Bank Math

PrepBank · পাতা ৮১ / ১৬১ · ৮,০০১–৮,১০০ / ১৬,১২৪

৮,০০১.

The average weight of 3 friends is 33 kg. None of the friends weights less than 31 kg. What can be the maximum weight of any three friends?

ক
ক) 37
খ
খ) 35
গ
গ) 33
ঘ
ঘ) 32

ব্যাখ্যা

Question: The average weight of 3 friends is 33 kg. None of the friends weights less than 31 kg. What can be the maximum weight of any three friends?

Solution:
তিনজনের গড় ওজন ৩৩ কেজি
মোট ওজন ৩৩ × ৩ কেজি
= ৯৯ কেজি

প্রতিজনের ওজন সর্বনিম্ন ৩১ কেজি
দুজনের সর্বনিম্ন ওজন ৩১ × ২ কেজি
= ৬২ কেজি

একজনের সর্বোচ্চ ওজন হতে পারে = ৯৯ - ৬২ কেজি
= ৩৭ কেজি

৮,০০২.

What would be the measure of the perimeter of a square whose area is equal to 225 square cm?

ক
60 cm
খ
55 cm
গ
50 cm
ঘ
45 cm

ব্যাখ্যা

Question: What would be the measure of the perimeter of a square whose area is equal to 225 square cm?

Solution:
দেওয়া আছে,
বর্গক্ষেত্রের ক্ষেত্রফল = 225 বর্গ সেমি
এক বাহুর দৈর্ঘ্য = a

প্রশ্নমতে,
a² = 225
⇒ a² = 15²
∴ a = 15

∴ বর্গক্ষেত্রের পরিসীমা = 4a
= 4 × 15 = 60 সেমি

৮,০০৩.

After traveling 35km, a train meets with an accident and travels at (3/4)^th of the usual Speed and reaches 45 min late. Had the accident happened 15km further on it would have reached 30 min late. Find the usual Speed?

ক
20 km/h
খ
10 km/h
গ
30 km/h
ঘ
35 km/h

ব্যাখ্যা

Question: After traveling 35km, a train meets with an accident and travels at (3/4)th of the usual Speed and reaches 45 min late. Had the accident happened 15km further on it would have reached 30 min late. Find the usual Speed?

Solution:
Here there are 2 cases
Case 1: accident happens at 35 km
Case 2: accident happens at 50 km
Difference between two cases is only for the 15 kms between 35 and 50.
Time difference of 15 minutes is only due to these 15 kms.

In case 1, 15 kms between 35 and 50 is covered at (3/4)th Speed.
In case 2, 15 kms between 35 and 50 is covered at usual Speed.

So the usual Time 't' taken to cover 15 kms, can be found out as below.
(4/3)t - t = 15 mins [s = vt]
⇒ t = 45 mins, d = 15kms

so usual
Speed = Distance/Time
= 15/45min
= 15/(3/4) km/hr.
= 20 km/hr.

৮,০০৪.

If θ = 60° , then what is the value of (1 - sec²θ)/(1 + sec²θ)?

ক
4/5
খ
0
গ
1/2
ঘ
- 3/5

ব্যাখ্যা

Question: If θ = 60° , then what is the value of (1 - sec²θ)/(1 + sec²θ)?

Solution:
Here, θ = 60°

Now,
(1 - sec²θ)/(1 + sec2θ)
= {1 - (sec60°)²}/{1 + (sec60°)²}
= (1 - 2²)/(1 + 2²)
= (1 - 4)/(1 + 4)
= - 3/5

৮,০০৫.

A can complete a piece of work in 36 days, B in 54 days and C in 72 days. All the three began the work together but A left 8 days before the completion of the work and B 12 days before the completion of work. Only C worked up to the end. In how many days was the work completed?

ক
20
খ
22
গ
24
ঘ
30

ব্যাখ্যা

Question: A can complete a piece of work in 36 days, B in 54 days and C in 72 days. All the three began the work together but A left 8 days before the completion of the work and B 12 days before the completion of work. Only C worked up to the end. In how many days was the work completed?

Solution:
Let the work be completed in y days. C works for y days
Therefore, A works for (y - 8) days
B works for (y - 12) days.

According to the question,
{(y - 8)/36} + {(y -12)/54} + (y/72) = 1
or, 6(y - 8) + 4(y - 12) + 3y = 216
or, 6y - 48 + 4y - 48 + 3y = 216
or, 13y = 216 + 96 = 312
or, y = 312/13
or, y = 24

৮,০০৬.

Find the single discount equivalent to two successive discounts of 20% and 10%

ক
18%
খ
24%
গ
30%
ঘ
28%

ব্যাখ্যা

Question: Find the single discount equivalent to two successive discounts of 20% and 10%

Solution:

Let the marked price of an article be Tk.100

Then first discount on it = Tk. 20

Price after discount = Tk. (100 - 20) = Tk. 80

Second discount on it = 10% of 80
= 80 × 10/100
= 8

Price after second discount = Tk. (80 - 8) = 72

Single discount equivalent to given successive discounts

=(100 - 72)%
= 28%

৮,০০৭.

A train 400 meters long passes a pole in 16 seconds. How long will it take to pass a platform that is 800 meters long?

ক
48 seconds
খ
36 seconds
গ
42 seconds
ঘ
44 seconds

ব্যাখ্যা

Question: A train 400 meters long passes a pole in 16 seconds. How long will it take to pass a platform that is 800 meters long?

Solution:
Train's speed = Distance/Time
= 400/16 = 25m/s

Total distance to pass the platform,
= Length of train + Length of platform
= 400 + 800
= 1200 meters

∴ Required time to pass platform = Distance/Speed
= 1200/25
= 48 seconds

৮,০০৮.

A train 175 m long crosses a bridge which is 125 m long in 1 min 40 seconds. What is the speed of the train?

ক
2.5 m/s
খ
3 m/s
গ
3.5 m/s
ঘ
4 m/s

ব্যাখ্যা

Question: A train 175 m long crosses a bridge which is 125 m long in 1 min 40 seconds. What is the speed of the train?

Solution:
মোট অতিক্রান্ত দূরত্ব = ১৭৫ + ১২৫ মিটার
= ৩০০ মিটার

সময় = ১ মিনিট ৪০ সেকেন্ড
= ৬০ + ৪০ সেকেন্ড
= ১০০ সেকেন্ড

বেগ = ৩০০/১০০ মিটার/সেকেন্ড
= ৩ মিটার/সেকেন্ড

৮,০০৯.

Complete the following series:
130, 222, ?, 520

ক
510
খ
484
গ
441
ঘ
350

ব্যাখ্যা

Question: Complete the following series:
130, 222, ?, 520

Solution:
130 = 5³ + 5,
222 = 6³ + 6,
?
520 = 8³ + 8,

As the following series we get:
7³ + 7 = 343 + 7 = 350,
So missing number is 350.

৮,০১০.

If √(2ⁿ) = 128, what will be the value of n?

ক
16
খ
14
গ
12
ঘ
18

ব্যাখ্যা

√(2ⁿ) = 128 = 2⁷
⇒ 2ⁿ = (2⁷)² = 2¹⁴
⇒ n = 14

৮,০১১.

The cost price of an article is Tk 7840. What should be the selling price of the article so that there is a profit of 7%?

ক
ক) Tk 8388.80
খ
খ) Tk 8300
গ
গ) Tk 8000
ঘ
ঘ) Tk 8500.50

ব্যাখ্যা

Question: The cost price of an article is Tk 7840. What should be the selling price of the article so that there is a profit of 7%?

Solution:
৭% লাভে বিক্রয়মূল্য = ১০০ + ৭ =১০৭ টাকা

ক্রয়মূল্য ১০০ টাকা হলে বিক্রয়মূল্য ১০৭ টাকা
∴ ক্রয়মূল্য ১ টাকা হলে বিক্রয়মূল্য ১০৭/১০০ টাকা
∴ ক্রয়মূল্য ৭৮৪০ টাকা হলে বিক্রয়মূল্য (১০৭ × ৭৮৪০)/১০০ টাকা
= ৮৩৮৮.৮০ টাকা

৮,০১২.

Disease : Health : : Freedom : ?

ক
ক) Slavery
খ
খ) Pleasure
গ
গ) Plight
ঘ
ঘ) Beauty

ব্যাখ্যা

Disease is the antonym of Health and the antonym of Freedom will be Slavery.

৮,০১৩.

Three numbers are in the ratio 3 : 4 : 6 and their products is 1944. The largest number is -

ক
ক) 18
খ
খ) 24
গ
গ) 12
ঘ
ঘ) 6

ব্যাখ্যা

Question: Three numbers are in the ratio 3 : 4 : 6 and their products is 1944. The largest number is -

Solution:
Let the number be 3x, 4x, 6x.

