Bank Math

PrepBank · পাতা ৮৮ / ১৬১ · ৮,৭০১–৮,৮০০ / ১৬,১২৪

৮,৭০১.

Find the value of x, if (x/7) - (x/9) = 2

ক
53
খ
75
গ
47
ঘ
63

ব্যাখ্যা

Question: Find the value of x, if (x/7) - (x/9) = 2

Solution:
Given that,
(x/7) - (x/9) = 2
⇒ (9x - 7x)/63 = 2
⇒ 2x = 2 × 63
∴ x = 63

৮,৭০২.

An train runs twice as fast as a bus which covers 60 kilometers in 80 minutes. What distance will the train cover in 1 hour?

ক
ক) 120 km
খ
খ) 40 km
গ
গ) 90 km
ঘ
ঘ) 100 km

ব্যাখ্যা

Question: An train runs twice as fast as a bus which covers 60 kilometers in 80 minutes. What distance will the train cover in 1 hour?

Solution:
Time taken to cover 60 km by bus in 80 min = 80/60 = 4/3 hour

∴ speed of bus = 60 × (3/4) = 45 kmph

Speed of train = 2 × 45 = 90 kmph

So, distance covered by train in 1 hour = 90 km

৮,৭০৩.

If difference and product of two numbers are 5 and 36 respectively, then find the difference of their reciprocals.

ক
ক) 8/33
খ
খ) 5/36
গ
গ) 9/33
ঘ
ঘ) 7/36

ব্যাখ্যা

Question: If difference and product of two numbers are 5 and 36 respectively, then find the difference of their reciprocals.
Solution:
ধরি, সংখ্যা দুটি হলো 'a' এবং 'b'

a, b এর বিপরীতক হলো যথাক্রমে 1/a এবং 1/b
(b - a) = 5 and ab = 36 ----(দেওয়া আছে)

এখন, 1/a - 1/b = (b - a)/ab = 5/36

৮,৭০৪.

Of 24 kg of salt water 8% is salt; of another 4% is salt, how many kgs of the second mixture should be added to the first mixture in order to get a mixture that is 5% salt?

ক
48
খ
56
গ
64
ঘ
72
ঙ
None of these

ব্যাখ্যা

(C1 × Q1 + C2 × Q2)/(Q1 + Q2) = P
or, (8 × 124 + 4 × Q2)/(24 + Q2) = 5
or, 192 + 4 × Q2 = 120 + 5 × Q2
or, Q2 = 72

৮,৭০৫.

If each side of the square is increased by 50%, what will be the ratio between the new area and the original area of the square?

ক
5 : 4
খ
9 : 4
গ
4 : 5
ঘ
4 : 9

ব্যাখ্যা

Question: If each side of the square is increased by 50%, what will be the ratio between the new area and the original area of the square?

Solution:
Let,
The side of original square is x
∴ The area of original square is x²

The side of new square is x + 50% of x = x + x/2 = 3x/2
∴ The area of new square is (9x²)/4

∴ The ratio between the new area and the original area of the square = (9x²)/4 : x²
= 9/4 : 1
= 9 : 4

৮,৭০৬.

The number of multiples of 4 between 10 and 250 is:

ক
30
খ
40
গ
50
ঘ
60

ব্যাখ্যা

Question: The number of multiples of 4 between 10 and 250 is:

Solution:
10 এবং 250 এর মধ্যে 4-এর প্রথম গুণিতক হলো 12 (যেহেতু 12 > 10)
10 এবং 250 এর মধ্যে 4-এর শেষ গুণিতক হলো 248 (যেহেতু 248 < 250)

এখন, এটি একটি সমান্তর ধারা, যার প্রথম পদ (a) = 12, শেষ পদ (p) = 248 এবং সাধারণ অন্তর (d) = 4।

আমরা জানি,
পদসংখ্যা (n) = {(শেষ পদ - প্রথম পদ)/সাধারণ অন্তর} + 1
= {(248 - 12)/4} + 1
= {236 / 4} + 1
= 59 + 1
∴ n = 60

৮,৭০৭.

How many kg of Sugar at Tk. 50 per kg must a man mix with 25 kg of sugar at Tk. 34 per kg so that by selling the mixture at Tk. 44 per kg he gains 10% on the outlay?

ক
20 kg
খ
18 kg
গ
16 kg
ঘ
15 kg

ব্যাখ্যা

Question: How many kg of Sugar at Tk. 50 per kg must a man mix with 25 kg of sugar at Tk. 34 per kg so that by selling the mixture at Tk. 44 per kg he gains 10% on the outlay?

Solution:
Cost price of mixture = (44/110) × 100 = 40

Ratio of sugar of Tk. 50 and Tk. 34 is = 6 : 10 = 3 : 5
∴ sugar of Tk. 50 per kg to be mixed = (3/5) × 25 = 15kg

৮,৭০৮.

- 5x - [4y - {9x - (3y - 7x)}] simplifies to

ক
- 21x + 7y
খ
1
গ
11x - 7y
ঘ
21x - 7y

ব্যাখ্যা

Question: - 5x - [4y - {9x - (3y - 7x)}] simplifies to

Solution:
- 5x - [4y - {9x - (3y - 7x)}]
= - 5x - [4y - {9x - 3y + 7x}]
= - 5x - [4y - 9x + 3y - 7x]
= - 5x - [7y - 16x]
= - 5x - 7y + 16x
= 11x - 7y

৮,৭০৯.

If m : 5 = n : 7 = r : 8, then (m + n + r)/m =?

ক
ক) 2
খ
খ) 3
গ
গ) 4
ঘ
ঘ) 5

ব্যাখ্যা

Question: If m : 5 = n : 7 = r : 8, then (m + n + r)/m =?

Solution:
m : 5 = n : 7
⇒ m/5 = n/7
⇒ n = (7/5)m

m : 5 = r : 8
⇒ m/5 = r/8
∴ r = 8m/5

(m + n + r)/m = {m + (7/5)m + 8m/5}/m
= 1 + (7/5) + (8/5)
= 20/5
= 4

৮,৭১০.

On dividing a number by 5, we get 3 as remainder. What will the remainder when the square of this number is divided by 5?

ক
0
খ
1
গ
2
ঘ
4

ব্যাখ্যা

Question: On dividing a number by 5, we get 3 as remainder. What will the remainder when the square of this number is divided by 5?

Solution:
Let,
The number be x
And on dividing x by 5, we get p as quotient and 3 as remainder
∴ x = 5p + 3
⇒ x² = (5p + 3)²
= 25p² + 30p + 9
= 25p² + 30p + 5 + 4
= 5(5p² + 6p + 1) + 4
∴ On dividing the square of this number by 5, we get the remainder as 4.

৮,৭১১.

(64x³/27a ^{- 3})^-2/3= ?

ক
ক) 16/9a³x³
খ
খ) 9/16a²x²
গ
গ) 5/16a³x²
ঘ
ঘ) 16/5a²x⁴

ব্যাখ্যা

Question: (64x³/27a ^{- 3})^-2/3= ?

Solution:
64x³/27a^{- 3})^-2/3
= (4³x³a³/27)^-2/3
= {(4xa)³/3³}^-2/3
= {(4ax/3)³}^-2/3
= (4ax/3)^{- 2}
= 1/(4ax/3)²
= (3/4ax)²
= 9/16a²x²

৮,৭১২.

The daily rate for a hotel room that sleeps 4 people is Tk. 350 each for first two person and X taka for each additional person. If 4 people take the room for one day and each pays Tk 250 for the room, then what is the value of X?

ক
70
খ
120
গ
150
ঘ
160

ব্যাখ্যা

Question: The daily rate for a hotel room that sleeps 4 people is Tk. 350 each for first two person and X taka for each additional person. If 4 people take the room for one day and each pays Tk 250 for the room, then what is the value of X?
(একটি হোটেল রুমে সর্বোচ্চ চারজন থাকতে পারে। প্রথম দুইজনের জন্য প্রতিদিনের ভাড়া জন প্রতি ৩৫০ টাকা এবং পরবর্তী প্রত্যেক অতিরিক্ত ব্যক্তির জন্য ভাড়া X টাকা। যদি চারজন ব্যক্তি ঐ রুমটি একদিনের জন্য ভাড়া নেয় এবং প্রত্যেকেই ২৫০ টাকা করে প্রদান করে, তবে X-এর মান নির্ণয় করুন।)

Solution:
একজনের দৈনিক ভাড়া = ৩৫০ টাকা
দুইজনের জন্য = ৭০০ টাকা
প্রতি অতিরিক্ত ব্যক্তির জন্য দৈনিক ভাড়া = X টাকা

৪ জনের জন্য মোট খরচ = ৭০০ + ২X

যদি ৪ জন একদিনের জন্য রুম নেয় এবং প্রত্যেকে ২৫০ টাকা করে দেয়
তাহলে মোট খরচ = ২৫০ × ৪ = ১০০০ টাকা

প্রশ্ন অনুযায়ী,
৭০০ + ২X = ১০০০
⇒ ২X = ১০০০ - ৭০০
⇒ ২X = ৩০০
∴ X = ১৫০

৮,৭১৩.

A village has a population of 8,000. It increases by 10% in the first year and then by 20% in the second year.What will be the population of the village after 2 years?

ক
10560
খ
8976
গ
10500
ঘ
10650

ব্যাখ্যা

Question: A village has a population of 8,000. It increases by 10% in the first year and then by 20% in the second year.What will be the population of the village after 2 years?

