0.3 kg. Determine the new average weight of those left.

Solution:
let,
The average weight of 49 people is x kg
Total weight of 49 people = 49x
Total weight of 50 people = 49x + 60

ATQ,
50(x - 0.3) = (49x + 60)
⇒ 50x - 15 = 49x + 60
⇒ 50x - 49x = 60 + 15
∴ x = 75

∴ the new average weight of the remaining 49 people is 75 kg.

১০,৭০৪.

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

ক
320 cm³
খ
250 cm³
গ
350 cm³
ঘ
480 cm³

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

১০,৭০৫.

A committee of 5 members is to be formed by selecting out of 7 men and 6 women. In how many different ways the committee can be formed if it should have at least 3 men?

ক
ক) 564
খ
খ) 645
গ
গ) 756
ঘ
ঘ) 735

ব্যাখ্যা

Question: A committee of 5 members is to be formed by selecting out of 7 men and 6 women. In how many different ways the committee can be formed if it should have at least 3 men?

Solution:
Men (7) Women (6)
1) 3 2
2) 4 1
3) 5 0

From (1) Number of ways = ⁷C₃ × ⁶C₂ = 35 × 15 = 525
From (2) Number of ways = ⁷C₄ × ⁶C₁ = 35 × 6 = 210
From (3) Number of ways = ⁷C₅ × ⁶C₀ = 21 × 1 = 21

Total number of ways = 525 + 210 + 21 = 756

১০,৭০৬.

If 2^{n - 1} + 2^{n + 1} = 160, then the value of n is = ?

ক
4
খ
5
গ
6
ঘ
7

ব্যাখ্যা

Question: If 2^{n - 1} + 2^{n + 1} = 160, then the value of n is = ?

Solution:
2^{n - 1} + 2^{n + 1} = 160
⇒ 2^{n - 1} (1 + 2²) = 160
⇒ 2^{n - 1} · 5 = 160
⇒ 2^{n - 1} = 160/5
⇒ 2^{n - 1} = 32
⇒ 2^{n - 1} = 2⁵
⇒ n - 1 = 5
∴ n = 6

১০,৭০৭.

If a trapezium has two parallel sides measuring 4 cm and 6 cm, and its area is 100 square centimeters, what is the perpendicular distance (height) between the parallel sides?

ক
5 cm
খ
6 cm
গ
8 cm
ঘ
20 cm

ব্যাখ্যা

Question: If a trapezium has two parallel sides measuring 4 cm and 6 cm, and its area is 100 square centimeters, what is the perpendicular distance (height) between the parallel sides?

Solution:
Given,
Parallel sides of a trapezium = 4 cm, and 6 cm

We know,
Area of trapezium = (1/2)(sum of the parallel sides) × distance between the parallel sides
100 = (1/2)(4 + 6) × distance
⇒ 100 = 5 × distance
⇒ distance = 100/5
∴ distance = 20 cm

So, the distance between the parallel lines of trapezium = 20 cm.

১০,৭০৮.

A cistern can be filled by three pipes A, B and C alone 12 hrs, 24hrs and 48 hrs respectively. There is an opening D in the cistern that empties the cistern at the rate of 6m/hr. If the cistern is 96m deep then, in how much time will it be filled upto 72hrs of its depth if all the pipes are opened together at the start but B is closed after an hour?

ক
ক) 17 hours
খ
খ) 20 hours
গ
গ) 12 hours
ঘ
ঘ) 20 hours

ব্যাখ্যা

Tank filled by A alone in 1 hr = 1/12
Tank filled by B alone in 1 hr = 1/24
Tank filled by C alone in 1 hr = 1/48
D empty the tank at the rate of 6m/hr

So,
Tank empty by D in 1hr = 6/96 = 1/16
Now, tan is to be filled up to 72m i.e., 72/96 =3/4 of tank
So,
Let the 3/4^th of the tank to be filled in 't' hours time

For 1 hr all are opened then B closed

So, for (t - 1) hr A, C and D opened
(1/12 + 1/24+ 1/48 - 1/16) + (t - 1) (1/12 + 1/48 - 1/16) = 3/4
⇒ (1/12) + (t - 1) (1/24) = 3/4
⇒ (t - 1)/24 = (3/4) - (1/12)
⇒ (t - 1)/24 = 2/3
⇒ 3t - 3 = 48
⇒ 3t = 51
⇒ t = 17 hours.

১০,৭০৯.

A man goes downstream with a boat to some destination and returns upstream to his original place in 20 hours. If the speed of the boat in still water and the stream is 40 km/hr and 20 km/hr respectively, the distance of the destination from the starting place?

ক
400 km
খ
250 km
গ
450 km
ঘ
300 km
ঙ
None of these

ব্যাখ্যা

Question: A man goes downstream with a boat to some destination and returns upstream to his original place in 20 hours. If the speed of the boat in still water and the stream is 40 km/hr and 20 km/hr respectively, the distance of the destination from the starting place?

Solution:
Let the distance of the destination from the starting point = x km.
Rate downstream= (40 + 20) km/hr = 60 km/hr
Rate upstream = (40 - 20) km/hr = 20 km/hr
According to the question,
x/60 + x/20 = 20
⇒ 4x = 20 × 60
⇒ 4x = 1200
∴ x = 300 km

Hence, A distance of the destination from the starting point = 300 km

১০,৭১০.

In a row of trees, a tree is 9th from the left end and 17th from the right end. How many trees are there in the row?

ক
26
খ
22
গ
23
ঘ
25

ব্যাখ্যা

Question: In a row of trees, a tree is 9th from the left end and 17th from the right end. How many trees are there in the row?

Solution:
number of trees = 9 + 17 - 1
= 25

১০,৭১১.

P can complete a piece of work in 18 days, while Q can complete the same work in 36 days. If both P and Q work together, in how many days will they be able to complete the entire work?

ক
18 days
খ
12 days
গ
36 days
ঘ
6 days

ব্যাখ্যা

Question: P can complete a piece of work in 18 days, while Q can complete the same work in 36 days. If both P and Q work together, in how many days will they be able to complete the entire work?

Solution:
Given that,
P can do a piece of the work in 18 days and Q can do same work in 36 days

Now,
Let the total work = LCM(18, 36) = 36

Hence, total work = 36 units

Thus,
P does each day = 36 ÷ 18 = 2 units
Q does each day = 36 ÷ 36 = 1 unit

Hence, time taken by them to finish the work together = 36 ÷ (1 + 2) = 12 days.

∴ In 12 days working together they will complete the entire work.

১০,৭১২.

If a + b = -1 and a² + b² = 25, then find the value of (a - b)².

ক
ক) 7
খ
খ) 14
গ
গ) 21
ঘ
ঘ) 49

ব্যাখ্যা

Question: If a + b = -1 and a² + b² = 25, then find the value of (a - b)².

Solution:
Given that
(a + b) = -1
a² + b² = 25

We know
(a + b)² = a² + b² + 2ab
(- 1)² = 25 + 2ab
1 = 25 + 2ab
1 - 25 = 2ab
2ab = - 24
ab = - 12

Now
(a - b)² = (a + b)² - 4ab
=(- 1)² - 4 × (- 12)
= 1 + 48
= 49

১০,৭১৩.

An iron rod that weighs 24 kg is cut into two pieces so that one of these pieces weighs 16 kg and is 34 m long. If the weight of each piece is proportional to its length, how long is the other piece?

ক
ক) 11 m
খ
খ) 17 m
গ
গ) 34 m
ঘ
ঘ) 68 m

ব্যাখ্যা

Given, 16 kg rod = 34m
ATQ, 24 kg rod = (34×24)/16 = 51m
∴ Length of the other part is = 51 - 34 = 17m∴

১০,৭১৪.

Which term of the sequence (1/√2), 1, √2,.............. will be 8√2?

ক
11
খ
9
গ
12
ঘ
6

ব্যাখ্যা

Question: Which term of the sequence (1/√2), 1, √2... will be 8√2?

Solution:
দেওয়া আছে,
ধারার প্রথম পদ, a = 1/√2
সাধারন অনুপাত, r = 1/( 1/√2 ) = √2
n তম পদ = ar^{n - 1}

প্রশ্নমতে,
ar^n-1 = 8√2
⇒ (1/√2) × (√2)^{n - 1} = 8√2
⇒ (√2)^{n - 1} = 8√2 × √2 = 16
⇒ (√2)^{n - 1} = (√2)⁸
⇒ n - 1 = 8
⇒ n = 8 + 1 = 9

অর্থাৎ ধারাটির 9 তম পদ হলো 8√2

১০,৭১৫.

A souvenir vendor purchased 1,000 shirts for a special event at a price of Tk. 500 each. The vendor sold 600 of the shirts on the day of the event for Tk. 1200 each and 300 of the shirts in the week following the event for Tk. 400 each. The vendor was unable to sell the remaining shirts. What was the vendor's gross profit on the sale of these shirts?

ক
Tk. 100000
খ
Tk. 220000
গ
Tk. 270000
ঘ
Tk. 300000
ঙ
Tk. 340000

ব্যাখ্যা

Question: A souvenir vendor purchased 1,000 shirts for a special event at a price of Tk. 500 each. The vendor sold 600 of the shirts on the day of the event for Tk. 1200 each and 300 of the shirts in the week following the event for Tk. 400 each. The vendor was unable to sell the remaining shirts. What was the vendor's gross profit on the sale of these shirts?

