Bank Math

PrepBank · পাতা ৮৫ / ১৬১ · ৮,৪০১–৮,৫০০ / ১৬,১২৪

৮,৪০১.

If a certain coin is flipped, the probability that the coin will land heads is 1/2. If the coin is flipped 4 times, what is the probability that it will land heads up on the first 3 flips and not on the last flip?

ক
1/8
খ
1/12
গ
1/16
ঘ
1/18

সঠিক উত্তর: গ

1/16

উত্তর

সঠিক উত্তর: গ

1/16

গ

ব্যাখ্যা

Question: If a certain coin is flipped, the probability that the coin will land heads is 1/2. If the coin is flipped 4 times, what is the probability that it will land heads up on the first 3 flips and not on the last flip?

Solution:
The probability of landing heads and not landing on heads is same = 1/2
The probability of first three heads =(1/2) × (1/2) × (1/2)
The probability of last landing not on heads = 1/2
The total probability =(1/2) × (1/2) × (1/2) × (1/2)
= 1/ 2⁴
= 1/16

৮,৪০২.

If a + b = √7 and a - b = √5, what is the value of 8ab(a² + b²)?

ক
ক) 20
খ
খ) 22
গ
গ) 26
ঘ
ঘ) 24

সঠিক উত্তর: ঘ

ঘ) 24

উত্তর

সঠিক উত্তর: ঘ

ঘ) 24

ঘ

ব্যাখ্যা

Question: If a + b = √7 and a - b = √5, what is the value of 8ab(a² + b²)?

Solution:
Given that,
a + b = √7
a - b = √5

8ab(a² + b²) = 4ab. 2(a² + b²)
= {(a + b)² - (a - b)²}{(a + b)² + (a - b)²}
= {(√7)² - (√5)²}{(√7)² + (√5)²}
= (7 - 5)(7 + 5)
= 2 × 12
= 24

৮,৪০৩.

A wire can be bent in the form of a circle of radius 7cm. If it is bent in the form of a square, then what will be its area?

ক
ক) 243 cm²
খ
খ) 121 cm²
গ
গ) 247 cm²
ঘ
ঘ) 224 cm²

সঠিক উত্তর: খ

খ) 121 cm²

উত্তর

সঠিক উত্তর: খ

খ) 121 cm²

খ

ব্যাখ্যা

প্রশ্ন: A wire can be bent in the form of a circle of radius 7cm. If it is bent in the form of a square, then what will be its area?

সমাধান:

দেওয়া আছে,
বৃত্তের ব্যাসার্ধ r = 7 cm
বৃত্তের পরিধি = 2πr
= 2 × (22/7) × 7
= 2 × 22 × 1
= 44 cm
বর্গের এক বাহুর দৈর্ঘ্য = 44/4 cm
= 11 cm
বর্গের ক্ষেত্রফল = (11)² cm²
= 121 cm²

৮,৪০৪.

If a = 0.202, then what is the value of

ক
0.808
খ
1.424
গ
1.202
ঘ
1.808

সঠিক উত্তর: গ

1.202

উত্তর

সঠিক উত্তর: গ

1.202

গ

ব্যাখ্যা

Question: If a = 0.202, then what is the value of

Solution:

৮,৪০৫.

A sum of Tk. 36,000 is invested at 8% per annum simple interest. Find the amount after 5 years.

ক
Tk. 14,400
খ
Tk. 44,400
গ
Tk. 50,400
ঘ
Tk. 54,400

সঠিক উত্তর: গ

Tk. 50,400

উত্তর

সঠিক উত্তর: গ

Tk. 50,400

গ

ব্যাখ্যা

Question: A sum of Tk. 36,000 is invested at 8% per annum simple interest. Find the amount after 5 years.

Solution:
Principal, P = Tk. 36,000
Rate of interest, r = 8%
Time, n = 5 years

We know,
I = Pnr
⇒ I = 36,000 × 5 × 8/100
= 36,000 × 40/100
= 36,000 × 2/5
= Tk. 14,400

Amount, A = P + I
= 36,000 + 14,400
= Tk. 50,400

৮,৪০৬.

If A can do 1/4 of a work in 3 days and B can do 1/9 of the same work in 4 days, how much will A get if both work together and paid Tk 800 in all?

ক
650 Tk
খ
600 Tk
গ
550 Tk
ঘ
500 Tk

সঠিক উত্তর: খ

600 Tk

উত্তর

সঠিক উত্তর: খ

600 Tk

খ

ব্যাখ্যা

Question: If A can do 1/4 of a work in 3 days and B can do 1/9 of the same work in 4 days, how much will A get if both work together and paid Tk 800 in all?

Solution:
Whole work is done by A in (3 × 4) = 12 days
∴ A's 1 day's work = 1/12 part
Whole work is done by B in (4 × 9) = 36 days
∴ B's 1 day's work = 1/36 part

A's 1 day's work : B's 1 day's work
= A's wages : B's wages
= 1/12 : 1/36
= 3 : 1

∴ A's share = (800 × 3/4) Tk
= 600 Tk

৮,৪০৭.

Currently, the age of P is twice the age of Q. After 6 years, P's age will be 3/2 times the age of Q. What is the present age of P?

ক
12 years
খ
18 years
গ
24 years
ঘ
8 years
ঙ
6 years

সঠিক উত্তর: ক

12 years

উত্তর

সঠিক উত্তর: ক

12 years

ক

ব্যাখ্যা

Question: Currently, the age of P is twice the age of Q. After 6 years, P's age will be 3/2 times the age of Q. What is the present age of P?

Solution:
Let the present age of Q = x years
Then the present age of P = 2x years

After 6 years, age of P = 2x + 6
And age of Q = x + 6

According to the question,
After 6 years, P's age will be 3/2 times Q's age. So,
⇒ 2x + 6 = (3/2) × (x + 6)
⇒ 2(2x + 6) = 3(x + 6)
⇒ 4x + 12 = 3x + 18
⇒ 4x - 3x = 18 - 12
∴ x = 6
Therefore, Present age of Q = 6 years
And present age of P = 2x = 12 years.

So the present age of P is 12 years.

৮,৪০৮.

Find the value of a and b if (x - 1) and (x + 1) are factors of x⁴ + ax³ - 3x² + 2x + b = ?

ক
ক) 2, -1
খ
খ) -2, 1
গ
গ) -2, 2
ঘ
ঘ) 1, -1

সঠিক উত্তর: গ

গ) -2, 2

উত্তর

সঠিক উত্তর: গ

গ) -2, 2

গ

ব্যাখ্যা

If (x - 1) and (x + 1) are the factors y equation then,

x - 1 = 0
x = 1
Put x = 1 we get,
1 + a - 3 + 2 + b = 0
a + b = 0 ........(i)

x + 1 = 0
x = -1
Put x = -1 we get,
1 - a - 3 - 2 + b = 0
b - a = 4 ........(ii)

(i) - (ii) we get,
a + b - b + a = 0 - 4
2a = -4
a = -2
∴ b = 2
a, b = -2, 2.

৮,৪০৯.

= ?

ক
ক) 1/xyz
খ
খ) 1
গ
গ) xyz
ঘ
ঘ) 3xyz

সঠিক উত্তর: গ

গ) xyz

উত্তর

সঠিক উত্তর: গ

গ) xyz

গ

ব্যাখ্যা

Question: = ?

Solution:

৮,৪১০.

In a class there are 50 students, their average weight is 45 kg. When a student leaves the class, the average is reduced by 100 g. Find the weight of the student who left class.

ক
43.90
খ
44.90
গ
46.90
ঘ
49.90
ঙ
None of above

সঠিক উত্তর: ঘ

49.90

উত্তর

সঠিক উত্তর: ঘ

49.90

ঘ

ব্যাখ্যা

Total weight = 45 × 50 = 2250 kg
New average = 45 - 0.1 = 44.9
The total weight of 49 = 49 × 44.9 = 2200.1
The weight of the student who left the class = (2250 - 2200.1) kg
= 49.9 kg

৮,৪১১.

If the product of 3 different positive integers is 6, then twice the sum of the integers is:

ক
ক) 12
খ
খ) 14
গ
গ) 18
ঘ
ঘ) 26
ঙ
ঙ) 36

সঠিক উত্তর: ক

ক) 12

উত্তর

সঠিক উত্তর: ক

ক) 12

ক

ব্যাখ্যা

Product of three different positive integer is 6 = 1 × 2 × 3
Twice the sum of the integers is = 2(1 + 2 + 3) = 12

৮,৪১২.

By working 5 hours a day, A can complete work in 8 days, and working 6 hours a day, B can complete the same work in 10 days. Working 8 hours a day, they can jointly complete the work in -

ক
2 days
খ
3 days
গ
4 days
ঘ
5 days

সঠিক উত্তর: খ

3 days

উত্তর

সঠিক উত্তর: খ

3 days

খ

ব্যাখ্যা

Question: By working 5 hours a day, A can complete work in 8 days, and working 6 hours a day, B can complete the same work in 10 days. Working 8 hours a day, they can jointly complete the work in -

Solution:
Working 5 hours a day,
A can complete the work in 8 days.
A can complete the work in = 5 × 8 = 40 hours.

Working 6 hours a day, B can complete the work in 10 days.
B can complete the work in = 6 × 10 = 60 hours.
(A + B)'s 1 hour's work,
= 1/40 + 1/60
= (3 + 2)/120
= 5/120
= 1/24

Hence, A and B can complete the work in 24 hours.
Working 8 hours a day
They require = 24/8 = 3 days to complete the work.

৮,৪১৩.

If 6 workers can build a wall in 10 days, how many workers would be needed to build the same wall in 5 days?

ক
12 workers
খ
15 workers
গ
18 workers
ঘ
20 workers

সঠিক উত্তর: ক

12 workers

উত্তর

সঠিক উত্তর: ক

12 workers

ক

ব্যাখ্যা

Question: If 6 workers can build a wall in 10 days, how many workers would be needed to build the same wall in 5 days?

Solution:
6 workers can build a wall in 10 days,
workers would be needed to build the same wall in 5 days = (6 × 10)/5
= 12 workers

৮,৪১৪.

If x = a secθ.cosφ, y = b secθ.sinφ and z = c tanθ. then the value of (x²/a²) + y²/b²) - (z²/c²) is?

ক
0
খ
1/3
গ
1
ঘ
3
ঙ
1/2

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: If x = a secθ.cosφ, y = b secθ.sinφ and z = c tanθ. then the value of (x²/a²) + y²/b²) - (z²/c²) is?