ATQ,
3x × 4x × 6x = 1944
⇒ 72x³ = 1944
⇒ x³ = 1944/72
⇒ x³ = 27
∴ x = 3

So largest number = 6x = 6 × 3 = 18

৮,০১৪.

If the cost price of an item is 5/9 of its marked price and the profit is 20%, then the percentage of discount is =?

ক
33.33%
খ
29.67%
গ
35.67%
ঘ
30.33%

ব্যাখ্যা

Question: If the cost price of an item is 5/9 of its marked price and the profit is 20%, then the percentage of discount is =?

Solution:
let the Marked price of product = 900
∴ cost price will be = 900 × (5/9) = 500

Profit = 20%
Selling price will be = 500 + 20% of 500 = 600
Discounted value = 900 - 600 = 300

Discount % = (100 × 300)/900

৮,০১৫.

Five students are to be arranged on five chairs for a photograph. Three of these are girls and the rest are boys. Find out the number of ways in which all three girls do not occupy consecutive seats.

ক
ক) 120
খ
খ) 36
গ
গ) 84
ঘ
ঘ) 76

ব্যাখ্যা

As per the question, three girls can’t occupy consecutive seats but two can.

Therefore, if we find the number of ways in which all three girls occupy consecutive seats and subtract this number from the total number of ways in which the five people can be arranged among themselves, we will get the required answer.

5 students can be arranged among themselves in ⁵p₅ ways = 120 ways.

Assume that the 3 girls are one entity. The total number of ways in which they can be arranged among themselves = 3! = 6
Also, the set of three girls and the other students can be arranged among themselves in 3! = 6 ways.
Thus, the total number of ways in which three girls are together = 6 × 6 = 36

Thus, a number of ways in which all 3 girls will not occupy consecutive seats = 120 – 36 = 84.

৮,০১৬.

Two taps, P and Q, can fill a tank in 6 and 8 minutes respectively. A leak (outlet pipe) R can empty 20 liters of water per minute. If all three are opened together, the tank is filled in 12 minutes. What is the capacity of the tank in liters?

ক
100 liters
খ
96 liters
গ
120 liters
ঘ
136 liters

ব্যাখ্যা

Question: Two taps, P and Q, can fill a tank in 6 and 8 minutes respectively. A leak (outlet pipe) R can empty 20 liters of water per minute. If all three are opened together, the tank is filled in 12 minutes. What is the capacity of the tank in liters?

সমাধান:
প্রথম নল P এর 1 মিনিটে পূর্ণ করে = 1/6 অংশ
দ্বিতীয় নল Q এর 1 মিনিটে পূর্ণ করে = 1/8 অংশ
তিনটি নল একত্রে 1 মিনিটে পূর্ণ করে = 1/12 অংশ।

ছিদ্র নল R এর 1 মিনিটে খালি করার অংশ = (P + Q এর কাজ - সম্মিলিত কাজ)
= (1/6 + 1/8) - 1/12 অংশ
= (4 + 3)/24 - 1/12 অংশ
= 7/24 - 1/12 অংশ
= (7 - 2)/24 অংশ
= 5/24 অংশ।

অর্থাৎ, ছিদ্র নল R একা 5/24 অংশ খালি করে 1 মিনিটে।

ছিদ্র নল R একা সম্পূর্ণ ট্যাঙ্কটি খালি করতে সময় নেয় = 1/(5/24) মিনিট
= 4.8 মিনিট।

ছিদ্র নল R 1 মিনিটে খালি করে 20 লিটার।
∴ ট্যাঙ্কটির মোট ধারণ ক্ষমতা = (মোট সময় × প্রতি মিনিটের নির্গমনের হার)
= (4.8 × 20) লিটার
= 96 লিটার।

৮,০১৭.

The perimeter of a rectangle is 104 inches. The width is 6 inches less than 3 times the length. Find the width of the rectangle.

ক
ক) 13.5 inches
খ
খ) 14.5 inches
গ
গ) 15 inches
ঘ
ঘ) 37.5 inches

ব্যাখ্যা

Let, length = x and Width = 3x - 6
ATQ,
2(x + 3x - 6) = 104
Or, 4x - 6 = 104/2 = 52
Or, 4x = 58
Or, x = 14.5
So, width = 3 × 14.5 - 6= 37.5

৮,০১৮.

A, B and C play cricket. The ratio of A’s runs to B’s runs is 5 : 3 and B’s runs to C’s is 3 ∶ 2. They made a total of 350 runs. How many runs did "B" make?

ক
95
খ
105
গ
120
ঘ
150

ব্যাখ্যা

Question: A, B and C play cricket. The ratio of A’s runs to B’s runs is 5 : 3 and B’s runs to C’s is 3 ∶ 2. They made a total of 350 runs. How many runs did "B" make?

Solution:
Given,
A + B + C = 350
and,
A : B = 5 : 3
B : C = 3 : 2

∴ A : B : C = 5 : 3 : 2

∴ Runs made by "B" = 3/10 × 350 = 105

৮,০১৯.

A blend consists of an equal amount of lemon juice and sugar syrup. When mixed with extra sugar syrup in a 1 : 3 ratio, what is the final ratio of lemon juice to sugar syrup?

ক
1 : 2
খ
1 : 7
গ
1 : 4
ঘ
1 : 9
ঙ
None of the above

ব্যাখ্যা

Question: A blend consists of an equal amount of lemon juice and sugar syrup. When mixed with extra sugar syrup in a 1 : 3 ratio, what is the final ratio of lemon juice to sugar syrup?

Solution:
Let, the new mixture is 12 litres

The old mixture = 12 × (1/4) = 3 liters
The sugar syrup = 9 liters

The new sugar syrup = 9 + (3/2) = 10.5

∴ The final ratio of lemon juice and sugar syrup = 1.5 : 10.5
= 15 : 105
= 1 : 7

[The statement "Let, the new mixture is 12 liters" is used as an assumption to make the calculations easier. By assuming the total amount of the new mixture is 12 liters, it simplifies the process of determining the amounts of lemon juice and sugar syrup in the mixture. This assumption is just a way to set up the problem in a manageable way.

12 liters is a convenient number because it is divisible by 4 (the ratio of lemon juice and sugar syrup in the original mixture), and it helps make the calculation easier for the final ratio.]

৮,০২০.

How many 4 letter code can be formed using the first 9 letters of the English alphabets, if no letter can be repeated?

ক
3024
খ
3036
গ
3021
ঘ
3034

ব্যাখ্যা

Question: How many 4 letter code can be formed using the first 9 letters of the English alphabets, if no letter can be repeated?

Solution:
By fundamental principle, it is (9 × 8 × 7 × 6) ways
= 3024 ways

Alternative:
⁹P₄ = 9!/(9 - 4)!
= 9!/5!
= (9 × 8 × 7 × 6 × 5!)/5!
= 3024 ways

৮,০২১.

If log_x196 = 2 , then x =?

ক
ক) 14
খ
খ) 16
গ
গ) 24
ঘ
ঘ) 26

ব্যাখ্যা

Question: If log_x196 = 2 , then x=?

Solution:
log_x196 = 2
⇒ x² = 196
∴ x = 14

৮,০২২.

Adults can do a job twice as fast as children. If x adults can complete a job in d days, how many children can do the same job in d + 2 days?

ক
ক) 2dx/(d + 2)
খ
খ) (d + 2)x/d
গ
গ) dx/2(d + 2)
ঘ
ঘ) (2x + d)/(d - x)

ব্যাখ্যা

x adults = 2x children
to complete in d days 2x children is required.
∴ to complete in (d + 2) days (2x × d)/(d + 2) children is required.

৮,০২৩.

There are 5 red, 4 black and 3 white color bulbs. If a bulb is picked at random, what is the probability of having either a red or a black bulb?

ক
ক) 2/3
খ
খ) 5/12
গ
গ) 3/4
ঘ
ঘ) none of the above

ব্যাখ্যা

Question: There are 5 red, 4 black and 3 white color bulbs. If a bulb is picked at random, what is the probability of having either a red or a black bulb?

Solution:
Red color bulb = 5
Black color bulb = 4
White color bulb = 5

Total bulb = 5 + 4 + 3 = 12

Total red and black bulb = (5 + 4) = 9
The probability of having either a red or a black bulb = 9/12 = 3/4

৮,০২৪.

A and B are partners in a business. A invests Taka 25,000 for 8 months and B invests Taka 30,000 for 6 months. If the profit at the end of the year is Taka 15,200, what is B's share in the profit?

ক
5600 Taka
খ
5000 Taka
গ
6450 Taka
ঘ
7200 Taka

ব্যাখ্যা

Question: A and B are partners in a business. A invests Taka 25,000 for 8 months and B invests Taka 30,000 for 6 months. If the profit at the end of the year is Taka 15,200, what is B's share in the profit?

Solution:
A’s Contribution = 25000×8 = 200000 Taka
B’s Contribution = 30000×6 = 180000 Taka

The ratio of A's and B's investment = 200000 : 180000
= 10 : 9

So, B's share in profit = {9/(10+9)} × 15200
= 7200 Taka

৮,০২৫.

