Bank Math

PrepBank · পাতা ১৪৫ / ১৬১ · ১৪,৪০১–১৪,৫০০ / ১৬,১২৪

১৪,৪০১.

A boat goes 13 km upstream (against current) in 39 minutes. The speed of stream (current) is 3 km/hr. What is the speed of the boat in still water?

ক
ক) 23km/hr
খ
খ) 27 km/hr
গ
গ) 17 km/hr
ঘ
ঘ) 20 km/hr

ব্যাখ্যা

Question: A boat goes 13 km upstream (against current) in 39 minutes. The speed of stream (current) is 3 km/hr. What is the speed of the boat in still water?

Solution:
স্রোতের বিপরীতে, ৩৯ মিনিটে বা ৩৯/৬০ ঘণ্টায় যায় ১৩ কিমি
১ ঘণ্টায় যায় = (১৩ × ৬০)/৩৯ কিমি
= ২০ কিমি

∴ স্থির পানিতে নৌকার বেগ - স্রোতের বেগ = ২০ কিমি /ঘন্টা

স্রোতের বেগ = ৩ কিমি/ঘন্টা

স্থির পানিতে নৌকার বেগ - ৩ = ২০
∴ স্থির পানিতে নৌকার বেগ = ২০ + ৩
= ২৩ কিমি/ঘণ্টা

১৪,৪০২.

If you toss a coin twice, what is the probability that you will get heads the second time?

ক
30%
খ
40%
গ
50%
ঘ
60%

ব্যাখ্যা

Question: If you toss a coin twice, what is the probability that you will get heads the second time?

Solution:
When we toss a coin the sample outcomes are = {HH, HT, TH, TT}
Total number of outcome = 4

Heads in the second time {HH, TH} = 2

∴ Probability that I will get heads the second time = 2/4 = 1/2 = (1 × 100)/2 % = 50%

১৪,৪০৩.

The average of the ages of a man and his son is 36 years. If the respective ratio of their ages four years from now is 13 : 7 then what his son's present age?

ক
ক) 48 years
খ
খ) 18 years
গ
গ) 24 years
ঘ
ঘ) 26 years

ব্যাখ্যা

Question: The average of the ages of a man and his son is 36 years. If the respective ratio of their ages four years from now is 13 : 7 then what his son's present age?

Solution:
The total age of man and his son = 36 × 2 = 72 years
Let man's age be x years
Then son's age = 72 - x
ATQ,
(x + 4)/{(72 - x) + 4} = 13/7
7x + 28 = 936 - 13x + 52
20x = 960
x = 48

Son's present age = 72 - 48 = 24 years

১৪,৪০৪.

A man covers half of his journey at 15 km/h and the remaining half at 5 km/h. His average speed is-

ক
3.25 km/h
খ
5 km/h
গ
7.5 km/h
ঘ
10 km/h
ঙ
15 km/h

ব্যাখ্যা

Question: A man covers half of his journey at 15 km/h and the remaining half at 5 km/h. His average speed is-

Solution:
Here, x = 15 km/h and y = 5 km/h

We know,
Average speed = 2xy/(x + y)
= (2 × 15 × 5)/(15 + 5)
= 150/20
= 7.5 km/h

১৪,৪০৫.

The ratio between the perimeter and the length of a rectangle is 7 : 2. If the area of the rectangle is 108 square centimeter, what is the perimeter of the rectangle?

ক
12 cm
খ
9 cm
গ
21 cm
ঘ
42 cm

ব্যাখ্যা

Let, the breadth and the length be x and y.
xy = 108 --- --- --- (1)
and 2(x + y)/y = 7/2
4(x + y) = 7y
4x + 4y = 7y
4x = 3y
x = 3y/4
From (1), we get,
3y/4 × y = 108
y2 = 108 × 4/3 = 144
y = 12
Therefore, breadth, x = 3y/4 = 9
The perimeter of the rectangle = 2(9 + 12) = 42 cm

১৪,৪০৬.

The radius of two cylinders are in the ratio of 4 : 5 and their heights are in the ratio of 3 : 2. The ratio of their volume is-

ক
17 : 15
খ
28 : 23
গ
24 : 25
ঘ
18 : 23
ঙ
21 : 17

ব্যাখ্যা

Question: The radius of two cylinders are in the ratio of 4 : 5 and their heights are in the ratio of 3 : 2. The ratio of their volume is-

Solution:
Let the radius of both cylinders be 4x and 5x.
Let the height of both cylinders be 3y and 2y.

Ratio of the volume of two cylinders = {π × (4x)² × (3y)}/{π × (5x)² × (2y)}
= (16x² × 3)/(25x² × 2)
= 24/25

∴ Ratio = 24 : 25

১৪,৪০৭.

There are three rooms in a Hotel: one single, one double and one for four persons. How many ways are there to house seven persons in these rooms?

ক
ক) 105
খ
খ) 7! x 6!
গ
গ) 7!/5!
ঘ
ঘ) 420
ঙ
ঙ) None of these

ব্যাখ্যা

Choose 1 person for the single room & from the remaining choose 2 for the double room & from the remaining choose 4 people for the four person room,
Then, 7C1 x 6C2 x 4C4
= 7 x 15 x 1
= 105

১৪,৪০৮.

একটি দৌড় প্রতিযোগিতায় A এবং B এর গতিবেগের অনুপাত 3 : 4। গন্তব্যে পৌঁছাতে A এর চেয়ে B এর সময় 30 মিনিট কম লাগে। তাহলে A গন্তব্যে পৌঁছাতে কত ঘণ্টা সময় নেয়?

ক
1
খ
2
গ
1.5
ঘ
3
ঙ
কোনোটিই নয়

ব্যাখ্যা

প্রশ্ন: একটি দৌড় প্রতিযোগিতায় A এবং B এর গতিবেগের অনুপাত 3 : 4। গন্তব্যে পৌঁছাতে A এর চেয়ে B এর সময় 30 মিনিট কম লাগে। তাহলে A গন্তব্যে পৌঁছাতে কত ঘণ্টা সময় নেয়?

সমাধান:
ধরি,
A ও B এর গতিবেগ যথাক্রমে 3x ও 4x কিমি/ঘণ্টা
এবং উভয়ের জন্য দূরত্ব = D কিমি

আমরা জানি,
সময় = দূরত্ব/গতি

∴ A এর সময় = D/3x
∴ B এর সময় = D/4x

প্রশ্নমতে,
D/3x - D/4x = 30 মিনিট
⇒ D/3x - D/4x = 30/60 ঘণ্টা
⇒ (4D - 3D)/12x = 1/2
⇒ D/12x = 1/2
⇒ D = 6x

তাহলে, A এর সময় = D/3x = 6x/3x = 2 ঘণ্টা

১৪,৪০৯.

Ashik buys a calculator for Tk. 600 and sells it to Neaz at 10% profit. Neaz sells it to Mahfuj for 5 % profit. Mahfuz after using it for certain time, sells it to Faysal at a loss of 20%. For how much Mahfuz sell the calculator to Faysal.

ক
Tk. 550.50
খ
Tk. 564.40
গ
Tk. 554.40
ঘ
None of these

ব্যাখ্যা

Question: Ashik buys a calculator for Tk. 600 and sells it to Neaz at 10% profit. Neaz sells it to Mahfuj for 5 % profit. Mahfuz after using it for certain time, sells it to Faysal at a loss of 20%. For how much Mahfuz sell the calculator to Faysal.

Solution:
SP for Mahfuz = 600 × (110/100) × (105/100) × (80/100)
= 600 × (924/1000)
= Tk. 554.40

১৪,৪১০.

In a bag, there are three types of coins- 1-tk, 50 paise and 25-paise in the ratio of 3 : 8 : 20. Their total value is 372. The total number of coins is

ক
738
খ
836
গ
1002
ঘ
961

ব্যাখ্যা

Question: In a bag, there are three types of coins- 1-tk, 50 paise and 25-paise in the ratio of 3 : 8 : 20. Their total value is 372. The total number of coins is

Solution:
Ratio of the number of coins of Re. 1, 50 paise and 25 paise = 3 : 8 : 20
Ratio of the values of these coins = 3 : (8/2) : (20/4) = 3 : 4 : 5

Value of 1 tk coins = (3/12) × 372 = 93 tk
Value of 50 paise coins = (4/12) × 372 = 124 tk
Value of 25 paise coins = (5/12) × 372 = 155 tk

Number of coins = 93 + (124 × 2) + (155 × 4)
= 93 + 248 + 620
= 961

১৪,৪১১.

Find the solution to the equation (x² + 4x + 4)/(x + 2) = 0.

ক
x = - 2
খ
x = - 4
গ
x = 2
ঘ
x = 4

ব্যাখ্যা

Question: Find the solution to the equation (x² + 4x + 4)/(x + 2) = 0.