Solution:
Given that,
x = a secθ.cosφ
∴ x/a = secθ.cosφ

y = b secθ.sinφ
∴ y/b = secθ.sinφ

And, z = c tanθ
z/c = tanθ

Now,
(x²/a²) + y²/b²) - (z²/c²)
= (x/a)² + (y/b)² - (z/c)²
= (secθ.cosφ)² + (secθ.sinφ)² - (tanθ)²
= sec²θ.cos²φ + sec²θ.sin²φ - tan²θ
= sec²θ(cos²φ + sin²φ) - tan²θ ; [cos²θ + sin²θ = 1]
= sec²θ - tan²θ
= 1

৮,৪১৫.

If the sum two numbers is 31 and their product is 240, then find the absolute difference between the numbers.

ক
1
খ
3
গ
4
ঘ
5

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: If the sum two numbers is 31 and their product is 240, then find the absolute difference between the numbers.

Solution:
Let two numbers be x and y
We are given that, sum of two numbers x + y = 31 and product = xy = 240
Therefore,
x - y = √{(x + y)² - 4xy}
Substituting the values, we get
x - y = √{(31)² - 4 × 240}
= √(961 - 960)
= √1
= 1
The required difference between the numbers is 1.

৮,৪১৬.

The complement of an angle exceeds the angle by 40° Then the angle is equal to-

ক
40°
খ
35°
গ
30°
ঘ
25°

সঠিক উত্তর: ঘ

25°

উত্তর

সঠিক উত্তর: ঘ

25°

ঘ

ব্যাখ্যা

Question: The complement of an angle exceeds the angle by 40° Then the angle is equal to-

Solution:
Let, the angle be x
complement of the angle 90 - x

ATQ,
90 - x = x + 40°
⇒ 2x = 90 - 40°
⇒ x = 50°/2
∴ x = 25°

৮,৪১৭.

If a train runs at 40 km/hr, it reaches its destination late by 11 minutes but if it runs at 50 km/hr, it is late by 5 min only. The correct time for the train to complete its journey is:

ক
ক) 21 min
খ
খ) 19 min
গ
গ) 17 min
ঘ
ঘ) 15 min

সঠিক উত্তর: খ

খ) 19 min

উত্তর

সঠিক উত্তর: খ

খ) 19 min

খ

ব্যাখ্যা

Question: If a train runs at 40 km/hr, it reaches its destination late by 11 minutes but if it runs at 50 km/hr, it is late by 5 min only. The correct time for the train to complete its journey is:

Solution:
Let the correct time to complete the journey be x min

ATQ,
Distance covered in (x + 11) min at 40 km/hr = Distance covered in (x + 5) min at 50 km/hr
⇒ {(x + 11) × 40}/60 = {(x + 5) × 50}/60
⇒ 2(x + 11)/3 = 5(x + 5)/6
⇒ (2x + 22)/3 = (5x + 25)/6
⇒ 15x + 75 = 12x + 132
⇒ 15x - 12x = 132 - 75
⇒ 3x = 57
⇒ x = 57/3
∴ x = 19

The correct time for the train to complete its journey is 19 min.

৮,৪১৮.

The cost of a book was Tk. 75. The cost was first increased by 20% and later on it was reduced by 20%. The present cost of the book is:

ক
Tk. 65
খ
Tk. 72
গ
Tk. 75
ঘ
Tk. 80

সঠিক উত্তর: খ

Tk. 72

উত্তর

সঠিক উত্তর: খ

Tk. 72

খ

ব্যাখ্যা

Question: The cost of a book was Tk. 75. The cost was first increased by 20% and later on it was reduced by 20%. The present cost of the book is:

Solution:
Initial Cost = Tk. 75
After 20% increase in the cost, it becomes,
(75 + 20% of 75)
= 75 + 15
= Tk. 90
Now, Cost is decreased by 20%, So cost will become,
(90 - 20% of 90)
= 90 - 18
=Tk. 72
So, present cost is Tk. 72.

৮,৪১৯.

The difference of ages between two brothers is 6 years. If the ratio of their ages is 7 : 5, find their ages.

ক
Elder 21, Younger 15
খ
Elder 28, Younger 22
গ
Elder 35, Younger 29
ঘ
Elder 14, Younger 8

সঠিক উত্তর: ক

Elder 21, Younger 15

উত্তর

সঠিক উত্তর: ক

Elder 21, Younger 15

ক

ব্যাখ্যা

Question: The difference of ages between two brothers is 6 years. If the ratio of their ages is 7 : 5, find their ages

Solution:
Let
The ages of the two brothers be 7x and 5x. (elder : younger).
Difference of ages = 6

Accordingly,
7x - 5x = 6
⇒ 2x = 6
⇒ x = 3.

∴ Elder = 7x = 7 × 3 = 21
∴ Younger = 5x = 5 × 3 = 15

So Elder brother 21 years and younger brother 15 years.

৮,৪২০.

2/5 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
60 liters
খ
65 liters
গ
70 liters
ঘ
75 liters

সঠিক উত্তর: ঘ

75 liters

উত্তর

সঠিক উত্তর: ঘ

75 liters

ঘ

ব্যাখ্যা

Question: 2/5 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:
Let us consider,
The tank has 5x liters of total capacity and holds 2x litres of water.
And if 30 liters of water is taken out, then the tank becomes empty.

It means 2x litres of water is taken out.
∴ 2x = 30 liters
⇒ x = 15 liters

Capacity of tank = 5x
= 5 × 15
= 75 liters

৮,৪২১.

In a school, students may bring breakfast, buy it, or may not eat breakfast. If 1/4 of the students bring breakfast, 1/7 don't eat breakfast, and 187 buy it, how many students bring breakfast?

ক
77
খ
68
গ
58
ঘ
49
ঙ
None

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: In a school, students may bring breakfast, buy it, or may not eat breakfast. If 1/4 of the students bring breakfast, 1/7 don't eat breakfast, and 187 buy it, how many students bring breakfast?

Solution:
Let,
Total number of student = x
The students bring breakfast = x/4
The students don't eat breakfast = x/7
The students buy breakfast = 187

ATQ,
x/4 + x/7 + 187 = x
⇒ x - x/4 - x/7 = 187
⇒ (28x - 7x - 4x)/28 = 187
⇒ 28x - 11x = 187 × 28
⇒ 17x = 187 × 28
⇒ x = (187 × 28)/17
∴ x = 308

∴ The students bring breakfast = x/4 = 308/4 = 77

৮,৪২২.

What is the amount for a sum of money Tk. 7500 at 6% rate of interest compound interest for 2 years?

ক
Tk. 8427
খ
Tk. 8417
গ
Tk. 8400
ঘ
Tk. 8390

সঠিক উত্তর: ক

Tk. 8427

উত্তর

সঠিক উত্তর: ক

Tk. 8427

ক

ব্যাখ্যা

Question: What is the amount for a sum of money Tk. 7500 at 6% rate of interest compound interest for 2 years?

Solution:
C = P(1 + r)ⁿ
= 7500(1 + 6/100)²
= 7500 × 1.06 × 1.06
= 8427

৮,৪২৩.

In how many different ways can the letters of the word 'CORRUPTION' be arranged so that the vowels always come together?

ক
ক) 30240
খ
খ) 30200
গ
গ) 30420
ঘ
ঘ) None of the above

সঠিক উত্তর: ক

ক) 30240

উত্তর

সঠিক উত্তর: ক

ক) 30240

ক

ব্যাখ্যা

Question: In how many different ways can the letters of the word 'CORRUPTION' be arranged so that the vowels always come together?

Solution:
In the word 'CORRUPTION', we treat the vowels OUIO as one letter.

Thus, we have CRRPTN (OUIO).

This has 7 (6 + 1) letters of which R occurs 2 times and the rest are different.

Number of ways arranging these letters = 7!/2! = 2520.

Now, 4 vowels in which O occurs 2 times and the rest are different, can be arranged in 4!/2! = 12 ways.

Required number of ways = (2520 x 12) = 30240.

৮,৪২৪.

NUMBER : UNBMRE : : GHOST : ?

ক
ক) HGSOT
খ
খ) TSOGH
গ
গ) OGHST
ঘ
ঘ) SOTGH

সঠিক উত্তর: ক

ক) HGSOT

উত্তর

সঠিক উত্তর: ক

ক) HGSOT

ক

ব্যাখ্যা

Every pair of letters in the terms are in reverse order as NU = UN, MB = BM and ER = RE.

৮,৪২৫.

The radius of a circle is the same as the diagonal of a square whose area is 25 sq. cm. The area of the circle is -

ক
75π sq. cm.
খ
50π sq. cm.
গ
100π sq. cm.
ঘ
65π sq. cm.

সঠিক উত্তর: খ

50π sq. cm.

উত্তর

সঠিক উত্তর: খ

50π sq. cm.

খ

ব্যাখ্যা

Question: The radius of a circle is the same as the diagonal of a square whose area is 25 sq. cm. The area of the circle is -

Solution:
Area of square = 25
Side of square = √25 = 5

Diagonal of square = 5√2
So, the radius of the circle is 5√2 cm

Area of circle = πr²
= π(5√2)²
= 50π cm²
The area of the circle is 50π sq. cm.

৮,৪২৬.

In a 500 m race, the speeds of two runners, A and B are in the ratio 5 : 6. If A is given a start of 100m, by how many meters does A win the race?

ক
45 meters
খ
25 meters
গ
30 meters
ঘ
20 meters

সঠিক উত্তর: ঘ

20 meters

উত্তর

সঠিক উত্তর: ঘ

20 meters

ঘ

ব্যাখ্যা

Question: In a 500 m race, the speeds of two runners, A and B are in the ratio 5 : 6. If A is given a start of 100m, by how many meters does A win the race?

Solution:
Total race length = 500 meters.
A is given a start of 100 meters, so A runs 500 - 100 = 400 meters.

Speed ratio A : B = 5 : 6.

Let, B runs = X meter

Therefore,
400/X = 5/6
⇒ X = (6 × 400)/5
∴ X = 480m

Remaining distance for B = 500 - 480 = 20 meters.
Therefore, A wins by 20 meters.

৮,৪২৭.

The average price of 10 books is Tk.12 while the average price of 8 of these books is Tk.11.75. Of the remaining two books, if the price of one book is 60% more than the price of the other, what is the price of each of these two books?

ক
Tk. 5, Tk. 17
খ
Tk. 8.50, Tk. 12.75
গ
Tk. 16, Tk. 10
ঘ
Tk. 12, Tk. 14

সঠিক উত্তর: গ

Tk. 16, Tk. 10

উত্তর

সঠিক উত্তর: গ

Tk. 16, Tk. 10

গ

ব্যাখ্যা

As per question,
Total cost of 10 books = Tk. 120
Total cost of 8 books = Tk. 94
∴ The cost of 2 books = Tk. 26

Let the price of one book is x TK.
∴ The other book must be 160% of x or 1.6x TK.

∴ x + 1.6x = 26
Or, x = 10.