Solution:

৮,৭১৪.

A man walks 20 km in 5 hours. How much time will it take for him to walk 32 km?

ক
3 hours
খ
4 hours
গ
6 hours
ঘ
8 hours
ঙ
9 hours

ব্যাখ্যা

Man walks 20 km in → 5 hours

That means it will take more time to walk 32 km.
This is the case of direct proportion.

20/5 = 32/x
x = 32/4
= 8

৮,৭১৫.

A pipe can fill a cistern in 16 hours. After half the tank is filled, three more similar taps are opened. What is the total time to fill the cistern completely?

ক
8 hours
খ
10 hours
গ
12 hours
ঘ
14 hours

ব্যাখ্যা

Question: A pipe can fill a cistern in 16 hours. After half the tank is filled, three more similar taps are opened. What is the total time to fill the cistern completely?

Solution:
Time is taken to fill half of the tank = (1/2) × 16 = 8 hrs
In 1 hour pipe can fill = 1/16 part filled by 4 pipes in1 hour = 4 × (1/16) = 1/4 part
So, remaining half part = 4 × (1/2) = 2 hours
∴ Total time = 8 + 2 = 10 hours.

৮,৭১৬.

If (x + 3)² = 64, which of the following can be the value of (x + 2)?

ক
5
খ
7
গ
9
ঘ
11

ব্যাখ্যা

Question: If (x + 3)² = 64, which of the following can be the value of (x + 2)?

Solution:
Given, (x + 3)² = 64
⇒ (x + 3)² = 8²
∴ x + 3 = ± 8

Case 1: x + 3 = 8
x = 8 - 3 = 5
x + 2 = 5 + 2 = 7

Case 2: x + 3 = - 8
x = - 8 - 3 = - 11
x + 2 = - 11 + 2 = - 9

Possible values of (x + 2) are 7 or - 9.

৮,৭১৭.

If 3√5 + √125 = 17.88, then what will be the value of √80 + 6√5= ?

ক
ক) 13.41
খ
খ) 20.46
গ
গ) 21.66
ঘ
ঘ) 22.35

ব্যাখ্যা

দেয়াআছে
3√5 + √125 = 17.88
3√5 + √5 × 25 = 17.88
3√5 + 5√5 = 17.88
8√5 = 17.88
√5 = 17.88/8
√5 = 2.235
এখন
√80 + 6√5 = √(16 × 5) + 6√5
= 4√5 + 6√5
= 10√5
= 10 × 2.235
= 22.35

৮,৭১৮.

Find the value of sin⁵θ + cosec⁵θ if sinθ + cosecθ = 2.

ক
32
খ
0
গ
1
ঘ
2
ঙ
None of the above

ব্যাখ্যা

Question: Find the value of sin⁵θ + cosec⁵θ if sinθ + cosecθ = 2.

Solution:
sinθ + cosecθ = 2
or, sinθ + 1/sinθ = 2
or, sin²θ + 1 = 2sinθ
or, sin²θ - 2sinθ + 1 = 0
or, (sinθ - 1)² = 0
or, sinθ - 1 = 0
∴ sinθ = 1

cosecθ = 1/sinθ = 1/1 = 1

∴ sin⁵θ + cosec⁵θ = (1)⁵ + (1)⁵
= 2

৮,৭১৯.

If isosceles ΔDEF has sides of length 11.5 and 13.7, which of the following could be the perimeter of the triangle ?

ক
ক) 2.1
খ
খ) 12.0
গ
গ) 25.2
ঘ
ঘ) 36.7

ব্যাখ্যা

Question: If isosceles ΔDEF has sides of length 11.5 and 13.7, which of the following could be the perimeter of the triangle ?

Solution:

ΔDEF এ
DE = DF = 11. 5
ΔDEF এর পরিসীমা = 11.5 + 11.5 + 13.7 = 36.7

আবার,
DE = DF = 13.7 হলে
ΔDEF এর পরিসীমা = 11.5 + 13.7 + 13.7 = 38.9 যা অপশনে নেই।
অতএব সঠিক উত্তর হবে 36.7

৮,৭২০.

একজন বিক্রেতা প্রতি সেট বিশ্বকোষ বিক্রির উপর ২০% কমিশন পান। তিনি যদি ১২টি সেট বিক্রি করে মোট ১৮০০ টাকা কমিশন উপার্জন করেন, তবে প্রতিটি সেটের বিক্রয় মূল্য কত হবে?

ক
৭৫০ টাকা
খ
৬০০ টাকা
গ
৫০০ টাকা
ঘ
৩০০ টাকা
ঙ
কোনটি নয়

ব্যাখ্যা

প্রশ্ন: একজন বিক্রেতা প্রতি সেট বিশ্বকোষ বিক্রির উপর ২০% কমিশন পান। তিনি যদি ১২টি সেট বিক্রি করে মোট ১৮০০ টাকা কমিশন উপার্জন করেন, তবে প্রতিটি সেটের বিক্রয় মূল্য কত হবে?

সমাধান:
ধরি,
১টি বিশ্বকোষের মূল্য = ক টাকা

১টি বিশ্বকোষের কমিশন = (২০/১০০) × ক = (১/৫) × ক
প্রদত্ত মোট কমিশন = ১৮০০ টাকা

অতএব, ১২টি বিশ্বকোষের কমিশন = ১২ × (১/৫) × ক =১৮০০
∴ ক = (১৮০০ × ৫)/১২ = ৭৫০ টাকা

৮,৭২১.

A person travels from one place to another at 30 km/hr and returns at 120 km/hr. If the total time taken is 5 hours, then find the Distance.

ক
100 km
খ
150 km
গ
140 km
ঘ
120 km

ব্যাখ্যা

Question: A person travels from one place to another at 30 km/hr and returns at 120 km/hr. If the total time taken is 5 hours, then find the Distance.

Solution:
Here the Distance is constant, so the Time taken will be inversely proportional to the Speed.
Ratio of Speed is given as 30 : 120 = 1 : 4

So the ratio of Time taken will be 4 : 1.
Total Time taken = 5 hours;
Time taken while going is 4 hours and returning is 1 hour.

Hence, Distance = 30 × 4 = 120 km

৮,৭২২.

A number is tripled, then 7 is subtracted from it. If the result is then doubled, it becomes 58. What is the number?

ক
11
খ
12
গ
15
ঘ
10
ঙ
19

ব্যাখ্যা

Question: A number is tripled, then 7 is subtracted from it. If the result is then doubled, it becomes 58. What is the number?

Solution:
Let,
the number be x

ATQ,
2(3x - 7) = 58
⇒ 6x - 14 = 58
⇒ 6x = 58 + 14
⇒ 6x = 72
⇒ x = 72/6
∴ x = 12

So the number is 12.

৮,৭২৩.

A sum of money at simple interest amounts to TK. 5600 in 2 years and Tk. 6500 in 5 years at the rate of -

ক
ক) 3%
খ
খ) 4%
গ
গ) 5%
ঘ
ঘ) 6%

ব্যাখ্যা

Question: A sum of money at simple interest amounts to TK. 5600 in 2 years and Tk. 6500 in 5 years at the rate of -

Solution:
S.I. in 3 years = 6500 - 5600 = 900
Interest every year = 900/3 = 300
So, principal = 5600 - 2 × 300 = 5000

We know, I = pnr
Or, Interest rate, r = I/pn
= 600/(5000×2) × 100
= 6%

৮,৭২৪.

When six fair coins are tossed simultaneously, in how many of the outcomes will at most three of the coins turn up as heads?

ক
25
খ
41
গ
42
ঘ
22

ব্যাখ্যা

The question requires you to find a number of the outcomes in which at most 3 coins turn up as heads.
i.e., 0 coins turn heads or 1 coin turns head or 2 coins turn heads or 3 coins turn heads.

The number of outcomes in which 0 coins turn heads is,
⁶C₀ = 1 outcome.

The number of outcomes in which 1 coin turns head is,
⁶C₁ = 6 outcomes.

The number of outcomes in which 2 coins turn heads is,
⁶C₂ = 15 outcomes.

The number of outcomes in which 3 coins turn heads is,
⁶C₃

Therefore, total number of outcomes
= 1 + 6 + 15 + 20
= 42 outcomes.

৮,৭২৫.

If a man were to sell his chair for Tk. 720, he would lose 25%. To gain 25% he should sell it for:

ক
Tk. 1100
খ
Tk. 1200
গ
Tk. 1300
ঘ
Tk. 1400

ব্যাখ্যা

Question: If a man were to sell his chair for Tk. 720, he would lose 25%. To gain 25% he should sell it for:

Solution:
Let the Cost price of the Chair is x.
Selling price = x - 25% of x
⇒ 720 = x - (25x/100)
⇒ 720 = 75x/100
⇒ 75x = 72000
∴ x = 960

To gain 25% = 960 + 25% of 960
= Tk. 1200

৮,৭২৬.

A man bought some eggs of which 10% are rotten. He gives 80% of the remainder to his neighbors. Now he is left out with 36 eggs. Then he ate two eggs. How many eggs did he buy?

ক
200
খ
190
গ
180
ঘ
210

ব্যাখ্যা

Question: A man bought some eggs of which 10% are rotten. He gives 80% of the remainder to his neighbors. Now he is left out with 36 eggs. Then he ate two eggs. How many eggs did he buy?