Solution:
(x² + 4x + 4)/(x + 2) = 0
⇒ (x² + 2.x.2 + 2²)/(x + 2) = 0
⇒ (x + 2)²/(x + 2) = 0
⇒ x + 2 = 0
∴ x = - 2

১৪,৪১২.

Which number will complete the series:
155, 153, 149, 141, 125, ........?

ক
105
খ
101
গ
93
ঘ
85

ব্যাখ্যা

Question: Which number will complete the series:
155, 153, 149, 141, 125, ........?

Solution:
The pattern is followed by
155 - 2 = 153
153 - 4 = 149
149 - 8 = 141
141 - 16 = 125
125 - 32 = 93

Hence the number= 93.

১৪,৪১৩.

The population of a village is 15000, and it increases by 6% each year. What will the population be after 2 years?

ক
18250
খ
15884
গ
16854
ঘ
17935
ঙ
None of these

ব্যাখ্যা

Question: The population of a village is 15000, and it increases by 6% each year. What will the population be after 2 years?

Solution:
Here we can use the compound interest based formula,
Population after n years
= P × [1 + (r/100)]ⁿ
∴ Population after 2 years = 15000 × [1 + (6/100)]²
= 15000 × (106/100)²
= 15000 × 1.1236
= 16854

∴ The population of the village after 2 years will be 16854

১৪,৪১৪.

If rsinθ = 5/2 and rcosθ = (5√3)/2, what is the value of r?

ক
± 3
খ
± 5
গ
± 7
ঘ
± 11

ব্যাখ্যা

Question: If rsinθ = 5/2 and rcosθ = (5√3)/2, what is the value of r?

Solution:
দেওয়া আছে,
rsinθ = 5/2 ……………..(i)
rcosθ = (5√3)/2 …………….(ii)

এখন, (i) এবং (ii) এর বর্গ করে যুক্ত করে পাই,
(rsinθ)² + (rcosθ)² = (5/2)² + ((5√3)/2)²
⇒ r²(sin²θ + cos²θ) = (25/4) + (75/4)
⇒ r²(sin²θ + cos²θ) = 100/4
⇒ r² = 25 [∵ sin²θ + cos²θ = 1]
∴ r = ± 5

১৪,৪১৫.

How many different words can be formed from the alphabets of the word 'SCISSORS'?

ক
1440
খ
1680
গ
1800
ঘ
2100

ব্যাখ্যা

Question: How many different words can be formed from the alphabets of the word 'SCISSORS'?

Solution:
The word SCISSORS consists of 8 alphabets in which S repeat 4 times and remaining alphabets are C, I, O, R and occuring only once.

Number of words can be formed 8!/4! = 1680

১৪,৪১৬.

Mita can answer 10 out of 13 questions in an examination such that she must choose at least 4 from the first five questions. The number of choices available to her is-

ক
196
খ
220
গ
235
ঘ
255

ব্যাখ্যা

Question: Mita can answer 10 out of 13 questions in an examination such that she must choose at least 4 from the first five questions. The number of choices available to her is-

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

∴ Number of choices available to Mita = ⁵C₄ × ⁸C₆ + ⁵C₅ × ⁸C₅
= 5 × 28 + 1 × 56
= 196

১৪,৪১৭.

If a quarter kg of potato costs 20 Tk, how much Tk will 200 gm cost?

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

ব্যাখ্যা

Question: If a quarter kg of potato costs 20 Tk, how much Tk will 200 gm cost?

Solution:
A quarter of 1 kg = 1/4 kg
200 gram = 200/1000 = 1/5 kg

1/4 kg of potato's cost is 20 Tk
1 kg of potato's cost is 20 × 4 Tk
1/5 kg of potato's cost is (20 × 4)/5 Tk
= 16 Tk

১৪,৪১৮.

Two numbers are respectively 20% and 50% of the third number. What percent is the first number of the second ?

ক
ক) 10%
খ
খ) 20%
গ
গ) 30%
ঘ
ঘ) 40%

ব্যাখ্যা

Let the 3rd number is 100
According to the question,
1st :2nd: 3rd= 20:50:100
Required % = (20/50) × 100 = 40%

১৪,৪১৯.

A alone can do a piece of work in 6 days and B alone in 8 days. A and B undertook to do it for 3200 Tk. With the help of C they completed the work in 3 days. How much is to be paid to C?

ক
ক) 1600 Tk.
খ
খ) 1200 Tk.
গ
গ) 600 Tk.
ঘ
ঘ) 400 Tk.

ব্যাখ্যা

Question: A alone can do a piece of work in 6 days and B alone in 8 days. A and B undertook to do it for 3200 Tk. With the help of C they completed the work in 3 days. How much is to be paid to C?

Solution:
A ১ দিনে করে ১/৬ অংশ
B ১ দিনে করে ১/৮ অংশ

A, B, C ১ দিনে করে ১/৩ অংশ কাজ

C ১ দিনে করে (১/৩) - (১/৬) - (১/৮)
= ১/২৪ অংশ

তাদের কাজের অংশের অনুপাত = (১/৬) : (১/৮) : (১/২৪)
= ৪ : ৩ : ১

C পাবে = ৩২০০ × ১/(৪ + ৩ + ১)
= ৩২০০/৮
= ৪০০ টাকা

১৪,৪২০.

Mr. Khan's salary is Tk. 5000.00 and he gets 10% commission of his salary. If his salary increased by 10%, by what percent his commission will increase?

ক
ক) 5%
খ
খ) 10%
গ
গ) 20%
ঘ
ঘ) 25%

ব্যাখ্যা

Question: Mr. Khan's salary is Tk. 5000.00 and he gets 10% commission of his salary. If his salary increased by 10%, by what percent his commission will increase?

Solution:
পূর্বের কমিশন:
= 5000 এর 10% টাকা
= 5000 এর 10/100
= 500 টাকা

১০% বৃদ্ধিতে
বর্তমান বেতন = 5000 + 5000 এর 10%
= 5000 + 5000 এর 10/100
= 5000 + 500
= 5500 টাকা

নতুন কমিশন = 5500 এর 10%
5500 এর 10/100
= 550 টাকা

কমিশন বৃদ্ধি পায় = 550 - 500 = 50 টাকা

শতকরা কমিশন বৃদ্ধি পায় = {(50/500) × 100}%
= 10%

১৪,৪২১.

A refrigerator is on sale for 20% off the original price. A store-wide sale results in an additional reduction of 25%. What is the total discount based on the original price?

ক
40%
খ
60%
গ
55%
ঘ
45%

ব্যাখ্যা

Question: A refrigerator is on sale for 20% off the original price. A store-wide sale results in an additional reduction of 25%. What is the total discount based on the original price?

Solution:
Let
x equal the original price.
First discount: x(1 - 0.20) = 0.80x (sale price).
Second discount: 0.80x(1 - 0.25) = 0.60x (final sale price)

Therefore, the final price is 60% of the original price. To calculate the discount:
1 - 0.60 = 0.40 or 40%

১৪,৪২২.

Applied to a bill for Tk. 100000, the difference between a discount of 40% and two successive discounts of 36% and 4% is-

ক
Tk. 1440
খ
Tk. 2500
গ
Tk. 4000
ঘ
Tk. 4200

ব্যাখ্যা

Question: Applied to a bill for Tk. 100000 the difference between a discount of 40% and two successive discounts of 36% and 4% is-

Solution:
40% of Tk. 100000 = Tk. 40000

36% of Tk. 100000 = 36000
4% of 36000 = Tk. 2560.
Therefore, two successive discounts on Tk 100000 = 36000 + 2560 = Tk. 38560

Difference between a discount of 40% and two successive discounts of 36% and 4%
= 40000 - 38560
= 1440

১৪,৪২৩.

The angle of elevation of the top of a tower of height h meter at point Q is θ. If the distance between point Q and base of the tower is equal to the height of the tower, find the value of θ?

ক
60°
খ
30°
গ
45°
ঘ
Any acute angle

ব্যাখ্যা

Question: The angle of elevation of the top of a tower of height h meter at point Q is θ. If the distance between point Q and base of the tower is equal to the height of the tower, find the value of θ?

Solution:

Let the height of tower PR be h.
PQ = h as point Q is at a distance of h meter from the base of tower.

PR/PQ = tanθ
⇒ h/h = tanθ
⇒ tanθ = 1
⇒ tanθ = tan45°
∴ θ = 45°

১৪,৪২৪.

30 women can do a work in 15 days. 18 men can complete the same work in 10 days. What is the ratio between the capacity of a man and a woman?

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

ব্যাখ্যা

Question: 30 women can do a work in 15 days. 18 men can complete the same work in 10 days. What is the ratio between the capacity of a man and a woman?

Solution:
( 30 × 15 ) women can complete the work in 1 day
1 woman's 1 day's work = 1/450

( 18 × 10 ) men can complete the work in 1 day
1 man's 1 day's work = 1/180

So, required ratio = 1/180 : 1/450
= 1/2 : 1/5
= 5 : 2

১৪,৪২৫.

