Bank Math

PrepBank · পাতা ৪০ / ১৬১ · ৩,৯০১–৪,০০০ / ১৬,১২৪

৩,৯০১.

The average of the ages of a man and his daughter is 34 years. If the respective ratio of their ages after four years from now is 14 : 5. what is daughter's present age?

ক
ক) 20 years
খ
খ) 52 years
গ
গ) 16 years
ঘ
ঘ) 34 years

ব্যাখ্যা

Average age of man and his daughter = 34 years
Their total age = (34 × 2) years = 68 years
Let man's age be x years,
Then daughter age = (68 - x) years
∴(x + 4)/(68 - x + 4) =14/5
⇒5(x + 4) = 14(72 −x)
⇒5x + 20= 1008 − 14x
⇒19x =988
⇒ x =52

∴ Daughter's present age
= (68 - 52) years
= 16 years

৩,৯০২.

In 24 seconds, an athlete covers 200 meters. What is his speed in kilometres per hour?

ক
15
খ
20
গ
30
ঘ
40

ব্যাখ্যা

Question: In 24 seconds, an athlete covers 200 meters. What is his speed in kilometres per hour?

Solution:
Speed of athlete
= 200 m/24 sec
= (200 × 18)/(5 × 24)
= 30 km/hour

৩,৯০৩.

A 1300 m long train crosses a tree in 130 sec, how much time will it take to pass a platform 800 m long?

ক
ক) 150 sec.
খ
খ) 160 sec.
গ
গ) 170 sec.
ঘ
ঘ) 210 sec.

ব্যাখ্যা

Here
Length of a train is 1300m
Train took 130 sec to cross a tree
Length of a platform is 800m

Speed of the train = 1300/130 = 10 m/sec
Total distance = 1300 + 800 = 2100 m

Time = distance/speed
= 2100/10 = 210 sec

∴ Time required to cross a platform is 210 sec.

৩,৯০৪.

3/7 part of the tank is full of water. When 42 liters of water is taken out, the tank becomes empty. The capacity of the tank is -

ক
82 liters
খ
66 liters
গ
78 liters
ঘ
98 liters

ব্যাখ্যা

Question: 3/7 part of the tank is full of water. When 42 liters of water is taken out, the tank becomes empty. The capacity of the tank is -

Solution:
Let us consider,
The tank has 7x liters of total capacity and holds 3x litres of water.
And if 42 liters of water is taken out, then the tank becomes empty.

It means 3x litres of water is taken out.
∴ 3x = 42 liters
⇒ x = 14 liters

∴ Capacity of tank = 7x = 7 × 14 = 98 liters

৩,৯০৫.

The circumference of the circle and the perimeter of the square is equal and the ratio between the diameter of the circle and the side of the square is 7 : 11. What is the area of the circle?

ক
ক) 154 cm²
খ
খ) 160 cm²
গ
গ) 132 cm²
ঘ
ঘ) Can’t be determined

ব্যাখ্যা

Let, the side of the square = 11x
and, diameter of the circle = 7x
ATQ,
2π(7x/2) = 4×11x
Or, 7πx = 44x
x omits from both side.
So, the radius of the circle can't be determined from the given information.

৩,৯০৬.

If 5^{(a + 3)} = 25^{(3a - 4)} then the value of a is = ?

ক
11/5
খ
5/2
গ
9/2
ঘ
7/2

ব্যাখ্যা

Question: If 5^{(a + 3)} = 25^{(3a - 4)} then the value of a is = ?

Solution:
5^{(a + 3)} = 25^{(3a - 4)}
⇒ 5^{(a + 3)} = (5²)^{(3a - 4)}
⇒ 5^{(a + 3)} = 5^{(6a - 8)}
⇒ a + 3 = 6a - 8
⇒ 5a = 11
∴ a = 11/5

৩,৯০৭.

Three pipes A, B, and C would fill a tank in 10 hours, 15 hours, and 20 hours respectively. If all these pipes are opened simultaneously, how much time will be taken to fill the tank?