Solution:
let, the man bought x eggs
10% are rotten

so eggs remained = x - x × 10%
= x - x/10
= 9x/10

80% of 9x/10
= (9x/10) × (80/100)
= 18x/25

he is left with = (9x/10) - (18x/25)
= (45x - 36x)/50
= 9x/50

So, (9x/50) = 36
⇒ 9x = 36 × 50
⇒ x = (36 × 50)/9
= 200

∴ He bought 200 eggs

৮,৭২৭.

A buys a product for Tk. 500 and sells it to B at a profit of 25%. B then sells it to C at a profit of 20%. How much does C pay to B?

ক
Tk. 600
খ
Tk. 650
গ
Tk. 720
ঘ
Tk. 750

ব্যাখ্যা

Question: A buys a product for Tk. 500 and sells it to B at a profit of 25%. B then sells it to C at a profit of 20%. How much does C pay to B?

সমাধান:
A এর 25% লাভে বিক্রয়মূল্য = 500 + 500 এর 25%
= 500 + (500 × 25 / 100)
= 500 + 125
= 625

A এর বিক্রয়মূল্য = B এর ক্রয়মূল্য

B এর 20% লাভে বিক্রয়মূল্য = 625 + 625 এর 20%
= 625 + (625 × 20 / 100)
= 625 + 125
= 750

সুতরাং, B এর বিক্রয়মূল্য = C এর ক্রয়মূল্য = Tk. 750

৮,৭২৮.

In a village of 100 households, 75 have at least one DVD player, 80 have at least one cell phone, and 55 have at least one MP3 player. If x and y are respectively the greatest and lowest possible number of households that have all three of these devices, x - y is:

ক
25
খ
35
গ
45
ঘ
55

ব্যাখ্যা

Question: In a village of 100 households, 75 have at least one DVD player, 80 have at least one cell phone, and 55 have at least one MP3 player. If x and y are respectively the greatest and lowest possible number of households that have all three of these devices, x - y is:

Solution:
The obvious maximum that have all 3 is 55, because you are limited by the smallest number.
The minimum is simply the sum of the max of each people who dont have the product, so:
100 - 80 = 20 don't have Cell
100 - 75 = 25 don't have DVD
and 100 - 55 = 45 don't have MP3

So a total of 20 + 25 + 45 = 90 combined who might not have some combination of the 3 products.
So subtract that from 100, to give you the minimum of the people who could have all 3 and you get 100 - 90 = 10.
55 - 10 = 45

৮,৭২৯.

What is the average of 0.36, 4.6, 0.64, and 2.4?

ক
1
খ
2
গ
2.5
ঘ
10
ঙ
None of these

ব্যাখ্যা

Question: What is the average of 0.36, 4.6, 0.64, and 2.42?

Solution:
Average = (0.36 + 4.6 + 0.64 + 2.4)/4
= 8.00/4
= 2.00

৮,৭৩০.

The curved surface area of a cylindrical pillar is 264 m² and its volume is 924 m³. What is the radius of the pilllar?

ক
ক) 5 m
খ
খ) 7 m
গ
গ) 9 m
ঘ
ঘ) 11 m

ব্যাখ্যা

Question: The curved surface area of a cylindrical pillar is 264 m² and its volume is 924 m³. What is the radius of the pilllar?

Solution:
let, radius r m and height h m

The curved surface area of a cylindrical pillar is 264 m2
2πrh = 264 m²

its volume is 924 m³
πr²h = 924

πr²h /2πrh = 924/264
⇒ r/2 = 924/264
∴ r = 7 m

৮,৭৩১.

A train travels 720 km at a uniform speed. If the speed had been 5 km/hr more, it would have taken 2 hour less for the same journey. Find the speed of the train.

ক
60 km/hr
খ
50 km/hr
গ
30 km/hr
ঘ
40 km/hr

ব্যাখ্যা

Question: A train travels 720 km at a uniform speed. If the speed had been 5 km/hr more, it would have taken 2 hour less for the same journey. Find the speed of the train.

Solution:
Given distance = 720 km.
Let,
The speed of the train be x km/hr.
Speed when increased by 5 km/hr = (x + 5) km/hr
ATQ,
(720/x) - {720/(x + 5)} = 2
⇒ (360/x) - {360/(x + 5)} = 1 [Divided by 2]
⇒ [360x + 1800 - 360x]/{x(x + 5)} =1
⇒ 1800/(x² + 5x) = 1
⇒ x² + 5x = 1800
⇒ x² + 5x - 1800 = 0
⇒ x² + 45x - 40x - 1800 = 0
⇒ x(x + 45) - 40(x + 45) = 0
⇒ (x - 40)(x + 45) = 0
x = 40, - 45
Here we ignore the negative value.

∴ The speed of the train is 40 km/hr.

৮,৭৩২.

In a right-angled triangle, if the two angles other than the right angle differ by 4 degrees, what is the value of the smaller angle?

ক
43°
খ
33°
গ
60°
ঘ
55°

ব্যাখ্যা

Question: In a right-angled triangle, if the two angles other than the right angle differ by 4 degrees, what is the value of the smaller angle?
(একটি সমকোণী ত্রিভুজের সমকোণ ব্যতীত অপর দুইটি কোণের পার্থক্য ৪ ডিগ্রি হলে ক্ষুদ্রতম কোণটি কত ডিগ্রি হবে?)

Solution:
ধরি,
ক্ষুদ্রতম কোণ = ক
বৃহত্তম কোণ = ক + ৪°

প্রশ্নমতে,
ক + ক + ৪° = ৯০°
⇒ ২ক = ৯০° - ৪°
⇒ ক = ৮৬°/২
∴ ক = ৪৩°

৮,৭৩৩.

A person rowing against the current can go 2 km per hour. If the speed of the current is 3 km per hour, how much time will he take to cover 32 km, rowing along the current?

ক
ক) 4 hr
খ
খ) 5 hr
গ
গ) 4.5 hr
ঘ
ঘ) 6 hr

ব্যাখ্যা

Question: A person rowing against the current can go 2 km per hour. If the speed of the current is 3 km per hour, how much time will he take to cover 32 km, rowing along the current?

Solution:
ধরি, ব্যক্তির বেগ x কিমি/ঘণ্টা
স্রোতের বেগ ৩ কিমি/ঘণ্টা

ব্যক্তি স্রোতের বিপরীতে ২ কিমি/ঘণ্টা বেগে যায়।

x - ৩ = ২
∴ x = ৫ কিমি/ঘণ্টা

স্রোতের অনুকূলে বেগ = ৩ + ৫ কিমি/ঘণ্টা
= ৮ কিমি/ঘণ্টা

স্রোতের অনুকূলে যেতে সময় লাগে = ৩২/৮ ঘণ্টা
= ৪ ঘন্টা

৮,৭৩৪.

A shopkeeper has sufficient money to buy 50 books. On reduction in the price of each book by Tk. 4, he could buy 10 books more. How much money does he has?

ক
Tk. 1200
খ
Tk. 1250
গ
Tk. 1300
ঘ
Tk. 1400

ব্যাখ্যা

Question: A shopkeeper has sufficient money to buy 50 books. On reduction in the price of each book by Tk. 4, he could buy 10 books more. How much money does he has?

Solution:
১টি বইয়ে দাম কমে ৪ টাকা
∴ ৫০টি বইয়ে দাম কমে (৫০ × ৪) টাকা
= ২০০ টাকা

সে মোট বই কিনে (৫০ + ১০) টি
= ৬০টি

১০টি বইয়ের দাম ২০০ টাকা
∴ ৬০টি বইয়ের দাম (২০০ × ৬০)/১০ টাকা
= ১২০০ টাকা

∴ তার কাছে ১২০০ টাকা আছে।

৮,৭৩৫.

If the interest of Tk. M at M% in 4 years is Tk. M, then M =?

ক
20
খ
25
গ
30
ঘ
32

ব্যাখ্যা

Question: If the interest of Tk. M at M% in 4 years is Tk. M, then M =?

Solution:
Here,
P = M
r = M%
I = M
n = 4

We know that,
I = Pnr
M = M × 4 × (M/100)
⇒ 1 = M/25
∴ M = 25

৮,৭৩৬.

The marked price of a chair was Tk. 12,800/-. The shopkeeper was offering it for a discount of 20% but on further bargaining agreed to offer a successive discount and finally sold the chair for Tk 9,216/- What was the second discount offered by him?

ক
ক) 5%
খ
খ) 10%
গ
গ) 20%
ঘ
ঘ) 25%

ব্যাখ্যা

Price after 1^stdiscount of 20% = (100-20)%
= 80% of Marked Price
∴ Price = (80/100) × 12800
= Tk. 10240

SP = (100 - Discount) % of Price

∴ 9216 = (100 - Discount) % x 10240
⇒ (100 - Discount)/100 = 9216/10240
⇒ 100 - Discount = (9216 × 100)/10240
⇒ 100 - Discount = 90
⇒ Discount = 100 - 90 = 10%

10% is the percent of 2^nddiscount offered.

৮,৭৩৭.

A cube with side length 6 cm fits perfectly inside a hollow spherical ball. What is the total surface area of the sphere?

ক
112π cm²
খ
92π cm²
গ
120π cm²
ঘ
108π cm²

ব্যাখ্যা

Question: A cube with side length 6 cm fits perfectly inside a hollow spherical ball. What is the total surface area of the sphere?