A woman says, ''if you reverse my own age, the figure represents my husband's age. He is, of course, senior to me and the difference between our ages is one eleventh of our sum. ''What is the age of the woman?

ক
ক) 23
খ
খ) 34
গ
গ) 45
ঘ
ঘ) 54

ব্যাখ্যা

Let x and y be the ten's and unit's digits respectively of the numeral denoting the woman's age.

Then, woman's age = (10X + y) years;
So, husband's age = (10y + x) years.
Therefore (10y + x) - (10X + y) = (1/11) (10y + x + 10x + y)
⇒ (9y - 9x) = (1/11)(11y + 11x) = (x + y)
⇒ 10x = 8y
⇒ x = (4/5)y

Clearly, y should be a single-digit multiple of 5, which is 5.
So, x = 4, y = 5.
Hence, woman's age = 10x + y = 45 years.

১৪,৪২৬.

Find the compound principal of Tk. 6250 in 2 years at the profit of Tk. 8 percent per annum.

ক
Tk. 7150
খ
Tk. 7290
গ
Tk. 7385
ঘ
Tk. 7560

ব্যাখ্যা

Question: Find the compound principal of Tk. 6250 in 2 years at the profit of Tk. 8 percent per annum.

Solution:
Given,
Principal P = Tk. 6250
percentage of profit r = 8% = 8/100 = 2/25
and time n = 2 years

We know,
C = P(1 + r)ⁿ
= 6250(1 + 2/25)²
= 6250(27/25)²
= (6250 × 27 × 27)/(25 × 25)
= Tk. 7290

১৪,৪২৭.

Working alone, pump A can empty a pool in 3 hours. Working alone, pump B can empty the same pool in 2 hours. Working together, how many minutes will it take pump A and pump B to empty the pool?

ক
ক) 72
খ
খ) 75
গ
গ) 84
ঘ
ঘ) 96

ব্যাখ্যা

Question: Working alone, pump A can empty a pool in 3 hours. Working alone, pump B can empty the same pool in 2 hours. Working together, how many minutes will it take pump A and pump B to empty the pool?

Solution:
Pump A can empty the pool in 3 hours therefore the rate at which it empties is 1/3 pool/hour
Pump b can empty the pool in 2 hours therefore the rate at which it empties is 1/2 pool/hour.
If they work together, the resulting rate is the addition of both rates
(1/3 +1/2)pool/hour
= 5/6 pool/hour

So, time taken by both pipe to empty the pool = 6/5 hr
= (6/5) × 60 minutes
= 72 minutes

১৪,৪২৮.

Solve

ক
ক) - 4 ≤ x ≤ 7
খ
খ) - 1 ≤ x ≤ 11
গ
গ) - 1 ≤ x ≤ 13
ঘ
ঘ) 1 ≤ x ≤ 13

ব্যাখ্যা

Question: Solve

Solution:
- 1 ≤ (3x - 4)/7 ≤ 5
⇒ - 7 ≤ (3x - 4) ≤ 35
⇒ - 7 + 4 ≤ 3x - 4 + 4 ≤ 35 + 4
⇒ - 3 ≤ 3x ≤ 39
⇒ - 1 ≤ x ≤ 13

১৪,৪২৯.

How many kgs of wheat costing Tk 8 per kg must be mixed with 36 kg of rice costing Tk 5.4 per kg so that 20% gain may be obtained by selling the mixture at Tk 7.2 per kg?

ক
ক) 10.2 kg
খ
খ) 10.5 kg
গ
গ) 10 kg
ঘ
ঘ) 10.8 kg

ব্যাখ্যা

Question: How many kgs of wheat costing Tk 8 per kg must be mixed with 36 kg of rice costing Tk 5.4 per kg so that 20% gain may be obtained by selling the mixture at Tk 7.2 per kg?

Solution:
Cost Price = (7.2 × 100)/120 = Tk 6
Let, the quantity of wheat of Tk 8 per kg be x

ATQ,
8x + 36 × 5.4 = 6x + 216
⇒ 8x + 210 = 10x + 216
⇒ 8x + 194.4 = 10x + 216
⇒ 2x = 21.6
⇒ x = 10.8

১৪,৪৩০.

How much interest will Tk. 1,000 earn in one year at an annual rate of 8% if interest is compounded every 6 months?

ক
ক) 82.4
খ
খ) 82.0
গ
গ) 81.6
ঘ
ঘ) 80.0

ব্যাখ্যা

Question: How much interest will Tk. 1,000 earn in one year at an annual rate of 8% if interest is compounded every 6 months?

Solution:
interest = 1000 {1 + (8/(2 × 100))}^{1 × 2} - 1000
= 1000 (1 + (1/25))² - 1000
= 1000 × 26²/25² - 1000
= 1000 {(26²/25²) - 1}
= 1000 (26² - 25²)/25²
= 1000 (26 + 25)(26 - 25)/625
= 1000 × 51/625
= 81.6 tk.

১৪,৪৩১.

The average of six numbers is p and the average of three of these is q. If the average of the remaining three is r, then which one is correct?

ক
p = q + r
খ
p = 2q + 2r
গ
2p = q + r
ঘ
None of these

ব্যাখ্যা

Question: The average of six numbers is p and the average of three of these is q. If the average of the remaining three is r, then which one is correct?

Solution:
ATQ,
6p = 3q + 3r
⇒ p = 3(q + r)/6
⇒ p = (q + r)/2
∴ 2p = q + r

১৪,৪৩২.

The value of sec30° is:

ক
1/√3
খ
2/√3
গ
1/3
ঘ
4/√3

ব্যাখ্যা

Question: The value of sec30° is:

Solution:
sec30°
= 1/cos30°
= 1/(√3/2)
= 2/√3

১৪,৪৩৩.

If 1 - sinθ = n cosθ, then find the value of cotθ.

ক
(n² - 1)/2n
খ
2n/(n² + 1)
গ
n/(n² - 1)
ঘ
2n/(1 - n²)

ব্যাখ্যা

Question: If 1 - sinθ = n cosθ, then find the value of cotθ.

Solution:
Given,
1 - sinθ = n cosθ
⇒ (1 - sinθ)/cosθ = n
⇒ (1/cosθ) - (sinθ/cosθ) = n
∴ secθ - tanθ = n ...............(i)

We know,
(secθ + tanθ)(secθ - tanθ) = 1
⇒ (secθ + tanθ) × n = 1
⇒ secθ + tanθ = 1/n ................(ii)

Now, (ii) - (i) ⇒
(secθ + tanθ) - (secθ - tanθ) = 1/n - n
⇒ secθ + tanθ - secθ + tanθ = 1/n - n
⇒ 2tanθ = (1 - n²)/n
⇒ tanθ = (1 - n²)/2n
⇒ 1/cotθ = (1 - n²)/2n
∴ cotθ = 2n/(1 - n²)

১৪,৪৩৪.

একটি ঘড়ির মিনিটের কাঁটা ১০ মিনিটে কত ডিগ্রি কোণ অতিক্রম করে?

ক
৫৫°
খ
৩০°
গ
৬৬°
ঘ
৬৫°
ঙ
৬০°

ব্যাখ্যা

প্রশ্ন: একটি ঘড়ির মিনিটের কাঁটা ১০ মিনিটে কত ডিগ্রি কোণ অতিক্রম করে?

সমাধান:
আমরা জানি,
মিনিটের কাঁটা ৬০ মিনিটে অতিক্রম করে = ৩৬০°
∴ ১০ মিনিটে অতিক্রান্ত কোণ = (৩৬০ × ১০)/৬০
= ৩৬০০/৬০
= ৬০°

১৪,৪৩৫.

If a + b + c = 0, the value of (a²/bc) + (b²/ca) + (c²/ab) is -

ক
1
খ
3
গ
1/3
ঘ
3abc

ব্যাখ্যা

Question: If a + b + c = 0, the value of (a²/bc) + (b²/ca) + (c²/ab) is -

Solution:
We know that,
a³ + b³ + c³ - 3abc = (a + b + c) (a² + b² + c² - ab - bc - ca)

Here, a + b + c = 0,
then, a³ + b³ + c³ - 3abc = 0
∴ a³ + b³ + c³ = 3abc

(a²/bc) + (b²/ca) + (c²/ab)
= (a³ + b³ + c³)/abc
= 3abc/abc
= 3

১৪,৪৩৬.

Find the amount of compound interest on Tk. 2000 for 1.5 years at a 20% annual rate, with interest added semi-annually.

ক
680
খ
600
গ
662
ঘ
720
ঙ
660

ব্যাখ্যা

Question: Find the amount of compound interest on Tk. 2000 for 1.5 years at a 20% annual rate, with interest added semi-annually.