ক
ক) 5 hr.
খ
খ) 60/13 hr.
গ
গ) 50/13 hr.
ঘ
ঘ) 6 hr.

ব্যাখ্যা

Question: Three pipes A, B, and C would fill a tank in 10 hours, 15 hours, and 20 hours respectively. If all these pipes are opened simultaneously, how much time will be taken to fill the tank?

Solution:
A, 1 ঘন্টায় পূর্ণ করে (1/10) অংশ
B, 1 ঘন্টায় পূর্ণ করে (1/15) অংশ
C, 1 ঘন্টায় পূর্ণ করে (1/20) অংশ

তিনটি পাইপ একত্রে পূর্ণ করে = (1/10) + (1/15) + (1/20) = 13/60 অংশ

আবার,
13/60 অংশ পূর্ণ করে 1 ঘণ্টায়
∴ 1 অংশ পূর্ণ করে 60/13 ঘণ্টায়

৩,৯০৮.

There are 10 orange sodas, 15 cream sodas, and 7 cherry sodas in an ice chest. How many sodas must be removed from the ice chest to guarantee that one of each type of sodas had been chosen?

ক
ক) 16
খ
খ) 18
গ
গ) 23
ঘ
ঘ) 26

ব্যাখ্যা

The answer is that you have to pick up enough sodas until you are guaranteed to have at least 1 of each.
So if you pick up 15 it could be only cream sodas but if you pick up 16 it must be 2 kinds.
If you pick up 25 it could be only 10 orange sodas and 15 cream sodas.
But if you pick up 26 it must be 3 kinds which ensures that one of each type of soda has been chosen.

৩,৯০৯.

What is the difference between the simple interest and the compound interest on a sum of Tk. 8000 for 2 years at the rate of 10% per annum when the interest is compounded yearly?

ক
Tk. 70
খ
Tk. 85
গ
Tk. 75
ঘ
Tk. 80

ব্যাখ্যা

Question: What is the difference between the simple interest and the compound interest on a sum of Tk. 8000 for 2 years at the rate of 10% per annum when the interest is compounded yearly?

Solution:
Given that,
Principal, P = Tk. 8000
Rate, r = 10% per annum
Time, n = 2 years

Wee know,
SI = (P × r × n)/100
= (8000 × 10 × 2)/100
= 160000/100
= Tk. 1600

And,
Compound Interest (CI) – compounded annually
= P(1 + r)ⁿ- P
= P × (1 + 10/100)² - 8000
= 8000 × (1 + 1/10)² - 8000
= 8000 × (1.1)² - 8000
= 8000 × 1.21
= 9680 - 8000
= Tk. 1680

Difference between CI and SI
= 1680 - 1600
= Tk. 80

৩,৯১০.

A sum of money amounts to Tk. 9800after 5 years and Tk. 12005 after 8 years at the same rate of simple interest. The rate of interest per annum is -

ক
15%
খ
5%
গ
12%
ঘ
8%

ব্যাখ্যা

Simple interest for 3 years
= 12005 - 9800
= 2205
Simple interest for 5 years
= (2205/3) × 5
= 3675
Some of money = 9800 - 3675
= 6125
R = (100 × 2205)/(6125 × 3)
= 12%

৩,৯১১.

A sum of money placed at compound interest doubles itself in 4 years. In how many years will it amount of 8 times itself?

ক
ক) 12 years
খ
খ) 8 years
গ
গ) 16 years
ঘ
ঘ) 64 years

ব্যাখ্যা

ধরি,
আসল P = x
সময় n = বছর
সুদের হার = r %

প্রশ্নমতে,
P{1 + (r/100)}⁴ = 2x
x{1 + (r/100)}⁴ = 2x
{1 + (r/100)}⁴ = 2

ধরি,
n বছর পর চক্রবৃদ্ধি মূলধন 8 গুণ হবে।

x{1 + (r/100)}ⁿ = 8x
{1 + (r/100)}ⁿ = 8
{1 + (r/100)}ⁿ = 2³
{1 + (r/100)}ⁿ =[{1 + (r/100)}⁴]³
{1 + (r/100)}ⁿ ={1 + (r/100)}¹²
n = 12

৩,৯১২.