If 5% more is gained by selling an article for Tk. 250 then by selling it for Tk. 240. The cost of the article is?

ক
ক) Tk. 200
খ
খ) Tk. 150
গ
গ) Tk. 250
ঘ
ঘ) Tk. 300

ব্যাখ্যা

Question: If 5% more is gained by selling an article for Tk. 250 then by selling it for Tk. 240. The cost of the article is?

Solution:
Selling price more = (250 - 240) = Tk. 10

If tk. 5 more gained then cost price 100
If tk. 1 more gained then cost price 100/5
If tk.10 more gained then cost price (100 × 10)/5
= Tk. 200

৮,০২৬.

A group consists of two male, two female and three children. The average age of the male is 67 years, that of the female is 35 years, and that of the children is six years. What is the average age of the group?

ক
30.71
খ
31.71
গ
28.71
ঘ
35.45

ব্যাখ্যা

Question: A group consists of two male, two female and three children. The average age of the male is 67 years, that of the female is 35 years, and that of the children is six years. What is the average age of the group?

Solution:
Total age of two male = 67 × 2 = 134
Total age of two female = 35 × 2 = 70
Total age of three children = 6 × 3 = 18

∴ Average age of the group = (134 + 70 + 18)/(2 + 2 + 3)
= 222/7
= 31.71

৮,০২৭.

A man takes twice as long to row a distance against the stream as to row the same distance in favor of the stream. The ratio of the speed of the boat (in still water) and the stream is:

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

ব্যাখ্যা

Let man's rate upstream be x kmph
Then, his rate downstream = 2x kmph
∴ (speed in still water) : (Speed of stream)
=(2x+x)/2 : (2x−x)/2
= 3x/2 : x/2 = 3 : 1

৮,০২৮.

The height of a tree is √3 times the length of its shadow. Find the angle of elevation of the sun.

ক
30°
খ
45°
গ
60°
ঘ
90°
ঙ
50°

ব্যাখ্যা

Question: The height of a tree is √3 times the length of its shadow. Find the angle of elevation of the sun.

Solution:

Given,
the height of the tree (h) AB = √3 and BC=1
let angle C be Ф

in ΔABC
tanФ= AB/BC
tanФ= √3
∵ Ф = 60°

so, the angle of elevation of the sun is 60°

৮,০২৯.

Find the rate of interest if the amount after 2 years of simple interest on a capital of tk 1200 is tk 1680.

ক
20%
খ
16%
গ
10%
ঘ
12%
ঙ
None of these

ব্যাখ্যা

Question: Find the rate of interest if the amount after 2 years of simple interest on a capital of tk 1200 is tk 1680.

Solution:
given,
Principal, P = 1200 taka
Time, n = 2
Interest, I = 1680 - 1200 = 480
Rate of interest, r = ?

We know that,
I = pnr
⇒ 480 = (1200 × 2 × r)/100
⇒ 480 = 2400 × r/100
⇒ 480 = 24 × r
⇒ r = 480/24
∴ r = 20

So, the rate of interest is 20%

৮,০৩০.

If 3x + 2y = 8 and 2x - y = 3, find the value of 4y - 3x.

ক
2
খ
3
গ
1/2
ঘ
- 2

ব্যাখ্যা

Question: If 3x + 2y = 8 and 2x - y = 3, find the value of 4y - 3x.

Solution:

3x + 2y = 8

2x - y = 3

⇒ 2 (2x - y) = 2 × 3

⇒ 4x - 2y = 6

3x + 2y + 4x - 2y = 8 + 6

⇒ 7x = 14

∴ x = 2

3 × 2 + 2y = 8

⇒ 6 + 2y = 8

⇒ 2y = 8 - 6 = 2

∴ y = 1

4y - 3x = 4 × 1 - 3 × 2

= 4 - 6

= - 2

৮,০৩১.

If cosecθ + cotθ = 2, Find the value of cotθ.

ক
4/3
খ
3/4
গ
3/2
ঘ
2/3

ব্যাখ্যা

Question: If cosecθ + cotθ = 2, Find the value of cotθ.

Solution:
given,
cosecθ + cotθ = 2.......(i)

cosec²θ - cot²θ = 1
or, (cosecθ + cotθ)(cosecθ - cotθ) = 1
or, cosecθ - cotθ = 1/2.......(ii)

subtracting (ii) from (i) we get,
2cotθ = 2 - 1/2
2cotθ = 3/2
cotθ = 3/4

৮,০৩২.

The H.C.F of three numbers is 24. If they are in the ratio 35 : 55 : 77, then the numbers are-

ক
280, 440, 615
খ
105, 175, 231
গ
840, 1320, 1848
ঘ
900, 1400, 1900

ব্যাখ্যা

Question: The H.C.F of three numbers is 24. If they are in the ratio 35 : 55 : 77, then the numbers are-

Solution:
H.C.F of 3 number = 24
Ration of three number = 35 : 55 : 77
Let the three numner be 35x, 55x, and 77x respectively.
H.C.F of 35x, 55x and 77x = x

Also H.C.F of 3 number = 24.
∴ x = 24
35x = 35 × 24 = 840
55x = 55 × 24 = 1320
77x = 77 × 24 = 1848

∴ Three numbers are 840, 1320 and 1848.

৮,০৩৩.

r > s > 0; Quantity A = (rs)/r and Quantity B = (rs)/s

ক
ক) Quantity in A is greater
খ
খ) Quantity in B is greater
গ
গ) The two quantities are equal
ঘ
ঘ) The relationship indeterminate
ঙ
ঙ) None of these

ব্যাখ্যা

Question: r > s > 0; Quantity A = (rs)/r and Quantity B = (rs)/s

Solution:
Given that,
r > s > 0

Let,
r = 3
s = 2

∴ Quantity A = (rs)/r = (3 × 2)/3 = 2

∴ Quantity B = (rs)/s = (3 × 2)/2 = 3

∴ Quantity in B is greater.

৮,০৩৪.

Find the number of factors of 360.

ক
12
খ
18
গ
24
ঘ
30

ব্যাখ্যা

Question: Find the number of factors of 360.

Solution:
Factorize 360 into prime factors:
360 = 2³ × 3² × 5¹
The formula for the number of factors of a number n =p^aq^br^c…n is:
Number of factors = (a + 1) (b + 1)(c + 1)
Apply the formula:
(3 + 1) (2 + 1) (1 + 1) = 4 × 3 × 2 = 24

৮,০৩৫.

A person's present age is two-fifth of the age of his mother. After 8 years, he will be one half of the age of his mother. What is the present age of the mother?

ক
ক) 30
খ
খ) 35
গ
গ) 40
ঘ
ঘ) 50

ব্যাখ্যা

Question: A person's present age is two-fifth of the age of his mother. After 8 years, he will be one half of the age of his mother. What is the present age of the mother?

Solution:
Let the age of the person’s mother be ‘x years
Age of the person = (2x/5)

After 8 years, age of the person’s mother = ‘x + 8’ years
Age of the person after 8 years = (2x/5) + 8 years

According to question:
(2x/5) + 8 = (1/2)(x + 8)
(2x/5) + 8 = (x/2) + 4
(x/2) - (2x/5) = 8 - 4
(5x - 4x)/10 = 4
x/10 = 4
x = 40

So, the present age of his mother is 40 years

৮,০৩৬.

If n is a prime number greater than 3, then (n² - 1) is always divisible by :

ক
12 but not 24
খ
24
গ
6 but not 12
ঘ
None of these

ব্যাখ্যা

Question: If n is a prime number greater than 3, then (n² - 1) is always divisible by :

Solution:
Let, n = 5
⇒ (n² - 1) = (25 - 1)
⇒ (n² - 1) = 24, which is divisible by 24

Let n = 7
⇒ (n²- 1) = (49 - 1)
⇒ (n² - 1) = 48, which is divisible by 24

Let, n = 11
⇒ (n² - 1) = (121 - 1)
⇒ (n² - 1) = 120, which is divisible by 24

Hence, (n² - 1) is always divisible by 24.

৮,০৩৭.

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

ক
32 ways
খ
60 ways
গ
80 ways
ঘ
110 ways

ব্যাখ্যা

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

Solution:
CYCLE whereas total 5 letters and C comes two times.

So, arrangements are = 5!/2!
= 60 ways

৮,০৩৮.

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

ক
ক) 5000Tk.
খ
খ) 10000Tk.
গ
গ) 15000Tk.
ঘ
ঘ) 20000Tk.

ব্যাখ্যা

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

Solution:
Let initially the investment of Tofael is = X
So, the investment of Himel is = 2X

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

hence, the initial investment of Himel is = (2 × 5000) = 10000 Tk.

৮,০৩৯.