∴ Cost of a book is = 10 TK
and cost of the other book is = 26 - 10 = 16 TK.

৮,৪২৮.

One third of Aman's marks in Mathematics exceeds a half of his marks in English by 30. If he got 240 marks in the two subjects together, how many marks did he get in English?

ক
180
খ
120
গ
90
ঘ
60

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: One third of Aman's marks in Mathematics exceeds a half of his marks in English by 30. If he got 240 marks in the two subjects together, how many marks did he get in English?

Solution:
Let, Aman's marks in Mathematics be x and marks in English be y.
Total marks in both the subject = 240

∴ x + y = 240
⇒ 2(x + y) = 240 × 2
⇒ 2x + 2y = 480 ..................(1)

(x/3) - (y/2) = 30
⇒ (2x - 3y)/6 = 30
⇒ 2x - 3y = (30 × 6)
⇒ 2x - 3y = 180 ..................(2)

Subtracting equation (1) from equation (2) we get,
2x - 3y = 180
2x + 2y = 480
- 5y = - 300
⇒ y = - 300/- 5
∴ y = 60

∴ He scored 60 marks in English.

৮,৪২৯.

A school has a total of 90 students. There are 30 students taking Physics, 25 taking English, and 13 taking both. How many students are taking either Physics or English?

ক
55
খ
68
গ
42
ঘ
81

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: A school has a total of 90 students. There are 30 students taking Physics, 25 taking English, and 13 taking both. How many students are taking either Physics or English?

Solution:
Students taking physics n(P) = 30 (these 30 include those 13 that take both)
Students taking english n(E) = 25 (these 25 also include those 13)
Students taking both n(P ∩ E) = 13
Students taking either Physics or English n(P ∪ E) = ?

We know
n(P ∪ E) = n(P) + n(E) - n(P ∩ E)
= 30 + 25 - 13 = 42

৮,৪৩০.

The length of a room is 5.5 m and the width is 3.75 m. Find the cost of paving the floor with slabs at the rate of Tk. 800 per square metre.

ক
15500 Tk
খ
16500 Tk
গ
17500 Tk
ঘ
18500 Tk

সঠিক উত্তর: খ

16500 Tk

উত্তর

সঠিক উত্তর: খ

16500 Tk

খ

ব্যাখ্যা

Question: The length of a room is 5.5 m and the width is 3.75 m. Find the cost of paving the floor with slabs at the rate of Tk. 800 per square metre.

Solution:
Area of the floor
= (5.5 × 3.75)m²
= 20.625m²

∴ Cost of paying
= (800 × 20.625) Tk
=16500 Tk

৮,৪৩১.

A man is walking at a speed of 10 kmph. After every km, he takes rest for 5 minutes. How much time will he take to cover a distance of 5 km?

ক
ক) 40 min
খ
খ) 50 min
গ
গ) 56 min
ঘ
ঘ) 66 min

সঠিক উত্তর: খ

খ) 50 min

উত্তর

সঠিক উত্তর: খ

খ) 50 min

খ

ব্যাখ্যা

Time taken by man if he did not stop = 5 km/10 kmph = 1/2 hr = 30 min
∵Man takes rest for 5 minutes on each km Total rest time= 5×4= 20 min
Total travelling time: = 30 min+20 min = 50 min

৮,৪৩২.

The average of the first five multiples of 7 is -

ক
ক) 20.5
খ
খ) 21
গ
গ) 25
ঘ
ঘ) 26.2

সঠিক উত্তর: খ

খ) 21

উত্তর

সঠিক উত্তর: খ

খ) 21

খ

ব্যাখ্যা

Question: The average of the first five multiples of 7 is-

Solution:
the first five multiples of 7, 14, 21, 28, 35

the average is = (7 + 14 + 21 + 28 + 35)/5
= 21

৮,৪৩৩.

The slant height of a right circular cone is 13 m and its height is 5 m. Find area of the curved surface.

ক
490.28 m²
খ
288.28 m²
গ
450 m²
ঘ
200 m²
ঙ
None of these

সঠিক উত্তর: ক

490.28 m²

উত্তর

সঠিক উত্তর: ক

490.28 m²

ক

ব্যাখ্যা

Question: The slant height of a right circular cone is 13 m and its height is 5 m. Find area of the curved surface.

Solution:
Area of curved surface = πrl
Now
r = √(13² - 5²)
= √(169 - 25)
= √144
= 12m

∴ Required Area= (22/7) × 13 × 12
= 490.28 m²

৮,৪৩৪.

For a research purpose 2500 individuals were interviewed. Among them 750 persons have bank accounts in State Owned Commercial Banks (SOCBs) and 2250 persons have bank accounts in Private Commercial Banks (PCBs). How many of them have bank accounts in both SOCBs and PCBs?

ক
600
খ
500
গ
300
ঘ
250

সঠিক উত্তর: খ

500

উত্তর

সঠিক উত্তর: খ

500

খ

ব্যাখ্যা

Question: For a research purpose 2500 individuals were interviewed. Among them 750 persons have bank accounts in State Owned Commercial Banks (SOCBs) and 2250 persons have bank accounts in Private Commercial Banks (PCBs). How many of them have bank accounts in both SOCBs and PCBs?

Solution:
Total persons 2500
Persons have bank accounts in State Owned Commercial Banks (SOCBs) 750
Persons have bank accounts in Private Commercial Banks (PCBs) 2250

Individuals who have bank accounts in both SOCBs and PCBs = 750 + 2250 - 2500
= 3000 - 2500
= 500

৮,৪৩৫.

A fighter jet covers a certain distance at a speed of 1200 km/h in 5 hours. What speed must it maintain to cover the same distance in 250 minutes?

ক
1220 km/h
খ
1440 km/h
গ
1650 km/h
ঘ
1050 km/h

সঠিক উত্তর: খ

1440 km/h

উত্তর

সঠিক উত্তর: খ

1440 km/h

খ

ব্যাখ্যা

Question: A fighter jet covers a certain distance at a speed of 1200 km/h in 5 hours. What speed must it maintain to cover the same distance in 250 minutes?

Solution:
Total distance = Speed × Time
= (1200 × 5) km
= 6000 km

Given time = 250 minutes = (250/60) hours= 25/6 hours

∴ Required speed = Distance/Time
= {6000/(25/6)} km/h
= {6000 × (6/25)} km/h
= (240 × 6) km/h
= 1440 km/h

৮,৪৩৬.

If a person walks at 22 km/hr instead of 16 km/hr, he would have walked 33 km more. The actual distance travelled by him is:

ক
48 km
খ
88 km
গ
64 km
ঘ
58 km

সঠিক উত্তর: খ

88 km

উত্তর

সঠিক উত্তর: খ

88 km

খ

ব্যাখ্যা

Question: If a person walks at 22 km/hr instead of 16 km/hr, he would have walked 33 km more. The actual distance travelled by him is:

Solution:
Let the actual distance travelled be x km.

ATQ,
Or, x/16 = (x + 33)/22
Or, 22x = 16x + 528
Or, 22x − 16x = 528
Or, 6x = 528
Or, x = 88

∴ distance = 88 km.

৮,৪৩৭.

A man can row upstream at 10 kmph and downstream at 20 kmph. Find the man's rate in still water and the rate of the stream.

ক
12 kmph , 7 kmph
খ
15 kmph , 5 kmph
গ
15 kmph , 12 kmph
ঘ
10 kmph , 5 kmph

সঠিক উত্তর: খ

15 kmph , 5 kmph

উত্তর

সঠিক উত্তর: খ

15 kmph , 5 kmph

খ

ব্যাখ্যা

Question: A man can row upstream at 10 kmph and downstream at 20 kmph. Find the man's rate in still water and the rate of the stream.

Solution:
If a is rate downstream and b is rate upstream
Rate in still water = (a + b)/2
Rate of current = (a - b)/2

Rate in still water = (20 + 10)/2 = 15 kmph
Rate of current = (20 - 10)/2 = 5 kmph

৮,৪৩৮.

A number is increased by 15% and then decreased by 25% and the number becomes 22 less than the original number. The original number is:

ক
150
খ
160
গ
175
ঘ
200

সঠিক উত্তর: খ

160

উত্তর

সঠিক উত্তর: খ

160

খ

ব্যাখ্যা

Question: A number is increased by 15% and then decreased by 25% and the number becomes 22 less than the original number. The original number is :

Solution:
Let the number is = 100x

Now after 15% of increase = 100x + 15% of 100x = 115x
Now 25% decrease = 115x - 25% of 115x = 115x - 28.75x = 86.25x

Actual decreased = 100x - 86.25x = 13.75x

According to the question,
13.75x = 22
⇒ x = 22/13.75
⇒ 100x = (22/13.75) × 100
⇒ 100x = 160.

∴ Original number = 160.

৮,৪৩৯.

The average of the first and the second of three numbers is 15 more than the average of the second and the third of these numbers. What is the difference between the first and the third of these three numbers?

ক
ক) 15
খ
খ) 45
গ
গ) 60
ঘ
ঘ) None of these

সঠিক উত্তর: ঘ

ঘ) None of these

উত্তর

সঠিক উত্তর: ঘ

ঘ) None of these

ঘ

ব্যাখ্যা

Let 1st no = x
2nd no = y
and, 3rd no = z
ATQ,
(x + y)/2 = 15 + (y + z)/2
Or, x + y = 30 + y + z
Or, x - z = 30
So, Difference of 1st and 3rd no is 30

৮,৪৪০.

Find the H.C.F. of p(x) = x³ - 4x and q(x) = x² + 3x - 10

ক
(x + 2)
খ
x(x - 2)
গ
(x + 5)
ঘ
(x - 2)
ঙ
none of these

সঠিক উত্তর: ঘ

(x - 2)

উত্তর

সঠিক উত্তর: ঘ

(x - 2)

ঘ

ব্যাখ্যা

Question: Find the H.C.F. of p(x) = x³ - 4x and q(x) = x² + 3x - 10

Solution:
Given that, p(x) = x³ - 4x and q(x) = x² + 3x - 10

Now,
The factors of p(x) = x³ - 4x
⇒ x(x² - 4)
⇒ x(x + 2)(x - 2)

And,
The factors of q(x) = x² + 3x - 10
⇒ x² + 5x - 2x - 10
⇒ x(x + 5) - 2(x + 5)
⇒ (x - 2)(x + 5)

∴ The required H.C.F. is (x - 2)

৮,৪৪১.