Solution:
Principal P = 2000
Rate, R = 20%
Time, T = 1.5 years

Now,
Compound interest for half-yearly:
A = P{1 + R/(2×100)}^2T
= 2000{1 + (20/200)}^2×1.5
= 2000 × {1 + (1/10)}³
= 2000 × (11/10)³
= 2000 × (1331/1000)
= 2662

Compound Interest = A – P
= 2662 – 2000 = 662 Tk

১৪,৪৩৭.

10 men completed half of a work in 8 days. How many extra men should be added to finish the rest of the work within 4 days?

ক
ক) 5 men
খ
খ) 8 men
গ
গ) 10 men
ঘ
ঘ) 15 men

ব্যাখ্যা

Question: 10 men completed half of a work in 8 days. How many extra men should be added to finish the rest of the work within 4 days?

Solution:
অর্ধেক কাজ করার পর বাকি থাকে (১ - ১/২) = ১/২

১/২ কাজ ৮ দিনে করে ১০ জন
১/২ কাজ ১ দিনে করে (১০ × ৮) জন
১/২ কাজ ৪ দিনে করে ৮০/৪ = ২০ জন

∴ অতিরিক্ত লোক লাগবে = ২০ - ১০ জন = ১০ জন

১৪,৪৩৮.

P sells a product to Q and makes a profit of 25%. Q sells it to R at a loss of 20%. If R buys it for Tk. 140, what price did P pay for it?

ক
ক) 211
খ
খ) 169
গ
গ) 140
ঘ
ঘ) 139

ব্যাখ্যা

Let the cost price of P be Tk. x
selling price of P = (x + 25% of x) = Tk. 5x/4
So, cost price of Q = Tk. 5x/4
selling price of Q = 80% of Tk. 5x/4 = Tk. x
So, cost price of R = Tk. x = Tk. 140
Therefore, the cost price of P is Tk. x = Tk. 140

১৪,৪৩৯.

If 21168 = 2^a × 3^b × 7^c, where a, b and c are natural numbers, then what is the value of (4a - 5b + c)?

ক
0
খ
1
গ
2
ঘ
More than one of the above
ঙ
None of the above

ব্যাখ্যা

Question: If 21168 = 2^a × 3^b × 7^c, where a, b and c are natural numbers, then what is the value of (4a - 5b + c)?

Solution:
21168 = 2^a × 3^b × 7^c
Factors of 21168 = 2⁴ × 3³ × 7²
∴ a = 4
⇒ b = 3
⇒ c = 2

According to the question:
(4a - 5b + c)
= {(4 × 4) - (5 × 3) + (2)}
= 16 - 15 + 2
= 3

১৪,৪৪০.

In how many ways can 3 students be chosen from a class of 15 students?

ক
ক) 455
খ
খ) 390
গ
গ) 180
ঘ
ঘ) 135

ব্যাখ্যা

Question: In how many ways can 3 students be chosen from a class of 15 students?

Solution:
total ways = ¹⁵C₃
= 15!/(3! × 12!)
= 455

১৪,৪৪১.

ক
ক) 13
খ
খ) 13.5
গ
গ) 12
ঘ
ঘ) None of these

ব্যাখ্যা

(27/4) × (40/3) + 4 × x = (520 × 30)/100
90 + 4x = 156
4x = 156 - 90
4x = 66
x = 66/4
x = 16.5

১৪,৪৪২.

A can do a piece of work in 12 days. B can do this work in 16 days. A started the work alone. After how many days should B join him, so that the work is finished in 9 days?

ক
2 days
খ
3 days
গ
4 days
ঘ
5 days
ঙ
1 day

ব্যাখ্যা

Question: A can do a piece of work in 12 days. B can do this work in 16 days. A started the work alone. After how many days should B join him, so that the work is finished in 9 days?

Solution:
A's work in 9 days = 9/12 = 3/4.
Remaining work = 1 - 3/4 = 1/4.

B can do full work in 16 days
1/4 portion of work was done by B in (1/4) × 16 = 4 days.
∴ B would have joined A after 9 - 4 = 5 days.

১৪,৪৪৩.

How many times are the hands of a clock at right angle in a day?

ক
ক) 22
খ
খ) 24
গ
গ) 44
ঘ
ঘ) 48

ব্যাখ্যা

In 12 hours, they are at right angles 22 times.
∴ In 24 hours, they are at right angles 44 times.

১৪,৪৪৪.

The area of the four walls of a room is 120 square meters and the length is twice the breadth. If the height of the room is 4 m, then the area of the floor is -

ক
20 square meters
খ
30 square meters
গ
50 square meters
ঘ
60 square meters

ব্যাখ্যা

Question: The area of the four walls of a room is 120 square meters and the length is twice the breadth. If the height of the room is 4 m, then the area of the floor is -

Solution:
Let the breadth = x meters and
length = (2x) metres

Area of 4 walls = 2(2x + x) × 4
= 24x

ATQ,
∴ 24x = 120
⇒ x = 5

So, length = 10 m, and breadth = 5 m

Area of the floor = 10 × 5 = 50 square meters

১৪,৪৪৫.

Find the number of factors of 600.

ক
18
খ
21
গ
24
ঘ
28

ব্যাখ্যা

Question: Find the number of factors of 600.

Solution:
600= 2³ × 3¹ × 5²
Number of factors = (3 + 1)(1 + 1)(2 + 1)
= 4 × 2 × 3
= 24

So, the number of factors of 600 is 24.

১৪,৪৪৬.

A woman buys 50 eggs for $6.60. Some cost 12 cents each and the rest 14 cents each. How many eggs has she bought at 14 cents each ?

ক
ক) 20
খ
খ) 25
গ
গ) 30
ঘ
ঘ) 35

ব্যাখ্যা

ধরি,
14 cents দরে কিনেছিল x টি
12 cents দরে কিনেছিল ( 50 - x ) টি.

প্রশ্নমতে,
14x + 12 × ( 50 - x ) = 6.60 × 100
14x + 600 - 12x = 660
2x + 600 = 660
2x = 660 - 600
2x = 60
x = 30

১৪,৪৪৭.

A football team is to be consisted out of 14 boys. In how many ways the team can be chosen so that the owner of the ball is always in the team?

ক
280
খ
282
গ
286
ঘ
289

ব্যাখ্যা

প্রশ্ন: A football team is to be consisted out of 14 boys. In how many ways the team can be chosen so that the owner of the ball is always in the team?

সমাধান:
14 জনের দল থেকে 1জন ঠিক রেখে বাকি 13জন থেকে (11 - 1) = 10 জনের টিম গঠন করা যাবে
= ¹³C₁₀
= 286

১৪,৪৪৮.

2 leak A and B can drain a cistern in 10 and 20 hours individually. If both the leaks work together, the time required to drain haft of the cistern's water is-

ক
10/3 hours
খ
4 hours
গ
5/2 hours
ঘ
10/7 hours

ব্যাখ্যা

Question: 2 leak A and B can drain a cistern in 10 and 20 hours individually. If both the leaks work together, the time required to drain haft of the cistern's water is-

Solution:
in one hour 2 leaks can drain = 1/10 + 1/20
= 3/20 of the cistern

so, time to drain the whole cistern is = 20/3 hours
∴ half of the cistern in = 20/6 = 10/3 hours

১৪,৪৪৯.

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

ক
- 6
খ
6
গ
2
ঘ
12

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

১৪,৪৫০.

If p ⊗ q = p - q + (p × q), what is the value of 5 ⊗ 2?

ক
10
খ
13
গ
7
ঘ
8

ব্যাখ্যা

Question: If p ⊗ q = p - q + (p × q), what is the value of 5 ⊗ 2?

Solution:
দেওয়া আছে, p ⊗ q = p - q + (p × q)

এখন, p = 5 এবং q = 2 বসিয়ে পাই,
⇒ 5 ⊗ 2 = 5 - 2 + (5 × 2)
⇒ 5 ⊗ 2 = 3 + 10
⇒ 5 ⊗ 2 = 13

∴ সুতরাং, নির্ণেয় মান হলো 13।

১৪,৪৫১.

Three numbers are in the ratio of 3 : 4 : 5 and their L.C.M is 2400. Their H.C.F is

ক
ক) 30
খ
খ) 40
গ
গ) 50
ঘ
ঘ) 60

ব্যাখ্যা

Let us consider the number be 3x,4x and 5x
The L.C.M of the three numbers be =60x
60x = 2400
x = 2400/60
x = 40

Then the number be
3x = 3 × 40 = 120
4x = 4 × 40 = 160
5x = 5 × 40 = 200

∴ The HCF of the three numbers be 40

১৪,৪৫২.

log₃(x² + 3x) - log₃(x + 3) = 2, Then what is the value of x?

ক
0
খ
3
গ
2
ঘ
9

ব্যাখ্যা

Question: log₃(x² + 3x) - log₃(x + 3) = 2, Then what is the value of x?