Solution:
If a cube fits perfectly inside a sphere, then the diameter of the sphere = space diagonal of the cube

Diagonal of the cube = √3a = 6√3
So, diameter of sphere = 6√3
Radius = 6√3/2 = 3√3

Surface area of the sphere = 4πr²
= 4π(3√3)²
= 108π cm²

৮,৭৩৮.

A box contains 5 detective and 15 non-detective bulbs. Two bulbs are chosen at random. Find the probability that both the bulbs are non-defective.

ক
ক) 5/19
খ
খ) 3/20
গ
গ) 21/38
ঘ
ঘ) 25/38
ঙ
ঙ) None of these

ব্যাখ্যা

n(S) = ²⁰C₂ = 190
n(E) = ¹⁵C₂ = 105
Therefore,
P(E) = 105/190
= 21/38

৮,৭৩৯.

5200 Taka is lent at compound interest of 10% per annum for 2 years. Find the amount after two years.

ক
Tk. 5280
খ
Tk. 6200
গ
Tk. 6292
ঘ
Tk. 7324
ঙ
None

ব্যাখ্যা

Question: 5200 Taka is lent at compound interest of 10% per annum for 2 years. Find the amount after two years.

Solution:
Compound amount = P (1 + r)ⁿ
= 5200 [1 + (10/100)]²
= 5200 × (110/100) × (110/100)
= 52 × 11 × 11
= 6292

Hence, The amount after two years is Tk. 6292

৮,৭৪০.

On selling 20 books for 800 Tk, a man incurs a loss equal to the cost price of 4 books. What is the cost price of one book?

ক
Tk. 45
খ
Tk. 50
গ
Tk. 60
ঘ
Tk. 48

ব্যাখ্যা

Question: On selling 20 books for 800 Tk, a man incurs a loss equal to the cost price of 4 books. What is the cost price of one book?

Solution:
Let,
cost price of 1 book is = Tk. x
cost price of 20 book is = Tk. 20x
We know,
Loss = Cost price - Selling price
⇒ 4x = 20x - 800
⇒ 20x - 4x = 800
⇒ 16x = 800
⇒ x = 800/16
∴ x = 50

So the cost price of one book is Tk. 50

৮,৭৪১.

A boat sails m miles upstream at r miles/hr. If the speed of the stream is s miles/hr, how long will it take the boat to return to its starting point?

ক
m/(r + 2s)
খ
mr - s
গ
m(r + s)
ঘ
None

ব্যাখ্যা

Question: A boat sails m miles upstream at r miles/hr. If the speed of the stream is s miles/hr, how long will it take the boat to return to its starting point?

Solution:
মনেকরি
নৌকার গতিবেগ = x কিমি/ঘণ্টা
স্রোতের গতিবেগ = s কিমি/ঘণ্টা

এখানে
x - s = r
x = r + s

আবার
স্রোতের অনুকূলে বেগ = x + s কিমি/ঘণ্টা
= r + s + s কিমি/ঘণ্টা
= r + 2s

স্রোতের অনুকূলে ফিরে আসতে সময় লাগবে = দূরত্ব/বেগ
= m/(r + 2s)

৮,৭৪২.

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

ক
ক) 4 hours
খ
খ) 6 hours
গ
গ) 7 hours
ঘ
ঘ) 3 hours

ব্যাখ্যা

The outlet pipe empties the one complete cistern in 9 hours
Time taken to empty (2/3) × 9 = 6 hours

৮,৭৪৩.

Tickets numbered 1 to 30 are mixed up and then a ticket is drawn at random. What is the probability that the ticket drawn has a number which is a multiple of 2 and 3?

ক
ক) 2/3
খ
খ) 1/6
গ
গ) 1/2
ঘ
ঘ) none of these

ব্যাখ্যা

Question: Tickets numbered 1 to 30 are mixed up and then a ticket is drawn at random. What is the probability that the ticket drawn has a number which is a multiple of 2 and 3?

Solution:
Number which are multiple of 2 and 3 between 1 to 30 = {6, 12, 18, 24, 30}

∴ The probability is = 5/30 = 1/6

৮,৭৪৪.

If January 1, 2026 is on a Thursday, what day will December 31, 2027 fall on?

ক
Friday
খ
Saturday
গ
Sunday
ঘ
Thursday

ব্যাখ্যা

Question: If January 1, 2026 is on a Thursday, what day will December 31, 2027 fall on?

Solution:
We know,
Except for a leap year, the first and last days of every year fall on the same day of the week. In a leap year, one extra day must be added.

The year 2026 is not a leap year.
If January 1st, 2026, is a Thursday, then the last day of the year will also be a Thursday.

January 1st, 2027, will be a Friday.
The last day of December 31, 2027 will also be a Friday.

December 31, 2027, will be a Friday.

৮,৭৪৫.

The ages of Sabiha and Suriya are in the ratio of 7 : 3 respectively. After 6 years, the ratio of their ages will be 5 : 3. What is the difference in their ages?

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

ব্যাখ্যা

Question: The ages of Sabiha and Suriya are in the ratio of 7 : 3 respectively. After 6 years, the ratio of their ages will be 5 : 3. What is the difference in their ages?

Solution:
Let, their ages are 7x, 3x

ATQ,
(7x + 6)/(3x + 6) = 5/3
⇒ 3(7x + 6) = 5(3x + 6)
⇒ 21x + 18 = 15x + 30
⇒ 21x - 15x = 30 - 18
⇒ 6x = 12
∴ x = 2

The difference in their ages is = 7x - 3x
= 4x
= 4 × 2
= 8 years

৮,৭৪৬.

On Dhaka- Sylhet highway 7% of the drivers are fined for exceeding the speed limit. However, 80% of the drivers who exceed the speed limit are not fined. What percentage of drivers on this highway exceed the speed limit?

ক
28
খ
31
গ
33
ঘ
35

ব্যাখ্যা

Question: On Dhaka- Sylhet highway 7% of the drivers are fined for exceeding the speed limit. However, 80% of the drivers who exceed the speed limit are not fined. What percentage of drivers on this highway exceed the speed limit?

Solution:
ধরি,
হাইওয়েতে 100 জন চালক ছিল।

এখন,
গতিসীমা অতিক্রম করার জন্য 7% চালককে জরিমানা করা হয়।
∴ 100 জন চালকের 7% বা 7 জন চালককে জরিমানা করা হয়।

আবার,
চালকদের মধ্যে যারা গতিসীমা অতিক্রম করেছিল তাদের মধ্যে 80% চালককে জরিমানা করা হয়নি।
∴ চালকদের মধ্যে যারা গতিসীমা অতিক্রম করেছিল তাদের মধ্যে 20% চালককে জরিমানা করা হয়।

প্রশ্নমতে,
20% = 7
∴ 1% = 7/20
∴ 100% = (7 × 100)/20
= 35

৮,৭৪৭.

By selling 90 mangoes for Tk.160 a person loses 20%. What is the cost price of nine mangoes?

ক
ক) Tk. 20
খ
খ) Tk.15
গ
গ) Tk. 21
ঘ
ঘ) Tk.18

ব্যাখ্যা

প্রশ্ন: By selling 90 mangoes for Tk.160 a person loses 20%. What is the cost price of nine mangoes?

সমাধান:
২০% ক্ষতিতে,
বিক্রয়মূল্য ৮০ টাকা হলে ক্রয়মূল্য ১০০ টাকা
∴ বিক্রয়মূল্য ১৬০ টাকা হলে ক্রয়মূল্য (১০০ × ১৬০)/৮০ টাকা
= ২০০ টাকা

৯০টি আমের ক্রয়মূল্য ২০০ টাকা
∴ ৯টি আমের ক্রয়মূল্য (২০০ × ৯)/৯০ টাকা
= ২০ টাকা

৮,৭৪৮.

If 30% of 1520 + 40% of 800 = x % of 5000, find the value of x.

ক
14.42%
খ
15.52%
গ
12.22%
ঘ
18.82%

ব্যাখ্যা

Question: If 30% of 1520 + 40% of 800 = x % of 5000, find the value of x.

Solution:
30% of 1520 + 40% of 800 = x% of 5000
⇒ (30/100) × 1520 + (40/100) × 800 = (x/100) × 5000
⇒ 456 + 320 = 50x
⇒ 50x = 776
⇒ x = 776/50
∴ x = 15.52

৮,৭৪৯.

If Z, D and A occupy three consecutive chairs - in the same order, then C must occupy

ক
ক) 2nd
খ
খ) 3rd
গ
গ) 4th
ঘ
ঘ) 5th

ব্যাখ্যা

Question: If Z, D, and A occupy three consecutive chairs - in the same order, then C must occupy

সমাধান:
শর্তানুযায়ী,
Z D A C B Y X এভাবে বসাতে হবে।
- তাই C এর অবস্থান হবে ৪র্থ।

৮,৭৫০.

If two coins are tossed, what is the probability of getting at least one head?

ক
1/2
খ
1/4
গ
2/3
ঘ
3/4

ব্যাখ্যা

Question: If two coins are tossed, what is the probability of getting at least one head?