A and B are two positive integers such that AB = 60. Which of the following cannot be the value of A + B?

ক
16
খ
32
গ
61
ঘ
20

ব্যাখ্যা

Question: A and B are two positive integers such that AB = 60. Which of the following cannot be the value of A + B?

Solution:
Factor pairs of 60:
(1, 60) → A + B = 61
(2, 30) → A + B = 32
(3, 20) → A + B = 23
(4, 15) → A + B = 19
(5, 12) → A + B = 17
(6, 10) → A + B = 16

So, possible values of A + B are: 61, 32, 23, 19, 17, 16.

Among the options, 20 is not possible.

৩,৯১৩.

The average weight of A, B and C is 50 kg. If the average weight of A and B be 46 kg and that of B and C be 52 kg, then the weight of B is-

ক
36 kg
খ
40 kg
গ
48 kg
ঘ
56 kg
ঙ
46 kg

ব্যাখ্যা

Question: The average weight of A, B and C is 50 kg. If the average weight of A and B be 46 kg and that of B and C be 52 kg, then the weight of B is-

Solution:
Let A, B, C represent their respective weights.
Then, we have,
A + B + C = (50 × 3) = 150 ........(i)
A + B = (46 × 2) = 92 ........(ii)
B + C = (52 × 2) = 104 .........(iii)

Adding (ii) and (iii), we get:
A + 2B + C = 92 + 104 = 196 ........(iv)
And,
Subtracting (iv) from (i),
A + 2B + C - (A + B + C) = 196 - 150
B = 46

∴ B's weight = 46 kg

৩,৯১৪.

(√7 + √7)² =

ক
ক) 98
খ
খ) 49
গ
গ) 28
ঘ
ঘ) 21

ব্যাখ্যা

(√7 + √7)² = (2√7)² = 4 × 7 = 28

৩,৯১৫.

A man can row three-quarters of a kilometre against the stream in 11.25 minutes and down the stream in 7.5 minutes. The speed (in km/hr) of the man in still water is:

ক
2 km/hr
খ
3 km/hr
গ
4 km/hr
ঘ
5 km/hr

ব্যাখ্যা

Question: A man can row three-quarters of a kilometre against the stream in 11.25 minutes and down the stream in 7.5 minutes. The speed (in km/hr) of the man in still water is:

Solution:
Three - quarters of a kilometer = (3 × 1000)/4 meters = 750 meters

11.25 minutes = 11.25 × 60 = 675 sec
7.5 minutes = 7.5 × 60 = 450sec

Rate upstream = 750/ 675 m/sec
= 10/9 m/sec

Rate downstream =750/450 m/sec
= 5/3 m/sec

∴Rate in still water = (1/2){(10/9) + (5/3)} m/sec
= (1/2){(10 + 15)/9}
= (1/2)(25/9)
= 25/18
= (25/18)(18/5) km/hr.
= 5 km/hr

৩,৯১৬.

10 years ago, the average age of a family of 4 members was 24 years. Since then, two children have been born. Still, the average age of the family is the same today. If the two children differ in age by 2 years, find the present age of the younger child.

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

ব্যাখ্যা

Question: 10 years ago, the average age of a family of 4 members was 24 years. Since then, two children have been born. Still, the average age of the family is the same today. If the two children differ in age by 2 years, find the present age of the younger child.

Solution:
Total age of 4 members, 10 years ago = (24 × 4) years = 96 years.
Total age of 4 members now = 4 × (24 + 10) = 136 years
Total age of 4 members now = (24 × 6) years = 144 years.