Solution:
Given that,
log₃(x² + 3x) - log₃(x + 3) = 2
⇒ log₃[(x² + 3x)/(x + 3)] = 2 ; [log₃A - log₃B = log₃(A/B)]
⇒ log₃[x(x + 3)/(x + 3)] = 2
⇒ log₃x = 2
⇒ x = 3²
∴ x = 9

১৪,৪৫৩.

A company increases salary of an officer at 15% per year. In 2024 an employee receives 28500. what was his salary in 2022?

ক
21000 Tk.
খ
21500 Tk.
গ
21550 Tk.
ঘ
21850 Tk.

ব্যাখ্যা

Question: A company increases salary of an officer at 15% per year. In 2024 an employee receives 28500. what was his salary in 2022?

Solution:
let, his saray in 2022 was 100 Tk.
at 15% increment,
salary of 2023 is = 100(1 + 15/100)
= 115 Tk.
salary of 2024 is = 115(1 + 15/100)
= 132.25
∴ his salary in 2022 was = (28500 × 100)/ 132.25
= 21550 Tk.

১৪,৪৫৪.

How many metres of carpet 63 cm wide will be required to cover the floor of a room 14 metres by 9 metres?

ক
150 metres
খ
200 metres
গ
250 metres
ঘ
300 metres

ব্যাখ্যা

Question: How many metres of carpet 63 cm wide will be required to cover the floor of a room 14 metres by 9 metres?
(একটি ১৪ মিটার দীর্ঘ এবং ৯ মিটার প্রশস্ত রুমের মেঝে ঢাকতে ৬৩ সেন্টিমিটার চওড়া কার্পেটের কত মিটার প্রয়োজন?)

Solution:
মেঝের ক্ষেত্রফল = (14 × 9) m² = 126 m²
কার্পেট = 63 cm = 0.63 m∴ কার্পেটের দৈর্ঘ্য = (126/0.63)m = 200 m

১৪,৪৫৫.

What is the profit percentage if the price for 120 bananas equals the selling price of 100 bananas?

ক
20%
খ
30%
গ
35%
ঘ
40%

ব্যাখ্যা

Question: What is the profit percentage if the price for 120 bananas equals the selling price of 100 bananas?

Solution:
Let the C.P. of 120 bananas be Tk. 120
As per question,
S.P. of 100 bananas = Tk. 120

C.P. of 100 bananas would be = Tk. 100
∴ Profit = (S.P. - C.P.) = 120 - 100 = 20

Profit percent = (20/100) × 100%
= 20%

১৪,৪৫৬.

All of the students of Music High School are in the band, the orchestra, or both. 80 percent of the students are in only one group. There are 119 students in the band. If 50 percent of the students are in the band only, how many students are in the orchestra only?

ক
30
খ
51
গ
60
ঘ
100

ব্যাখ্যা

Question: All of the students of Music High School are in the band, the orchestra, or both. 80 percent of the students are in only one group. There are 119 students in the band. If 50 percent of the students are in the band only, how many students are in the orchestra only?

Solution:
only in orchestra = 80% - 50% = 30%
students in both band and orchestra = 100 - 80 = 20%

total students in band = 50% + 20% = 70%

70% of total students = 119
total students = 119/0.7
= 170

students in orchestra = 30% of 170 = 170 × 0.3 = 51

১৪,৪৫৭.

A man invested Tk. 5280 in Tk.10 shares quoted at Tk. 8.25. If the rate of dividend be 12%, his annual income is:

ক
ক) Tk.788
খ
খ) Tk.678
গ
গ) Tk.768
ঘ
ঘ) Tk.748

ব্যাখ্যা

Number of shares =(5280/8.25) = 640
Face value = Tk.(640 × 10)= Tk.6400
Annual income = {(12/100) × 6400} = 768

১৪,৪৫৮.

A boy agrees to work at the rate of one Taka on the first day, two Taka on the second day, and four Taka on third day and so on. How much will the boy get if he started working on the 1^st of February and finishes on the 20^th of February?

ক
2²⁰
খ
2¹⁹ -1
গ
2¹⁹
ঘ
2²⁰ - 1
ঙ
None of these

ব্যাখ্যা

Question: A boy agrees to work at the rate of one Taka on the first day, two Taka on the second day, and four Taka on third day and so on. How much will the boy get if he started working on the 1^st of February and finishes on the 20^th of February?

Solution:
Given that,
1st term, a = 1
Common ratio, r = 2 ; r > 1

We know,
Sum Sn = a × {(rⁿ - 1)/(r - 1)}
= 1 × {(2²⁰ - 1)/(2 - 1)}. ; [Putting, a = 1, r = 2 and n = 20]
= 2²⁰ - 1

So the boy will get 2²⁰ - 1 Takas if he works from February 1^st to February 20^th.

১৪,৪৫৯.

A and B can do a piece of work in 9 days, B and C in 12 days, A and C in 18 days. If all of them work together, then how much time will they take to finish the same work?

ক
6
খ
8
গ
7
ঘ
10

ব্যাখ্যা

Question: A and B can do a work in 9 days, B and C can do it in 12 days and A and C can do it in 18 days. If all of them work together, in how many days they can finish the work?

Solution:
A + B can do in 1 day = 1/9 part
B + C can do in 1 day = 1/12 part
A + C can do in 1 day = 1/18 part
2(A + B + C ) can do in 1 day = ((1/9) + (1/12) + (1/18)) = 1/4 part
∴ A + B + C can do in 1 day = 1/(4 × 2) part = 1/8 part

∴ Total days needed = 1/(1/8) = 8 days

১৪,৪৬০.

A girl bought a book for Tk. 450 and sold it at 20% profit. By using that amount she bought another book and sold it at 5% loss. Then the overall profit amount is -

ক
Tk. 102
খ
Tk. 132
গ
Tk. 58
ঘ
Tk. 63

ব্যাখ্যা

Cost Price of 1st book = Tk. 450.
Profit % of 1st book = 20% profit
Selling price of 1st book = Cost Price of 1st book + Profit % of 1st book
= Tk. 450 + 20% of 450
= Tk. 540
Cost price of 2nd book = Tk. 540
Loss % of 2nd book = 5%
Selling price of 2nd book = Cost Price of 1st book + Profit % of 1st book
= Tk. 540 - 5% of 540 = Tk. 513
Overall profit = Tk. 513 - Tk. 450
= Tk. 63.

১৪,৪৬১.

The length of the rectangular hall is 5 m more than its breadth. The area of the hall is 750 m². The length of the hall is:

ক
15 m
খ
22.5 m
গ
25 m
ঘ
30 m

ব্যাখ্যা

Question: The length of the rectangular hall is 5 m more than its breadth. The area of the hall is 750 m2. The length of the hall is:

Solution:
আয়তাকার ক্ষেত্রের প্রস্থ = x মিটার
আয়তাকার ক্ষেত্রের দৈর্ঘ্য = x + 5 মিটার

প্রশ্নমতে
x(x + 5) = 750
⇒ x² + 5x - 750 = 0
⇒ x² - 25x + 30x - 750 = 0
⇒ x(x - 25) + 30(x - 25) = 0
(x - 25)(x + 30) = 0

হয়
x - 25 = 0
x = 25

অথবা
x + 30 = 0
x = - 30

আয়তাকার ক্ষেত্রের দৈর্ঘ্য = 25 + 5 মিটার
= 30 মিটার

১৪,৪৬২.

The sum of two numbers is 45. Their difference is 1/9 of their sum. Their LCM is -

ক
ক) 110
খ
খ) 100
গ
গ) 120
ঘ
ঘ) 70

ব্যাখ্যা

Question: The sum of two numbers is 45. Their difference is 1/9 of their sum. Their LCM is -
Solution:
ধরি সংখ্যা দুটি x এবং y যেখানে x > y
প্রশ্নমতে,
x + y = 45.............(1)

x - y = 1/9(x + y)...........(1)
বা, x - y = 45/9 [From equation (1)]
বা, x - y =5...........(2)

(1) এবং (2) থেকে পাই,
x = 25; y = 20

অতএব, 20 এবং 25 এর লসাগু = 100

১৪,৪৬৩.

The ratio of speeds of A and B is 2 : 3 and therefore A takes 20 minutes more time than B. What is the ratio of time taken by A and B?

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

ব্যাখ্যা

Question: The ratio of speeds of A and B is 2 : 3 and therefore A takes 20 minutes more time than B. What is the ratio of time taken by A and B?

Solution:
When distance is constant then speed is inversely proportional to time.
ST = D

When distance is constant,
S ∝ 1/T

So, the ratio of time taken by A and B = 3 : 2

১৪,৪৬৪.

A man travelled a distance of 61 km in 9 hours. He travelled partly on foot at 4 km/hr and partly on bicycle at 9 km/hr. What is the distance (in km) travelled on foot?