Solution:
purchased 1000 shirts each for 500, we have a total cost of 500000

600 shirts were sold for 1200 each so 600 × 1200 = 720000

300 shirts were sold for 400 each so 300 × 400 = 120000

total revenue= 720000 + 120000 = 840000

∴ gross profit = 840000 - 500000 = 340000

১০,৭১৬.

A sum of money amounts to Tk. 18000 in 5 years at 20% simple interest per annum. Find the sum.

ক
Tk. 8500
খ
Tk. 9000
গ
Tk. 10500
ঘ
Tk. 12000

ব্যাখ্যা

Question: A sum of money amounts to Tk. 18000 in 5 years at 20% simple interest per annum. Find the sum.

Solution:
দেওয়া আছে,
সুদ-আসল (Amount), A = 18000 টাকা
সময় (Time), n = 5 বছর
সুদের হার (Rate), r = 20%
মূলধন (Principal), P = ?

আমরা জানি,
সুদ (Interest), I = A - P
আবার, I = (Pnr)/100
সুতরাং, A - P = (Pnr)/100
⇒ 18000 - P = (P × 5 × 20)/100
⇒ 18000 - P = (100P)/100
⇒ 18000 - P = P
⇒ 18000 = P + P
⇒ 18000 = 2P
⇒ P = 18000/2
⇒ P = 9000
সুতরাং, নির্ণেয় মূলধন 9000 টাকা।

১০,৭১৭.

There are 145 students in the first three standards. The ratio of number of students in the first and the second standards is 2 : 3, while that of students in standards second and third is 4 : 3. Find the number of students in 2nd standard.

ক
40
খ
45
গ
60
ঘ
65

ব্যাখ্যা

Question: There are 145 students in the first three standards. The ratio of number of students in the first and the second standards is 2 : 3, while that of students in standards second and third is 4 : 3. Find the number of students in 2nd standard.

Solution:
Total students = 145.
Ratio of students in 1st and 2nd standards = 2 : 3 = (2 × 4) : (3 × 4) = 8 : 12

Ratio of students in 2nd and 3rd standards = 4 : 3 = (4 × 3) : (3 × 3) = 12 : 9
Hence combined ratio i.e. 1st : 2nd: 3rd is = 8 : 12 : 9.

∴ Number of students in 2nd standard = (145 × 12)/29 = 60

১০,৭১৮.

If log₁₀4 + log₁₀(3x + 30)= log₁₀(2x + 8) + 1, then what is the value of x?

ক
10
খ
5
গ
15
ঘ
8

ব্যাখ্যা

Question: If log₁₀4 + log₁₀(3x + 30)= log₁₀(2x + 8) + 1, then what is the value of x?

Solution:
Given equation:
⇒ log₁₀4 + log₁₀(3x + 30) = log₁₀(2x + 8) + 1
⇒ log₁₀4 + log₁₀(3x + 30) = log₁₀(2x + 8) + log₁₀10 [যেহেতু, log₁₀10 = 1]
⇒ log₁₀[4(3x + 30)] = log₁₀[10(2x + 8)] [যেহেতু, logA + logB = log(AB)]
⇒ 4(3x + 30) = 10(2x + 8)
⇒ 12x + 120 = 20x + 80
⇒ 20x − 12x = 120 − 80
⇒ 8x = 40
⇒ x = 5

১০,৭১৯.

Fahim and Robin invested in a business where the investment of Robin is double of Fahim. But Fahim immediately invested 15000 Tk. that brings him double the profit of Robin after one year. Robin's investment was -

ক
ক) Tk. 8000
খ
খ) Tk. 10000
গ
গ) Tk. 12000
ঘ
ঘ) Tk. 15000

ব্যাখ্যা

Question: Fahim and Robin invested in a business where the investment of Robin is double of Fahim. But Fahim immediately invested 15000 Tk. that brings him double the profit of Robin after one year. Robin's investment was -

Solution:
Let initially the investment of Fahim is = X
So, the investment of Robin is = 2X

ATQ,
(X + 15000) : 2X = 2 : 1
⇒ 4X = X + 15000
⇒ 3X = 15000
∴ X = 5000 Tk.

Hence, the initial investment of Robin is = (2 × 5000) = Tk. 10000

১০,৭২০.

Printer P, Printer Q and Printer R can print a batch of flyers in 4,8 and 16 hours respectively. How many Printer R are needed with One Printer P and three Printer Q to complete the lot in 1 hour?

ক
12
খ
8
গ
6
ঘ
3

ব্যাখ্যা

Question: Printer P, Printer Q and Printer R can print a batch of flyers in 4,8 and 16 hours respectively. How many Printer R are needed with One Printer P and three Printer Q to complete the lot in 1 hour?

Solution:
Here, Total work LCM(4, 8, 16) = 16 units
So, Efficiency of P = 4, Q = 2 and R = 1

Let, x number of Printer R required to complete the lot in 1 hour.
Now,
16 = (1 × 4 + 3 × 2) + 1 × x
⇒ 16 = 10 + x
⇒ x = 16 - 10
∴ x = 6

Thus, 6 Printer R are needed.

১০,৭২১.

5 year ago Sushma was 5 times as old as her Son. 5 year hence her age will be 8 less than three times the corresponding age of her Son. Find their ages?

ক
ক) 24 and 13 year
খ
খ) 48 and 24 year
গ
গ) 35 and 11 year
ঘ
ঘ) 33 and 15 year

ব্যাখ্যা

Let the age of sushma be x and
the age of her son is y
Then five year before x-5=5(y-5) ...(1)
Five year hence x+5 = 3(y+5)-8 .....(2)
By soving (1) & (2), we get
5y - 15 = 3y + 7
y = 11
=> x = 35
Therefore, the age of Sushma = 35 and her son = 11.

১০,৭২২.

What will be the ratio of simple interest earned by certain amount at the same rate of interest for 9 years and that for 6 years?

ক
3 : 4
খ
3 : 1
গ
3 : 2
ঘ
4 : 5

ব্যাখ্যা

Question: What will be the ratio of simple interest earned by certain amount at the same rate of interest for 9 years and that for 6 years?

Solution:
Let
The principal be P and rate of interest be r%

Required ratio = [(P × r × 9)/100]/[(P × r × 6)/100]
=9Pr/6Pr
= 9/6
= 3/2
= 3 : 2

১০,৭২৩.

A Tank is normally filled in 8 hours but takes 5 hours longer to fill because of a leak in its bottom. If the tank is full, the leak will empty it in?

ক
ক) 20.8
খ
খ) 19.9
গ
গ) 20.5
ঘ
ঘ) 19.8

ব্যাখ্যা

Question: A Tank is normally filled in 8 hours but takes 5 hours longer to fill because of a leak in its bottom. If the tank is full, the leak will empty it in?

Solution:
Let the leak will empty the tank in x hrs.
Total time = 8 + 5 = 13 hrs.

Then,
Or, 1/8 - 1/x = 1/13
Or, (x − 8)/8x = 1/13
Or, 13x − 104 = 8x
Or, 13x − 8x = 104
Or, 5x = 104
Or, x = 20.8

১০,৭২৪.

The sum of the weights of A and B is 80 kg. Half of the weight of A is equal to 5/6 times the weight og B. Find the weight of B.

ক
40 Kg
খ
30 Kg
গ
25 Kg
ঘ
20 Kg

ব্যাখ্যা

Question: The sum of the weights of A and B is 80 kg. Half of the weight of A is equal to 5/6 times the weight og B. Find the weight of B.

Solution:
A + B = 80 and
A/2 = 5B/6
∴ A = 5B/3

A + B = 80
or, 5B/3 + B = 80
or, 8B/3 = 80
∴ B = 30 Kg

১০,৭২৫.

What is the difference between simple and compound interest at 10% per annum on a sum of Tk. 3000 at the end of 2 years?

ক
Tk. 80
খ
Tk. 60
গ
Tk. 30
ঘ
Tk. 50

ব্যাখ্যা

Question: What is the difference between simple and compound interest at 10% per annum on a sum of Tk. 3000 at the end of 2 years?

Solution:
Principal (P) = Tk. 3000
Rate (r) = 10% per annum
Time (n) = 2 years

Simple Interest (SI):
SI = (P × R × T)/100
= (3000 × 10 × 2)/100
= 60000/100
= Tk. 600

Compound Interest (CI):
Amount (A) = P × (1 + r/100)ⁿ
= 3000 × (1 + 10/100)²
= 3000 × (1.1)²
= 3000 × 1.21
= Tk. 3630

∴ CI = A - P = 3630 - 3000
= Tk. 630

∴ Difference between CI and SI = 630 - 600
= Tk. 30

১০,৭২৬.

In a camp, provisions are sufficient for 200 persons for 35 days. After 20 days, 50 persons depart. Determine the number of days the remaining food will last.

ক
10 days
খ
15 days
গ
20 days
ঘ
25 days

ব্যাখ্যা

Question: In a camp, provisions are sufficient for 200 persons for 35 days. After 20 days, 50 persons depart. Determine the number of days the remaining food will last.

Solution:
10 দিন পর 50 জন চলে যাওয়ায়,
অবশিষ্ট দিন = (35 - 20) = 15 দিন
এবং অবশিষ্ট লোক = (200 - 50) = 150 জন

এখন,
হোস্টেলে 200 জনের খাদ্য মজুদ আছে = 15 দিনের
∴ 1 জনের খাদ্য মজুদ আছে = (15 × 200) দিনের
∴ 150 জনের খাদ্য মজুদ আছে = (15 × 200)/150 দিনের = 20 দিনের

১০,৭২৭.