The average of the reciprocals of x and y is:

ক
ক) 2xy/(x + y)
খ
খ) (x + y)/2xy
গ
গ) xy/2(x + y)
ঘ
ঘ) 2(x + y)/xy

ব্যাখ্যা

Question: The average of the reciprocals of x and y is:

Solution:
The reciprocals of x and y are = 1/x and 1/y
Required average
= {(1/x) + (1/y)}/2
= {(y + x)/xy}/2
= {(y + x)/xy} × (1/2)
=(x + y)/2xy

৮,০৪০.

Three friends A, B, and C started a business, each investing Tk. 10000. After 5 months A withdrew Tk. 3000, B withdrew Tk. 2000 and C invested Tk. 3000 more. At the end of the year, a total profit of Tk. 34600 was recorded. Find the share of C.

ক
Tk. 9900
খ
Tk. 10600
গ
Tk. 14100
ঘ
Tk. 13500
ঙ
None of these

ব্যাখ্যা

Question: Three friends A, B, and C started a business, each investing Tk. 10000. After 5 months A withdrew Tk. 3000, B withdrew Tk. 2000 and C invested Tk. 3000 more. At the end of the year, a total profit of Tk. 34600 was recorded. Find the share of C.

Solution:
We know that if the period of investment is not uniform, the gains/losses from the business are divided in the ratio of their inputs, where input is calculated as the product of an amount of investment and the time period of investment.
So, input = value of investment × period of investment, and here, the period of investment would be broken into parts as the investment is not uniform throughout the time period.
A’s input = (10000 × 5) + (7000 × 7) = 99000
B’s input = (10000 × 5) + (8000 × 7) = 106000
C’s input = (10000 × 5) + (13000 × 7) = 141000
∴ A : B : C = 99000 : 106000 : 141000
⇒ A : B : C = 99 : 106 : 141

C’s share = (141/346) × 34600 = Tk. 14100

৮,০৪১.

My brother is 3 years elder to me. My father was 28 years of age when my sister was born while my mother was 26 years of age when I was born. if my sister was 4 years of age when my brother was born, then what was the age of my father and mother respectively when my brother was born?

ক
32 years, 29 years
খ
35 years, 33 years
গ
32 years, 23 years
ঘ
35 years, 29 years

ব্যাখ্যা

Question: My brother is 3 years elder to me. My father was 28 years of age when my sister was born while my mother was 26 years of age when I was born. if my sister was 4 years of age when my brother was born, then what was the age of my father and mother respectively when my brother was born?

Solution:
My Mother was 26 years of age when I was born
My brother is 3 years elder to me
Mother's age when my brother was born = 26 - 3 = 23 years

My father was 28 years of age when my sister was born
my sister was 4 years of age when my brother was born
Father's age when my brother was born = 28 + 4 = 32 years

৮,০৪২.

What should be the value of "P" so that the expression (16 − 24x + Px²) becomes a perfect square?

ক
10
খ
9
গ
16
ঘ
8

ব্যাখ্যা

Question: What should be the value of "P" so that the expression (16 − 24x + Px²) becomes a perfect square?

Solution:
(16 − 24x + Px²)
= (4)² − 2 × 4 × 3x + (3x)²+ Px² − (3x)²
= (4 − 3x)² + Px² − 9x²

∴ The expression becomes a perfect square if,
Px² − 9x² = 0
⇒ Px² = 9x²
∴ P = 9

৮,০৪৩.

A machine is sold at a profit of 10%. Had it been sold for Tk. 40 less, there would have been a loss of 10%. What was the cost price?

ক
Tk.175
খ
Tk. 200
গ
Tk. 225
ঘ
Tk. 250

ব্যাখ্যা

Question: A machine is sold at a profit of 10%. Had it been sold for Tk. 40 less, there would have been a loss of 10%. What was the cost price?

Solution:
Let, cost price of the machine is x taka

Selling price = 1.1x taka

ATQ,
1.1x - 40 = 0.9x
⇒ 1.1x - 0.9x = 40
⇒ 0.2x = 40
⇒ x = 40/.2
= Tk. 200

৮,০৪৪.

76 men complete a job in 33 days. Due to some reason, some men did not join the work and therefore it was completed in 44 days. The number of men who did not report for the work is-

ক
19
খ
20
গ
21
ঘ
23

ব্যাখ্যা

Question: 76 men complete a job in 33 days. Due to some reason, some men did not join the work and therefore it was completed in 44 days. The number of men who did not report for the work is-

Solution:
Let L be the rate of work per men.

the total work = 76L × 33

Let, with x men absent, the total work is = (76 - x)L × 44

76L × 33 = (76 - x)L × 44
⇒ 76 × 33 = 76 × 44 - 44L
⇒ 44L = 76 × 44 - 76 × 33 = 76 × 11
⇒ L = 19

৮,০৪৫.

The average of fourteen numbers is 20 and the average of the first eight is 16. What is the average for the rest?

ক
26.25
খ
25.33
গ
28
ঘ
22.75

ব্যাখ্যা

Question: The average of fourteen numbers is 20 and the average of the first eight is 16. What is the average for the rest?

Solution:
The average of the fourteen numbers is = 20
Sum of 14 numbers = 14 × 20 = 280

The average of the first 8 numbers is = 16
Sum of first 8 numbers = 8 × 16 = 128

Total of remaining six numbers = 280 - 128 = 152
Average of the rest = 152/6 = 25.33

৮,০৪৬.

A large tanker can be filled by two pipes A and B in 60 and 40 minutes respectively. How many minutes will it take to fill the tanker from empty state if B is used for half the time and A and B fill it together for the other half ?

ক
ক) 20 minutes
খ
খ) 30 minutes
গ
গ) 35 minutes
ঘ
ঘ) 40 minutes

ব্যাখ্যা

Question: A large tanker can be filled by two pipes A and B in 60 and 40 minutes respectively. How many minutes will it take to fill the tanker from empty state if B is used for half the time and A and B fill it together for the other half ?

Solution:
A ৬০ মিনিটে পূর্ণ করে ১ অংশ
১ মিনিটে পূর্ণ করে ১/৬০ অংশ

B ৪০ মিনিটে পূর্ণ করে ১ অংশ
১ মিনিটে পূর্ণ করে ১/৪০ অংশ

A,B একসাথে ১ মিনিটে পূর্ণ করে ১/৬০ + ১/৪০
= ৫/১২০
= ১/২৪ অংশ

ধরি,
সময় লাগে x মিনিট

B ৪০ মিনিটে পূর্ণ করে ১ অংশ
১ মিনিটে পূর্ণ করে ১/৪০ অংশ
x/২ মিনিটে পূর্ণ করে x/৮০ অংশ

A,B একসাথে ১ মিনিটে পূর্ণ করে ১/২৪ অংশ
x/২ মিনিটে পূর্ণ করে x/৪৮ অংশ

(x/৮০) + (x/৪৮) = ১
⇒ (৩x + ৫x)/২৪০ = ১
⇒ ৮x = ২৪০
∴ x = ৩০ মিনিট

৮,০৪৭.

Question: Which number replaces the question mark?

ক
6
খ
12
গ
8
ঘ
30

ব্যাখ্যা

Question: Which number replaces the question mark?

Solution:
এই পিরামিডের ক্ষেত্রে,
উপরে থাকা প্রতিটি সংখ্যা তার সরাসরি নিচের দুটি সংখ্যার পার্থক্যের সমান।
অর্থাৎ, প্রতিটি সংখ্যা = নিচের বাম সংখ্যা - নিচের ডান সংখ্যা
এখানে,
65 = 110 - 45
27 = 45 - 18
38 = 65 - 27
15 = 27 - 12
23 = 38 - 15

যেহেতু, 12 = 18 - ?
∴ ? = 18 - 12 = 6
সুতরাং, প্রশ্নবোধক চিহ্নের স্থানে 6 বসবে।

৮,০৪৮.

Find the sum of 2 + 4 + 6 + …………. + 38?

ক
360
খ
380
গ
400
ঘ
420

ব্যাখ্যা

Question: Find the sum of 2 + 4 + 6 + …………. + 38?

Solution:
2 + 4 + 6 + …………. + 38
= 2(1 + 2 + 3 + …………. + 19)
Sum of first 19 natural numbers = (19 × 20)/2
= 190

∴ 2 + 4 + 6 + …………. + 38 = 2 × 190 = 380

৮,০৪৯.

A person make a profit of 10% on 25% of the quantity and a loss of 20% on the rest of the quantity. What is the gain or loss in percentage on the whole?

ক
10%
খ
- 12%
গ
12.5%
ঘ
20%

ব্যাখ্যা

Question: A person make a profit of 10% on 25% of the quantity and a loss of 20% on the rest of the quantity. What is the gain or loss in percentage on the whole?

Solution:
Let Cost Price = 100
Profit = 25 x 10%
= 25/10
= 2.5

Loss = 75 x 20%
= 75/5
= 15

Net loss = 15 - 2.5
= 12.5

∴ Net Loss as % = 12.5/100 x 100%
= 12.5%

৮,০৫০.