Solution:
দুটি মুদ্রা নিক্ষেপ করলে নমুনা ক্ষেত্রটি হলো,
S = {HH, HT, TH, TT}
এখানে মোট ফলাফল সংখ্যা, n(S) = 4

"at least one head" বলতে বোঝায় কমপক্ষে 1টি head অর্থাৎ 1 টি অথবা 2 টিও হতে পারে।

1টি head আছে এমন ফলাফল: {HT, TH}
2টি head আছে এমন ফলাফল: {HH}

∴ অনুকূল ফলাফল, n(E) = {HT, TH, HH}
∴ n(E) = 3

∴ সম্ভাবনা P(E) = n(E)/n(S)
∴ P(E) = 3/4

৮,৭৫১.

What reminder of any perfect square is divided by 3?

ক
ক) 0
খ
খ) 1
গ
গ) 0 or 1
ঘ
ঘ) 0 and 1

ব্যাখ্যা

2²= 4/3 (remainder 1)

3² = 9/3 (remainder 0)

4²= 16/3 ( remainder 1)

5²= 25/3 (remainder 1)

6²= 36/3 ( remainder 0)

7²= 49/3 (remainder 1)
So what we understood from the above examples is that perfect square which were multiples of 3 they give 0 as remainder
whereas numbers which were not multiples of 3 they gave 1 as remainder

৮,৭৫২.

30% of a number when subtracted from 91, gives the number itself. Find the number.

ক
50
খ
55
গ
60
ঘ
65
ঙ
None

ব্যাখ্যা

Question: 30% of a number when subtracted from 91, gives the number itself. Find the number.

Solution:

Let the number be p

According to the question,

91 - 30p/100 = p
⇒ 9100 - 30p = 100p
⇒ 9100 = 130p
∴ p = 70

Hence, the number = 70

৮,৭৫৩.

A train 340 m long passes a pole in 17 seconds. How long will it take to pass a platform 660 m long?

ক
30 seconds
খ
39 seconds
গ
45 seconds
ঘ
50 seconds

ব্যাখ্যা

Question: A train 340 m long passes a pole in 17 seconds. How long will it take to pass a platform 660 m long?

Solution:
Speed of the train = (Distance ÷ Time)
= (340 ÷ 17) m/sec
= 20 m/sec

∴ Required time = (340 + 660)/20 seconds
= 50 seconds

৮,৭৫৪.

What number is midway between 1/2 and 1?

ক
ক) 3/4
খ
খ) 4/9
গ
গ) 7/15
ঘ
ঘ) None

ব্যাখ্যা

প্রশ্নঃ What number is midway between 1/2 and 1?
সমাধানঃ
1 = 1.0
3/4 = o.75
1/2 = 0.5

4/9 = 0.44
7/15 = 0.46

৮,৭৫৫.

A, B, and C invest a total of TK. 80000 in a business. A invests TK. 7000 more than B, and B invests TK. 5000 more than C. Out of a total profit of TK. 48000. How much profit does A receive?

ক
TK. 20800
খ
TK. 19800
গ
TK. 31600
ঘ
TK. 49800
ঙ
TK. 89800

ব্যাখ্যা

Question: A, B, and C invest a total of TK. 80000 in a business. A invests TK. 7000 more than B, and B invests TK. 5000 more than C. Out of a total profit of TK. 48000. How much profit does A receive?

Solution:
Let C = x.
Then, B = x + 5000
and A = x + 5000 + 7000
= x + 12000

So,
x + x + 5000 + x + 12000 = 80000
⇒ 3x + 17000 = 80000
⇒ 3x = 80000 - 17000
⇒ 3x = 63000
⇒ x = 21000
∴ x = 21000

so, C = 21000, B = 21000 + 5000 = 26000, A = 21000 + 12000 = 33000

A : B : C = 33000 : 26000 : 21000
= 33 : 26 : 21

∴ Total ratio = 33 + 26 + 21 = 80

So A's Share
= TK. 48000 × (33/80)
= TK. 19800

৮,৭৫৬.

Find the remainder when p(x) = x⁴ - 3x² - 10x + 2 is divided by (x - 3).

ক
0
খ
22
গ
26
ঘ
32

ব্যাখ্যা

Question: Find the remainder when p(x) = x⁴ - 3x² - 10x + 2 is divided by (x - 3).

Solution:
The remainder is p(3)

p(x) = x⁴ - 3x² - 10x + 2
∴ p(3) = 3⁴ - 3.3² - 10.3 + 2
= 81 - 27 - 30 + 2
= 26

৮,৭৫৭.

If a is 150 percent of B, then B is what percent of (A + B)?

ক
ক) 40%
খ
খ) 33(1/3)%
গ
গ) 66(2/3)%
ঘ
ঘ) 75%

ব্যাখ্যা

A = 150% of B
⇒ A = 150/100 B
⇒ A/B = 3/2
⇒ A/B + 1 = 3/2 + 1
⇒ (A + B)/B = 5/2
⇒ B/(A + B) = 2/5

∴ Required percentage:
= {B / (A + B)} × 100 %
= (2/5 × 100) %
= 40%

৮,৭৫৮.

If a 1.5 m tall boy stands at a distance of 3 m from a lamp-post and casts a shadow of length 4.5 m on the ground, then the height of the lamp-post is

ক
2.5 meters
খ
4.5 meters
গ
3.5 meters
ঘ
1.5 meters

ব্যাখ্যা

Let AB be boy
Therefore, AB = 1.5m
CD be lamp
Let CD be 'h' meter.
'EB' be shadow cast by boy
Therefore, EB = 4.5meter.
BD be distance between boy and lamp
Therefore, BD = 3meters.

Therefore
In ΔAEB,
=> tanθ = AB/EB
In ΔCED,
=> tanθ = CD/ED = CD/(EB+BD)
∴ AB/EB = CD/(EB+BD)
=> 1.5/4.5 = CD/(4.5+3)
=> CD = 7.5/3 = 2.5m

৮,৭৫৯.

At what rate percent of simple interest will a sum of money double itself in 20 years?

ক
6%
খ
8%
গ
5%
ঘ
4%

ব্যাখ্যা

Let sum be x
T = 20 years

Since the money doubles, simple interest = x
R = (100 × SI)/PT
= (100 × x)/(x × 20)
= 5%

৮,৭৬০.

The average temperature for Wednesday, Thursday and Friday was 40°C. The average for Thursday, Friday and Saturday was 41° C. If temperature on Saturday was 44° C, what was the temperature on Wednesday?

ক
41° C
খ
39° C
গ
42° C
ঘ
38° C
ঙ
None of these

ব্যাখ্যা

Question: The average temperature for Wednesday, Thursday and Friday was 40°C. The average for Thursday, Friday and Saturday was 41° C. If temperature on Saturday was 44° C, what was the temperature on Wednesday?

Solution:
Average temperature for Wednesday, Thursday and Friday = 40° C
∴ Total temperature = 3 × 40 = 120° C

Average temperature for Thursday, Friday and Saturday = 41° C
∴ Total temperature = 41 × 3 = 123° C

And,
Temperature on Saturday = 44° C

Now,
(Thursday + Friday + Saturday) - (Wednesday + Thursday + Friday) = 123 - 120
⇒ Saturday - Wednesday = 3
∴ Wednesday = 44 - 3 = 41° C

৮,৭৬১.

What amount of money will amount to 750 in 6 years and to 800 in 8 years at simple interest?

ক
Tk. 550
খ
Tk. 700
গ
Tk. 650
ঘ
Tk. 600

ব্যাখ্যা

Question: What amount of money will amount to 750 in 6 years and to 800 in 8 years at simple interest?

Solution:
In two years the interest becomes = 800 - 750 = 50Tk.
∴ Interest in 1 year = 25
∴ Interest in 6 year = 25 × 6 = 150

Principal = Amount - SI
= 750 - 150 = Tk. 600

৮,৭৬২.

If a typist can type 125 pages, 36 lines each, 11 words to each line in 5 days, how many pages of 30 lines each and 12 words to each line can he type in 6 days?

ক
ক) 165
খ
খ) 160
গ
গ) 145
ঘ
ঘ) 120

ব্যাখ্যা

প্রতি লাইনে 11 words, প্রতি page এ 36 লাইন বিশিষ্ট 125 pages এ
word আছে 11× 36 × 125
5 দিনে 11 × 36 × 125 words হয়
1 দিনে (11 × 36 × 125)/5 words হয়
∴ 6 দিনে (11 × 36 × 125 × 6)/5 words হয়
= (11 × 36 × 125 × 6)/(5 × 12) lines
= (11 × 36 × 125 × 6)/(5 × 12 ×30) pages
= 165 pages.

৮,৭৬৩.

If there are 15 dots on a circle,how many triangles can be formed?

ক
ক) 455
খ
খ) 450
গ
গ) 469
ঘ
ঘ) 500
ঙ
ঙ) None of these

ব্যাখ্যা

There are 15 dots in total,and to make a triangle we need to select any three of those dots.
So, ¹⁵C₃ = 455

৮,৭৬৪.

How many terms of arithmetic progression (A. P.) 21, 18, 15, 12, … must be taken to give the sum zero?

ক
ক) 10
খ
খ) 15
গ
গ) 22
ঘ
ঘ) 27

ব্যাখ্যা

এখানে,
21, 18, 15, 12,
১ম পদ, a = 21
সাধারন অন্তর, d = 18 - 21 = -3
∴ সমষ্টি = n/2{2a + (n - 1)d}
⇒ 0 = n/2{(2 × 21) + (n - 1)(-3)}
⇒ 0 = n/2(42 - 3n + 3)
⇒ 45n - 3n² = 0
⇒ 3n(n - 15) = 0
⇒ n - 15 = 0
∴ n = 15.