The sum of the ages of 2 children = (144 - 136) years
= 8 years

Let, the age of the younger child be x years
Then, age of elder child = (x + 2) years
So, x + x + 2 = 8
⇒ 2x = 6
∴ x = 3

So, the age of the younger child 3 years.

৩,৯১৭.

Due to a 10% reduction in the price of tea, a dealer can purchase 25 kg more tea for TK. 22500. Find the reduced price per kg.

ক
Tk. 75
খ
Tk. 80
গ
Tk. 95
ঘ
Tk. 90

ব্যাখ্যা

Question: Due to a 10% reduction in the price of tea, a dealer can purchase 25 kg more tea for TK. 22500. Find the reduced price per kg.

Solution:
Let the original price per kg be x.
After a 10% reduction, New price per kg = x - 10% of x = 0.9x
Total money = TK. 22500

According to the question, the dealer can buy 25 kg more tea after the reduction.
So,
⇒ 22500/0.9x = (22500/x) + 25
⇒ 25000/x = (22500/x) + 25
⇒ (25000/x) - (22500/x) = 25
⇒ (25000 - 22500)/x = 25
⇒ 2500/x = 25
⇒ x = 2500/25
∴ x = 100

So, original price per kg = Tk. 100

∴ Reduced price per kg = 0.9 × 100 = Tk. 90

So the reduced price per kg of tea is Tk. 90.

৩,৯১৮.

A 120 m long train passed a pole in 12 seconds. How long will it take to pass a 450 m long platform?

ক
ক) 89 sec
খ
খ) 57 sec
গ
গ) 59 sec
ঘ
ঘ) 61 sec

ব্যাখ্যা

Train’s speed = 120/12 = 10 m/s
The train has to cover = (120 + 450) = 570 m.
∴ Required time = 570/10 = 57 seconds

৩,৯১৯.

Two numbers are in the ratio of 21 : 26. If 8 is added in each, the new numbers are in ratio of 5 : 6. Find the ratio of numbers, if 6 is subtracted from each number?

ক
ক) 19 : 25
খ
খ) 18 : 23
গ
গ) 9 : 16
ঘ
ঘ) 6 : 7

ব্যাখ্যা

Let the numbers be 21x and 26x.
⇒ (21x + 8)/(26x + 8) = 5/6
⇒ 6(21x + 8) = 5(26x + 8)
⇒ 126x + 48 = 130x + 40
⇒ x = 2.
So, the numbers will be 42 and 52.
If 6 is subtracted, then numbers will be 36 and 46.
Required ratio = 36 : 46.
i.e. 18 : 23.

৩,৯২০.

A is 30% more efficient than B. How much time will they, working together, take to complete a job which A alone could have done in 23 days?

ক
ক) 13 days
খ
খ) 15 days
গ
গ) 10 days
ঘ
ঘ) None

ব্যাখ্যা

Ratio of times taken by A and B = 100 : 130 = 10 : 13.
Let, B takes x days to do the work.
ATQ, x : 23 = 13 : 10
Or, x = (23×13 / 10)
Or, x = 299/10
A’s 1 day’s work = 1/23
B’s 1 day’s work = 10/299
(A + B)’s 1 day’s work = ( 1/23 + 10/299 ) = 23/299 = 1/13 part
So, A and B together can complete the work in 13 days.

৩,৯২১.

The sum of the three consecutive even numbers is 48 more then the average of these numbers. Which of the following is the third largest of these numbers?

ক
26
খ
18
গ
22
ঘ
24

ব্যাখ্যা

Question: The sum of the three consecutive even numbers is 48 more then the average of these numbers. Which of the following is the third largest of these numbers?

Solution:
Let, the numbers be = x, (x + 2) and (x + 4)

Then,
(x + x + 2 + x + 4) - (x + x + 2 + x + 4)/3 = 48
⇒ (3x + 6) - (3x + 6)/3 = 48
⇒ 9x + 18 - 3x - 6 = 144
⇒ 6x + 12 = 144
⇒ 6x = 144 - 12
⇒ 6x = 132
∴ x = 22

The third largest of these numbers is = x + 4 = 22 + 4 = 26

৩,৯২২.