Which of the following is not a leap year?

ক
1200
খ
800
গ
1700
ঘ
2000

ব্যাখ্যা

Question: Which of the following is not a leap year?

Solution:

To determine if a year is a leap year, you can use the following rules:

(1) If the year is evenly divisible by 4, go to step 2. If not, it is not a leap year.
(2) If the year is divisible by 100, go to step 3. If not, it is a leap year.
(3) If the year is divisible by 400, go to step 4. If not, it is not a leap year.
(4) If the year is divisible by 400, it is a leap year. Otherwise, it is not.

Now, let's apply these rules to the year 1700:

(1) 1700 is divisible by 4.
(2) 1700 is divisible by 100.
(3) 1700 is not divisible by 400.

According to the rules, since 1700 is divisible by 4 but not by 400, it is not a leap year.

১০,৭২৮.

The area of the base of a cylinder is 100π m². The volume of the cylinder is 900π m³. What is the height of the cylinder?

ক
ক) 9m
খ
খ) 10m
গ
গ) 11m
ঘ
ঘ) 12m

ব্যাখ্যা

Area of the base of a cylinder, πr² = 100πThe volume of the cylinder, πr²h = 900π
∴ h = πr²h/πr²
= 900/100
= 9 m

১০,৭২৯.

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

ক
ক) 400
খ
খ) 900
গ
গ) 1600
ঘ
ঘ) 2500

ব্যাখ্যা

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

সমাধান:
We need to find out the LCM of the given numbers.
12 = 2 × 2 × 3
15 = 3 × 5
18 = 2 × 3 × 3
20 = 2 × 2 × 5

Hence,
LCM = 2 × 2 × 3 × 5 × 3

Since, the soldiers are in the form of a solid square.
Hence, LCM must be a perfect square. To make the LCM a perfect square, We have to multiply it by 5,
hence,
The required number of soldiers
= 2 × 2 × 3 × 3 × 5 × 5
= 900

১০,৭৩০.

10 pens costs Tk. 100 each. If half of the pens are sold at 10% loss then find at what price remaining each pen should be sold for making no loss and no profit.

ক
Tk. 110
খ
Tk. 112
গ
Tk. 120
ঘ
Tk. 90

ব্যাখ্যা

Question: 10 pens costs Tk. 100 each. If half of the pens are sold at 10% loss then find at what price remaining each pen should be sold for making no loss and no profit.

Solution:
Total cost price of 10 pens = 10 × 100 = Tk. 1000

Selling price of 1 pen = 100 - (100 × 10%) = Tk. 90
Hence, selling price of 5 pens = Tk. 450

Now, selling price of remaining 5 pens = 1000 - 450 = Tk. 550
Hence, selling price of 1 pen = Tk. 110

১০,৭৩১.

A, B and C enter into partnership with investments in the ratio of 5 : 6 : 9. If, at the end of the year B's share of profit is tk 40,350, how much is the total profit?

ক
Tk. 134,000
খ
Tk. 134,500
গ
Tk. 125,500
ঘ
Tk. 243,500

ব্যাখ্যা

Question: A, B and C enter into partnership with investments in the ratio of 5 : 6 : 9. If, at the end of the year B's share of profit is tk 40,350, how much is the total profit?

Solution:
Let the total profit be P. The investments are in the ratio 5 : 6 : 9, so the total parts of the profit distribution will be,
5 + 6 + 9 = 20 parts

B's share corresponds to 6 parts,
⇒ 6/20 × P= 40,350
⇒ P =( 40350 × 20)/6
⇒ P = 6725 × 20
⇒ P = 134500

So the total profit is Tk 134,500.

১০,৭৩২.

A started a business with Tk. 42,000 and is joined afterwards by B with Tk. 72,000. After how many months did B join if the profits at the end of the year are divided equally?

ক
ক) 3 months
খ
খ) 4 months
গ
গ) 5 months
ঘ
ঘ) 6 months

ব্যাখ্যা

Suppose B joined after x months
then,
42000 × 12 = 72000 × (12 - x)
42 × 12 = 72 × (12 - x)
42 × 12 = 72 × 12 - 72x
72x = 864 - 504
⇒ 72x = 360
⇒ x = 5

১০,৭৩৩.

In the series 1, 3, 4, 8, 15, 27, ... what will be the next number?

ক
38
খ
42
গ
50
ঘ
60

ব্যাখ্যা

Question: In the series 1, 3, 4, 8, 15, 27, ... what will be the next number?

Solution:
Here,
1 + 3 + 4 = 8
3 + 4 + 8 = 15
8 + 15 + 27 = 50

So, the sum of any three consecutive numbers of the series gives the next number.
Hence, missing number = 8 + 15 + 27
= 50

১০,৭৩৪.

The marked price of a ceiling fan is Tk. 1250 and the shopkeeper allows a discount of 6% on it. Find the selling price of the fan.

ক
Tk. 920
খ
Tk. 1250
গ
Tk. 1065
ঘ
Tk. 1175

ব্যাখ্যা

Question: The marked price of a ceiling fan is Tk. 1250 and the shopkeeper allows a discount of 6% on it. Find the selling price of the fan.

Solution:
Marked price = Tk. 1250 and discount = 6%.

Discount = 6% of Marked Price
= (6% of Tk. 1250)
= Tk. {1250 × (6/100)}
= Tk. 75

Selling price = (Marked Price) - (discount)
= Tk. (1250 - 75)
= Tk. 1175.

∴ Hence, the selling price of the fan is Tk. 1175.

১০,৭৩৫.

If x > 5 and y < - 1, then which of the following statements is true?

ক
(x + 4y) > 1
খ
x > 4y
গ
- 4x < 5y
ঘ
None

ব্যাখ্যা

Question: If x > 5 and y < - 1, then which of the following statements is true?

Solution:
⇒ (x + 4y) > 1
let, x = 6 and y = -10
x + 4y = 6 + 4 × -10= -34 < 1

⇒ x > 4y

x > 5, x is a positive number

y < -1, y is a negative number. 4y is also a negative number.

x > 4y is always true.

⇒ −4x < 5y
5y+4x > 0

let, x = 6 and y = -10
4x + 5y = 4 × 6 + 5 × -10 = 24 - 50 = -26<0

১০,৭৩৬.

A person buys certain number of gems at 20 per taka and an equal number at 30 per taka. He mixes them and sells them at 25 per taka. His gain or loss in the transaction is-

ক
5% gain
খ
3% gain
গ
4% loss
ঘ
None of the above is correct

ব্যাখ্যা

Question: A person buys certain number of gems at 20 per taka and an equal number at 30 per taka. He mixes them and sells them at 25 per taka. His gain or loss in the transaction is-

Solution:
The cost price of each gem is 20 per Tk. = Tk. 1/20
The price of each gem is 30 per Tk. = Tk. 1/30

∴ Average cost of gem = {(1/20) + (1/30)}/2
= (3 + 2)/(2 × 60)
= 5/120
= 1/24

The selling price of mixed gem = Tk. 1/25

∴ Loss = (1/24) - (1/25) = 1/600

Cost price 1/24 then loss Tk. 1/600
Cost price 1 then loss Tk. (1/600)/(1/24)

∴ Cost price 1 then loss Tk. {(1/600)/(1/24)} × 100
= 4%

∴ There is a loss of 4%

১০,৭৩৭.

The H.C.F of 2⁴ × 3² × 5³ × 7, 2³ × 3³ × 5² × 7² and 3 × 5 × 7 × 11 is = ?

ক
95
খ
115
গ
120
ঘ
105
ঙ
None of the above

ব্যাখ্যা

Question: The H.C.F of 2⁴ × 3² × 5³ × 7, 2³ × 3³ × 5² × 7² and 3 × 5 × 7 × 11 is = ?

Solution:
H.C.F. = Product of lowest powers of common factors
= 3 × 5 × 7
= 105

১০,৭৩৮.

A man invested 1/3 of his capital at 7%, 1/4 at 8% and the remainder at 10%. If his annual income is Tk.561, The capital is-

ক
ক) Tk. 4600
খ
খ) Tk. 4800
গ
গ) Tk. 6600
ঘ
ঘ) Tk. 6400

ব্যাখ্যা

Let the capital be Tk. x
Then according to the question, we have
[{(x/3)× (7/100) × 1} + [{(x/4)×(8/100) × 1} + [{(5x/12)× (10/100) × 1}= 561
⇒(7x/300) + (8x/400) + (50x/1200) = 561
⇒(28x + 24x + 50x)/1200 = 561
⇒102x/1200 = 561
x = (561 × 1200)/102
x = 6600

১০,৭৩৯.

Two pipes A and B can fill a tank in 8 minutes and 20 minutes respectively. Both the pipes are opened together but after 4 minutes, pipe A is turned off. What is the total time required to fill the tank?

ক
8 minutes
খ
10 minutes
গ
16 minutes
ঘ
20 minutes

ব্যাখ্যা

Question: Two pipes A and B can fill a tank in 8 minutes and 20 minutes respectively. Both the pipes are opened together but after 4 minutes, pipe A is turned off. What is the total time required to fill the tank?

Solution:
Given,
Pipe A can fill the tank in 8 minutes,
so its rate is 1/8 of the tank per minute.

Pipe B can fill the tank in 20 minutes,
so its rate is 1/20 of the tank per minute.