Find the HCF of x² - 4, x⁴ - 16, and x³ - 2x² - 4x + 8 is-

ক
x² + 4
খ
x² - 3
গ
x² - 4
ঘ
x³ - 3

সঠিক উত্তর: গ

x² - 4

উত্তর

সঠিক উত্তর: গ

x² - 4

গ

ব্যাখ্যা

Question: Find the HCF of x² - 4, x⁴ - 16, and x³ - 2x² - 4x + 8 is-

Solution:
১ম রাশি = x² - 4 = (x + 2)(x - 2)
২য় রাশি = x⁴ - 16 = (x² + 4)(x² - 4) = (x² + 4)(x + 2)(x - 2)
৩য় রাশি = x³ - 2x² − 4x + 8 = (x - 2)(x² - 4)= (x + 2)(x - 2)(x - 2)

So HCF is = (x + 2)(x - 2) = x² - 4

৮,৪৪২.

If a giraffe has two eyes, a monkey has two eyes, and an elephant has two eyes, how many eyes do we have?

ক
3
খ
4
গ
1
ঘ
2

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: If a giraffe has two eyes, a monkey has two eyes, and an elephant has two eyes, how many eyes do we have?

Solution:
4 eyes.
Here in the question, it is asked how many Eyes We have so that means here the person who has asked the question is also including the person who is suppose to give the answer. In a clear understanding, the Conversation is happening between 2 people 1st who asked the question and 2nd to whom it has been asked, which means there are 4 eyes.

এখানে প্রশ্ন করা হয়েছে আমাদের কতগুলো চোখ আছে তার মানে এখানে যিনি প্রশ্ন করেছেন তিনি নিজেকে অন্তর্ভুক্ত করেছেন যিনি উত্তর দিবেন তাঁর সাথে। এটি স্পষ্ট বুঝা যাচ্ছে, কথোপকথনটি ২ জনের মধ্যে ঘটছে ১ম জন যিনি প্রশ্নটি করেছে এবং ২য় জন যার কাছে প্রশ্নটি জিজ্ঞাসা করা হয়েছে, যার অর্থ ২ জনের ৪টি চোখ রয়েছে৷

৮,৪৪৩.

Two persons Choton and Billal started a business in which Choton invested Tk. 50000, Billal invested Tk. 80000, after 4 months Robiul joined them with a certain amount. At the end of the year, a total profit of Tk. 40000 was recorded. Robiul's share in the profit was Tk. 15000, then find Robiul's investment in the business.

ক
Tk. 120000
খ
Tk. 116000
গ
Tk. 117000
ঘ
Tk. 100000

সঠিক উত্তর: গ

Tk. 117000

উত্তর

সঠিক উত্তর: গ

Tk. 117000

গ

ব্যাখ্যা

Question: Two persons Choton and Billal started a business in which Choton invested Tk. 50000, Billal invested Tk. 80000, after 4 months Robiul joined them with a certain amount. At the end of the year, a total profit of Tk. 40000 was recorded. Robiul's share in the profit was Tk. 15000, then find Robiul's investment in the business.

Solution:
Let,
Robiul's investment in the business be Tk. 1000x
Profit ratio of Choton, Billal and Robiul = (50000 × 12) : (80000 × 12) : {1000x × (12 - 4)}
= 75 : 120 : x

Total profit = 75 + 120 + x = 195 + x

Robiul's share in profit = x
ATQ,
(195 + x)/x = 40000/15000
⇒ (195 + x)/x = 8/3
⇒ 3(195 + x) = 8x
⇒ 585 + 3x = 8x
⇒ 5x = 585
⇒ x = 117

Investment by Robiul = 1000 × 117 = Tk. 117000

৮,৪৪৪.

A shopkeeper incurs a loss by selling an article for Tk 550. If he had sold it for Tk 850, he would have made a profit which is four times the initial loss. At what price should he sell the article to make 10% profit?

ক
Tk. 650
খ
Tk. 671
গ
Tk. 690
ঘ
Tk. 710

সঠিক উত্তর: খ

Tk. 671

উত্তর

সঠিক উত্তর: খ

Tk. 671

খ

ব্যাখ্যা

Question: A shopkeeper incurs a loss by selling an article for Tk 550. If he had sold it for Tk 850, he would have made a profit which is four times the initial loss. At what price should he sell the article to make 10% profit?

Solution:
ধরি, পণ্যের ক্রয়মূল্য = x টাকা
550 টাকায় বিক্রি করলে ক্ষতি = x - 550 টাকা
850 টাকায় বিক্রি করলে লাভ = 850 - x টাকা

প্রশ্নমতে,
850 - x = 4(x - 550)
⇒ 850 - x = 4x - 2200
⇒ 850 + 2200 = 4x + x
⇒ 3050 = 5x
∴ x = 610 টাকা

এখন,
10% লাভে,
ক্রয়মূল্য 100 টাকা হলে বিক্রয়মূল্য 110 টাকা
ক্রয়মূল্য 1 টাকা হলে বিক্রয়মূল্য (110/100) টাকা
∴ ক্রয়মূল্য 610 টাকা হলে বিক্রয়মূল্য (110 × 610)/100 টাকা
= 671 টাকা

৮,৪৪৫.

The equation 2x² + kx + 3 = 0 has two equal roots, then the value of k is-

ক
± √6
খ
± 4
গ
± 3√2
ঘ
± 2√6

সঠিক উত্তর: ঘ

± 2√6

উত্তর

সঠিক উত্তর: ঘ

± 2√6

ঘ

ব্যাখ্যা

Question: The equation 2x² + kx + 3 = 0 has two equal roots, then the value of k is-

Solution:
Here a = 2, b = k, c = 3
Since the equation has two equal roots
∴ b² - 4ac = 0
⇒ (k)² - 4 × 2 × 3 = 0
⇒ k² = 24
⇒ k = ± √24
∴ k = ± √(4 × 6) = ± 2√6

৮,৪৪৬.

Two circles of equal radii touch externally at a point P. From a point T on the tangent at P, tangents TQ and TR are drawn to the circles with points of contact Q and R respectively. The relation of TQ and TR is-

ক
ক) TQ < TR
খ
খ) TQ > TR
গ
গ) TQ = 2 TR
ঘ
ঘ) TQ = TR

সঠিক উত্তর: ঘ

ঘ) TQ = TR

উত্তর

সঠিক উত্তর: ঘ

ঘ) TQ = TR

ঘ

ব্যাখ্যা

- স্পর্শবিন্দুতে স্পর্শকের ওপর অঙ্কিত লম্ব কেন্দ্রগামী।
- বৃত্তের কোনো বিন্দু দিয়ে ঐ বিন্দুগামী ব্যাসার্ধের ওপর অঙ্কিত লম্ব উক্ত বিন্দুতে বৃত্তটির স্পর্শক হয়।
- বৃত্তের বহিঃস্থ কোনো বিন্দু থেকে বৃত্তে দুইটি স্পর্শক টানলে, ঐ বিন্দু থেকে স্পর্শ বিন্দুদ্বয়ের দূরত্ব সমান।

এখানে
দুটি বৃত্ত A ও B পরস্পর P বিন্দুতে বহিঃস্থভাবে স্পর্শ করেছে।
T বিন্দু থেকে A ও B বৃত্তে দুইটি স্পর্শক TQ এবং TR
TQ = TR

৮,৪৪৭.

Having scored 98 runs in the 19th innings a cricketer increase his average score by 4.what will be his average score after the 19th innings?

ক
ক) 24
খ
খ) 28
গ
গ) 26
ঘ
ঘ) 22

সঠিক উত্তর: গ

গ) 26

উত্তর

সঠিক উত্তর: গ

গ) 26

গ

ব্যাখ্যা

Question: Having scored 98 runs in the 19th innings a cricketer increase his average score by 4.what will be his average score after the 19th innings?

Solution:
Let, his average score is x
His total score after 19th innings = 19x

ATQ,
(19x - 98)/18 = x - 4
⇒ 19x - 98 = 18x - 72
⇒ 19x - 18x = 98 - 72
∴ x = 26

His average score is 26

৮,৪৪৮.

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

ক
128
খ
196
গ
346
ঘ
280

সঠিক উত্তর: খ

196

উত্তর

সঠিক উত্তর: খ

196

খ

ব্যাখ্যা

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

Solution:
The student can choose 4 questions from first 5 questions or he can also choose 5 questions from the first five questions.

∴ No. of choices available to the student = ⁵C₄ × ⁸C₆ + ⁵C₅ × ⁸C₅
= 5 × 28 + 1 × 56
= 196

৮,৪৪৯.

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

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

সঠিক উত্তর: ঘ

Tk. 38,000

উত্তর

সঠিক উত্তর: ঘ

Tk. 38,000

ঘ

ব্যাখ্যা

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

Solution:
Let the original salaries of Rakib and Sumon be Tk. 2x and Tk. 3x respectively.

Then,
(2x + 4000)/(3x + 4000) = 40/57
⇒ 57(2x + 4000) = 40(3x + 4000)
⇒ 6x = 68,000
⇒ 3x = 34,000

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

৮,৪৫০.

Three numbers which are co-prime to each other are such that the product of the first two is 551 and that of the last two is 1073. The sum of the three numbers is:

ক
ক) 75
খ
খ) 81
গ
গ) 85
ঘ
ঘ) 89
ঙ
ঙ) None of the above

সঠিক উত্তর: গ

গ) 85

উত্তর

সঠিক উত্তর: গ

গ) 85

গ

ব্যাখ্যা

Since the numbers are co-prime, they contain only 1 as the common factor.

Also, the given two products 551 and 1073 have the middle number in common.

So, middle number = H.C.F. of 551 and 1073 = 29

First number = 551/29 = 19
Third number = 1073/29 = 37

Required sum = (19 + 29 + 37) = 85

৮,৪৫১.

If 62 = 34 + 4x. What is x?

ক
ক) x = 16
খ
খ) x = -16
গ
গ) x = 7
ঘ
ঘ) x = 5

সঠিক উত্তর: গ

গ) x = 7

উত্তর

সঠিক উত্তর: গ

গ) x = 7

গ

ব্যাখ্যা

প্রশ্ন: If 62 = 34 + 4x. What is x?

সমাধান:
62 = 34 + 4x
⇒ 4x = 62 - 34
⇒ 4x = 28
⇒ x = 28/4
⇒ x = 7

৮,৪৫২.

A is greater than B by 13; C is less than B by 7. If the ratio of A and C is 9 : 5, what is the value of B?

ক
ক) 25
খ
খ) 32
গ
গ) 26
ঘ
ঘ) 36

সঠিক উত্তর: খ

খ) 32

উত্তর

সঠিক উত্তর: খ

খ) 32

খ

ব্যাখ্যা

Question: A is greater than B by 13; C is less than B by 7. If the ratio of A and C is 9 : 5, what is the value of B?