A person's present age is two-fifth of the age of his mother. After 8 year, he will be one-half of the age of his mother. How old is the mother at present?

ক
ক) 36 years
খ
খ) 38 years
গ
গ) 40 years
ঘ
ঘ) 42 years

ব্যাখ্যা

Let the mother's present age be x year.
Then, the person's present age = 2/5 x year.
(2x/5 + 8) = 1/2 (x + 8)
2(2x + 40) = 5(x + 8)
=> x = 40.

৮,০৫১.

120% of 45 + 45% of 120 = ?

ক
90
খ
108
গ
98
ঘ
100

ব্যাখ্যা

Question: 120% of 45 + 45% of 120 = ?

Solution:
120% of 45 + 45% of 120
= {(120/100) × 45} + {(45/100) × 120}
= 54 + 54
= 108

৮,০৫২.

Gold is 19 times as heavy as water and copper 9 times as heavy as water. The ratio in which these two metals be mixed so that the mixture is 15 times as heavy as water is :

ক
ক) 1:2
খ
খ) 2:3
গ
গ) 3:2
ঘ
ঘ) 19:135
ঙ
ঙ) None of the above

ব্যাখ্যা

Required ratio 6 : 4 = 3 : 2

৮,০৫৩.

Three metal wires of lengths 90 cm, 126 cm, and 162 cm are provided. Calculate the maximum length of wire segments that can be cut, ensuring no waste.

ক
18 cm
খ
12 cm
গ
30 cm
ঘ
24 cm

ব্যাখ্যা

Question: Three metal wires of lengths 90 cm, 126 cm, and 162 cm are provided. Calculate the maximum length of wire segments that can be cut, ensuring no waste.

Solution:
The maximum length of wire segments that can be cut (ensuring no waste) = (H.C.F. of 90, 126, 162) cm = 18 cm.

Elaborately,

The prime factorization of each number:
90 = 2 × 3² × 5
126 = 2 × 3² × 7
162 = 2 × 3⁴

Common factors of all three numbers are: 2 (common), 3² = 9 (common)

∴ HCF = 2 × 3² = 18

৮,০৫৪.

40 litre of mixture of milk and water contains 25% of water. When 10 liters of water is added, what will be the percentage of milk in the final mixture?

ক
40%
খ
45%
গ
55%
ঘ
60%

ব্যাখ্যা

Question: 40 litre of mixture of milk and water contains 25% of water. When 10 liters of water is added, what will be the percentage of milk in the final mixture?

Solution:
25% of 40 litre = 10 litre.
10 liter water is present in the mixture.
When 10 liters more water is added in the mixture then the final mixture contains 20 liters water and 30 liters milk.
% of milk in final mixture= (30/50) × 100 = 60%

৮,০৫৫.

A pipe can fill an empty tank in 15 minutes. Another pipe drains water at a rate of 10 liters per minute. If both pipes are opened simultaneously, the tank gets filled in 60 minutes. What is the capacity of the tank (in liters)?

ক
178 liter
খ
200 liter
গ
224 liter
ঘ
250 liter
ঙ
400 liter

ব্যাখ্যা

Question: A pipe can fill an empty tank in 15 minutes. Another pipe drains water at a rate of 10 liters per minute. If both pipes are opened simultaneously, the tank gets filled in 60 minutes. What is the capacity of the tank (in liters)?

Solution:
মনে করি, ট্যাঙ্কটির মোট ধারণক্ষমতা V লিটার।

প্রথম পাইপটি 15 মিনিটে খালি ট্যাঙ্কটি পূর্ণ করে।
∴ প্রথম পাইপ দিয়ে প্রতি মিনিটে পানি পূর্ণ হয় = V/15 লিটার।

দ্বিতীয় পাইপ দিয়ে প্রতি মিনিটে পানি বের হয়ে যায় = 10 লিটার।

দুটি পাইপ একসাথে খোলা থাকলে ট্যাঙ্কটি 60 মিনিটে পূর্ণ হয়।

∴ প্রতি মিনিটে ট্যাঙ্কটি কার্যকরভাবে পূর্ণ হয় = V/60 লিটার।

প্রশ্নমতে,
(V/15) - 10 = V/60
⇒ (V/15) - (V/60) = 10
⇒ (4V - V)/60 = 10
⇒ 3V/60 = 10
⇒ 3V = 10 × 60
⇒ 3V = 600
⇒ V = 600/3
∴ V = 200

সুতরাং, ট্যাঙ্কটিতে 200 লিটার পানি ধরে।

৮,০৫৬.

If the least common multiple of two numbers is twelve times their highest common factor, and their sum (HCF + LCM) equals 403, then what is the other number when one number is 93?

ক
105
খ
124
গ
85
ঘ
115

ব্যাখ্যা

Question: If the least common multiple of two numbers is twelve times their highest common factor, and their sum (HCF + LCM) equals 403, then what is the other number when one number is 93?

Solution:
Let HCF be h and LCM be l
Then l = 12h and
l + h = 403

∴12h + h = 403
⇒ h = 31

So, l = (403 - 31) = 372

Hence, the other number = (31 × 372)/93 = 124

৮,০৫৭.

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
17 days
খ
20 days
গ
15 days
ঘ
24 days
ঙ
21 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 = 30/1.25 = 24 days to get the work done.

৮,০৫৮.

In how many years the compound interest on Tk. 10000 at the rate of 10% will be Tk. 2100?

ক
1.5 years
খ
2 years
গ
2.5 years
ঘ
3 years

ব্যাখ্যা

Question: In how many years the compound interest on Tk. 10000 at the rate of 10% will be Tk. 2100?

Solution:
P = 10000
r = 10% = 10/100 = 1/10 = 0.1
compound principal C = 10000 + 2100 = 12100
time = n years

C = P(1 + r)ⁿ
⇒ 12100 = 10000(1 + 0.1)ⁿ
⇒ 12100/10000 = (1.1)ⁿ
⇒ 121/100 = (11/10)ⁿ
⇒ (11/10)² = (11/10)ⁿ
∴ n = 2

৮,০৫৯.

If one number exceeds another number by 14 and the larger number is 3/2 times the smaller number, then the smaller number is -

ক
ক) 13
খ
খ) 26
গ
গ) 28
ঘ
ঘ) 31

ব্যাখ্যা

Let small number is x, so large number is (3/2)x
ATQ, x + 14 = (3/2)x
Or, 2x + 28 = 3x
So, x = 28

৮,০৬০.

The angle of elevation of a ladder leaning against a wall is 60º and the foot of the ladder is 5.5 m away from the wall. The length of the ladder is-

ক
12.25 m
খ
11 m
গ
9 m
ঘ
15.75 m

ব্যাখ্যা

Question: The angle of elevation of a ladder leaning against a wall is 60º and the foot of the ladder is 5.5 m away from the wall. The length of the ladder is-

Solution:

Let AB be the wall and BC be the ladder.
Then, ∠ACB = 60° = AC = 5.5 m
AC/BC = cos⁡60^∘= 1/2
⇒ BC = 2 × AC = 2 × 5.5 = 11 m

৮,০৬১.

In how many ways can you rearrange the word JUMBLE such that the rearranged word starts with a vowel?

ক
120
খ
230
গ
250
ঘ
360
ঙ
None of these

ব্যাখ্যা

Question: In how many ways can you rearrange the word JUMBLE such that the rearranged word starts with a vowel?

Solution:
JUMBLE is a six-lettered word.
Since the rearranged word has to start with a vowel, the first letter can be either U or E.
The balance 5 letters can be arranged in ⁵P₅ or 5! ways.

∴ Total number of words = 2 × 5! = 240.

৮,০৬২.

Monthly income of B is Tk. 3000 more than the monthly income of A. Monthly savings of A and B are Tk. 8000 and Tk. 17000 respectively. Find the monthly income of A if the monthly expenditures of A and B are in the ratio 4 : 3 respectively.

ক
Tk. 27000
খ
Tk. 32000
গ
Tk. 35000
ঘ
Tk. 37000

ব্যাখ্যা

Question: Monthly income of B is Tk. 3000 more than the monthly income of A. Monthly savings of A and B are Tk. 8000 and Tk. 17000 respectively. Find the monthly income of A if the monthly expenditures of A and B are in the ratio 4 : 3 respectively.

Solution:
Let,
The monthly income of A is Tk. x
∴ The monthly income of B is Tk. x + 3000

Monthly expenditures of A is Tk. x - 8000
Monthly expenditures of B is Tk. x + 3000 - 17000 = Tk. x - 14000

ATQ,
(x - 8000)/(x - 14000) = 4/3
⇒ 3x - 24000 = 4x - 56000
⇒ 4x - 3x = 56000 - 24000
∴ x = 32000

৮,০৬৩.