ক
ক) 10
খ
খ) 12
গ
গ) 14
ঘ
ঘ) 16

ব্যাখ্যা

মনে করি,
হেঁটে যায় = x km
∴ সাইকেলে যায় = (61 - x) km
প্রশ্নমতে, x/4 + (61−x)/9 = 9
⇒ (9x + 244 - 4x)/36 = 9
⇒ 5x + 244 = 324
⇒ 5x = 80
⇒ x = 16 km

১৪,৪৬৫.

Sumi can type 10 pages in 5 minutes. Maria can type 5 pages in. 10 minutes. Working together, how many pages can they type in 30 minutes?

ক
70 pages
খ
75 pages
গ
78 pages
ঘ
80 pages

ব্যাখ্যা

Question: Sumi can type 10 pages in 5 minutes. Maria can type 5 pages in. 10 minutes. Working together, how many pages can they type in 30 minutes?

Solution:
Sumi can type in 1 min = 10/5 = 2 pages
Maria can type in 1 min = 5/10 = 1/2 page

Working together they can type in 1 min = (2 + 1/2) pages
= 5/2 pages

∴ They can type in 30 min = (5 × 30)/2 pages
= 75 pages

১৪,৪৬৬.

In a metro train there are 600 passengers out of which 34% are females. Fare of each male is Tk. 20 and each female's fare is 25% less than each male. What is the total revenue generated by all the passengers together?

ক
Tk. 10772
খ
Tk. 10850
গ
Tk. 10891
ঘ
Tk. 10980

ব্যাখ্যা

Question: In a metro train there are 600 passengers out of which 34% are females. Fare of each male is Tk. 20 and each female's fare is 25% less than each male. What is the total revenue generated by all the passengers together?

Solution:
Total Passengers = 600
No. of females = (600 × 34)/100
= 204
No. of male passengers = (600 - 204)
= 396
Fare of each male = Tk. 20
Fare of female, 15% less,
= (20 × 75)/100
= Tk. 15 each

∴ Total revenue generated by male = Tk. (396 × 20)
= Tk. 7920

∴ Total revenue generated by female = (204 × 15)
= 3060

So, Total Revenue = (7920 + 3060)
= Tk. 10980

১৪,৪৬৭.

A and B can do a work a piece of work in 12 days, B and C in 10 days and C and A in 6 days. How long would B take to do the same work alone?

ক
ক) 40 days
খ
খ) 48 days
গ
গ) 60 days
ঘ
ঘ) 120 days

ব্যাখ্যা

(A + B)'s 1 day's work=1/12
(B + C)'s 1 day's work=1/10
(A + C)'s 1 day's work=1/6

[ (A + B)'s 1 day's work + (B + C)'s 1 day's work ] - (A + C)'s 1 day's work
= (1/12) + (1/10) - (1/6)
= (5 + 6 - 10)/60
⇒2(B's 1 day's work)=1/60
⇒B's 1 day's work=1/120

Hence, B alone can do the work in 120 days.

১৪,৪৬৮.

A man can row at 6 kmph in still water. If the velocity of current is 2 kmph and it takes his 3 hours to row to a place and come back, how far is the place?

ক
ক) 6 km
খ
খ) 8 km
গ
গ) 10 km
ঘ
ঘ) 12 km

ব্যাখ্যা

Question: A man can row at 6 kmph in still water. If the velocity of current is 2 kmph and it takes his 3 hours to row to a place and come back, how far is the place?

Solution:
Speed of downstream = (6 + 2) kmph = 8 kmph
Speed of upstream = (6 - 2) kmph = 4 kmph

Let the required distance be x km

ATQ,
(x/8) + (x/4) = 3
⇒ (3x + 6x)/24 = 3
⇒ 9x = 3 × 24
⇒ 9x = 72
⇒ x = 72/9
∴ x = 8

Hence, the place is 8 km away

১৪,৪৬৯.

x² - 25 = 12, x + 5 = 4, x - 5 = ?

ক
ক) 2
খ
খ) 3
গ
গ) 4
ঘ
ঘ) 6

ব্যাখ্যা

Question: x² - 25 = 12, x + 5 = 4, x - 5 = ?

Solution:
দেয়া আছে,
x² - 25 = 12, x + 5 = 4

এখন,
x² - 25 = 12
x² - 5² = 12
(x + 5)(x - 5) = 12
4(x - 5) = 12
(x - 5) = 12/4
x - 5 = 3

১৪,৪৭০.

What is the solution of x² - 5x + 6 < 0?

ক
2 > x > 3
খ
x < 3
গ
2 ≤ x < 3
ঘ
2 < x < 3

ব্যাখ্যা

Question: What is the solution of x² - 5x + 6 < 0?

Solution:
x² - 5x + 6 < 0
⇒ (x - 2)(x - 3) < 0

The inequality will be true if x - 2 > 0 and x - 3 < 0.
x - 2 > 0
⇒ x > 2

x - 3 < 0
⇒ x < 3

The inequality will be true if 2 < x < 3.
The solution of the inequality is 2 < x < 3

১৪,৪৭১.

Which value of x will satisfy the given inequality,
2(x - 4) ≥ 3x - 5 ?

ক
x ≥ 3
খ
x ≤ - 8
গ
x ≥ - 4
ঘ
x ≤ -3

ব্যাখ্যা

Question: Which value of x will satisfy the given inequality,
2(x - 4) ≥ 3x - 5 ?

Solution:
Given,
2(x - 4) ≥ 3x - 5
⇒ 2x - 8 ≥ 3x - 5
⇒ 2x - 3x ≥ -5 + 8
⇒ - x ≥ 3
⇒ x ≤ - 3 [ Multiplying both sides of an inequality by a negative number reverses the inequality sign ]

১৪,৪৭২.

A certain number when divided by 899 gives a remainder 63. What is the remainder when the same number is divided by 29?

ক
ক) 5
খ
খ) 25
গ
গ) 27
ঘ
ঘ) 28

ব্যাখ্যা

Let the number be x and the quotient is q.
Then, x = 899q + 63 = (29 × 31q) + (29 × 2) + 5
= 29(31q + 2) + 5
So, the given number when divided by 29 gives 5 as remainder.

১৪,৪৭৩.

The ratio of the speed of a boat in still water to the speed of the stream is 5 : 3. The boat covers 16 km upstream in 4 hours and x km downstream in 2 hours. Find the value of x.

ক
32 km
খ
36 km
গ
40 km
ঘ
48 km
ঙ
None of these

ব্যাখ্যা

Question: The ratio of the speed of a boat in still water to the speed of the stream is 5 : 3. The boat covers 16 km upstream in 4 hours and x km downstream in 2 hours. Find the value of x.

Solution:
ধরি, স্থির পানিতে নৌকার গতিবেগ = 5k কিমি/ঘণ্টা এবং স্রোতের গতিবেগ = 3k কিমি/ঘণ্টা।
স্রোতের প্রতিকূলে (Upstream) নৌকার গতিবেগ = (5k - 3k) = 2k কিমি/ঘণ্টা।
স্রোতের অনুকূলে (Downstream) নৌকার গতিবেগ = (5k + 3k) = 8k কিমি/ঘণ্টা।

প্রশ্ন অনুযায়ী,
স্রোতের প্রতিকূলে 16 কিমি যেতে সময় লাগে 4 ঘণ্টা।
∴ স্রোতের প্রতিকূলে গতিবেগ = 16 / 4 = 4 কিমি/ঘণ্টা।

শর্তমতে,
2k = 4
⇒ k = 4/2
⇒ k = 2

এখন, স্রোতের অনুকূলে নৌকার গতিবেগ = 8k = 8 × 2 = 16 কিমি/ঘণ্টা।
∴ স্রোতের অনুকূলে 2 ঘণ্টায় অতিক্রান্ত দূরত্ব, x = 16 কিমি/ঘণ্টা × 2 ঘণ্টা
= 32 কিমি

∴ x = 32 কিমি

১৪,৪৭৪.

If 40 men can make 30 boxes in 20 days. How many more men are needed to make 60 boxes in 25 days?

ক
ক) 21
খ
খ) 24
গ
গ) 25
ঘ
ঘ) 27

ব্যাখ্যা

M1 = 40 Men , D1 = 20 days , W1 = 30 Boxes
M2 = 40 + M , D2 = 25 days , W2 = 60 Boxes

১৪,৪৭৫.

A man and a boy received Tk. 800 as wages for 5 days for the work they did together. The man's efficiency in the work was thrice times that of the boy. What are the daily wages of the boy?

ক
Tk. 30
খ
Tk. 35
গ
Tk. 40
ঘ
Tk. 42

ব্যাখ্যা

Question: A man and a boy received Tk. 800 as wages for 5 days for the work they did together. The man's efficiency in the work was thrice times that of the boy. What are the daily wages of the boy?

Solution:
Man's wage = 3 boy's wage
Daily wages for them = 800/5
= 160

A man and a boy = 4 boys
4 boys' wage = 160
4 boys' wag = 160
⇒ boy's wage = Tk. 40

১৪,৪৭৬.