Solution:
Given,
A - B = 13
B - C = 7
and A : C = 9 : 5
⇒ A/C = 9/5
⇒ A = 9C/5

Now,
A - B + B - C = 13 + 7
⇒ A - C = 20
⇒ 9C/5 - C = 20
⇒ (9C - 5C)/5 = 20
⇒ 4C = 100
⇒ C = 25

So, B - 25 = 7
⇒ B = 32

৮,৪৫৩.

The value of the polynomial 5x³ - 4x² + 3 when x = - 1 is

ক
6
খ
8
গ
- 6
ঘ
- 12

সঠিক উত্তর: গ

- 6

উত্তর

সঠিক উত্তর: গ

- 6

গ

ব্যাখ্যা

Question: The value of the polynomial 5x³ - 4x² + 3 when x = - 1 is

Solution:
5x³ - 4x² + 3
If x = - 1,
then replace x with - 1,

We get,
5x³ - 4x² + 3 = 5 × (- 1)³ - 4(- 1)² + 3
= - 5 - 4 + 3
= - 6

৮,৪৫৪.

A worker earns Tk. 500 on the first day and spends Tk. 200 on the second day, earns Tk. 500 on the third day and again spends Tk. 200 on the fourth day, and so on. On which day would he have had Tk. 2000?

ক
9th day
খ
11th day
গ
13th day
ঘ
15th day

সঠিক উত্তর: খ

11th day

উত্তর

সঠিক উত্তর: খ

11th day

খ

ব্যাখ্যা

Question: A worker earns Tk. 500 on the first day and spends Tk. 200 on the second day, earns Tk. 500 on the third day and again spends Tk. 200 on the fourth day, and so on. On which day would he have had Tk. 2000?

Solution:
1ম দিনে আয় = 500 টাকা
2য় দিনে ব্যয় = 200 টাকা
∴ প্রতি 2 দিনে প্রকৃত জমা হয় = (500 - 200) = 300 টাকা

শেষ দিনে আয় করার পর কাঙ্ক্ষিত লক্ষ্যে পৌঁছাবে, তাই শেষ দিনের আয় বাদ দিলে থাকে = (2000 - 500) = 1500 টাকা

এখন, 300 টাকা জমা হয় = 2 দিনে
∴ 1500 টাকা জমা হয় = (2 × 1500) / 300 = 10 দিনে

অর্থাৎ, 10 দিনের শেষে তার হাতে জমা থাকবে 1500 টাকা।
পরের দিন অর্থাৎ 11-তম দিনে তিনি আবার 500 টাকা আয় করবেন।
∴ মোট সঞ্চয় হবে = 1500 + 500 = 2000 টাকা

∴ 11-তম দিনে তার কাছে মোট 2000 টাকা থাকবে।

৮,৪৫৫.

Find the missing letter marked (?) in the series:
B, E, I, ?, T, A

ক
L
খ
N
গ
O
ঘ
P

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: Find the missing letter marked (?) in the series:
B, E, I, ?, T, A

Solution:

এখানে প্যাটার্নটি হলো +3, +4, +5, +6, ...
B (2) + 3 = E (5)
E (5) + 4 = I (9)
I (9) + 5 = 14তম অক্ষর, অর্থাৎ N
N (14) + 6 = T (20)
T (20) + 7 = 27তম অক্ষর (20 + 7), অর্থাৎ 1 (26 + 1) বা A

∴ সুতরাং প্রশ্নবোধক চিহ্নিত স্থানে N বসবে।

৮,৪৫৬.

All possible three digit numbers are formed by 1, 2, 3. If one number is chosen randomly; the probability that it would be divisible by 111 is -

ক
ক) 0
খ
খ) 2/9
গ
গ) 1/3
ঘ
ঘ) 1/4

সঠিক উত্তর: ক

ক) 0

উত্তর

সঠিক উত্তর: ক

ক) 0

ক

ব্যাখ্যা

1,2,3 অংকগুলো দ্বারা গঠিত সংখ্যাগুলো হলোঃ 123, 132, 213, 231, 312, 321 (6টি)
এখন এর মধ্যে 111 দ্বারা বিভাজ্য সংখ্যা একটিও নেই।
Probability = 0/6 = 0

৮,৪৫৭.

Selling a dozen of bananas for a taka causes a loss of 25%. How many bananas do you have to sell for a taka to get 50% profit?

ক
8
খ
6
গ
10
ঘ
5

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Let x bananas be purchased for a Taka, causing 25% loss.
x/12 = 75/100
x = (75/100) * 12
x = 9

Let y bananas be sold for a Taka to get 50% profit.
y/9 = 100/150
y = (100/150)*9
y = 6

বিকল্প:
ক্রয়মূল্য ১০০ টাকা হলে,
২৫% ক্ষতিতে বিক্রয়মূল্য = ১০০-২৫ = ৭৫ টাকা।
৫০% লাভে বিক্রয়মূল্য = ১০০+৫০ = ১৫০ টাকা।
পূর্বের বিক্রয়মূল্য ৭৫ টাকা হলে বিক্রয় করতে হবে ১৫০ টাকায়।
পূর্বের বিক্রয়মূল্য ১ টাকা হলে বিক্রয় করতে হবে ১৫০/৭৫ = ২ টাকায়।
২ টাকায় বিক্রি করতে হবে = ১২ টি কলা।
১ টাকায় বিক্রি করতে হবে = ১২/২ = ৬ টি কলা।

৮,৪৫৮.

Four men and three women can do a job in 6 days. When five men and six women work on the same job, the work gets completed in 4 days. How long will a woman take to do the job, if she works alone on it?

ক
30 days
খ
40 days
গ
54 days
ঘ
60 days

সঠিক উত্তর: গ

54 days

উত্তর

সঠিক উত্তর: গ

54 days

গ

ব্যাখ্যা

Question: Four men and three women can do a job in 6 days. When five men and six women work on the same job, the work gets completed in 4 days. How long will a woman take to do the job, if she works alone on it?

Solution:
ধরি, ১ জন পুরুষ ১ দিনে করে x অংশ এবং ১ জন মহিলা ১ দিনে করে y অংশ

4x + 3y = 1/6
⇒ 5 (4x + 3y) = 5/6
⇒ 20x + 15y = 5/6

5x + 6y = 1/4
⇒ 4(5x + 6y) = 1
⇒ 20x + 24y = 1

20x + 24y - 20x - 15y = 1 - (5/6)
⇒ 9y = 1/6
⇒ y = 1/54

অর্থাৎ, ১ জন মহিলা ১/৫৪ অংশ সম্পন্ন করে ১ দিনে
১ অংশ সম্পন্ন করে ৫৪ দিনে।

৮,৪৫৯.

In a group of 144 persons, 50% people contributed tk. 50 each, 25% contributed Tk. 60 each and the remaining persons contributed Tk. 70 each. Find the average contribution for the group?

ক
57.5
খ
58.5
গ
59.5
ঘ
60.5

সঠিক উত্তর: ক

57.5

উত্তর

সঠিক উত্তর: ক

57.5

ক

ব্যাখ্যা

Question: In a group of 144 persons, 50% people contributed tk. 50 each, 25% contributed Tk. 60 each and the remaining persons contributed Tk. 70 each. Find the average contribution for the group?

Solution:
total people = 144

Since
50% of them contributed Tk. 50
∴ amount contributed by these 50% = (144/2) × 50 = 3600

Since 25% of them contributed TK. 60
∴ amount contributed by these 25% = (25/100) × (144) × 60 = 2160

and
remaining 25% of them contributed Tk. 70
∴ amount contributed by these 25% = (25/100) × 144 × 70 = 2520

∴ Total amount contributed = 3600 + 2160 + 2520 = 8280

∴ Average contribution = 8280/144 = 57.5

৮,৪৬০.

Which one of the following is a rational number?

ক
√7 × √2
খ
√5 × √6
গ
√5 × √20
ঘ
√3 × √8

সঠিক উত্তর: গ

√5 × √20

উত্তর

সঠিক উত্তর: গ

√5 × √20

গ

ব্যাখ্যা

Question: Which one of the following is a rational number?

Solution:
ক) √7 × √2 = √14 .......... irrational
খ) √5 × √6 = √30 .......... irrational
গ) √5 × √20 = √100 = 10 ......... rational
ঘ) √3 × √8 = √24 ................ irrational

৮,৪৬১.

The sum of the fourth and twelfth term of an arithmetic progression is 20. What is the sum of the first fifteen terms of that arithmetic progression?

ক
ক) 300
খ
খ) 120
গ
গ) 150
ঘ
ঘ) 130

সঠিক উত্তর: গ

গ) 150

উত্তর

সঠিক উত্তর: গ

গ) 150

গ

ব্যাখ্যা

Let, first number of that series = a
Common difference = d
So, 4th term = a + 3d and 12th term = a + 11d

ATQ, a + 3d + a + 11d = 20
⇒ 2a + 14d = 20

∴ S₁₅ = 15/2{2a + (15 - 1)d}
= 15/2(2a + 14d)
= 15/2 × 20
= 150

৮,৪৬২.

If x = a(b - c), y = b(c - a), z = c(a - b), then the value of (x/a)³ + (y/b)³ + (z/c)³ is -

ক
ক) 2xyz/abc
খ
খ) xyz/abc
গ
গ) 0
ঘ
ঘ) 3xyz/abc

সঠিক উত্তর: ঘ

ঘ) 3xyz/abc

উত্তর

সঠিক উত্তর: ঘ

ঘ) 3xyz/abc

ঘ

ব্যাখ্যা

x = a(b - c),
y = b(c - a),
z = c(a - b)

Let,
x/a = b - c = A
y/b = c - a = B
z/c = a - b = C

∴ A + B + C = b - c + c - a + a - b
= 0
A³ + B³ + C³ = 3ABC
= (x/a)³ + (y/b)³ + (z/c)³
= 3 × (x/a) × (y/b) × (z/c)
= 3xyz/abc

৮,৪৬৩.

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

ক
ক) 3 hr
খ
খ) 3 hr 20 min
গ
গ) 3 hr 45 min
ঘ
ঘ) 4 hr 20 min

সঠিক উত্তর: গ

গ) 3 hr 45 min

উত্তর

সঠিক উত্তর: গ

গ) 3 hr 45 min

গ

ব্যাখ্যা

Question: A tap can fill a tank in 6 hours. After half the tank is filled, three more similar taps are opened. What is the total time taken to fill the tank completely?

Solution:
সম্পূর্ণ অংশ পূর্ণ হতে সময় নেয় ৬ ঘণ্টা
অর্ধেক অংশ পুর্ণ করতে সময় লাগে ৩ ঘণ্টা

বাকি অর্ধেক অংশ চারটি নল মিলে পূর্ণ করে।
ফলে ১ ঘণ্টায় পুর্ণ হয় ৪ × (১/৬) অংশ
= ২/৩ অংশ

২/৩ অংশ পুর্ণ হতে সময় লাগে ১ ঘন্টা
১/২ অংশ পুর্ণ হতে সময় লাগে (৩/২) × (১/২) ঘণ্টা
= ৩/৪ ঘণ্টা
= (৩/৪) × ৬০ মিনিট
= ৪৫ মিনিট

মোট সময় লাগবে = ৩ ঘণ্টা ৪৫ মিনিট

৮,৪৬৪.