If the length of a side of a regular pentagon is 4 cm, the area of the pentagon is approximately-

ক
ক) 25 cm²
খ
খ) 27 cm²
গ
গ) 29 cm²
ঘ
ঘ) 32 cm²

ব্যাখ্যা

The area of a regular polygon with n sides of length s is A =

For a regular pentagon with sides 4 cm, n = 5, s = 4. The area is =

= 27.527
The area of the regular pentagon is 27.527 square cm ≅ 27 cm²

৮,০৬৪.

If y = 5, then what is the value of 10y√(y³ - y²)?

ক
25
খ
125
গ
625
ঘ
500

ব্যাখ্যা

Question: If y = 5, then what is the value of 10y√(y³ - y²)?

Solution:
Given,
y = 5

∴ 10y√(y³ - y²)
= 10. 5. √(5³ - 5²)
= 50√(125 - 25)
= 50√(100)
= 50 × 10
= 500

৮,০৬৫.

The sum of 7 consecutive natural numbers is 77. Find how many of these are prime numbers?

ক
2
খ
5
গ
1
ঘ
3

ব্যাখ্যা

Question: The sum of 7 consecutive natural numbers is 77. Find how many of these are prime numbers?

Solution:
Given that,
The sum of seven consecutive natural numbers = 77

Now,
Let the numbers be n, n + 1, n + 2, n + 3, n + 4, n + 5, n + 6 respectively
∴ 7n + 21 = 77
⇒ 7n = 77 - 21
⇒ 7n = 56
⇒ n = 56/7
∴ n = 8

So the numbers is 8, 9, 10, 11, 12, 13, 14
Out of these 11, 13 are prime numbers

∴ Required prime numbers is 2

৮,০৬৬.

A whole number is added to 200 and the same number is subtracted from 200. The sum of the resulting numbers is-

ক
300
খ
400
গ
500
ঘ
350
ঙ
None of the above

ব্যাখ্যা

Question: A whole number is added to 200 and the same number is subtracted from 200. The sum of the resulting numbers is-

Solution:
Let the whole number be x.

∴ (x + 200) + (200 - x)
= x + 200 + 200 - x
= 400

৮,০৬৭.

If x = y = 2z and xyz = 500, then y = ?

ক
ক) 5
খ
খ) 8
গ
গ) 10
ঘ
ঘ) 12

ব্যাখ্যা

Given that
x = y = 2z
xyz = 500

Now
⇒ (2z) (2z) z = 500
⇒ 4z³= 500
⇒ z³ =125
⇒ z³ =5³
⇒ z = 5

∴ y = 2z
= (2 × 5)
= 10

৮,০৬৮.

Rakibul has saved 1250 Taka by purchasing a laptop with a 5% discount on it. what is the quoted price of the laptop in Taka?

ক
20000 Tk.
খ
25000 Tk.
গ
30000 Tk.
ঘ
35000 Tk.

ব্যাখ্যা

প্রশ্ন: Rakibul has saved 1250 Taka by purchasing a laptop with a 5% discount on it. what is the quoted price of the laptop in Taka?

সমাধান :
5% ছাড়ে
5 টাকা ছাড় পাওয়া যায় যখন তালিকামূল্য 100 টাকা
1 টাকা ছাড় পাওয়া যায় যখন তালিকামূল্য 100/5 টাকা
∴ 1250 টাকা ছাড় পাওয়া যায় যখন তালিকামূল্য (100 × 1250)/5 টাকা
= 25,000 টাকা

৮,০৬৯.

Pipe X can fill a cistern thrice as fast as another pipe Y and the pipe Y is thrice as fast as pipe Z. If X, Y and Z together fill the cistern in 10 minutes then the time taken by X to fill the cistern is -

ক
ক) 1 hour and 42 minutes
খ
খ) 2 hours and 10 minutes
গ
গ) 1 hour and 23 minutes
ঘ
ঘ) none of these

ব্যাখ্যা

Let,
The pipe Z alone takes A minutes to fill the tank.
Given that,
Y is thrice as fast as Z.
Then, Y takes A/3 minutes to fill the tank.
And, X is thrice as fast as Y.
X takes (A/3)/3 = A/9 minutes to fill the tank.
Now,
Part filled by X in 1 minute = 9/A
Part filled by Y in 1 minute = 3/A
Part filled by Z in 1 minute = 1/A
Net part filled by (X+Y+Z) in 1 minute = 9/A + 3/A + 1/A
= 13/A
(X+Y+Z) take 10 minutes to fill the cistern.
Part filled by (X+Y+Z) in 1 minute = 1/10
Thus,
We have,
1/10 = 13/A
⇒ A = 13 × 10
⇒ A = 130
Therefore, Z alone takes 130 minutes
So, X can fill the cistern in 130/9 minutes.
Hence, the correct answer is - ঘ) none of these

৮,০৭০.

What is the least square number which is exactly divisible by 2, 3, 10, 18 and 20 ?

ক
180
খ
900
গ
720
ঘ
360

ব্যাখ্যা

Question: What is the least square number which is exactly divisible by 2, 3, 10, 18 and 20?

Solution:
The least or smallest number which is exactly divisible by 2, 3, 10, 18, and 20 is the LCM of 2, 3, 10, 18 and 20.

LCM of (2, 3, 10, 18, 20) = 180
∴ 180 = 2² × 3² × 5

So, to become a perfect square 180 needs to be multiplied by 5 .

Now, the least square number which is exactly divisible by 2, 3, 10, 18 and 20
⇒ (180 × 5) = 900

৮,০৭১.

The perimeter of a circle measures 16πcm, what is the area of the circle in sq.cm?

ক
32√2
খ
64π
গ
256π
ঘ
128π

ব্যাখ্যা

Question: The perimeter of a circle measures 16πcm, what is the area of the circle in sq.cm?

Solution:
মনেকরি
বৃত্তের ব্যাসার্ধ r
বৃত্তের পরিধি = 2πr
বৃত্তের ক্ষেত্রফল = πr²

প্রশ্নমতে
2πr = 16π
2r = 16
r = 8

বৃত্তের ক্ষেত্রফল = πr²
=π8²
=64π

৮,০৭২.

A trader purchases an item for Tk. 540 and sets it marked price at 20% above the cost price. He then sells it at a discount of 10% on the marked price. What is his profit percentage?

ক
10%
খ
12%
গ
15%
ঘ
8%

ব্যাখ্যা

Question: A trader purchases an item for Tk. 540 and sets it marked price at 20% above the cost price. He then sells it at a discount of 10% on the marked price. What is his profit percentage?

Solution:
Let the cost price be Tk. 100
Therefore the marked price is Tk. 120 (20% above CP)
Discount is Tk. 12 (10% of marked price)
Selling price is = (Tk. 120 - Tk. 12) = Tk. 108

Therefore, profit percentage is 8%.

The given cost price of Tk. 540 is unnecessary and is only for creating confusion.

৮,০৭৩.

A football team has a ratio of win to loss of 3 : 1. After winning 6 games in a row, the team's ratio of win to loss became 5 : 1. How many games had the team won before it played the last six games?

ক
9
খ
12
গ
15
ঘ
6
ঙ
None of these

ব্যাখ্যা

Quesion: A football team has a ratio of win to loss of 3 : 1. After winning 6 games in a row, the team's ratio of win to loss became 5 : 1. How many games had the team won before it played the last six games?

Solution:
Let,
3x be the number of games won and x be the number of games lost.

According to the question,
⇒ (3x + 6) : x = 5 : 1
⇒ (3x + 6)/x = 5/1
⇒ 3x + 6 = 5x
⇒ 5x - 3x = 6
⇒ 2x = 6
∴ x = 3

∴ Initial wins = 3x = 3 × 3 = 9
So, the team had won 9 games before the last 6 wins.

৮,০৭৪.

The population of a city grows from 175,000 to 262,500. What is the percentage growth in the city's population?

ক
40%
খ
43.5%
গ
50%
ঘ
60%

ব্যাখ্যা

Question: The population of a city grows from 175,000 to 262,500. What is the percentage growth in the city's population?

Solution:
জনসংখ্যা বৃদ্ধি পেয়েছে
= (262500 - 175000) = 87500

175000 জনে বৃদ্ধি পায় = 87500 জন
∴ 1 জনে বৃদ্ধি পায় = 87500/175000 জন
∴ 100 জনে বৃদ্ধি পায় = (87500 × 100)/175000 = 50 জন

∴ জনসংখ্যা বৃদ্ধির শতকরা হার = 50%

৮,০৭৫.

From the top of a lighthouse which is 90 m above the sea, the angle of depression of a ship is 60°. How far is the ship from the lighthouse?

ক
33.29 m
খ
30√3 m
গ
28√2 m
ঘ
17.34 m

ব্যাখ্যা

Question: From the top of a lighthouse which is 90 m above the sea, the angle of depression of a ship is 60°. How far is the ship from the lighthouse?