Tea worth Tk. 126 per kg and Tk. 135 per kg are mixed with a third variety in the ratio 1 : 1 : 2. If the mixture is worth Tk. 153 per kg, the price of the third variety per kg will be:

ক
Tk. 169.50
খ
Tk. 170
গ
Tk. 175.50
ঘ
Tk. 180

ব্যাখ্যা

Question: Tea worth Tk. 126 per kg and Tk. 135 per kg are mixed with a third variety in the ratio 1 : 1 : 2. If the mixture is worth Tk. 153 per kg, the price of the third variety per kg will be:

Solution:
let, price of third variety x tk per kg

126y + 135 y + x × 2y = 153 (y + y + 2y)
⇒ 126y + 135y + 2xy = 153 × 4y
⇒ 126 + 135 + 2x = 612
⇒ 2x + 261 = 612
⇒ 2x = 351
∴ x = 175.5 tk

৩,৯২৩.

A train crosses a platform 90 m long in 50 seconds at a speed of 36 km/hr. What is the time taken by the train to cross an electric pole?

ক
ক) 31 sec
খ
খ) 33 sec
গ
গ) 37 sec
ঘ
ঘ) 41 sec

ব্যাখ্যা

Question: A train crosses a platform 90 m long in 50 seconds at a speed of 36 km/hr. What is the time taken by the train to cross an electric pole?

Solution:
Speed of the train = (36 × 1000)/(60 × 60) = 10 m/sec
So, in 50 sec it goes = (10 × 50) = 500 m

Since the platform is 90 m so, only train = (500 - 90) m = 410 m [ খুঁটি অতিক্রম করতে হলে শুধু এতটুকু যেতে হবে ]

∴ Time taken by the train to cross an electric pole = (410/10) sec = 41 sec

৩,৯২৪.

At what annual compound interest rate will Tk. 5000 become Tk. 6050 in 2 years?

ক
10%
খ
12.5%
গ
15%
ঘ
8.25%

ব্যাখ্যা

Question: At what annual compound interest rate will Tk. 5000 become Tk. 6050 in 2 years?

Solution:
Given that,
Principal, P = Tk. 5000
Amount, A = Tk. 6050
Time, n = 2 years

We know,
A = P(1 + r/100)ⁿ
⇒ 6050 = 5000 × (1 + r/100)²
⇒ (1 + r/100)² = 6050/5000
⇒ (1 + r/100)² = 121/100
⇒ 1 + (r/100) = √(121/100) = 11/10
⇒ r/100 = (11/10) - 1
⇒ r/100 = (11 - 10)/10
⇒ r/100 = 1/10
⇒ r = 100/10
∴ r = 10%

So the annual compound interest rate is 10%.

৩,৯২৫.

Two stories A and B mark the price of an item identically. A allows 3 successive discounts of 10% each. B allows 10% discount on the list price and a subsequent discount of 19%. Under the circumstances, which of the following is true?

ক
The price of the article is same at A and B
খ
The price of the article is cheaper at A
গ
The price cannot be determined
ঘ
The price of the article is cheaper at B

ব্যাখ্যা

Question: Two stories A and B mark the price of an item identically. A allows 3 successive discounts of 10% each. B allows 10% discount on the list price and a subsequent discount of 19%. Under the circumstances, which of the following is true?

Solution:
Let the M.P. of the item at each of the stories A and B be tk100.
The final price at A = 90% of 90% of 90% of tk 100
= {(90/100) × (90/100) × (90/100) × 100} tk
= 72.9 tk

The final price at B = 81% of 90% of Rs. 100
= {(81/100) × (90/100) × 100} tk
= 72.9 tk

Hence, the price of the article is the same at A and B.

৩,৯২৬.