Angles of a quadrilateral are in the ratio 3 : 4 : 5 : 8. The largest angle is -

ক
18°
খ
54°
গ
124°
ঘ
144°

সঠিক উত্তর: ঘ

144°

উত্তর

সঠিক উত্তর: ঘ

144°

ঘ

ব্যাখ্যা

Question: Angles of a quadrilateral are in the ratio 3 : 4 : 5 : 8. The largest angle is -

Solution:
Let First angle = 3x
Second angle = 4x
Third angle = 5x
and fourth angle = 8x
We know 3x + 4x + 5x + 8x = 360°
⇒ 20x = 360°
⇒ x = 18°

∴ Measure of largest angle = 8x
= (8 × 18°)
= 144°

৮,৪৬৫.

The difference between two positive numbers is 6 and the difference of their squares is 132. The smallest number is-

ক
6
খ
8
গ
9
ঘ
12

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: The difference between two positive numbers is 6 and the difference of their squares is 132. The smallest number is-

Solution:
Let
the numbers be x and (x + 6)

ATQ,
(x + 6)² - x² = 132
⇒ x² + 12x + 36 - x² = 132
⇒ 12x + 36 = 132
⇒ 12x = 96
∴ x = 8

∴ the smallest number is = 8

৮,৪৬৬.

A certain amount of money earning simple interest becomes 7/5^th of the initial amount in 4 years. What is the interest rate?

ক
10%
খ
8%
গ
6%
ঘ
4%

সঠিক উত্তর: ক

10%

উত্তর

সঠিক উত্তর: ক

10%

ক

ব্যাখ্যা

Question: A certain amount of money earning simple interest becomes 7/5^th of the initial amount in 4 years. What is the interest rate?

Solution:
ধরি,
আসল P = 100 টাকা
মুনফা-আসল A = 100 এর 7/5
= 140 টাকা

মুনাফা I = 140 - 100 = 40 টাকা
সময় n = 4 বছর
মুনাফার হার r = ?

আমরা জানি,
I = Pnr
⇒ r = I/Pn
= (40 × 100)/(100 × 4)
= 10%

৮,৪৬৭.

If 12a + 3b = 2 and 7b – 2a = 8 , what is the average of a and b ?

ক
ক) 2
খ
খ) 2/3
গ
গ) 1/2
ঘ
ঘ) 3

সঠিক উত্তর: গ

গ) 1/2

উত্তর

সঠিক উত্তর: গ

গ) 1/2

গ

ব্যাখ্যা

Adding the given equations:
12a + 3b + 7b - 2a = 2 + 8
Or, 10a + 10b = 10
Or, 10(a + b) = 10
Or, a + b = 1
So, average of a and b is 1/2

৮,৪৬৮.

Which of the following is the solution to x² - 2x - 2 = 0?

ক
ক) 1 - √3, 1 + √2
খ
খ) 1 + √3, 1 - √3
গ
গ) 1 + √2, 1 - √2
ঘ
ঘ) 1 - √3, 1 - √3

সঠিক উত্তর: খ

খ) 1 + √3, 1 - √3

উত্তর

সঠিক উত্তর: খ

খ) 1 + √3, 1 - √3

খ

ব্যাখ্যা

Question: Which of the following is the solution to x² - 2x - 2 = 0?

Solution:
ax² + bx + c = 0 সমীকরণের সাথে তুলনা করে পাই,
a = 1, b = - 2, c = - 2

∴ x = {- ( - 2) ± √( - 2)² - 4 . 1 . (- 2)}/ 2 . 1
= {2 ± √(4 + 8)}/2
= (2 ± √12)/2
= {2(1 ± √3)}/2
= 1 ± √3

অর্থাৎ, x₁ = 1 + √3, x₂ = 1 - √3

৮,৪৬৯.

If (sinθ + cosθ)/(sinθ - cosθ) = 5 , then, tanθ =?

ক
1/2
খ
3/4
গ
3/2
ঘ
3/7

সঠিক উত্তর: গ

3/2

উত্তর

সঠিক উত্তর: গ

3/2

গ

ব্যাখ্যা

প্রশ্ন: If (sinθ + cosθ)/(sinθ - cosθ) = 5 , then, tanθ =?

সমাধান:
5(sinθ - cosθ) = sinθ + cosθ
বা, 5sinθ - 5cosθ = sinθ + cosθ
বা,5sinθ - sinθ = cosθ + 5cosθ
বা, 4sinθ = 6cosθ
বা, sinθ/cosθ = 6/4
∴ tanθ = 3/2

৮,৪৭০.

If Arif works alone he will take 20 more hours to complete a task than if he works with Babu to complete the task. If Babu work alone, he will take 5 more hours to complete the task than if he works with Arif to complete the task. What is the ratio of the time taken by Arif to than taken by Babu if each of them works alone to complete the task?

ক
5 : 2
খ
4 : 3
গ
3 : 1
ঘ
2 : 1
ঙ
None

সঠিক উত্তর: ঘ

2 : 1

উত্তর

সঠিক উত্তর: ঘ

2 : 1

ঘ

ব্যাখ্যা

Question: If Arif works alone he will take 20 more hours to complete a task than if he works with Babu to complete the task. If Babu work alone, he will take 5 more hours to complete the task than if he works with Arif to complete the task. What is the ratio of the time taken by Arif to than taken by Babu if each of them works alone to complete the task?

Solution:
ধরি
আরিফ ও বাবু কাজটি করে x ঘণ্টায়

আরিফ একা কাজটি করে (x + 20) ঘণ্টায়
বাবু একা কাজটি করে (x + 5) ঘণ্টায়

প্রশ্নমতে,
{1/(x + 20)} + {1/(x + 5)} = 1/x
⇒ 1/(x + 20) = (1/x) - {1/(x + 5)}
⇒ 1/(x + 20) = (x + 5 - x)/(x² + 5x)
⇒ 1/(x + 20) = 5/(x² + 5x)
⇒ x² + 5x = 5x + 100
⇒ x² = 100
∴ x = 10

আরিফ একা কাজটি করে (10 + 20) ঘণ্টা = 30 ঘণ্টায়
বাবু একা কাজটি করে (10 + 5) ঘণ্টা = 15 ঘণ্টায়

আরিফ ও বাবুর কাজের সময়ের অনুপাত = 30 : 15
= 2 : 1

৮,৪৭১.

To complete a work, P takes 25% more time than Q. If together they take 20 days to complete the work, how much time will Q take to do it?

ক
26 days
খ
30 days
গ
36 days
ঘ
25 days
ঙ
40 days

সঠিক উত্তর: গ

36 days

উত্তর

সঠিক উত্তর: গ

36 days

গ

ব্যাখ্যা

Question: To complete a work, P takes 25% more time than Q. If together they take 20 days to complete the work, how much time will Q take to do it?

Solution:
Let,
Q takes x days to complete the work

Then P will take 25% more time
i.e. 125% of x days
i.e. (5/4)x days

So, the one day’s work of P and Q together will be
(1/x) + {1/(5x/4)} = 1/20

⇒ (1/x) + (4/5x) = 1/20
⇒ (9/5x) = 1/20
⇒ x = 36

∴ Q takes 36 days to complete the work.

৮,৪৭২.

If 0 ≤ x ≤ 4 and y < 6, which of the following cannot be the value of xy?

ক
- 2
খ
0
গ
6
ঘ
None of these

সঠিক উত্তর: ঘ

None of these

উত্তর

সঠিক উত্তর: ঘ

None of these

ঘ

ব্যাখ্যা

Question: If 0 ≤ x ≤ 4 and y < 6, which of the following cannot be the value of xy?

Solution:
y < 6 হলে y এর মান 5, 4, 3, 2, 1, 0, - 1,..............
0 ≤ x ≤ 4 হলে x এর মান 0, 1, 2, 3, 4

এখন
x = 0, y = 1 হলে xy = 0 × 1 = 0
x = 2, y = - 1 হলে xy = 2 × (- 1) = - 2
x = 3, y = 2 হলে xy = 3 × 2 = 6

সঠিক উত্তর: None of these

৮,৪৭৩.

Two friends A and B started a business with an initial capital contribution of Tk. 1 lac and Tk. 2 lacs. At the end of the year, the business made a profit of Tk. 30,000. Find the share of A in the profit.

ক
Tk. 15000
খ
Tk. 20000
গ
Tk. 18000
ঘ
Tk. 10000

সঠিক উত্তর: ঘ

Tk. 10000

উত্তর

সঠিক উত্তর: ঘ

Tk. 10000

ঘ

ব্যাখ্যা

Question: Two friends A and B started a business with an initial capital contribution of Tk. 1 lac and Tk. 2 lacs. At the end of the year, the business made a profit of Tk. 30,000. Find the share of A in the profit.

Solution:
We know that if the time period of investment is the same, profit/loss is divided by the ratio of the value of the investment.
Ratio of value of investment of A and B = 100000 : 200000 = 1 : 2
Ratio of share in profit = 1 : 2

Share of A in profit = (1/3) × 30,000 = Tk. 10000

৮,৪৭৪.

In your wallet, there are Tk 500, Tk 200 and Tk 100. notes in the ratio 5 : 9 : 4, amounting to Tk 18,800. Find the number of each note respectively.

ক
20, 36, 16
খ
37, 1, 1
গ
10, 18, 8
ঘ
25, 10, 7

সঠিক উত্তর: ক

20, 36, 16

উত্তর

সঠিক উত্তর: ক

20, 36, 16

ক

ব্যাখ্যা

Question: In your wallet, there are Tk 500, Tk 200 and Tk 100. notes in the ratio 5 : 9 : 4, amounting to Tk 18,800. Find the number of each note respectively.

Solution:
In your wallet, there are Tk 500, Tk 200 and Tk 100. notes in the ratio 5 : 9 : 4
Let,
Number of Tk. 500 note is 5x
Number of Tk. 200 note is 9x
Number of Tk. 100 note is 4x

ATQ,
500 × 5x + 200 × 9x + 100 × 4x = 18800
⇒ 2500x + 1800x + 400x = 18800
⇒ 4700x = 18800
⇒ x = 18800/4700
∴ x = 4

Number of Tk. 500 note is 5x = 5 × 4 = 20
Number of Tk. 200 note is 9x = 9 × 4 = 36
Number of Tk. 100 note is 4x = 4 × 4 = 16

৮,৪৭৫.