∴ Part filled in 1 minutes = (1/8 + 1/20)
Part filled in 4 minutes = (1/8 + 1/20) × 4
= 7/10

∴ Remaining part = (1 - 7/10) = 3/10

1/20 Part filled by B in 1 minute
∴ Full Part filled by B in 20 minutes
∴ 3/10 Part filled by B in (20 × 3)/10 minutes
= 6 minutes

The tank will be full in (4 + 6) minutes
=10 minutes

১০,৭৪০.

A train running at a speed of 90 km/hr crosses a platform double its length in 36 seconds. What is the length of the platform in meters?

ক
300 m
খ
450 m
গ
600 m
ঘ
700 m

ব্যাখ্যা

Question: A train running at a speed of 90 km/hr crosses a platform double its length in 36 seconds. What is the length of the platform in meters?

Solution:
Let, the length of the train be x metres.
Then, the length of the platform = 2x metres.
Speed of the train = 90 × (1000/3600) m/sec
= 25 m/sec

ATQ,
(x + 2x)/25 = 36
⇒ 3x = 900
⇒ x = 300

Hence, the length of platform = 2x = (2 × 300) m
= 600 m

১০,৭৪১.

The ratio of speed of a motor-boat to that of the current of water is 17 : 5. The boat goes along with the current in 4 hours. It will come back in-

ক
7 hour
খ
7 hour 20 minute
গ
7 hour 30 minute
ঘ
7 hour 40 minute

ব্যাখ্যা

Question: The ratio of speed of a motor-boat to that of the current of water is 17 : 5. The boat goes along with the current in 4 hours. It will come back in-

Solution:
Since the ratio 17 : 5 is given.
Let the speed of boat in still water = 17 km/hr
and speed of stream = 5 km/hr

Downstream speed = 17 + 5 = 22 km/hr
Upstream speed = 17 - 5 = 12 km/hr

Distance = Downstream speed × downstream time
= 22 × 4 = 88 km

Upstream time = Distance/upstream speed
= 88/12
∴ Come back time = 7 hour 20 minute

১০,৭৪২.

In the triangle ABC, ∠B is equal to ∠C and D is a point strictly between B and C on side BC (i.e., BD < DC). Which of the following is true?

ক
AB > AC
খ
AC > AD
গ
AB < AC
ঘ
AC < AD
ঙ
All of them

ব্যাখ্যা

Question: In the triangle ABC, ∠B is equal to ∠C and D is a point strictly between B and C on side BC (i.e., D ≠ B, C). Which of the following is true?

Solution:
In △ABC, ∠B = ∠C.
∴ The triangle is isosceles.
AB = AC and point D lies strictly between B and C on side BC.

Now, In △ABC,
∠ADC = ∠ABD + ∠BAD
∴ ∠ADC > ∠ABD
∠ADC > ∠ACD [∵ ∠B = ∠C and D ≠ B, C]
∴ AC > AD

১০,৭৪৩.

The sum invested in scheme B is thrice the sum invested in scheme A. The investment in scheme A is made for 4 years at 8% p.a. simple interest and in scheme B for 2 years at 13% p.a. simple interest. The total interest earned from both the schemes is Tk. 1320. How much amount was invested in scheme A?

ক
ক) Tk. 1200
খ
খ) Tk. 1100
গ
গ) Tk. 960
ঘ
ঘ) Tk. 1500

ব্যাখ্যা

Let the amount invested in scheme A be Tk. x and that in B be Tk. 3x.
Then,
Then,
{(x × 4 × 8)/100 + (3x × 2 × 13)/100} = 1320
⇒ 32x/100 + 78x/100 = 1320
⇒ 110x/100 = 1320
⇒ x = (1320 × 100)/110
⇒ x = Tk. 1200

Hence, Tk. 1200 was invested in scheme A.

১০,৭৪৪.

A right angled triangle, whose perpendicular sides measure 1.8cm and 2.4cm, is inscribed in a circle. What is the circumference of the circle (in cm)?

ক
ক) π
খ
খ) 3π
গ
গ) 2π
ঘ
ঘ) None of these

ব্যাখ্যা

মনেকরি
ABC সমকোণী ত্রিভুজ যার লম্ব = 1.8 cm এবং ভূমি = 2.4 cm
অতিভুজ = √(1.8² + 2.4²)
= √(3.24 + 5.76)
= √9 = 3

বৃত্তটির ব্যাস 2r= 3
বৃত্তটির পরিধি = 2πr= 3π

১০,৭৪৫.

The angle of elevation of the sun when the length of the shadow of a tree is equal to the height of the tree is:

ক
ক) 45°
খ
খ) 30°
গ
গ) 60°
ঘ
ঘ) 90°

ব্যাখ্যা

Consider the diagram is shown above where QR represents the tree and PQ represents its shadow.
We have, QR = PQ
Let, ∠QPR = θ
tanθ = QR/PQ
= QR/QR [since QR = PQ]
= 1
= tan 45°
⇒ θ = 45°
i.e., the required angle of elevation = 45°

১০,৭৪৬.

The average age of three boys is 20 years and their age ratio is 4 : 5 : 6. What is the age of the youngest boy?

ক
ক) 12 years
খ
খ) 15 years
গ
গ) 16 years
ঘ
ঘ) 18 years

ব্যাখ্যা

Question: The average age of three boys is 20 years and their age ratio is 4 : 5 : 6. What is the age of the youngest boy?

Solution:
Sum of the three boy's age = 20 × 3 = 60 years
So, age of the youngest boy is = 60 × (4/15) = 16 years

১০,৭৪৭.

At present, the ratio between the ages of Nehal and Rahat is 5 : 4. After 6 years, Nehal's age will be 26 years. What is the age of Rahat at present?

ক
21 years
খ
19 years
গ
18 years
ঘ
16 years

ব্যাখ্যা

Question: At present, the ratio between the ages of Nehal and Rahat is 5 : 4. After 6 years, Nehal's age will be 26 years. What is the age of Rahat at present?

Solution:
Let,
the present ages of Nehal and Rahat be 5x years and 4x years respectively.

ATQ,
5x + 6 = 26
⇒ 5x = 20
∴ x = 4

∴ Rahat's age = 4x = (4 × 4) = 16 years

১০,৭৪৮.

13, 25, 51, 101, 203, ?

ক
306
খ
344
গ
405
ঘ
406

ব্যাখ্যা

Question: 13, 25, 51, 101, 203, ?

Solution:
Here,
Two series exist:
1st: 13, 51, 203
2nd: 25, 101, ?

1st:
13
13 × 4 - 1 = 52 - 1 = 51
51 × 4 - 1 = 204 - 1 = 203

2nd:
25
25 × 4 + 1 = 100 + 1 = 101
101 × 4 + 1 = 404 + 1 = 405

১০,৭৪৯.

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

ক
360
খ
720
গ
480
ঘ
120

ব্যাখ্যা

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

Solution:
TENNIS whereas total 6 letters and N comes two times.
So, arrangements are = 6!/2! = 720/2 = 360 ways

১০,৭৫০.

What is the slope of a line perpendicular to the line whose equation is 20x - 2y = 6?

ক
- (1/10)
খ
1/4
গ
- (2/5)
ঘ
1/12

ব্যাখ্যা

Question: What is the slope of a line perpendicular to the line whose equation is 20x - 2y = 6?

Solution:
The general equation of a straight line is
y = mx + c ......(1) (Where, m = slope)

If the slope of a line is m, then the slope of the line perpendicular to it is,
m' = - (1/m)

Now,
20x - 2y = 6
⇒ 2y = 20x - 6
∴ y = 10x - 3
Comparing with equation (1), we get,
∴ m = 10

∴ The slope of the perpendicular line is, m' = - (1/10)

১০,৭৫১.

The sum of the squares of three number is 83, while the sum of their products taken two at a time is 71. Their sum is-

ক
18
খ
25
গ
20
ঘ
15

ব্যাখ্যা

Question: The sum of the squares of three number is 83, while the sum of their products taken two at a time is 71. Their sum is-

Solution:
Given that,
Sum of squares, a² + b² + c² = 83
Sum of products two at a time, ab + bc + ca =71

We know that,
(a + b + c)² = a² + b² +c² + 2(ab + bc + ca)
⇒ (a + b + c)²= 83 + (2 × 71)
⇒ (a + b + c)²= 83 + 142
⇒ (a + b + c)²= 225
⇒ (a + b + c)² = 225
⇒ a + b + c = √225
∴ a + b + c = 15

১০,৭৫২.

Three numbers are in the ratio 2 : 3 : 5, and their H.C.F. is 15. What is their L.C.M.?

ক
450
খ
600
গ
300
ঘ
475

ব্যাখ্যা

Question: Three numbers are in the ratio 2 : 3 : 5, and their H.C.F. is 15. What is their L.C.M.?

Solution:
দেওয়া আছে,
সংখ্যা তিনটির গ. সা. গু ১৫
সংখ্যা তিনটি হচ্ছে (২ × ১৫) বা ৩০, (৩ × ১৫) বা ৪৫, (৫ × ১৫) বা ৭৫

এখন,
৩০, ৪৫, ৭৫ এর ল. সা. গু = ৪৫০

১০,৭৫৩.

A is the widow of B. B & C were the only children of E. C is unmarried and is a doctor. D is the grand-daughter of E and studies science. How is A related to D?

ক
ক) aunt
খ
খ) daughter
গ
গ) sister
ঘ
ঘ) sister-in-law
ঙ
ঙ) None of these

ব্যাখ্যা

Question: A is the widow of B. B & C were the only children of E. C is unmarried and is a doctor. D is the grand-daughter of E and studies science. How is A related to D?