If A³ is odd, which of the following is true?

ক
ক) A is odd only
খ
খ) A² is odd only
গ
গ) A² is even
ঘ
ঘ) Both A & B are true
ঙ
ঙ) Both A & C are true

ব্যাখ্যা

Question: If A³ is odd, which of the following is true?

Solution:
Let,
A³ = 27
∴ A = 3, here A is odd.
And,
A² = 3² = 9, here A² is odd.

So, both option 1 and 2 are correct.

১৪,৪৭৭.

A fruit seller buys lemons at 2 for Tk. 1 and sells them at 5 for Tk. 3. What is his gain percent?

ক
25%
খ
26.67%
গ
33.33%
ঘ
20%

ব্যাখ্যা

Question: A fruit seller buys lemons at 2 for Tk. 1 and sells them at 5 for Tk. 3. What is his gain percent?

Solution:
Cost price,
2 lemons = Tk. 1
So, cost price of 1 lemon = 1/2

Selling price,
5 lemons = Tk. 3
So, selling price of 1 lemon = 3/5

Profit per lemon = SP - CP
= (3/5) - (1/2)
= (6 - 5)/10
= 1/10

Gain percent = (Profit/CP) × 100%
= (1/10)/(1/2) × 100%
= (1/10) × (2/1) × 100%
= (2/10) × 100%
= 100/5%
= 20%

So his gain percent is 20%.

১৪,৪৭৮.

A man can row at 10 kmph in still water. If the speed of the current is 2 kmph and it takes him 1.5 hours to row to a place and come back, how far is the place?

ক
7.2 km away
খ
7 km away
গ
8.2 km away
ঘ
6 km away

ব্যাখ্যা

Question: A man can row at 10 kmph in still water. If the speed of the current is 2 kmph and it takes him 1.5 hours to row to a place and come back, how far is the place?

Solution:
Speed downstream = 10 + 2 = 12 kmph
Speed upstream = 10 - 2 = 8 kmph

Let the required distance = x km

ATQ,
(x/12) + (x/8) = 1.5
Or, (2x + 3x)/24 = 1.5
Or, 5x/24 = 1.5
Or, 5x = 36
∴ x = 36/5
∴ x = 7.2 km

∴ The place is 7.2 km away.

১৪,৪৭৯.

At present, the ratio of the age of Maya and Chhaya is 6 : 5 and fifteen years from now, the ratio will get changed to 9 : 8. Maya’s present age is-

ক
22 years
খ
25 years
গ
30 years
ঘ
34 years

ব্যাখ্যা

Question: At present, the ratio of the age of Muna and Riya is 6 : 5 and fifteen years from now, the ratio will get changed to 9 : 8. Muna’s present age is-

Solution:
Let, Muna’s present age be = 6x years
and Riya’s present age be = 5x years.

After 15 years, Muna’s age be = (6x + 15) years
After 15 years, Riya’s age be = (5x + 15) years

ATQ,
(6x + 15)/(5x + 15) = 9/8
⇒ 48x + 120 = 45x + 135
⇒ 48x - 45x = 135 - 120
⇒ 3x = 15
⇒ x = 5

∴ Muna’s present age = 6x
= 6 × 5) = 30 years

১৪,৪৮০.

50⁷ × 20⁷ is 10^x times larger than 1 × 10⁸, where x is:

ক
13
খ
6
গ
21
ঘ
29

ব্যাখ্যা

Question: 50⁷ × 20⁷ is 10^x times larger than 1 × 10⁸, where x is:

Solution:
(50⁷ × 20⁷) = 10^x × 1 × 10⁸
⇒ 1000⁷ = 10 ^{x + 8}
⇒ (10³)⁷ = 10^{x + 8}
⇒ 10 ²¹ = 10 ^{x + 8}
⇒ x + 8 = 21
⇒ x = 21 - 8 = 13

১৪,৪৮১.

Train A travels at a speed of 90 km/h and takes 9 seconds to pass a Man. It meets Train B, which is moving in the opposite direction at 72 km/h and length of the train B is x m. If Train A crosses Train B completely in 9 seconds. Find the value of x?

ক
220 m
খ
180 m
গ
160 m
ঘ
190 m

ব্যাখ্যা

Question: Train A travels at a speed of 90 km/h and takes 9 seconds to pass a Man. It meets Train B, which is moving in the opposite direction at 72 km/h and length of the train B is x m. If Train A crosses Train B completely in 9 seconds. Find the value of x?

Solution:
Given that,
Speed of Train A = 90 km/h
Speed of Train B = 72 km/h
Train A crosses a man in 9 sec
Train A crosses Train B in 9 sec

Now,
Train A speed = 90 × (5/18) = 25 m/s
∴ Length of Train A = 25 × 9 = 225 m

Relative speed = (90 + 72) × (5/18) = 162 × (5/18) = 45 m/s

∴ Total length = 45 × 9 = 405 m

∴ Length of Train B is, x = 405 - 225 = 180 m
∴ x = 180 m

১৪,৪৮২.

ক
ক) 1/2
খ
খ) 1
গ
গ) 0
ঘ
ঘ) 7

ব্যাখ্যা

Question:

Solution:

১৪,৪৮৩.

20 men complete one-third of a piece of work in 20 days. How many more men should be employed to finish the rest of the work in 25 more days ?

ক
ক) 32
খ
খ) 21
গ
গ) 24
ঘ
ঘ) 12

ব্যাখ্যা

Let the total number of men be x
Work done = 1/3
Remaining work = (1 - 1/3) = 2/3
More work, More men (Direct proportion)
More days, Less men (Indirect proportion)
Work 1/3 : 2/3
Days 25 : 20
1/3 × 25 × x = 2/3 × 20 × 20
x = 32
∴ More men to be employed = (32 - 20) = 12

১৪,৪৮৪.

Nasim starts climbing an 11 m high wall at 4 pm. In each minute he climbs up 1 m but slips down 50 cm. At what time will he climb the wall?

ক
ক) 4:18 pm
খ
খ) 4:20 pm
গ
গ) 4:21 pm
ঘ
ঘ) 4:22 pm

ব্যাখ্যা

Question: Nasim starts climbing an 11 m high wall at 4 pm. In each minute he climbs up 1 m but slips down 50 cm. At what time will he climb the wall?

Solution:
প্রতি মিনিটে উঠে ১ মিটার বা ১০০ সেমি, নামে ৫০ সেমি
প্রতি মিনিটে উঠে ১০০ - ৫০ সেমি = ৫০ সেমি বা ১/২ মিটার

১/২ মিটার উঠতে সময় লাগে ১ মিনিট
১ মিটার উঠতে সময় লাগে ২ মিনিট
১০ মিটার উঠতে সময় লাগে ২০ মিনিট

পরের ১ মিটার ১ মিনিটে উঠে যায়।

সময় লাগে = ২০ + ১ মিনিট
= ২১ মিনিট

∴ ৪ টা ২১ মিনিটে দেয়ালের উপরে উঠে যাবে।

১৪,৪৮৫.

How many three-digit numbers can be formed by using the digits in 735621, if repetition is not allowed?

ক
ক) 90
খ
খ) 120
গ
গ) 150
ঘ
ঘ) 210

ব্যাখ্যা

ⁿP_r = n!/(n-r)!
⁶P₃ = 6!/(6-3)!
⁶P₃ = 6!/3!
⁶P₃ = 120

১৪,৪৮৬.

Two runners start running together for a certain distance, one at 5 km/h and another at 8 km/h. The former arrives one and a half an hour before the latter. The distance in Km is -

ক
10 km
খ
15 km
গ
20 km
ঘ
25 km

ব্যাখ্যা

Question: Two runners start running together for a certain distance, one at 5 km/h and another at 8 km/h. The former arrives one and a half an hour before the latter. The distance in Km is -

Solution:
Let the distance be = P
time of the first runner = P/5 hour
second runner = P/8

ATQ,
P/5 - P/8 = 3/2
3P/40 = 3/2
3P = 60
P = 20 km

১৪,৪৮৭.

A shopkeeper buys 100 pencils for 500 taka. How many pencils should he sell for 400 taka to make a 25% profit?

ক
68
খ
56
গ
64
ঘ
58

ব্যাখ্যা

Question: A shopkeeper buys 100 pencils for 500 taka. How many pencils should he sell for 400 taka to make a 25% profit?

Solution:
Cost price of 100 pencils = Tk. 500
Cost price per pencil = 500/100 = Tk. 5

Since the seller wants 25% profit,
The selling price per pencil will be = 5 × (1 + 25/100) = (5 × 1.25) = 6.25 tk

Now, the number of pencils the seller can sell = 400/6.25 = 64

So the shopkeeper should sell 64 pencils for 400 taka to make a 25% profit.

১৪,৪৮৮.