A man can row at the rate of 4 km/hr in still water. If the time taken to row a certain distance upstream is 3 times as much as to row the same distance downstream, find the speed of the current?

ক
ক) 3 km/hr
খ
খ) 2 km/hr
গ
গ) 1 km/hr
ঘ
ঘ) 4 km/hr

সঠিক উত্তর: খ

খ) 2 km/hr

উত্তর

সঠিক উত্তর: খ

খ) 2 km/hr

খ

ব্যাখ্যা

Question: A man can row at the rate of 4 km/hr in still water. If the time taken to row a certain distance upstream is 3 times as much as to row the same distance downstream, find the speed of the current?

Solution:
Let,
speed of stream x
Time taken in downstream = t
Time taken in downstream = 3t

ATQ.
Distance in upstream = Distance in downstream
(4 - x)3t = (4 + x)t
⇒ (4 - x)3 = (4 + x)
⇒ 12 - 3x = 4 + x
⇒ 4x = 8
∴ x = 2

∴ Speed of stream = 2 km/hr

৮,৪৭৬.

ক
34
খ
32
গ
36
ঘ
28

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: If

Solution:

৮,৪৭৭.

A boat while downstream in a river covered a distance of 50 miles at an average speed of 60 miles per hour. While returning, because of the water resistance, it took 1 hour 15 minutes to cover the same distance. What was the average speed during the whole journey?

ক
24 mph
খ
36 mph
গ
48 mph
ঘ
64 mph
ঙ
80 mph

সঠিক উত্তর: গ

48 mph

উত্তর

সঠিক উত্তর: গ

48 mph

গ

ব্যাখ্যা

Question: A boat while downstream in a river covered a distance of 50 miles at an average speed of 60 miles per hour. While returning, because of the water resistance, it took 1 hour 15 minutes to cover the same distance. What was the average speed during the whole journey?

Solution:
Time taken to cover 50 miles downstream = (50/60)h
= 5/6 h
Time taken to cover 50 miles upstream = 1h 15m
= 5/4 h

Total time taken to cover 100 miles = (5/6) + (5/4)
= 25/12 h

∴ Average speed = 100/(25/12)
= (100 × 12)/25
= 48 mph

৮,৪৭৮.

If one root of x² - (q + 2)x + 15 = 0 is 5, then the value of q is:

ক
3
খ
4
গ
6
ঘ
8

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: If one root of x² - (q + 2)x + 15 = 0 is 5, then the value of q is:

Solution:
Given equation: x² - (q + 2)x + 15 = 0

One root is x = 3. Substitute:
(5)² - (q + 2)(5) + 15 = 0
⇒ 25 - 5(q + 2) + 15 = 0
⇒ 25 - 5q - 10 + 15 = 0
⇒ 30 - 5q = 0
⇒ 5q = 30
∴ q = 6

৮,৪৭৯.

If p = 10^0.9, q = 10^1.8 and p^r = q², then what is the value of r = ?

ক
4
খ
8
গ
9
ঘ
3
ঙ
None of these

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: If p = 10^0.9, q = 10^1.8 and p^r = q², then what is the value of r = ?

Solution:
Given that,
10^0.9, q = 10^1.8

Now,
p^r = q²
⇒ (10^0.9)^r = (10^1.8)²
⇒ 10^0.9r = 10^3.6
⇒ 0.9r = 3.6
⇒ r = 3.6/0.9
∴ r = 4

৮,৪৮০.

একটি কারখানার মহিলা কর্মচারীদের দৈনিক গড় মজুরি ৩০ টাকা এবং পুরুষ কর্মচারীদের দৈনিক গড় মজুরী ৪২ টাকা। সকল কর্মচারীর গড় মজুরী ৩৭ টাকা হলে পুরুষ ও মহিলা কর্মচারীর অনুপাত কত?

ক
৬ : ৫
খ
৫ : ৬
গ
৫ : ৭
ঘ
৭ : ৫
ঙ
কোনটিই নয়

সঠিক উত্তর: ঘ

৭ : ৫

উত্তর

সঠিক উত্তর: ঘ

৭ : ৫

ঘ

ব্যাখ্যা

প্রশ্ন: একটি কারখানার মহিলা কর্মচারীদের দৈনিক গড় মজুরি ৩০ টাকা এবং পুরুষ কর্মচারীদের দৈনিক গড় মজুরী ৪২ টাকা। সকল কর্মচারীর গড় মজুরী ৩৭ টাকা হলে পুরুষ ও মহিলা কর্মচারীর অনুপাত কত?

সমাধান:
মনে করি,
পুরুষের সংখ্যা = ক
মহিলার সংখ্যা = খ

প্রশ্নমতে,
৪২ক + ৩০খ = ৩৭(ক + খ)
বা, ৪২ক - ৩৭ক = ৩৭খ - ৩০খ
বা, ৫ক = ৭খ
বা, ক/খ = ৭/৫
ক : খ = ৭ : ৫
∴ পুরুষ ও মহিলা ক‍‍র্মচারীর অনুপাত ৭ : ৫

৮,৪৮১.

After 3 semesters in college, Ratul has a 3.0 GPA. What GPA must Ratul attain in his fourth semester if he wishes to raise his GPA to a 3.1?

ক
3.2
খ
3.3
গ
3.4
ঘ
3.5

সঠিক উত্তর: গ

3.4

উত্তর

সঠিক উত্তর: গ

3.4

গ

ব্যাখ্যা

Question: After 3 semesters in college, Ratul has a 3.0 GPA. What GPA must Ratul attain in his fourth semester if he wishes to raise his GPA to a 3.1?

Solution:
After 3 semesters in college, Ratul has a 3.0 GPA.

Ratul's Total points after 3 semester = 3.0 × 3 = 9.0

Let,
He must attain X points in his fourth semester.

ATQ,
(9.0 + X)/4 = 3.1
⇒ 9.0 + X = 12.4
⇒ X = 12.4 - 9.0
∴ X = 3.4

৮,৪৮২.

The expression (11.98 × 11.98 + 11.98 × Q + 0.02 × 0.02) will be a perfect square for Q =

ক
0.02
খ
0.2
গ
0.04
ঘ
0.4

সঠিক উত্তর: গ

0.04

উত্তর

সঠিক উত্তর: গ

0.04

গ

ব্যাখ্যা

Question: The expression (11.98 × 11.98 + 11.98 × Q + 0.02 × 0.02) will be a perfect square for Q =

Solution:
We know that,
(a + b)² = a² + 2ab + b² = a × a + 2ab + b × b

Now,
(11.98 × 11.98 + 2 × 11.98 × 0.02 + 0.02 × 0.02) = (11.98 + 0.02)²

We can say that, (11.98 × 11.98 + 11.98 × Q + 0.02 × 0.02) will be a perfect square if Q = 2 × 0.02 = 0.04

৮,৪৮৩.

The difference between the simple interest received from two different sources on Rs. 1500 for 3 years is Rs. 13.50. The difference between their rates of interest is ?

ক
0.1%
খ
0.2%
গ
0.3%
ঘ
0.4%

সঠিক উত্তর: গ

0.3%

উত্তর

সঠিক উত্তর: গ

0.3%

গ

ব্যাখ্যা

Let the rate of interest for two different sources is r₁ and r₂ respectively.

1500 × r₁ × 3/100 - 1500 × r₂ × 3/100 = 13.50
4500r₁ - 4500r₂ = 1350
(r₁ - r₂) = 1350/4500
Hence required difference in rates = 0.3%

৮,৪৮৪.

Find the income on 7(½) % stock of Tk. 2000 purchased at Tk. 80.

ক
ক) TK.160
খ
খ) TK.148
গ
গ) TK.150
ঘ
ঘ) TK.109
ঙ
ঙ) TK. 125

সঠিক উত্তর: গ

গ) TK.150

উত্তর

সঠিক উত্তর: গ

গ) TK.150

গ

ব্যাখ্যা

Face Value of the stock = Tk.2000
Dividend is 7(1/2)% = 15/2% of the face value
Dividend = (2000 × 15) ÷ (2 × 100)
= Tk. 150

৮,৪৮৫.

In a series of 5 consecutive odd numbers if 13 is the 5^th number, what is the 3^rd number in the series?

ক
5
খ
7
গ
9
ঘ
11

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: In a series of 5 consecutive odd numbers if 13 is the 5^th number, what is the 3^rd number in the series?

Solution:
5^th odd number = 13
4^th odd number = 11
3^rd odd number = 9
2^nd odd number = 7
1^st odd number = 5

৮,৪৮৬.

In the figure, lines m and n are parallel. If y - z = 60 then what is the value of x?

ক
60°
খ
120°
গ
100°
ঘ
135°

সঠিক উত্তর: খ

120°

উত্তর

সঠিক উত্তর: খ

120°

খ

ব্যাখ্যা

Question: In the figure, lines m and n are parallel. If y - z = 60 then what is the value of x?

Solution:
যেহেতু একটি সরলরেখার উপর উৎপন্ন কোণগুলোর সমষ্টি 180°,
তাই, z + y = 180........(1)

আবার,
দেওয়া আছে, y - z = 60..........(2)

সমীকরণ দুটি যোগ করে পাই,
z + y + y - z = 180 + 60
⇒ 2y = 240
⇒ y = 120°

যেহেতু m এবং n রেখাদ্বয় সমান্তরাল, তাই বিপরীত বহিঃস্থ কোণ x এবং y পরস্পর সমান। সুতরাং, x এর মান 120°

৮,৪৮৭.

A rope, 40 m long, is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole if the angle made by the rope with the ground level is 45°.

ক
20√2 m
খ
20 m
গ
40 m
ঘ
10√2 m

সঠিক উত্তর: ক

20√2 m

উত্তর

সঠিক উত্তর: ক

20√2 m

ক

ব্যাখ্যা

Question: A rope, 40 m long, is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole if the angle made by the rope with the ground level is 45°.

Solution:

ধরি, খুঁটির উচ্চতা (Height of the pole), AB = h
দড়ির দৈর্ঘ্য (Length of the rope), AC = 40 m
দড়ি ভূমির সাথে যে কোণ তৈরি করে, ∠ACB = 45°
আমরা জানি, sinθ = লম্ব/অতিভুজ
∴ sin 45° = AB / AC
⇒ 1/√2 = h / 40
⇒ h = 40/√2
⇒ h = (40√2)/(√2 × √2)
∴ h = 20√2 m

অতএব, খুঁটির উচ্চতা = 20√2 m

৮,৪৮৮.

What is the difference between the sixth and the fifth terms of the sequence 2, 4, 7, ____ whose n^th term is n + 2^{(n - 1)}?