Solution:
A is the widow of B ⇒ A's husband was B.

B & C were the only children of E ⇒ A is the daughter in law of E.

D is the grand-daughter of E ⇒ D is daughter of A [D can not be daughter of C because C is unmarried]

So, A is the mother of D.
So, the correct answer is - ঙ) None of these

১০,৭৫৪.

If 1 + sinθ = x cosθ, then cotθ is-

ক
ক) 2x/(x² + 1)
খ
খ) 2x/(x² - 1)
গ
গ) 3x/(x² - 1)
ঘ
ঘ) x/(x² - 1)

ব্যাখ্যা

Question: If 1 + sinθ = x cosθ, then cotθ is-

Solution:
Given,
1+ sinθ = xcos θ
⇒ (1 + sinθ)/cosθ = x
⇒ 1/cosθ + sinθ/cosθ = x
⇒ secθ + tanθ = x...............(i)

We know,
sec²θ - tan²θ = 1
⇒ (secθ + tanθ) (secθ - tanθ) = 1
⇒ x(secθ - tanθ) = 1
⇒ secθ - tanθ = 1/x.................(ii)

(i) - (ii) ⇒
secθ + tanθ - (secθ - tanθ) = x - 1/x
⇒ secθ + tanθ - secθ + tanθ = (x² - 1)/x
⇒ 2tanθ = (x² - 1)/x
⇒ tanθ = (x² - 1)/2x
∴ cotθ = 2x/(x² - 1)

১০,৭৫৫.

If 2log₄(x) = 1 + log₄(x - 1) find the value of x?

ক
1/2
খ
4
গ
0
ঘ
None of these

ব্যাখ্যা

Question: If 2log₄(x) = 1 + log₄(x - 1) find the value of x?

Solution:
⇒ 2log₄(x) = 1 + log₄(x - 1)
⇒ log₄(x²) = log₄4 + log₄(x - 1)
⇒ x² = 4(x - 1)
⇒ x² - 4x + 4 = 0
⇒ (x - 2)² = 0
⇒ x - 2 = 0
∴ x = 2

১০,৭৫৬.

Among students in a class, 30 are basketball players, 20 are volleyball players, and 8 play both games. If 12 students play neither game, how many students are in the class altogether?

ক
34
খ
66
গ
54
ঘ
45
ঙ
58

ব্যাখ্যা

Question: Among students in a class, 30 are basketball players, 20 are volleyball players, and 8 play both games. If 12 students play neither game, how many students are in the class altogether?

Solution:
Let the number of students who play basketball = 30
Number of students who play volleyball = 20
Number of students who play both basketball and volleyball = 8
Number of students who play neither = 12

First, calculate the number of students who play basketball or volleyball:
n(B ∪ V) = n(B) + n(V) − n(B ∩ V)
n(B ∪ V) = 30 + 20 − 8 = 42

Now, add the students who play neither sport to get total students:
Total students = n(B ∪ V) + neither

Total students = 42 + 12 = 54

১০,৭৫৭.

In how many different ways can the letters of the word 'EXTRA' be arranged so that the vowels are never together?

ক
108
খ
72
গ
120
ঘ
210

ব্যাখ্যা

Question: In how many different ways can the letters of the word 'EXTRA' be arranged so that the vowels are never together?

Solution:
Taking the vowels (EA) as one letter, the given word has the letters XTR (EA),
i.e., 4 letters.
These letters can be arranged in = 4! = 4 × 3 × 2 × 1 = 24 ways
The letters EA may be arranged amongst themselves in 2 ways.
Number of arrangements having vowels together = (24 × 2) = 48 ways

∴ Total arrangements of all letters = 5!
= (5 × 4 × 3 × 2 × 1)
= 120

∴ Number of arrangements not having vowels together,
= (120 - 48)
= 72

১০,৭৫৮.

What is the largest four-digit number that is exactly divisible by 12, 18, 24, and 36?

ক
9900
খ
9963
গ
9936
ঘ
9972

ব্যাখ্যা

Question: What is the largest four-digit number that is exactly divisible by 12, 18, 24, and 36?

Solution:
Greatest number of 4-digits is 9999.

L.C.M. of 12, 18, 24 and 36 is 72.

On dividing 9999 by 72, the remainder is 63.

Required number = (9999 - 63) = 9936.

১০,৭৫৯.

A shopkeeper purchased 70 kg of potatoes for Tk. 420 and sold the whole potatose at the rate of 7 tk per kg. What will be his gain percent?

ক
16.28%
খ
15%
গ
11.50%
ঘ
16.67%
ঙ
None of these

ব্যাখ্যা

Question: A shopkeeper purchased 70 kg of potatoes for Tk. 420 and sold the whole potatose at the rate of 7 tk per kg. What will be his gain percent?

Solution:
70 কেজি আলুর বিক্রয়মূল্য = (70 × 7) = 490 টাকা

∴ লাভ = 490 - 420 = 70

420 টাকা লাভ হয় = 70 টাকা
1 টাকা লাভ হয় = 70/420 টাকা
100 টাকা লাভ হয় = {(70 × 100)/420} টাকা
= 100/6
= 16.67 টাকা

১০,৭৬০.

A boat can travel with a speed of 11 km/hr in still water. If the speed of the stream is 5 km/hr. find the time taken by the boat to go 112 km downstream.

ক
3 hrs.
খ
6 hrs.
গ
7 hrs.
ঘ
11 hrs.

ব্যাখ্যা

Question: A boat can travel with a speed of 11 km/hr in still water. If the speed of the stream is 5 km/hr. find the time taken by the boat to go 112 km downstream.

Solution:
Speed downstream = Speed of Boat in still water + Speed of the stream =
(11 + 5)km/hr. = 16km/hr.

Time taken to travel 112 km downstream = Distance/Speed=(112/16)hrs = 7 hrs.

১০,৭৬১.

Three partners A, B and C invest Tk.1500, Tk.1200, and Tk.1800 respectively in a company and make a profit of Tk.900. What is the amount of profit of C?

ক
ক) Tk.200
খ
খ) Tk.240
গ
গ) Tk.300
ঘ
ঘ) Tk.360

ব্যাখ্যা

প্রশ্ন: Three partners A, B and C invest Tk.1500, Tk.1200, and Tk.1800 respectively in a company and make a profit of Tk.900. What is the amount of profit of C?

সমাধান:
A, B, C এর বিনিয়োগের অনুপাত = ১৫০০ : ১২০০ : ১৮০০
= ১৫ : ১২ : ১৮
= ৫ : ৪ : ৬

তারা লাভের পরিমাণকেও উক্ত অনুপাতে বণ্টন করবে।

অনুপাতের রাশিগুলোর যোগফল = ৫ + ৪ + ৬ = ১৫

C এর লাভের পরিমাণ = ৯০০ × (৬/১৫) টাকা
= ৩৬০ টাকা

১০,৭৬২.

The time taken by a train 180 m long, travelling at 42 kmph, in passing a person walking in the same direction at 6 kmph, will be -

ক
17 sec
খ
18 sec
গ
19 sec
ঘ
21 sec

ব্যাখ্যা

Question: The time taken by a train 180 m long, travelling at 42 kmph, in passing a person walking in the same direction at 6 kmph, will be -

Solution:
Speed of train relative to man
= (44 - 8) kmph
= 36 kmph
= (36 × 5)/18m/sec
= 10 m/sec

∴ Time taken to pass the man
= 180/10 sec
= 18 sec

১০,৭৬৩.

z% of z is the same as 10% of

ক
z³/10
খ
z²/10
গ
z⁵/10
ঘ
z^{- 2}/10
ঙ
None

ব্যাখ্যা

Question: z% of z is the same as 10% of

Solution:
Let z% of z = 10% of p
⇒ (z/100) × z = (10/100) × p
⇒ p = (z²/100) × 10
⇒ p = z²/10

১০,৭৬৪.

If α, β are the roots of the equation x² - 6x + 8 = 0, then α² + β² equals:

ক
16
খ
20
গ
13
ঘ
26

ব্যাখ্যা

Question: If α, β are the roots of the equation x² - 6x + 8 = 0, then α² + β² equals:

Solution:
x² - 6x + 8 = 0
=> x² - 4x - 2x + 8 = 0
=> x(x - 4) - 2(x - 4) = 0
=> (x - 4)(x - 2) = 0
=> x = 4, 2

Hence, α = 4, β = 2

Hence, The value of α² + β² = 4² + 2²
= 16 + 4
= 20

সুতরাং, α² + β² = 20

১০,৭৬৫.

In a shower, 5 cm of rain falls. The volume of water that falls on 1.5 hectares of ground is:

ক
75 m³
খ
750 m³
গ
175 m³
ঘ
7500 m³

ব্যাখ্যা

1hectare=10,000 m²
Area = (1.5 × 10,000) m²
= 15000 m²
Depth = 5/100 m
= 1/20 m
Volume = (15000 × 1/20) m³ = 750 m³

১০,৭৬৬.

Two trains of equal length take 8 seconds and 12 seconds respectively to cross a lamp post. If the length of each train is 160 meters, in how many seconds will they cross each other when traveling in opposite directions?

ক
9.2 seconds
খ
9.6 seconds
গ
10.2 seconds
ঘ
10.6 seconds

ব্যাখ্যা

Question: Two trains of equal length take 8 seconds and 12 seconds respectively to cross a lamp post. If the length of each train is 160 meters, in how many seconds will they cross each other when traveling in opposite directions?