What is the missing number in the following expression?
(5 + 2) ÷ 7 - 3 × 2 = 1 + ?

ক
- 6
খ
- 5
গ
- 4
ঘ
- 7

ব্যাখ্যা

Question: What is the missing number in the following expression?
(5 + 2) ÷ 7 - 3 × 2 = 1 + ?

Solution:
(5 + 2) ÷ 7 - 3 × 2 = 1 + ?
⇒ 7 ÷ 7 - 3 × 2 = 1 + ?
⇒ 1 - 3 × 2 = 1 + ?
⇒ 1 - 6 = 1 + ?
⇒ - 5 = 1 + ?
⇒ ? = - 5 - 1
∴ ? = - 6

১৪,৪৮৯.

If 45% of a certain number is 72, then find the number-

ক
140
খ
160
গ
172
ঘ
150

ব্যাখ্যা

Question: If 45% of a certain number is 72, then find the number-

Solution:
Let the number be x.
Then,
⇒ 45% of x = 72
⇒ (45/100) × x = 72
⇒ 9x/20 = 72
⇒ x = (20 × 72)/9
⇒ x = 1440/9
∴ x = 160

১৪,৪৯০.

The length of a rectangle is 25% more than its breadth. What will be the ratio of the area of the rectangle to that of a square whose side is equal to the breadth of the rectangle?

ক
5 : 6
খ
5 : 4
গ
2 : 3
ঘ
1 : 4

ব্যাখ্যা

Question: The length of a rectangle is 25% more than its breadth. What will be the ratio of the area of the rectangle to that of a square whose side is equal to the breadth of the rectangle?

Solution:
Let,
breadth = X metres
∴ length = 125% of X metres
= 125X/100 metres
= 5X/4 metres

∴ Area of the rectangle = (5x/4 × X) m²
∴ Area of the square = (X × X) m²

∴ The ratio = (5X/4 × X) : (X × X)
= 5 : 4

১৪,৪৯১.

Which of the following numbers is a prime number?

ক
213
খ
191
গ
275
ঘ
333

ব্যাখ্যা

Question: Which of the following numbers is a prime number?

Solution:
A prime number is a number that can only be divided by itself and 1 without remainders.

ক) 213 is divisible by 3.
খ) 191 has no divisors other than 1 and itself → prime
গ) 275 is divisible by 5.
ঘ) 333 is divisible by 3.

∴ 191 is the required prime number as it is not divisible by any prime number.

১৪,৪৯২.

A team of 8 students goes on an excursion. In two cars, of which one can seat 5 and the other only 4. In how many ways can they travel?

ক
56
খ
126
গ
224
ঘ
256

ব্যাখ্যা

Question: A team of 8 students goes on an excursion. In two cars, of which one can seat 5 and the other only 4. In how many ways can they travel?

Solution:
There are 8 students and the maximum capacity of the cars together is 9.
We may divide the 8 students as follows

Case I: 5 students in the first car and 3 in the second
Case II: 4 students in the first car and 4 in the second

Hence, in Case I: 8 students are divided into groups of 5 and 3 in ⁸C₃ ways.
Similarly, in Case II: 8 students are divided into two groups of 4 and 4 in⁸C₄ways.

Therefore, the total number of ways in which 8 students can travel is:
⁸C₃+⁸C₄= 56 + 70 = 126

১৪,৪৯৩.

The compound interest on Tk. 3000 at 10% per annum is Tk. 630. The period (in years) is-

ক
3 years
খ
4 years
গ
5 years
ঘ
2 years

ব্যাখ্যা

Question : The compound interest on Tk. 3000 at 10% per annum is Tk. 630. The period (in years) is:

Solution :
Amount= Tk. (3000 + 630)
= Tk. 3630

Let, the time = n years

Then,
3000(1 + 10/100)ⁿ = 3630
⇒ 3000(11/10)ⁿ = 3630
⇒ (11/10)ⁿ = 3630/3000
⇒ (11/10)ⁿ = 121/100
⇒ (11/10)ⁿ = (11/10)²
∴ n = 2

১৪,৪৯৪.

A 4% stock yields 5%. The market value of the stock of face value Tk. 100 is-

ক
Tk 80
খ
Tk 75
গ
Tk 120
ঘ
Tk 140

ব্যাখ্যা

Question: A 4% stock yields 5%. The market value of the stock of face value Tk. 100 is-

Solution:
For an income of tk. 5, investment = tk. 100.
For an income of tk. 4, investment = tk. (100/5) × 4 = tk. 80.
Market value of tk. 100 stock = tk. 80.

১৪,৪৯৫.

Which of the following pairs of crops will grow together in field 3 if no other crops are planted in the field and no fertilizers or pesticides are applied?

ক
ক) F and H
খ
খ) F and I
গ
গ) G and H
ঘ
ঘ) G and I

ব্যাখ্যা

Question: Which of the following pairs of crops will grow together in field 3 if no other crops are planted in the field and no fertilizers or pesticides are applied?

Solution:
ফিল্ড ৩ এ উৎপাদন করা যায় F, G, H, I
F উৎপাদনের জন্য সার x প্রয়োজন।
H উৎপাদনের জন্য সার y প্রয়োজন।

অতএব, এক্ষেত্রে G ও I ফিল্ড ৩ এ উৎপাদন করা যায়।

১৪,৪৯৬.

A rectangular tank is 8 m long, 4 m wide, and 3 m high. If it is filled with water up to 75% of its height, what is the volume of water in the tank?

ক
48 m³
খ
72 m³
গ
90 m³
ঘ
96 m³

ব্যাখ্যা

Question: A rectangular tank is 8 m long, 4 m wide, and 3 m high. If it is filled with water up to 75% of its height, what is the volume of water in the tank?

Solution:
A rectangular tank is 8 m long, 4 m wide, and 3 m high
75% of its height = 75% of 3 m = 2.25 m

∴ Volume of water is = 8 × 4 × 2.25 m³
= 72 m³

১৪,৪৯৭.

A man borrowed a certain sum of money at the rate of 6% per annum for the first two years, 9% per annum for the next three years, and 14% per annum for the period beyond 5 years. If he pays a total interest of Tk. 22800 at the end of 9 years, find the amount he borrowed.

ক
Tk. 20000
খ
Tk. 22000
গ
Tk. 24000
ঘ
Tk. 25000
ঙ
Tk. 26000

ব্যাখ্যা

Question: A man borrowed a certain sum of money at the rate of 6% per annum for the first two years, 9% per annum for the next three years, and 14% per annum for the period beyond 5 years. If he pays a total interest of Tk. 22800 at the end of 9 years, find the amount he borrowed.

Solution:
Let the borrowed sum be P.
SI for first 2 years + SI for next 3 years + SI for next 4 years = 22800
⇒ (P × 6 × 2/100) + (P × 9 × 3/100) + (P × 14 × 4/100) = 22800
⇒ (95P)/100 = 22800
⇒ P = (22800 × 100)/95
⇒ P = 24000
Therefore, Borrowed sum = Tk. 24000

১৪,৪৯৮.

Three times a whole number is equal to four less than the square of the number. Find the number.

ক
5
খ
2
গ
4
ঘ
1

ব্যাখ্যা

Question: Three times a whole number is equal to four less than the square of the number. Find the number.

Solution:
Let the number be x.
Then, 3x = x² - 4
⇒ x² - 3x - 4 = 0
⇒ x²- 4x + x - 4 = 0
⇒ x (x - 4) + 1 (x - 4) = 0
⇒ (x - 4) (x + 1) = 0
⇒ x = 4, - 1
Solving, x = 4 is the only whole number solution.

১৪,৪৯৯.

If arcs AXB and CYD of a circle are congruent, find the ratio of AB and CD.

ক
1 : 2
খ
2 : 3
গ
3 : 2
ঘ
1 : 1

ব্যাখ্যা

Question: If arcs AXB and CYD of a circle are congruent, find the ratio of AB and CD.

Solution:
Given that arcs AXB and CYD of a circle are congruent,
i.e. arc AXB ≅ arc CYD.

We know that if two arcs of a circle are congruent, their corresponding chords are also equal.
i.e. chord AB = chord CD

Thus, AB/CD = 1

AB/CD = 1/1
∴ AB : CD = 1 : 1

১৪,৫০০.

If tan²A - 6tanA + 9 = 0, what is the value of 6cotA =?

ক
1/2
খ
5
গ
2/3
ঘ
2

ব্যাখ্যা

Question: If tan²A - 6tanA + 9 = 0, what is the value of 6cotA =?

Solution:
Given that,
tan²A - 6tanA + 9 = 0
⇒ tan²A - 2 . tanA . 3 + 3² = 0
⇒ (tanA - 3)² = 0
⇒ (tanA - 3) = 0
⇒ tanA = 3
⇒ cotA = 1/3

∴ 6cotA = 6 × (1/3) = 2