ক
ক) 3
খ
খ) 6
গ
গ) 16
ঘ
ঘ) 17

সঠিক উত্তর: ঘ

ঘ) 17

উত্তর

সঠিক উত্তর: ঘ

ঘ) 17

ঘ

ব্যাখ্যা

Given that, a_n = n + 2ⁿ⁻¹

Thus:
a₆ = 6 + 2⁶⁻¹ = 38;
a₅ = 5 + 2⁵⁻¹ = 21;

∴ The difference is = 38 - 21 = 17.

৮,৪৮৯.

A sum of Tk. 1000 was lent to two people, one at the rate of 5% and the other at the rate of 8%. If the simple interest after one year is Tk. 62, find the sum lent at 5% rate.

ক
Tk. 400
খ
Tk. 600
গ
Tk. 700
ঘ
Tk. 650

সঠিক উত্তর: খ

Tk. 600

উত্তর

সঠিক উত্তর: খ

Tk. 600

খ

ব্যাখ্যা

Question: A sum of Tk. 1000 was lent to two people, one at the rate of 5% and the other at the rate of 8%. If the simple interest after one year is Tk. 62, find the sum lent at 5% rate.

Solution:
Let the sum lent at 5% be P.
∴ Sum lent at 8 % = 1000 - P

Now, according to the question,
SI for 5% + SI for 8% = 62
⇒ (P × 5 × 1/100) + ((1000 - P) × 8 × 1/100) = 62
⇒ 5P + 8(1000 - P) = 6200
⇒ 5P + 8000 - 8P = 6200
⇒ 3P = 1800
⇒ P = 600
Therefore, sum lent at 5 % = P = Tk. 600

৮,৪৯০.

A can finish a task in 36 days, B in 54 days, and C in 72 days. All three start working together, but A leaves 8 days before the work is completed, and B leaves 12 days before completion. C works alone until the task is finished. How many days does it take to complete the work?

ক
34 days
খ
20 days
গ
14 days
ঘ
24 days
ঙ
21 days

সঠিক উত্তর: ঘ

24 days

উত্তর

সঠিক উত্তর: ঘ

24 days

ঘ

ব্যাখ্যা

Question: A can finish a task in 36 days, B in 54 days, and C in 72 days. All three start working together, but A leaves 8 days before the work is completed, and B leaves 12 days before completion. C works alone until the task is finished. How many days does it take to complete the work?

Solution:
Let the work be completed in y days.
C works for y days

Therefore, A works for (y - 8) days
and, B works for (y - 12) days.

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

৮,৪৯১.

Rakib invested 10000 Tk. in two 5 years term. He received 5% for the first term and 8% for the second term. What is the average rate of interest did he received?

ক
6.5%
খ
5.78%
গ
6.25%
ঘ
6.278%

সঠিক উত্তর: ক

6.5%

উত্তর

সঠিক উত্তর: ক

6.5%

ক

ব্যাখ্যা

Question: Rakib invested 10000 Tk. in two 5 years term. He received 5% for the first term and 8% for the second term. What is the average rate of interest did he received?

Solution:
in first term, he received = 10000 × 5 × 5%
= 2500 Tk.
in second term, he received = 10000 × 5 × 8%
= 4000 Tk.
so,
total interest after 10 years, I = 6500
P = 10000
n = 10
r = ?

we know,
I = Pnr/100
r = 100I / Pn
= 650000/(10000 × 10)
= 6.5%

৮,৪৯২.

If f(a) = a - 5 and g(b)= 5 − b, what is the value of the expression ।f(x)।− ।g(x)। + ।f[g(x)]।?

ক
।x - 5।
খ
2x + 15
গ
।- x।
ঘ
।2x - 3।
ঙ
x

সঠিক উত্তর: গ

।- x।

উত্তর

সঠিক উত্তর: গ

।- x।

গ

ব্যাখ্যা

Question: If f(a) = a - 5 and g(b)= 5 − b, what is the value of the expression ।f(x)।− ।g(x)। + ।f[g(x)]।?

Solution:
f(x) = x - 5 ⇒ so |f(x)| = |x - 5|
g(x) = 5 - x ⇒ so |g(x)| = |5 - x| = |- (x - 5)| = |x - 5|

f(g(x)) = f(5 - x) = (5 - x) - 5 = - x ⇒ so |f(g(x))| = |- x|

Now plug into the expression:

|f(x)| - g(x)| + |f(g(x))| = |x - 5| - |x - 5| + |- x| = 0 + |- x| = |- x| = |x|

So the correct answer is গ) |- x|.
--------------------
Note: Why not ঙ) x ?

Actually, |- x| is always the same as |x|, not the same as the plain x unless x ≥ 0.
For a negative x (say, x = −2), we get
|−(−2)| = |2| = 2,
while x = −2.
cannot reduce to the bare x for all x.

So, Option ঙ) x would only match when x happens to be non-negative, but it fails for negative values.

That is why the correct answer is গ) ।- x।

৮,৪৯৩.

A cylindrical tank with diameter 14 m and height 5 m is filled with water. If the water is transferred to a rectangular tank with base 10 m × 7 m, what will be the height of water in the rectangular tank?

ক
8 m
খ
11 m
গ
13 m
ঘ
17 m

সঠিক উত্তর: খ

11 m

উত্তর

সঠিক উত্তর: খ

11 m

খ

ব্যাখ্যা

Question: A cylindrical tank with diameter 14 m and height 5 m is filled with water. If the water is transferred to a rectangular tank with base 10 m × 7 m, what will be the height of water in the rectangular tank?

Solution:
Volume of the cylinder = π(7)²5
= π × 49 × 5
= 245π m³

Volume of the rectangle = 10 × 7 × h (Assuming, height of the rectangle is 'h')
= 70h m³

So, 70h = 245π
⇒ h = (245/70)(22/7)
∴ h = 11m

৮,৪৯৪.

James hired some workers to build his house in 20 days. But on the day of starting of the construction, 12 men did not come. Rest of the people then built the entire house in 32 days. How many workers had he initially hired?

ক
28
খ
32
গ
36
ঘ
40

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: James hired some workers to build his house in 20 days. But on the day of starting of the construction, 12 men did not come. Rest of the people then built the entire house in 32 days. How many workers had he initially hired?

Solution:
Initially hired workers = x
Total work = x × 20 (man-days)
After 12 workers didn't come, Remaining workers = x - 12
Total work = (x - 12) × 32 (man-days)

ATQ,
x × 20 = (x - 12) × 32
⇒ 20x = 32x - 384
⇒ 12x = 384
∴ x = 32

৮,৪৯৫.

How many terms are there in the Geometric Progression(GP) 7, 21, 63, 189,......…,15309?

ক
6
খ
8
গ
10
ঘ
7

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: How many terms are there in the Geometric Progression(GP) 7, 21, 63, 189,......…,15309?

Solution:
First term, a = 7
Common ratio, r = 21/7
= 3

Last term or nth term of GP = ar^{n - 1}
⇒ 15309 = 7 × (3^{n - 1})
⇒ 3^{n - 1} = 15309/7
⇒ 3^{n - 1} = 2187
⇒ 3^{n - 1} = 3⁷
So, comparing the power,
Thus, n - 1 = 7
∴ n = 8

Number of terms = 8

৮,৪৯৬.

If x² - √5.x + 1 = 0 then, x² + 1/x² = ?

ক
2
খ
3
গ
5
ঘ
9

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: If x² - √5.x + 1 = 0 then, x² + 1/x² = ?

Solution:
Given,
x² - √5.x + 1 = 0
⇒ x² + 1 = √5.x
⇒ x²/x + 1/x = (√5.x)/x
⇒ x + 1/x = √5

Now,
x² + 1/x² = (x + 1/x)² - 2. x.1/x
= (√5)² - 2
= 5 - 2
= 3

৮,৪৯৭.

25% of 25% is equal to-

ক
ক) 0.0625
খ
খ) 0.0210
গ
গ) 0.0525
ঘ
ঘ) 0.0725
ঙ
ঙ) 0.0825

সঠিক উত্তর: ক

ক) 0.0625

উত্তর

সঠিক উত্তর: ক

ক) 0.0625

ক

ব্যাখ্যা

25% of 25%
= 0.25 × 0.25
= 0.0625

৮,৪৯৮.

A certain machine produces 10,000 units of product P per hour. Working continuously at this constant rate, this machine will produce how many units of product P in 11 days?

ক
2,640,000
খ
168,000
গ
264,000
ঘ
1,680,000

সঠিক উত্তর: ক

2,640,000

উত্তর

সঠিক উত্তর: ক

2,640,000

ক

ব্যাখ্যা

7 days = 11 × 24 hours = 264 hours
In 1 hour, the machine produces 10,000 units
In 168 hours, the machine will produce = 10,000 × 264 units = 2640000 units

৮,৪৯৯.

In how many ways can 4 people from a group of 6 people be seated around a circle table?

ক
65
খ
90
গ
120
ঘ
144

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: In how many ways can 4 people from a group of 6 people be seated around a circle table?

Solution:
এখানে 6 জন মানুষের মধ্য থেকে 4 জনকে নিয়ে একটি বৃত্তাকার টেবিলে বসাতে হবে।

প্রথমে 6 জন থেকে 4 জনকে বাছাই করার উপায় = ⁶C₄
= 6!/(4! × 2!)
= (6 × 5)/(2 × 1)
= 15

আমরা জানি, n জন ব্যক্তিকে একটি বৃত্তাকার টেবিলে বসানোর উপায় = (n - 1)!
সুতরাং, বাছাইকৃত 4 জন ব্যক্তিকে বৃত্তাকার টেবিলে বসানোর উপায় = (4 - 1)! = 3!
= 3 × 2 × 1 = 6 উপায়।

∴ মোট বিন্যাস সংখ্যা = (বাছাই করার উপায়) × (বৃত্তাকারে বসানোর উপায়)
= 15 × 6
= 90 ways

৮,৫০০.

What is the wet surface area of a cistern of 8m in length, 4.5m in width, and 3.5m in height?

ক
127.5 m²
খ
115.5 m²
গ
122.5 m²
ঘ
123.5 m²

সঠিক উত্তর: ঘ

123.5 m²

উত্তর

সঠিক উত্তর: ঘ

123.5 m²

ঘ

ব্যাখ্যা

Question: What is the wet surface area of a cistern of 8m in length, 4.5m in width, and 3.5m in height?

Solution:
here,
l = 8m
b = 4.5m
h = 3.5m
total surface area is = 2 ( lb + bh + lh )
= 2 {(8 × 4.5) + (4.5 × 3.5) + (8 × 3.5)}
= 2 (36 + 15.75 + 28)
= 159.5 m²

one of the surface is unwet,
unwet area = 8 × 4.5 = 36m²

∴ Wet surface area = 159.5 - 36
= 123.5 m²