Solution:
Speed of the first train = 160/8 m/sec
= 20 m/sec

Speed of the second train = 160/12 m/sec
= 40/3 m/sec

∴ Relative speed = 20 + 40/3
= (60 + 40)/3
= 100/3 m/sec

∴ Required time = (160 + 160)/(100/3) sec
= 320 × 3/100
= 960/100
= 9.6 seconds

১০,৭৬৭.

Q (33 - 56): Read the following questions carefully and choose the right answer.
৩৩. If m and n are whole numbers such that mⁿ = 121, the value of (m - 1)^{n + 1} is:

ক
1
খ
10
গ
121
ঘ
1,000

ব্যাখ্যা

Question: If m and n are whole numbers such that mⁿ = 121, the value of (m - 1)^{n + 1} is:

Solution:
mⁿ = 121
mⁿ = (11)²
∴m = 11 , n = 2

∴(m - 1)^{n + 1} =(11 - 1)^{2 + 1}
=(10)³
=1000

১০,৭৬৮.

The price of raw materials has gone up by 15%, labor cost has also increased from 25% of the cost of raw material to 30% of the cost of raw material. By how much percentage should there be reduction in the usage of raw materials so as to keep the cost same?

ক
ক) 28%
খ
খ) 17%
গ
গ) 27%
ঘ
ঘ) 24%

ব্যাখ্যা

Let the initial cost of raw material be 100. So, initial labor cost was 25 and net cost was 125
Now,
15% increment in raw materials cost and labor cost has gone up to 30% from 25 %
So, Raw material cost = 115 And Labor cost = (115 × 30%) = 34.5
New net cost = 115 + 34.5 = 149.5
Difference of labor cost = 149.5 - 125 = 24.5
Price has to be reduced by = (24.5×100) / 149.5 = 16.387
∴ Price has to be reduced by approximately 17%

১০,৭৬৯.

The circumference of the back-sided wheel of a vehicle is 1 m greater than that of the front-side wheel. To travel 600 m, the front wheel rotates 30 times more than the back wheel. The circumference of the back-sided wheel is:

ক
5 m
খ
6 m
গ
4 m
ঘ
3 m

ব্যাখ্যা

Question: The circumference of the back-sided wheel of a vehicle is 1 m greater than that of the front-side wheel. To travel 600 m, the front wheel rotates 30 times more than the back wheel. The circumference of the back-sided wheel is:

Solution:
ধরি, সামনের চাকার পরিধি x m
পেছনের চাকার পরিধি = x + 1 m

600 m যেতে সামনের চাকা ঘোরে 600/x বার
600 m যেতে পেছনের চাকা ঘোরে 600/x + 1 বার

(600/x) - (600/x + 1) = 30
⇒ (x + 1 - x)/x (x + 1) = 30 / 600
⇒ 1/x² + x = 1/20
⇒ x² + x = 20
⇒ x² + x - 20 = 0
⇒ x² + 5x - 4x - 20 = 0
⇒ x (x + 5) - 4 (x + 5) = 0
⇒ (x + 5) (x - 4) = 0

x এর ঋণাত্মক মান গ্রহণযোগ্য নয়।
∴ x - 4 = 0
x = 4

পেছনের চাকার পরিধি = 4 + 1 = 5 m

১০,৭৭০.

If x² + 1/x² = 34, then x + 1/x is equal to-

ক
3
খ
4
গ
5
ঘ
None of these

ব্যাখ্যা

Question: If x² + 1/x² = 34, then x + 1/x is equal to-

Solution:
x² + 1/x² = 34
⇒ (x + 1/x)² - 2.x.(1/x) = 34
⇒ (x + 1/x)² = 36
∴ (x +1/x) = ± 6

১০,৭৭১.

A student bought a bag for Tk. 350 and later sold it for Tk. 400. Find the profit percentage he earned.

ক
17.29%
খ
14.29%
গ
16.57%
ঘ
11.11%

ব্যাখ্যা

Question: A student bought a bag for Tk. 350 and later sold it for Tk. 400. Find the profit percentage he earned.

Solution:
Here,
CP = 350, and SP = 400

As SP > CP,

∴ Profit = SP - CP = 400 - 350 = 50.

Profit% = (50/350) ×100%

=100/7 %
= 14.29%

১০,৭৭২.

A certain number of men can complete a piece of work in 180 days. If there are 30 men less, it will take 20 days more for the work to be completed. How many men were there originally?

ক
135
খ
165
গ
150
ঘ
180
ঙ
300

ব্যাখ্যা

Question: A certain number of men can complete a piece of work in 180 days. If there are 30 men less, it will take 20 days more for the work to be completed. How many men were there originally?

Solution:
Let there be x men originally.
They were to complete the work in 180 days but as the number of persons is reduced to x - 30.
∴ Work takes 20 more days.

So the equation is
180x = (x - 30)200
⇒ 180x = 200x - 6000
⇒ 20x = 6000
⇒ x = 300

১০,৭৭৩.

A Triangle has a perimeter 13. The two shorter sides have integer lengths equal to x and x + 1. Which of the following could be the length of the other side?

ক
ক) 3
খ
খ) 4
গ
গ) 10
ঘ
ঘ) 6

ব্যাখ্যা

The SHORTER sides have integral lengths equal to x and x + 1

Let the longest side be 'a'

So, a + x + (x +1) = 13
Or, a + 2x = 12 .......(1)

We know that the sum of the lengths of the shorter sides has to be more than the length of the longer one
Looking at the options, we can't have 8 or 10 as values for 'a'

Similarly, we can't have 2 or 4 as values for 'a' as it wouldn't be the longest side then.

So, the correct length of other side is 6

১০,৭৭৪.

An outlet pipe can empty a cistern in 30 min, then what part of the cistern will it empty in 1 min?

ক
ক) 1/20
খ
খ) 1/15
গ
গ) 1/25
ঘ
ঘ) 1/30
ঙ
ঙ) 1/18

ব্যাখ্যা

We know that, when a pipe empties a cistern in 'n' min, then the part emptied by the pipe in 1 min = 1/n
Here, n = 30
Therefore, Required part of the tank emptied in 1 min = 1/30part

১০,৭৭৫.

A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number of choices available to him is

ক
192
খ
196
গ
189
ঘ
183
ঙ
194

ব্যাখ্যা

If the student answers 4 questions out of the first five questions he can choose 6 questions from the remaining 8 questions.
Number of combinations will be = ⁵C₄ × ⁸C₆ = 140
If the student answers 5 questions from the first five questions he can choose 5 questions from the remaining 8 questions; Number of combinations will be = ⁵C₅ × ⁸C₅ = 56
So, total number of choices are = 140 + 56
= 196

১০,৭৭৬.

Find the midpoint of the line segment joining the points P₁ = (- 5, - 2) and P₂ = (1, 6).

ক
(2, - 2)
খ
(3, 4)
গ
(- 2, 2)
ঘ
(- 3, 1)

ব্যাখ্যা

Question: Find the midpoint of the line segment joining the points P₁ = (- 5, - 2) and P₂ = (1, 6).

Solution:
দেওয়া আছে, P₁ = (- 5, - 2) এবং P₂ = (1, 6)

আমরা জানি, দুটি বিন্দুর (x₁, y₁) এবং (x₂, y₂) সংযোগকারী রেখাংশের মধ্যবিন্দু নির্ণয়ের সূত্র হলো:
মধ্যবিন্দু = {(x₁ + x₂)/2 , (y₁+y₂)/2}

∴ মধ্যবিন্দু = {(- 5 + 1)/2 , (- 2 + 6)/2}
= ( - 4/2 , 4/2 )
= (- 2, 2)

সুতরাং, নির্ণেয় মধ্যবিন্দুটি হলো (- 2, 2)।

১০,৭৭৭.

Two trains of length 140 meters and 166 meters are moving towards each other on parallel tracks at a speed of 50 km/hr and 60 km/hr respectively. In what time the trains will cross each other from the moment they meet?

ক
8 seconds
খ
12 seconds
গ
9 seconds
ঘ
11 seconds
ঙ
None of these

ব্যাখ্যা

Question: Two trains of length 140 meters and 166 meters are moving towards each other on parallel tracks at a speed of 50 km/hr and 60 km/hr respectively. In what time the trains will cross each other from the moment they meet?

Solution:
In this problem, both the trains are moving so we will find the relative speed of the train. They are moving in the opposite direction, so the relative speed will be sum of their individual speeds.

Relative speed: (50 + 60) = 110 km/hr

Relative speed in m/s = 110 × (5/18) m/s = 550/18 m/s = 275/9 m/s

Distance covered is equal to the sum of the length of trains: 140 + 166= 306 meters

Time = 306 × (9/275) sec
= 10.01 sec
≈ 10 sec

১০,৭৭৮.

Some principal becomes Tk. 1,750 as profit-principal in 5 years and Tk. 2,250 as profit-principal in 10 years. Find the rate of profit.

ক
8%
খ
10%
গ
15%
ঘ
5%

ব্যাখ্যা

Question: Some principal becomes Tk. 1,750 as profit-principal in 5 years and Tk. 2,250 as profit-principal in 10 years. Find the rate of profit.

Solution:
Given,
Principal + Profit for 5 years = 1,750
Principal + Profit for 10 years = 2,250

So the profit of 5 years, I = 2250 - 1750 = Tk. 500
So the principal, P= 1750 - 500 = Tk. 1250

∴ The rate of profit, r = (I × 100)/(P × n)
= (500 × 100)/(1250 × 5)
= 50000/6250
= 8%

১০,৭৭৯.