Solution:

Let the height of the lighthouse above sea be AC and it is given 90 m.
Ship is at point B so the distance between the base of lighthouse A and ship is AB.

Then, From ΔABC,
AC/AB = tan 60° (√3)
⇒ 90/AB = √3
⇒ AB = 90/√3
⇒ AB = (30 · √3 · √3)/√3
∴ AB = 30√3 m

৮,০৭৬.

If x is an odd integer, then which of the following is true?

ক
5x - 2 is even
খ
5x² + 2 is odd
গ
5x² + 3 is odd
ঘ
None of these

ব্যাখ্যা

Question: If x is an odd integer, then which of the following is true?

Solution:
Now, suppose x = 1. So,
ক) 5.1 - 2 = 3 ; odd
খ) 5.1² + 2 = 7 ; odd
গ) 5.1² + 3 = 8 ; even

Shortcut: x is odd ⇒ x² is odd ⇒ 5x² is odd ⇒ (5x² + 2) is odd.

৮,০৭৭.

A car owner buys petrol at Tk.17, TK. 19 and TK. 20 per liter for three consecutive years. Compute the average cost per liter. If he spends Tk. 6460 per year.

ক
Tk. 12
খ
Tk. 18.58
গ
Tk. 24.5
ঘ
Tk. 28.92

ব্যাখ্যা

Question: A car owner buys petrol at Tk.17, TK. 19 and TK. 20 per liter for three consecutive years. Compute the average cost per liter. If he spends Tk. 6460 per year.

Solution:
Total quantity of petrol consumed in 3 years
= (6460/17 + 6460/19 + 6460/20) litres
= (380 + 340 + 323) litres
= 1043 litres

Total amount spent
= Tk. (3 × 6460)
= Tk. 19380

∴ Average cost
= Tk. (19380/1043)
= Tk. 18.58

৮,০৭৮.

40% of the employees in a factory are workers. All the remaining employees are executives. The annual income of each worker is 390 taka. The annual income of each executive is 420 taka. What is the average annual income of all the employees in the factory together?

ক
408
খ
360
গ
270
ঘ
410
ঙ
412

ব্যাখ্যা

Let X be the number of employees.
Now, 40% of X = 0.4X
Hence, the number of workers is 2X/5

All the remaining employees are executives,
so the number of executives = X - 2X/5 = 3X/5
The total annual income of all workers together = 2X/5 × 390 = 156X

The total income of all the executives together = 3X/5 × 420 = 252X

Hence, the total income of the employees = 156X + 252X = 408X
The average income of all the employees together = 408

৮,০৭৯.

The greatest number will divide 3026 and 5053 leaving remainders 11 and 13 respectively?

ক
15
খ
20
গ
35
ঘ
45

ব্যাখ্যা

Question: The greatest number will divide 3026 and 5053 leaving remainders 11 and 13 respectively?

Solution:
৩০২৬ - ১১ = ৩০১৫ ও ৫০৫৩ - ১৩ = ৫০৪০ এর গ.সা.গু

৩০১৫ = ৩ × ৩ × ৫ × ৬৭
৫০৪০ = ২ × ২ × ২ × ২ × ৩ × ৩ × ৫ × ৭

৩০১৫ ও ৫০৪০ এর গ সা গু = ৩ × ৩ × ৫
= ৪৫

৮,০৮০.

Arif and Babu worked together to paint a house. Arif worked for 1 hour 45 minutes and Babu worked for 45 minutes. Babu's hourly rate is double the rate of Arif's. If the two together earned Tk 71.50, what is hourly rate of Arif in Taka?

ক
20
খ
22
গ
22.50
ঘ
None of these

ব্যাখ্যা

Question: Arif and Babu worked together to paint a house. Arif worked for 1 hour 45 minutes and Babu worked for 45 minutes. Babu's hourly rate is double the rate of Arif's. If the two together earned Tk 71.50, what is hourly rate of Arif in Taka?

Solution:
ধরি
আরিফের প্রতি মিনিটের পারিশ্রমিক = x টাকা
বাবুর প্রতি মিনিটের পারিশ্রমিক = 2x টাকা

আরিফ কাজ করে = 1 ঘণ্টা 45 মিনিট
= (1 × 60)মিনিট + 45 মিনিট
= 60 মিনিট + 45 মিনিট
= 105 মিনিট

বাবু কাজ করে = 45 মিনিট

প্রশ্নমতে
105x + 45 × 2x = 71.50
105x + 90x = 71.5
x = 71.5/195

আরিফের প্রতি ঘণ্টার পারিশ্রমিক = (71.5 × 60)/195 টাকা
= 22 টাকা

৮,০৮১.

Two trains of equal length 200 meters are moving in opposite directions on parallel tracks. The trains cross each other in 10 seconds. If one train is moving twice as fast as the other train, find the speed of the faster train.

ক
48 km/hr
খ
52 km/hr
গ
60 km/hr
ঘ
65 km/hr

ব্যাখ্যা

Let the speed of the slower train = X m/sec
Then, the speed of the faster train will be = 2X m/sec

Relative Speed = X + 2X = 3X m/sec
Relative Speed is also = Sum of the length of the trains/Time taken to cross each other
= (200 + 200)/10
= 400/10
= 40.

So, 3X = 40
X = 40/3
X (speed) in Km/hr = (40/3) × (18/5)
= 720/15
= 48 km/hr.

৮,০৮২.

A train 300 meters long passes a man running at 6 km/h in the same direction as the train in 15 seconds. Find the speed of the train in km/h.

ক
92 km/h
খ
84 km/h
গ
81 km/h
ঘ
78 km/h

ব্যাখ্যা

Question: A train 300 meters long passes a man running at 6 km/h in the same direction as the train in 15 seconds. Find the speed of the train in km/h.

Solution:
Let,
the speed of train be x km/h
∴ Relative speed = (x - 6) km/h
Distance = 300 m = 3/10 km
Time = 15 sec = 15/3600 hr

We know,
Relative speed = Distance/Time

ATQ,
(x - 6) = (3/10)/(15/3600)
⇒ (x - 6) = (3 × 3600)/(10 × 15)
⇒ (x - 6) = 72
∴ x = 78

৮,০৮৩.

A man sitting in a train is counting the pillars of electricity. The distance between two pillars is 40 meters and the speed of the train is 35 km/hr. In 4 hours, how many pillars will be count?

ক
ক) 3500
খ
খ) 2807
গ
গ) 3209
ঘ
ঘ) 3501

ব্যাখ্যা

Question: A man sitting in a train is counting the pillars of electricity. The distance between two pillars is 40 meters and the speed of the train is 35 km/hr. In 4 hours, how many pillars will be count?

Solution:
Distance covered by the train in 4 hours
= (35 × 4) km
= 140 km × 1000
= 140000 m
Number of pillars counted by man = {(140000/40) + 1} = (3500 + 1) = 3501

[ Since the man start counting with pillar and end with pillar so 1 is added]

৮,০৮৪.

By selling a shirt for Tk 450 the shopkeeper incurs a loss of 10%, then what will be the selling price of the shirt when he gets a profit of 10%?

ক
480 tk
খ
500 tk
গ
520 tk
ঘ
550 tk

ব্যাখ্যা

Question: By selling a shirt for Tk 450 the shopkeeper incurs a loss of 10%, then what will be the selling price of the shirt when he gets a profit of 10%?

Solution:
ধরি,
শার্টটির ক্রয়মূল্য = ১০০ক টাকা
১০% ক্ষতিতে,
বিক্রয়মূল্য = ১০০ক - ১০০ক এর ১০%
= ১০০ক - {১০০ক × (১০/১০০)}
= ৯০ক

প্রশ্নমতে,
৯০ক = ৪৫০
⇒ ক = ৪৫০/৯০
∴ ক = ৫
∴ ক্রয়মূল্য = ১০০ × ৫ =৫০০ টাকা

এখন,
১০% লাভে,
বিক্রয়মূল্য = ৫০০ + ৫০০ এর ১০%
= ৫০০ + {৫০০ × (১০/১০০)}
= ৫৫০ টাকা

৮,০৮৫.

If 30% of x = y, then y% of 50 is equal to what percentage of x?

ক
15% of x
খ
10% of x
গ
25% of x
ঘ
None of these

ব্যাখ্যা

Question: If 30% of x = y, then y% of 50 is equal to what percentage of x?

Solution:
Here,
30% of x = y
⇒ 30x/100 = y
∴ y = 3x/10

Now,
y% of 50 = 50y/100
= 50 × (3x/10) × 1/100
= 15x/100
= 15% of x

৮,০৮৬.

In a 500 m race, the ratio of the speeds of two participants A and B is 3 : 4. A has a start of 140 m. Then, A wins by-

ক
60 m
খ
40 m
গ
10 m
ঘ
20 m

ব্যাখ্যা

Question: In a 500 m race, the ratio of the speeds of two participants A and B is 3 : 4. A has a start of 140 m. Then, A wins by-

Solution:
To reach the winning point, A has to cover a distance of (500 - 140) = 360 m

According to the question
When A covers 3 m, then B covers 4 m
And when A covers 360 m, then B covers (4 × 360)/3 m = 480 m

Thus, when A reaches the winning point, B covers 480 m and remains 20 m behind.
Therefore A wins from B by 20 m.