৮,৭৬৫.

In the xy-plane, a triangle has vertices (0, 0), (k, 0), and (k, - 5k), where k > 0. If the area of the region enclosed by the triangle is 40, what is the value of k?

ক
2
খ
4
গ
5
ঘ
6

ব্যাখ্যা

Question: In the xy-plane, a triangle has vertices (0, 0), (k, 0) and (k, - 5k), where k > 0. If the area of the region enclosed by the triangle is 40, what is the value of k?

Solution:

প্রদত্ত ত্রিভুজটির শীর্ষবিন্দুগুলো হলো (0, 0), (k, 0) এবং (k, - 5k)।

যেহেতু B এবং C বিন্দুর x-স্থানাঙ্ক একই (k), তাই BC রেখাটি y-অক্ষের সমান্তরাল।
যেহেতু A এবং B বিন্দুর y-স্থানাঙ্ক একই (0), তাই AB রেখাটি x-অক্ষের সমান্তরাল।
সুতরাং, ত্রিভুজটি B বিন্দুতে একটি সমকোণী ত্রিভুজ।

দুটি বিন্দুর স্থানাঙ্ক (x₁, y₁) এবং (x₂, y₂) হলে তাদের মধ্যবর্তী দূরত্ব = √{(x₂ - x₁)² + (y₂ - y₁)²}

∴ ভূমি = (0, 0) এবং (k, 0) বিন্দুর মধ্যবর্তী দূরত্ব = k (যেহেতু k > 0)।
∴ উচ্চতা = (k, 0) এবং (k, - 5k) বিন্দুর মধ্যবর্তী দূরত্ব = 5k (যেহেতু k > 0)।

ত্রিভুজের ক্ষেত্রফল = (1/2) × ভূমি × উচ্চতা

প্রশ্নমতে,
(1/2) × k × 5k = 40
⇒ (5/2)k²= 40
⇒ 5k² = 80
⇒ k² = 80/5
⇒ k² = 16
⇒ k = √16
⇒ k = ±4

যেহেতু প্রশ্নে দেওয়া আছে k > 0, তাই k এর মান হবে 4।
∴ k এর মান 4।

৮,৭৬৬.

A two-digit number becomes 54 more than the original number when its digits are reversed. What is the difference between the digits of the number?

ক
3
খ
6
গ
39
ঘ
7

ব্যাখ্যা

Question: A two-digit number becomes 54 more than the original number when its digits are reversed. What is the difference between the digits of the number?

Solution:
ধরি,
সংখ্যাটির একক স্থানীয় অঙ্ক = x
দশক স্থানীয় অঙ্ক = y
∴ সংখ্যাটি = x + 10y
এবং অঙ্কদ্বয় স্থান পরিবর্তন করলে সংখ্যাটি = 10x + y

সংখ্যাটির অঙ্কদ্বয়ের পার্থক্য = x - y

প্রশ্নমতে,
10x + y = x + 10y + 54
⇒ 10x - x + y - 10y = 54
⇒ 9x - 9y = 54
⇒ 9(x - y) = 54
⇒ x - y = 54/9
⇒ x - y = 6

৮,৭৬৭.

If x² - 10x + 25 = 0, then the value of x is:

ক
5
খ
- 5
গ
4
ঘ
None

ব্যাখ্যা

Question: If x² - 10x + 25 = 0, then the value of x is:

Solution:
দেওয়া আছে,
x² - 10x + 25 = 0
⇒ x² - 2. x. 5 + 25 = 0
⇒ (x - 5)² = 0
⇒ (x - 5)(x - 5) = 0
∴ x = 5 এবং x = 5 [যেহেতু সমীকরণটি একটি দ্বিঘাত সমীকরণ তাই এর মূল হবে দুইটি]

৮,৭৬৮.

A certain psychologist charges Tk. 30 more for the first hour of therapy than for each additional hour. If the total charge to a patient who receives 6 hours of therapy is Tk. 300, what is the total charge to a patient who receives only 3 hours of therapy?

ক
135
খ
150
গ
165
ঘ
192

ব্যাখ্যা

Question: A certain psychologist charges Tk. 30 more for the first hour of therapy than for each additional hour. If the total charge to a patient who receives 6 hours of therapy is Tk. 300, what is the total charge to a patient who receives only 3 hours of therapy?

Solution:
Let the charge for first hour = x + 30
then charge for each additional hour = x
x + 30 + 5x = 300
⇒ 6x = 270
⇒ x = 45

Total charge for patient for 3 hours of therapy = x + 30 + 2x = 3x+30
= 3 × 45 + 30
= 165

৮,৭৬৯.

Sum of the two number is 37. If their product is 336, then the subtraction of the two numbers will be -

ক
ক) 37
খ
খ) 25
গ
গ) 10
ঘ
ঘ) 5

ব্যাখ্যা

Question: Sum of the two number is 37. If their product is 336, then the subtraction of the two numbers will be -

Solution:
Let the number be x and y.
Then, x + y = 37 and xy = 336

We know,
(x - y)² = (x + y)² - 4xy
= (37)² - (4 × 336)
= 1369 - 1344
= 25
⇒ x - y = √25
∴ x - y = 5.

৮,৭৭০.

A man completes a certain journey by a car. If he covered 30% of the distance at the speed of 20kmph. 60% of the distance at 40km/h and the remaining of the distance at 10 kmph, his average speed is:

ক
44 km/h
খ
36 km/h
গ
25 km/h
ঘ
22 km/h
ঙ
None of the above

ব্যাখ্যা

Question: A man completes a certain journey by a car. If he covered 30% of the distance at the speed of 20kmph. 60% of the distance at 40km/h and the remaining of the distance at 10 kmph, his average speed is:

Solution:
Suppose, total distance = 100 km

He covered 30% of the distance at the speed of 20kmph
His time taken = 30/20 hours

He covered 60% of the distance at 40km/h
His time taken = 60/40 hours

He covered the remaining of the distance at 10 kmph
The remaining of the distance = 100% - (30 + 60)% = 10%
His time taken = 10/10 hour

∴ Average speed = total distance/time
= 100/(30/20 + 60/40 + 10/10)
= 25 km/h

৮,৭৭১.

Two dice are thrown simultaneously. What is the probability of getting two numbers whose product is even?

ক
6/5
খ
3/4
গ
1/2
ঘ
2/3

ব্যাখ্যা

Question: Two dice are thrown simultaneously. What is the probability of getting two numbers whose product is even?

Solution:
In a simultaneous throw of two dice,
we have n(S) = (6 × 6) = 36

Now, we find the odd product,
E = {(1, 1), (1, 3), (1, 5), (3, 1), (3, 3), (3, 5), (5, 1), (5, 3), (5 ,5)}

∴ n(E) = 9

∴ P(odd product)= 9/36 = 1/4

Now, we find the even product,
probability(product even) = 1 - (1/4) = 3/4

৮,৭৭২.

If a person has Tk. 2000 and he wants to distribute this to his five children in the manner that each son has Tk. 30 more than the younger one, what will be the share of the youngest child?

ক
ক) Tk. 175
খ
খ) Tk. 325
গ
গ) Tk. 340
ঘ
ঘ) Tk. 260
ঙ
ঙ) Tk. 300

ব্যাখ্যা

Assume first child (the youngest) get = Tk. x
According to the question ;
each son having Tk. 30 more than the younger one

Second child will get = Tk. x + 30
Third child will get = Tk. x + 30 + 30 = x + 60
Fourth child will get = Tk. x + 30 + 30 + 30 = x + 90
Fifth child will get = Tk. x + 30 + 30 + 30 + 30 = x + 120

Total amount they got = Tk. 2000

x + (x+30) + (x+60) + (x+90) + (x+120) = 2000
5x + 300 = 2000
5x = 1700
x = Tk. 340
So the youngest child will get Tk. 340.

৮,৭৭৩.

The simple interest on a sum of money in 5 years at 12% per annum is Tk. 400 less than the simple interest accrued on the same sum in 7 years at 10% per annum. Find the sum.

ক
Tk. 3800
খ
Tk. 4200
গ
Tk. 4000
ঘ
Tk. 5000

ব্যাখ্যা

Question: The simple interest on a sum of money in 5 years at 12% per annum is Tk. 400 less than the simple interest accrued on the same sum in 7 years at 10% per annum. Find the sum.

Solution:
Let the sum be P.
SI in 5 years at 12% per annum = P × 12 × (5/100) = 0.6 P
SI in 7 years at 10% per annum = P × 10 × (7/100) = 0.7 P

Now, according to the question,
0.7 P - 0.6 P = 400
⇒ 0.1 P = 400
⇒ P = 4000
Thus, the required sum is Tk. 4000

৮,৭৭৪.

A pipe can fill a cistern in 45 minutes while another pipe can empty it in 1 hour 30 minutes. If both pipes are opened at 8 : 15 A.M., at what time will the cistern be full?

ক
9 : 25 A. M
খ
9 : 45 A. M
গ
10 : 45 A. M
ঘ
9 : 45 P. M
ঙ
None of these

ব্যাখ্যা

Quesation: A pipe can fill a cistern in 45 minutes while another pipe can empty it in 1 hour 30 minutes. If both pipes are opened at 8 : 15 A.M., at what time will the cistern be full?