If a carton containing a dozen mirrors is dropped. which of the following cannot be the ratio of broken mirrors to unbroken mirrors?

ক
ক) 2 : 1
খ
খ) 3 : 1
গ
গ) 3 : 2
ঘ
ঘ) 7 : 5
ঙ
ঙ) None of these

ব্যাখ্যা

অনুপাতগুলোর যোগফল ১২ এর উৎপাদক হতে হবে। এখানে 3:2 এর যোগফল 12 এর কোনো উৎপাদক নয়। সুতরাং, সঠিক উত্তর হবে 3:2।

১০,৭৮০.

A milkman purchases the milk at Tk. x per liter and sells it a Tk. 2x per liter still he mixes 2 Iiters water with every 6 liters of pure milk. What is the profit percentage?

ক
ক) 116%
খ
খ) 166.66%
গ
গ) 60%
ঘ
ঘ) 100%

ব্যাখ্যা

Question: A milkman purchases the milk at Tk. x per liter and sells it a Tk. 2x per liter still he mixes 2 Iiters water with every 6 liters of pure milk. What is the profit percentage?

Solution:
the buying price of 6 litres of milk is = 6x Tk.

after mixing 2 litres of water the total amount of mixture is = 8 litres

selling 2x per litres the total selling price is = 16x
∴ profit = 16x - 6x = 10x

profit in percentage = (10x/6x)100%
= 166.66%

১০,৭৮১.

Discussing grocery rates with her husband, a wife says, Rice is 30% cheaper than wheat. Now, find out how much percent is the price of wheat more than that of rice?

ক
ক) 37.5%
খ
খ) 40.65%
গ
গ) 42.85%
ঘ
ঘ) 45%

ব্যাখ্যা

Suppose, 1 Kg Wheat = Tk. 100
So, 1 Kg Rice will be Tk. 100 - (30% of 100)
= 100 - 30
= Tk. 70

According to the question,
(100/70) × 100 = 142.85%
So, Wheat is 142.85% of Rice
⇒ It is (142.85 - 100)
= 42.85% more than that of Rice.

১০,৭৮২.

There is a leak at the bottom of a cistern. Due to this it takes 8 hours to fill the cistern. Had there not been a leak, it would take one hour less to fill the cistern. How much time does it take for the leak to completely empty the cistern?

ক
47 hours
খ
51 hours
গ
49 hours
ঘ
56 hours

ব্যাখ্যা

Question: There is a leak at the bottom of a cistern. Due to this it takes 8 hours to fill the cistern. Had there not been a leak, it would take one hour less to fill the cistern. How much time does it take for the leak to completely empty the cistern?

Solution:
Normally, the cistern gets filled in one hour less than 8 hours which means 7 hours.
So in 1 hour it fills = 1/7 parts
Due to leak, it takes 8 hours. So in 1 hour it actually fills = 1/8 parts

∴ Water removed by the leak in 1 hour = (1/7) - (1/8)
= (8 - 7)/56
= 1/56
∴ Leak empties the tank in 56 hours.

১০,৭৮৩.

A tap can fill a tank in 6 hours. After half the tank is filled another similar tap is opened. What is the total time taken to fill the tank completely?

ক
ক) 3 h 30 m
খ
খ) 3 h 45 m
গ
গ) 4 h 30 m
ঘ
ঘ) 4 h

ব্যাখ্যা

The tap in 1 hour can fill 1/6 part
After 3 hour, 1/2 of the tank is filled
Two tap in 1 hour can fill 1/3 part

So, two tap can fill 1/3 part in 1 hour
They can fill 1 part of the tap in 3/1 = 3 hour
Thus, can fill 1/2 part of the tap in 3×1/2 = 3/2 = 1 and 1/2 hour = 1 hour 30 minute

Total required time = 3 hour + 1 hour and 30 minute = 4 hour and 30 minute

১০,৭৮৪.

The cost of renting a small bus for a trip was Taka X which was to be shared equally by 16 persons. Actually 10 persons availed the trip. How much more Taka will be the cost per person?

ক
ক) X/6
খ
খ) X/10
গ
গ) 3X/40
ঘ
ঘ) X/16
ঙ
ঙ) 3X/80

ব্যাখ্যা

Question: The cost of renting a small bus for a trip was Taka X which was to be shared equally by 16 persons. Actually 10 persons availed the trip. How much more Taka will be the cost per person?

Solution:
For 16 persons,
Initial cost per person = X/16

For 10 persons
Actual cost per person = X/10

More cost will be = X/10 - X/16
= (8X - 5X)/80
= 3X/80

১০,৭৮৫.

If a² = b⁴ = c⁶ = d⁸ , then the value of log _a(abcd) is ?

ক
12/25
খ
1/12
গ
25/12
ঘ
5/12

ব্যাখ্যা

Question: If a² = b⁴ = c⁶ = d⁸ , then the value of log _a(abcd) is ?

Solution:
b⁴= a²
⇒ b = a^1/2

c⁶ = a²
⇒ c = a^1/3

d⁸ = a²
⇒ d = a^1/4

∴ log a(abcd)
= log_a(a × a^1/2 × a^1/3 × a^1/4)
= log _a (a^25/12)
= 25/12 log_aa
= 25/12

১০,৭৮৬.

In a boat race, a person rows a boat 8 km upstream and returns to the starting point in 3 hours. If the speed of the stream is 2 km/hr, find the speed of the boat in still water.

ক
4 km/hr
খ
6 km/hr
গ
5.5 km/hr
ঘ
4.5 km/hr

ব্যাখ্যা

Question: In a boat race, a person rows a boat 8 km upstream and returns to the starting point in 3 hours. If the speed of the stream is 2 km/hr, find the speed of the boat in still water.

সমাধান:
ধরি, স্থির পানিতে নৌকার গতিবেগ = x কিমি/ঘন্টা।
স্রোতের গতিবেগ = 2 কিমি/ঘন্টা।

স্রোতের প্রতিকূলে (Upstream) নৌকার গতিবেগ = (x - 2) কিমি/ঘন্টা।
স্রোতের অনুকূলে (Downstream) নৌকার গতিবেগ = (x + 2) কিমি/ঘন্টা।

মোট সময় = স্রোতের প্রতিকূলে যেতে সময় + স্রোতের অনুকূলে যেতে সময়
⇒ 3 = 8/(x - 2) + 8/(x + 2)
⇒ 3 = 8{1/(x - 2) + 1/(x + 2)}
⇒ 3/8 = (x + 2 + x - 2)/(x - 2)(x + 2)
⇒ 3/8 = 2x/(x² - 4)
⇒ 3(x² - 4) = 8(2x)
⇒ 3x² - 12 = 16x
⇒ 3x² - 16x - 12 = 0
⇒ 3x² - 18x + 2x - 12 = 0
⇒ 3x(x - 6) + 2(x - 6) = 0
⇒ (3x + 2)(x - 6) = 0

সুতরাং, x এর সম্ভাব্য মান হলো 6 অথবা - 2/3
যেহেতু গতিবেগ ঋণাত্মক হতে পারে না, তাই x = 6 কিমি/ঘন্টা।

সুতরাং, স্থির পানিতে নৌকার গতিবেগ হলো 6 কিমি/ঘন্টা।

১০,৭৮৭.

Which of the following numbers is divisible by 3?

ক
177
খ
182
গ
220
ঘ
331

ব্যাখ্যা

Question: Which of the following numbers is divisible by 3?

Solution:
কোন সংখ্যা 3 দ্বারা বিভাজ্য হলে সংখ্যাটির অঙ্কগুলোর যোগফল 3 দ্বারা বিভাজ্য হবে।

প্রদত্ত সংখ্যা গুলোর মধ্যে,
177 → 1 + 7 + 7 = 15, যা 3 দ্বারা বিভাজ্য।

182 → 1 + 8 + 2 = 11 , যা 3 দ্বারা বিভাজ্য নয়।

220 → 2 + 2 + 0 = 4 , যা 3 দ্বারা বিভাজ্য নয়।

331 → 3 +3 + 1 = 7 , যা 3 দ্বারা বিভাজ্য নয়।

সুতরাং, 177 সংখ্যাটি 3 দ্বারা বিভাজ্য।

১০,৭৮৮.

In how many ways can 5 examination papers be arranged so that the best and the worst papers never come together?

ক
58 ways
খ
66 ways
গ
72 ways
ঘ
76 ways

ব্যাখ্যা

Question: In how many ways can 5 examination papers be arranged so that the best and the worst papers never come together?

Solution:
Total ways = 5!
= 120 ways

if two papers come together, we can consider them one.
ways that they will come together = 4! × 2!
= 24 × 2
= 48 ways

∴ ways the best and the worst papers never come together = 120 - 48 ways
= 72 ways

১০,৭৮৯.

Tk. 1400 is divided among Aman, Belal and Shipon so that Aman receives half as much as Belal and Belal half as much as Shipon. Then Shipon's share is-

ক
Tk. 800
খ
Tk. 600
গ
Tk. 300
ঘ
Tk. 200

ব্যাখ্যা

Question: Tk. 1400 is divided among Aman, Belal and Shipon so that Aman receives half as much as Belal and Belal half as much as Shipon. Then Shipon's share is-

Solution:
Let Aman's share = Tk. x
Then
Belal's share = Tk. 2x
Shipon's share = Tk. 4x

Aman : Belal : Shipon = x : 2x : 4x = 1 : 2 : 4
Hence, Shipon's share = 1400 × (4/7) = Tk. 800

১০,৭৯০.