৮,০৮৭.

Given that 24 carat gold is pure gold, 18 carat gold is 3/4th of pure gold and 20 carat gold is 5/6th of pure gold, the ratio of the pure gold in 18 carat gold to the pure gold in 20 carat gold is

ক
ক) 5 : 8
খ
খ) 10 : 9
গ
গ) 8 : 5
ঘ
ঘ) 9 : 10

ব্যাখ্যা

Question: Given that 24 carat gold is pure gold, 18 carat gold is 3/4 of pure gold and 20 carat gold is 5/6 of pure gold, the ratio of the pure gold in 18 carat gold to the pure gold in 20 carat gold is-

Solution:
18 ক্যারেট সোনায় খাঁটি সোনার পরিমাণ = (24 × 3)/4 ক্যারেট
= 18 ক্যারেট

20 ক্যারেট সোনায় খাঁটি সোনার পরিমাণ = (24 × 5)/6 ক্যারেট
= 20 ক্যারেট

নির্ণেয় অনুপাত = 18 : 20
= 9 : 10

৮,০৮৮.

(289)^0.17 × (17)^0.16 = ?

ক
4
খ
√7
গ
√17
ঘ
√19

ব্যাখ্যা

Question: (289)^0.17 × (17)^0.16 = ?

Solution:
(289)^0.17 × (17)^0.16
= {(17)²}^0.17 × (17)^0.16
= 17^{(2 × 0.17)} × (17)0.16
= (17)^0.34 × (17)0.16
= (17)^{0.34 + 0.16}
= (17)^0.50
= (17)^50/100
= (17)^1/2
= √17

৮,০৮৯.

Tap P fills a tank in 4 hours whereas tap Q empties the tank in 24 hours. P and Q are opened alternately for 1 hour each. Every 2 hours the level of water is found to increase by 0.5m. The depth of the tank is -

ক
3 m
খ
1.2 m
গ
2.4 m
ঘ
3.6 m

ব্যাখ্যা

Question: Tap P fills a tank in 4 hours whereas tap Q empties the tank in 24 hours. P and Q are opened alternately for 1 hour each. Every 2 hours the level of water is found to increase by 0.5m. The depth of the tank is -

Solution:
let, the depth of the tank is = h

in two hours total fill-up = 1/4 - 1/24
= 5/24

∴ (5/24)h = 0.5
h = 2.4m

৮,০৯০.

If CODE is 31545 and BOOK is 2151511, then LIVE will be equal to-

ক
129225
খ
121925
গ
122925
ঘ
129255

ব্যাখ্যা

Question: If CODE is 31545 and BOOK is 2151511, then LIVE will be equal to-

Solution:
CODE শব্দটির জন্য অক্ষরগুলোর অবস্থান হলো:
C = 3
O = 15
D = 4
E = 5

∴ CODE = 31545

BOOK শব্দটির জন্য অক্ষরগুলোর অবস্থান হলো:
B = 2
O = 15
O = 15
K = 11

∴ BOOK = 2151511।

LIVE শব্দটির জন্য প্রয়োগ করি:
L = 12
I = 9
V = 22
E = 5

∴ LIVE = 129225

সুতরাং, LIVE এর কোড হবে 129225

৮,০৯১.

A sum of money is to be divided among P, Q, R, S in the ratio 7 : 3 : 5 : 2. If R gets Tk. 2000 more than S, what is Q's share?

ক
Tk. 2000
খ
Tk. 2200
গ
Tk. 2500
ঘ
Tk. 3000

ব্যাখ্যা

Question: A sum of money is to be divided among P, Q, R, S in the ratio 7 : 3 : 5 : 2. If R gets Tk. 2000 more than S, what is Q's share?

Solution:
Let their shares be 7x, 3x, 5x, and 2x respectively.

ATC,
5x - 2x = 2000
⇒ 3x = 2000
⇒ x = 2000/3

Therefore,
Q's share = 3x
= 3 × (2000/3)
= 2000 taka.

৮,০৯২.

A Stationery seller had some Pens, Sharpeners, Erasers & Pencils. He sells 65% of the total units and still has 175 units. Originally, he had-

ক
588 units
খ
400 units
গ
272 units
ঘ
500 units

ব্যাখ্যা

Question: A Stationery seller had some Pens, Sharpeners, Erasers & Pencils. He sells 65% of the total units and still has 175 units. Originally, he had-

Solution:
Suppose originally he had x units.
Then,
(100 - 65)% of x = 175.
⇒ (35/100) × x = 175
⇒ x = (175 × 100)/35
∴ x = 500

৮,০৯৩.

A man's present age is two-fifth of the ages of his mother. After 8 years, he will be one-half of the age of his mother. How old is the mother at present?

ক
ক) 36
খ
খ) 38
গ
গ) 40
ঘ
ঘ) 42

ব্যাখ্যা

Let present age of mother is x,
then present age of his son = 2x/5

According to the question
(2x/5) + 8 = (1/2)(x + 8)
(2x/5) + 8 = x/2 + 4
(2x/5) - (x/2) = 4 - 8
(4x - 5x)/10 = - 4
- x/10 = - 4
x = 40

Hence, the present age of mother is 40

৮,০৯৪.

The length of the side of a square whose area is four times the area of a square with a side 25m is-

ক
125 m
খ
100 m
গ
50 m
ঘ
25 m

ব্যাখ্যা

Question: The length of the side of a square whose area is four times the area of a square with a side 25m is-

Solution:
Area of given square = 25² = 625 m²
Area of new square = 625 × 4 = 2500 m²
Side of new square = √2500 = 50 m

৮,০৯৫.

A shopkeeper fixes the marked price of an item 35% above its cost price. The percentage of discount allowed to gain 8% is = ?

ক
ক) 20%
খ
খ) 27%
গ
গ) 31%
ঘ
ঘ) 43%

ব্যাখ্যা

Let C.P.=Tk. 100
Then,
M.P. = Tk. 135,
S.P. =Tk.108
∴Discount % =(27/135 ×100)% =20%

৮,০৯৬.

What is the value of tan240°?

ক
√3
খ
√5
গ
3
ঘ
√2

ব্যাখ্যা

Question: What is the value of tan240°?

Solution:
tan240°
= tan(180° + 60°)
= tan(180° + θ)
= tanθ
= tan60°
=√3

৮,০৯৭.

If sec²θ + tan²θ = 7/12, then sec⁴θ - tan⁴θ = ?

ক
7/12
খ
1
গ
0
ঘ
12/7
ঙ
1/2

ব্যাখ্যা

Question: If sec²θ + tan²θ = 7/12, then sec⁴θ - tan⁴θ = ?

Solution:
Given that,
sec²θ + tan²θ = 7/12

Now,
sec⁴θ - tan⁴θ
= (sec²θ)² - (tan²θ)²
= (sec²θ + tan²θ)(sec²θ - tan²θ) ; [sec²θ - tan²θ = 1]
= (7/12) × 1
= 7/12

৮,০৯৮.

In an examination, a student was asked to find 3/14 of a certain number. By mistake he found 3/4 of it, his answer is 75 more than the correct answer. The given number is -

ক
135
খ
140
গ
142
ঘ
145

ব্যাখ্যা

Question: In an examination, a student was asked to find 3/14 of a certain number. By mistake he found 3/4 of it, his answer is 75 more than the correct answer. The given number is -

Solution:
let, the number be x

(3x/4) - (3x/14) = 75
⇒ 21x - 6x/28 = 75
⇒ 15x = 28 × 75
⇒ x = (28 × 75)/15
= 140

৮,০৯৯.

The perimeter of an equilateral triangle is 96√3 cm. Find its height.

ক
32 cm
খ
48 cm
গ
16 cm
ঘ
64 cm
ঙ
24 cm

ব্যাখ্যা

Question: The perimeter of an equilateral triangle is 96√3 cm. Find its height.

Solution:
Perimeter of the equilateral triangle is 96√3 cm.
Each of the side of the equilateral triangle is (96√3/3) = 32√3 cm.
The height of the equilateral triangle will be = (√3/2) × (32√3) = 48 cm

৮,১০০.

A box contains 4 red, 5 green and 6 white balls. A ball is drawn at random from the box. What is the probability that the ball drawn is either red or green?

ক
ক) 2/5
খ
খ) 3/5
গ
গ) 1/7
ঘ
ঘ) 7/15

ব্যাখ্যা

Total number of balls = (4 + 5 + 6)
= 15.
P(drawing a red ball or a green ball)
= P(red) + P(green)
= (4/15 + 5/15)
= 9/15
= 3/5.