Solution:
1 মিনিটে পূর্ণ হয় = 1/45 অংশ
আবার,
1 মিনিটে খালি হয় = 1/(60 + 30) = 1/90 অংশ

∴ 1 মিনিটে পূর্ণ হয় = {(1/45) - (1/90)} = (2 - 1)/90 = 1/90 অংশ

∴ 1/90 অংশ পূর্ণ হয় = 1 মিনিটে
1 বা সম্পূর্ণ অংশ পূর্ণ হয় = 1/(1/90) = 90 মিনিটে

সুতরাং ট্যাংকটি পূর্ণ হবে = 8 : 15 A. M + 90 মিনিটে = 9 : 45 A. M

৮,৭৭৫.

3/4 part of the tank is full of water. When 30 liters of water is taken out, the tank becomes empty. The capacity of the tank is -

ক
ক) 45 liters
খ
খ) 42 liters
গ
গ) 36 liters
ঘ
ঘ) 40 liters

ব্যাখ্যা

Question: 3/4 part of the tank is full of water. When 30 liters of water is taken out, the tank becomes empty. The capacity of the tank is -

Solution:
If the tank has 4x liters of total capacity and its holds 3x liters of water.
If 30 liters of water is taken out, then the tank becomes empty.

It means 3x liters of water is taken out
3x = 30 liters
x = 10 liters

Capacity of tank = 4x = 4 × 10 = 40 liters

৮,৭৭৬.

If the areas of a circle and a square are equal then the ratio of their perimeters is-

ক
π : 2
খ
√π : 1
গ
√π : 3
ঘ
√π : 2

ব্যাখ্যা

Question: If the areas of a circle and a square are equal then the ratio of their perimeters is-

Solution:
Let the length of each side of the square = a cm and the radius of the circle = r cm.
Given that
area of square = area of circle
⇒ a² = πr²
⇒ a = r√π

∴ Required ratio = 2πr/4a
= 2πr/4r√π
= √π/2
= √π : 2

৮,৭৭৭.

= ?

ক
0.25
খ
0.5
গ
0.05
ঘ
None

ব্যাখ্যা

Question:
= ?

Solution:

৮,৭৭৮.

If log_xy = 100 and log₃x = 20; then the value of y is-

ক
3²⁰
খ
3²⁰⁰
গ
3²⁰⁰⁰
ঘ
3²⁰⁰⁰⁰

ব্যাখ্যা

Question: If log_xy = 100 and log₃x = 20; then the value of y is-

Solution:
Given,
log₃x = 20
∴ x = 3²⁰

And, log_xy = 100
⇒ y = x¹⁰⁰
⇒ y = (3²⁰)¹⁰⁰∴ y = 3²⁰⁰⁰

৮,৭৭৯.

The average daily wage of 10 workers is Tk. 400. if the lowest wage is Tk. 300, then what is the possible maximum wage?

ক
800
খ
1000
গ
1200
ঘ
1300

ব্যাখ্যা

Question: The average daily wage of 10 workers is Tk. 400. if the lowest wage is Tk. 300, then what is the possible maximum wage?

Solution:
10 জন লোকের মোট মজুরি = (10 × 400) টাকা = 4000 টাকা
300 টাকা করে 9 জনের মজুরি = (300 × 9) = 2700 টাকা
∴ সম্ভাব্য সর্বোচ্চ মজুরি = (4000 - 2700) টাকা = 1300 টাকা।

৮,৭৮০.

The speeds of three cars are the ratio 2 : 3 : 4. The ratio of the taken by these cars to travel the same distance is-

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

ব্যাখ্যা

Question: The speeds of three cars are the ratio 2 : 3 : 4. The ratio of the taken by these cars to travel the same distance is-

Solution:
Given that,
Speed ratio of three cars,
v₁ : v₂ : v₃ = 2 : 3 : 4
Let,
v₁ = 2k, v₂ = 3k, v₃ = 4k (for some constant k)

We know,
Time = distance/Speed
∴ t₁ = d/2k, t₂ = d/3k, t₃ = d/4k

∴ Ratio of time = t₁ : t₂ : t₃ = d/2k : d/3k : d/4k
= 1/2 : 1/3 : 1/4 ; [Cancel d and k (since d,k ≠ 0)]
= 12/2 : 12/3 : 12/4. ; [LCM of 2, 3, 4 = 12]
= 6 : 4 : 3

So the ratio of the time taken is 6 : 4 : 3

৮,৭৮১.

Of the 1000 students who entered College X as freshmen in September 1979, 112 did not graduate in May 1983. If 962 students graduated in May 1983, how many of the graduates did not enter College X as freshmen in September 1979?

ক
38
খ
74
গ
112
ঘ
150
ঙ
188

ব্যাখ্যা

Question: Of the 1000 students who entered College X as freshmen in September 1979, 112 did not graduate in May 1983. If 962 students graduated in May 1983, how many of the graduates did not enter College X as freshmen in September 1979?

Solution:
Freshmen graduated = 1000 - 112 = 888
No of persons who graduated but were not freshmen = 962 - 888 = 74.

৮,৭৮২.

If 0 ≤ θ ≤ 90° and 4 cos²θ - 4√3 cosθ + 3 = 0 then the value of θ is

ক
90°
খ
30°
গ
45°
ঘ
60°

ব্যাখ্যা

Question: If 0 ≤ θ ≤ 90° and 4 cos²θ - 4√3 cosθ + 3 = 0 then the value of θ is

Solution:
4 cos²θ – 4√3 cosθ + 3 = 0
⇒ (2cosθ)² – 2 · 2 cosθ · √3 + (√3)² = 0
⇒ (2cosθ – √3)² = 0
⇒ 2 cosθ – √3 = 0
⇒ 2 cosθ = √3
⇒ cosθ = √3/2
⇒ cosθ = cos 30°
∴ θ = 30°

৮,৭৮৩.

A number when divided by divisor leaves a remainder of 24. When twice the original number is divided by the same divisor, the remainder is 11. What is the value of the divisor?

ক
ক) 13
খ
খ) 35
গ
গ) 37
ঘ
ঘ) 59

ব্যাখ্যা

ATQ,
dx + 11 = 2dx + 48
⇒ dx = 37
So, the divisor is 37

৮,৭৮৪.

The largest four digit number which when divided by 4, 7 or 13 leaves a remainder of 3 in each case, is -

ক
8739
খ
9831
গ
9834
ঘ
9893

ব্যাখ্যা

Greatest number of four - digits is 9999.
L.C.M. of 4, 7 and 13 = 364
On dividing 9999 by 364, the remainder obtained is 171.
∴ Greatest number of 4 - digits divisible by 4, 7 and 13 = (9999 - 171)
= 9828.
Hence, required number = (9828 + 3)
= 9831.

৮,৭৮৫.

If the average of 31, x, 42, 39, 56, 78, 83 and 91 is 58.5, then what is the value of x?

ক
26
খ
34
গ
58
ঘ
48

ব্যাখ্যা

Question: If the average of 31, x, 42, 39, 56, 78, 83 and 91 is 58.5, then what is the value of x?

Solution:
(31 + x + 42 + 39 + 56 + 78 + 83 + 91)/8 = 58.5
⇒ 420 + x = 468
⇒ x = 468 - 420
∴ x = 48

৮,৭৮৬.

A small company employs 3 men and 5 women. If a team of 4 employees is to be randomly selected to organize the company retreat, what is the probability that the team will have exactly 2 women?

ক
1/7
খ
3/5
গ
3/7
ঘ
3/8

ব্যাখ্যা

Question: A small company employs 3 men and 5 women. If a team of 4 employees is to be randomly selected to organize the company retreat, what is the probability that the team will have exactly 2 women?

Solution:
the probability that the team will have exactly 2 women is = (⁵C₂ × ³C₂)/⁸C₄
= 30/70
= 3/7

৮,৭৮৭.

The ratio of the amount of work done by (x-1) labours in (x+1) days and (x+1) labours in (x+2) days is 5 : 6. Then the value of x is?

ক
ক) 16
খ
খ) 20
গ
গ) 17
ঘ
ঘ) 15

ব্যাখ্যা

From M1D1 = M2D2
⇒ M1D1 / M2D2 = 5/6
⇒ (x−1)(x+1)/(x+1)(x+2) = 5/6
⇒ (x−1)/(x+2) = 5/6
⇒ 6x−6 = 5x+10
⇒ x = 16

৮,৭৮৮.

A circuit has a resistor of 80Ω and a current of 2A is flowing through it. What is the voltage across the resistor?

ক
1600V
খ
160V
গ
8V
ঘ
40V

ব্যাখ্যা

Question: A circuit has a resistor of 80Ω and a current of 2A is flowing through it. What is the voltage across the resistor?

Solution:
To find the voltage across the resistor, I'll use Ohm's Law:
V = I × R
Where:

V is the voltage (in volts)
I is the current (in amperes)
R is the resistance (in ohms)

Given:

Resistance (R) = 80Ω
Current (I) = 2A

Substituting these values:
V = 2A × 80Ω
V = 160V
Therefore, the voltage across the 80Ω resistor is 160 volts.

৮,৭৮৯.

A sum of money lent out at simple interest amounts to Tk. 720 after 2 years and Tk. 1020 after a further period of 5 years. Find the principal?