A can finish a work in 5 days and B takes 4 days to do the same work. If the work is increased 8 times, how many days will it take for both of them to finish the work if they work together?

ক
ক) 10 days
খ
খ) 15 days
গ
গ) 20 days
ঘ
ঘ) 25 days

ব্যাখ্যা

Question: A can finish a work in 5 days and B takes 4 days to do the same work. If the work is increased 8 times, how many days will it take for both of them to finish the work if they work together?

Solution:
A 1 দিনে করতে পারে কাজটির = 1/5 অংশ
B 1 দিনে করতে পারে কাজটির = 1/4 অংশ

A এবং B 1 দিনে করতে পারে কাজটির = (1/5) + (1/4)অংশ
= (4 + 5)/20
= 9/20

8 গুণ বৃদ্ধিতে কাজের মোট পরিমাণ = (1 + 8) = 9 গুণ

9/20 অংশ কাজ করে = ১ দিনে
1 অংশ কাজ করে = 20/9 দিনে
9 অংশ কাজ করে = (20 × 9)/9 দিনে
= 20 দিন

১০,৭৯১.

A job can be done by 3 skilled workmen in 20 days or by 5 boys in 30 days. How many days will they take if they work together?

ক
6 days
খ
8 days
গ
10 days
ঘ
12 days

ব্যাখ্যা

Question: A job can be done by 3 skilled workmen in 20 days or by 5 boys in 30 days. How many days will they take if they work together?

Solution:
3 men's 1 days work = 1/20 and 5 boy's 1 day's work = 1/30
(3 men + 5 boy)'s 1 day's work = (1/20 + 1/30)
= 5/60 = 1/12

∴ 3 men and 5 boys will complete the work in 12 days.

১০,৭৯২.

What is the ratio of 4 inches to 7 feet?

ক
1 : 12
খ
1 : 18
গ
1 : 21
ঘ
1 : 25
ঙ
None of the above

ব্যাখ্যা

Question: What is the ratio of 4 inches to 7 feet?

Solution:
We know,
1 feet = 12 inches
So, 7 feet = 7 × 12
= 84 inches

Now,
4 inches : 7 feet = 4 : 84 = 1 : 21

১০,৭৯৩.

Solve |2x + 5| < 14.

ক
0 < x < 5
খ
(- 19/2) < x < 0
গ
(- 19/2) < x < (9/2)
ঘ
0 < x < (9/2)

ব্যাখ্যা

Question: Solve |2x + 5| < 14.

Solution:
We have,
|2x + 5| < 14
⇒ - 14 < 2x + 5 < 14
⇒ - 19 < 2x < 9
⇒ (- 19/2) < x < (9/2)

১০,৭৯৪.

Three numbers are in ratio 1 : 2 : 3 and HCF is 12. The numbers are:

ক
12, 24, 36
খ
11, 22, 33
গ
5, 10, 25
ঘ
5, 10, 15
ঙ
12, 24, 32

ব্যাখ্যা

Each one's common factor is HCF.
Here, HCF = 12,
Hence, the numbers are 12, 24 and 36.

১০,৭৯৫.

If p is the circumference of the circle Q and the area of the circle is 25π, what is the value of p?

ক
25
খ
10π
গ
35
ঘ
25π

ব্যাখ্যা

Question: If p is the circumference of the circle Q and the area of the circle is 25π, what is the value of p?

Solution:
বৃত্তের ক্ষেত্রফল (A) = πr²

প্রশ্নানুসারে, বৃত্তের ক্ষেত্রফল 25π।
∴ πr²= 25π
⇒ r²= 25
⇒ r = √25
⇒ r = 5

সুতরাং, বৃত্তের ব্যাসার্ধ (r) হলো 5।
এখন, বৃত্তের পরিধি (p) = 2πr
∴ p = 2π(5)
⇒ p = 10π

সুতরাং p-এর মান 10π

১০,৭৯৬.

A, B, C subscribe Tk. 50,000 for business. A subscribes Tk. 4000 more than B and B Tk. 5000 more than C. Out of a total profit of Tk. 35,000, A receives-

ক
ক) 15000
খ
খ) 15500
গ
গ) 17400
ঘ
ঘ) 14700

ব্যাখ্যা

Question: A, B, C subscribe Tk. 50,000 for business. A subscribes Tk. 4000 more than B and B Tk. 5000 more than C. Out of a total profit of Tk. 35,000, A receives-

Solution:
Let
C subscribes = x.

Then,
B = x + 5000 and
A = x + 5000 + 4000
= x + 9000

So,
x + x + 5000 + x + 9000 = 50000
⇒ 3x = 36000
⇒ x = 12000

A : B : C = 21000 : 17000 : 12000 = 21 : 17 : 12

So A's Share = 35000×(21/50)
= 14700

১০,৭৯৭.

A thief steals a scooter at 10:00 a.m. and drives away at 30 km/h. The theft is discovered at 11:00 a.m. and the owner immediately starts chasing the thief in a car at 50 km/h. At what time will the owner catch the thief?

ক
12 : 30 p.m.
খ
1 : 00 p.m.
গ
2 : 00 p.m.
ঘ
2 : 30 p.m.

ব্যাখ্যা

Question: A thief steals a scooter at 10:00 a.m. and drives away at 30 km/h. The theft is discovered at 11:00 a.m. and the owner immediately starts chasing the thief in a car at 50 km/h. At what time will the owner catch the thief?

Solution:
চোর কর্তৃক অতিক্রান্ত প্রারম্ভিক দূরত্ব (Head Start):
সময় পার্থক্য = 11:00 a.m. - 10:00 a.m. = 1 ঘন্টা।
চোর কর্তৃক অতিক্রান্ত দূরত্ব = 30 × 1 কিমি = 30 কিমি।

আপেক্ষিক গতিবেগ = (মালিকের গতিবেগ - চোরের গতিবেগ)
= (50 - 30) কিমি/ঘন্টা = 20 কিমি/ঘন্টা।

চোরকে ধরতে প্রয়োজনীয় সময় = দূরত্ব/আপেক্ষিক গতিবেগ
= 30/20 ঘন্টা
= 3/2 ঘন্টা
= 1 ঘন্টা 30 মিনিট।

ধাওয়া শুরু হয়েছিল 11 : 00 a.m. এ।
∴ চোরকে ধরার সময়সময় = 11 : 00 a.m. + 1 ঘন্টা 30 মিনিট
= 12 : 30 p.m.
∴ মালিক চোরটিকে 12 : 30 p.m. এ ধরে ফেলবে।

১০,৭৯৮.

Two trains running in opposite directions cross a man standing on the platform in 27 seconds and 17 seconds respectively and they cross each other in 23 seconds. The ratio of their speeds is:

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

ব্যাখ্যা

প্রশ্ন: Two trains running in opposite directions cross a man standing on the platform in 27 seconds and 17 seconds respectively and they cross each other in 23 seconds. The ratio of their speeds is:

সমাধান:
Let,
The speeds of first train be x m/sec
The speeds of second train be y m/sec

Then,
length of the first train = 27x metres,
length of the second train = 17y metres.

ATQ,
(27x + 17y)/(x+ y) = 23
⇒ 27x + 17y = 23x + 23y
⇒ 4x = 6y
⇒ x/y = 6/4
⇒ x/y = 3/2
∴ x : y = 3 : 2

১০,৭৯৯.

A man covers half of his journey at 6 km/h and the remaining half at 3 km/h. His average speed is-

ক
9 km/h
খ
3 km/h
গ
4 km/h
ঘ
3.5 km/h

ব্যাখ্যা

Question: A man covers half of his journey at 6 km/h and the remaining half at 3 km/h. His average speed is-

Solution:
Let the total distance be 2d km.
First half at 6 km/h.
Second half at 3 km/h.

Now,
Time for first half, t₁ = d/6 hours
Time for second half, t₂ = d/3 hours

∴ Total Time = (d/6) + (d/3) = 3d/6 = d/2

∴ Average speed = Total distance/Total time = 2d/(d/2) = 4 km/h

∴ Average speed = 4 km/h

১০,৮০০.

A 125-meter-long train overtakes a person moving at 5 km/h in 10 seconds, while both are moving in the same direction. What is the speed of the train?

ক
12.5 km/h
খ
24 km/h
গ
40 km/h
ঘ
50 km/h

ব্যাখ্যা

Question: A 125-meter-long train overtakes a person moving at 5 km/h in 10 seconds, while both are moving in the same direction. What is the speed of the train?

Solution:
ট্রেনটি ব্যক্তিকে অতিক্রম করতে ট্রেনের নিজের দৈর্ঘ্যের সমান দূরত্ব অতিক্রম করে।

∴ আপেক্ষিক গতিবেগ = 125m/10s
= 12.5 m/s
= (12.5/1000)/(1/3600) km/h
= (12.5 × 3600)/1000 km/h
= 45 km/h

ধরি,
ট্রেনের গতিবেগ = x km/h

দেওয়া আছে,
ব্যক্তির গতিবেগ = 5 km/h

আমরা জানি,
আপেক্ষিক গতিবেগ = ট্রেনের গতিবেগ - ব্যক্তির গতিবেগ
⇒ ট্রেনের গতিবেগ = আপেক্ষিক গতিবেগ + ব্যক্তির গতিবেগ
⇒ x = (45 + 5) km/h
⇒ x = 50 km/h