ক
Tk. 6000
খ
Tk. 620
গ
Tk. 600
ঘ
Tk. 120

ব্যাখ্যা

Question: A sum of money lent out at simple interest amounts to Tk. 720 after 2 years and Tk. 1020 after a further period of 5 years. Find the principal?

Solution:
According to the question,
Principal + Simple interest for 2 year = Tk. 720 ------------- (1)
Principal + Simple interest for 7 year = Tk. 1020 ------------ (2)

Subtracting equation (1) from (2)
Principal + Simple interest for 7 year = Tk. 1020
Principal + Simple interest for 2 year = Tk. 720
∴ Simple interest for 5 years = Tk. 300
⇒ Simple interest for 1 years = Tk. 300/5
⇒ Simple interest for 1 years = Tk. 60
⇒ Simple interest for 2 years = Tk. (60 × 2)
∴ Simple interest for 2 years = = Tk. 120

∴ Principal amount = (Amount after 2 years - 2 years Simple interest)
⇒ Principal amount = Tk. (720 - 120)
∴ Principal amount = Tk. 600

৮,৭৯০.

An examination paper consists of 8 questions divided into two parts 'A' and 'B'. Part A consists of 4 questions, and Part B consists of 4 questions. A candidate is required to attempt 5 questions selecting at least 2 questions from each part. In how many ways can the candidate select the question?

ক
56 ways
খ
96 ways
গ
72 ways
ঘ
48 ways
ঙ
120 ways

ব্যাখ্যা

Question: An examination paper consists of 8 questions divided into two parts 'A' and 'B'. Part A consists of 4 questions, and Part B consists of 4 questions. A candidate is required to attempt 5 questions selecting at least 2 questions from each part. In how many ways can the candidate select the question?

Solution:
Possibility 1: This can be done in ⁴C₂ × ⁴C₃ = 6 × 4 = 24 ways
Possibility 2: This can be done in ⁴C₃ × ⁴C₂ = 4 × 6 = 24 ways

∴ Total number of ways = 24 + 24 = 48 ways

৮,৭৯১.

In a two digit number, the digit in the unit's place is two more than the three times of the digit in ten's place. If the sum of the two digits is 6, the number is

ক
ক) 51
খ
খ) 24
গ
গ) 15
ঘ
ঘ) 42

ব্যাখ্যা

Question: In a two digit number, the digit in the unit's place is two more than the three times of the digit in ten's place. If the sum of the two digits is 6, the number is

Solution:
let the tens digit be x
ones digit will be 3x + 2

Now,
x + 3x + 2=6
4x + 2 = 6
4x = 4
x=1

Hence tens digit will be 1 and ones digit will be = 3 × 1 + 2 = 5
The number is 15

৮,৭৯২.

Dhaka and Mymensingh apart from each other 760 km. A train starts from Dhaka at 9 am and travels towards Mymensingh at speed 60 km/hr. Another train starts from Mymensingh at 10 am and travels towards Dhaka at speed 80 km/hr. At what time both will meet?

ক
2 pm
খ
2 : 45 pm
গ
3 : 15 pm
ঘ
3 pm

ব্যাখ্যা

Question: Dhaka and Mymensingh apart from each other 760 km. A train starts from Dhaka at 9 am and travels towards Mymensingh at speed 60 km/hr. Another train starts from Mymensingh at 10 am and travels towards Dhaka at speed 80 km/hr. At what time both will meet?

Solution:
Total distance between D and M = 760 km.
A travels 1 hour before B so it travels = 60 × 1 = 60 km
Now the remaining distance D and M= 760 - 60 = 700 km
Relative speed = 60 + 80 = 140 km/hr

Time = 700/140
= 5 hour.

So, the time when they meet = 10 am + 5 hour = 3 pm

৮,৭৯৩.

A boy rides his bicycle 10 km at an average speed of 12 km/hr. and again travels 12km at an average speed of 10km/hr. His average speed of for the entire trip is approximately -

ক
10 km/hour
খ
10.5 km/hour
গ
11.2 km/hour
ঘ
10.8 km/hour

ব্যাখ্যা

Question: A boy rides his bicycle 10 km at an average speed of 12 km/hr. and again travels 12km at an average speed of 10km/hr. His average speed of for the entire trip is approximately -

Solution:
Total Distance = 10 + 12 = 22 km
Total time = (10/12) + (12/10)
= (5/6) + (6/5)
= (25 + 36)/30
= 61/30 hours

Average speed = 22/(61/30) = 660/61 = 10.8 km/hour

৮,৭৯৪.

An observer 1.6 m tall is 20√3 away from a tower. The angle of elevation from his eye to the top of the tower is 30º. The height of the tower is:

ক
51.6 m
খ
41.6 m
গ
31.6 m
ঘ
21.6 m

ব্যাখ্যা

Question: An observer 1.6 m tall is 20√3 away from a tower. The angle of elevation from his eye to the top of the tower is 30º. The height of the tower is:

Solution:

Let AB be the observer and CD tower
Draw BE perpendicular to CD

Then CE = AB = 1.6 m
And BE = AC = 20√3 m

Then right angle triangle DEB
∴ tan30° = DE/BE
⇒ 1/√3 = DE/20√3
⇒ DE = 20√3m

Then CD = CE + DE = 1.6 + 20 = 21.6 m

৮,৭৯৫.

Cost of two types of dates is Tk.15 and Tk. 20 per KG, respectively. If both the dates are mixed together in the ratio 2 : 3, then what should be the price with Tk. 2 profit of mixed variety of dates per KG?

ক
ক) 22 Taka
খ
খ) 20 Taka
গ
গ) 15 Taka
ঘ
ঘ) 18 Taka

ব্যাখ্যা

প্রশ্ন: Cost of two types of dates is Tk.15 and Tk. 20 per KG, respectively. If both the dates are mixed together in the ratio 2 : 3, then what should be the price with Tk. 2 profit of mixed variety of dates per KG?

সমাধান:
ধরি,
১৫ টাকার খেজুর ছিল ২ক কিলোগ্রাম
২০ টাকার খেজুর ছিল ৩ক কিলোগ্রাম
∴ মোট খেজুর ৫ক কিলোগ্রাম

১৫ টাকার খেজুর আছে ৩০ক টাকার
২০ টাকার খেজুর আছে ৬০ক টাকার
∴ মোট দাম ৯০ক টাকা

৫ক কিলোগ্রামের দাম ৯০ক টাকা
∴ ১ কিলোগ্রামের দাম ৯০ক/৫ক টাকা
= ১৮ টাকা

২ টাকা লাভে প্রতি কেজি খেজুরের দাম = ১৮ + ২ = ২০ টাকা

৮,৭৯৬.

If 3√3 × 3³ ÷ 3^1/3 = 3^a/6 then the value of a is-

ক
4
খ
5
গ
25
ঘ
2

ব্যাখ্যা

Question: If 3√3 × 3³ ÷ 3^1/3 = 3^a/6 then the value of a is-

Solution:
Given that,
⇒ 3√3 × 3³ ÷ 3^1/3 = 3^a/6
⇒ 3¹ × 3^1/2 × 3³ ÷ 3^1/3 = 3^a/6
⇒ 3^{{1 + (1/2) + 3 - (1/3)}} = 3^a/6
⇒ {1+ (1/2) + 3 - (1/3)} = a/6
⇒ (6 + 3 + 18 - 2 )/6 = a/6
⇒ 25/6 = a/6
∴ a = 25

৮,৭৯৭.

What is the greatest number that will divide 96, 132, and 150 leaving remainders 6, 9, and 12 respectively?

ক
4
খ
5
গ
7
ঘ
8
ঙ
3

ব্যাখ্যা

Question: What is the greatest number that will divide 96, 132, and 150 leaving remainders 6, 9, and 12 respectively?

Solution:
We have to subtract the reminder first:
96 - 6 = 90
132 - 9 = 123
150 - 12 = 138

Using prime factorization:
For 123:
123 = 3 × 41

For 90:
90 = 2 × 3² × 5

For 138:
138 = 2 × 3 × 23

Digit (123, 90, 138) have at least 3 in common.

∴ The greatest number is 3

৮,৭৯৮.

A parallelogram has sides 60 m and 40 m and one of its diagonals is 80 m long. Its area is-

ক
500√15 m²
খ
600√15 m²
গ
400√15 m²
ঘ
450√15 m²

ব্যাখ্যা

Question: A parallelogram has sides 60 m and 40 m and one of its diagonals is 80 m long. Its area is-

Solution:

৮,৭৯৯.

A is two years older than B who is twice as old as C. If the total of the ages of A, B and C be 27, the how old is B?

ক
8 years
খ
9 years
গ
10 years
ঘ
11 years
ঙ
14 years

ব্যাখ্যা

Let C's age be x years.
Then, B's age = 2x years.
A's age = (2x + 2) years.
∴ (2x + 2) + 2x + x = 27
⇒ 5x = 25
⇒ x = 5.
Hence, B's age = 2x = 10 years.

৮,৮০০.

⁵P₂ -⁵C₂ =?

ক
ক) 0
খ
খ) 5
গ
গ) 10
ঘ
ঘ) 15

ব্যাখ্যা

প্রশ্ন: ⁵P₂ -⁵C₂ =?

সমাধান:
⁵P₂
= 5!/(5 - 2)!
= 5!/3!
= 20

⁵C₂
= 5!/2!(5 - 2)!
= 5!/2! 3!
= 10

∴ ⁵P₂ - ⁵C₂ = 20 - 10
= 10