A man invested Tk. 26000 in 5% stock at 104. He sold the stock when the price rose to Tk. 120 and invested the sale proceeds in 6% stock. By doing this his income increased by Tk. 2500. At what price did he purchase the second stock?

ক
ক) Tk. 48
খ
খ) Tk. 76
গ
গ) Tk. 125
ঘ
ঘ) Tk. 24

ব্যাখ্যা

Assuming that face value of the first stock = Tk. 100 as it is not given in the question.
Since it is a 5% stock, we can take the dividend per stock = Tk. 5
Market Value of the first stock = Tk. 104
Investment on the first stock = Tk. 26000
Number of stocks purchases = 26000/104 = 250
His total income from all these stocks = Tk. 250 × 5 = Tk. 1250
He sells each of these stocks at Tk. 120
ie, amount he earns = Tk. 120 × 250 = Tk. 30000
He invests this Tk. 30000 in 6% stock (here also face value is not given and hence take it as Tk. 100)
His new income = Tk. (1250 + 2500) = Tk. 3750
ie, By Tk. 30000 of investment, he earns an income of Tk. 3750
To get an income of Tk. 6, the investment needed = (30000 × 6)/3750
= Tk. 48.

৩,৯২৭.

The LCM and HCF of two numbers are 495 and 5. If the sum of the numbers is 100, what is the difference of the numbers?

ক
5
খ
15
গ
20
ঘ
10

ব্যাখ্যা

Question: The LCM and HCF of two numbers are 495 and 5. If the sum of the numbers is 100, what is the difference of the numbers?

Solution:
let,
one number is x
∴ another number is (100 - x)

∴ x(100 - x) = 5 × 495
or, 100x - x² = 2475
or, x² - 100x + 2475 = 0
or, x² - 55x - 45x + 2475 = 0
or, x(x - 55) - 45(x - 55) = 0
or, (x - 55)(x - 45) = 0
∴ x = 45, 55
∴ the numbers are 45 and 55.
difference = 55 - 45 = 10

৩,৯২৮.

(sin 30° + cos 60°) - (sin 60° + cos 30°) is equal to:

ক
0
খ
1 + 2√3
গ
1 - √3
ঘ
1 + √3
ঙ
1

ব্যাখ্যা

sin 30° = 1/2, sin 60° = √3/2, cos 30° = √3/2 and cos 60° = 1/2

Putting these values, we get:

(1/2 + 1/2) - (√3/2 + √3/2)
= 1 – [(2√3)/2]
= 1 – √3

৩,৯২৯.

What is the value of n?

ক
ক) 78°
খ
খ) 55°
গ
গ) 49°
ঘ
ঘ) 39°

ব্যাখ্যা

Question: What is the value of n?

Solution:
2m + 3m = 180°
⇒ 5m = 180°
⇒ m = 36°

3m = 2n + 30
⇒ 2n + 30 = (3 × 36°) = 108°
⇒ 2n = 108° - 30°
⇒ 2n = 78°
⇒ n = 39°

৩,৯৩০.

log 36/log6 =?

ক
1
খ
2
গ
3
ঘ
4

ব্যাখ্যা

Question: log 36/log6 =?

solution:
log 36/log6
= log6²/log6
= 2

৩,৯৩১.

a : b = 3 : 5 and c : b = 7 : 3 then what is the ratio of c : b : a?

ক
ক) 35 : 15 : 9
খ
খ) 9 : 15 : 35
গ
গ) 35 : 9 : 15
ঘ
ঘ) 9 : 35 : 15

ব্যাখ্যা

Question: a : b = 3 : 5 and c : b = 7 : 3 then what is the ratio of c : b : a?

Solution:
here,
a : b = 3 : 5 = 9 : 15 [multiplying by 3]
c : b = 7 : 3 = 35 : 15 [multiplying by 5]

c : b : a = 35 : 15 : 9

৩,৯৩২.

A's income is Tk. 140 more than B's income and C's income is Tk 80 more than D's. If the ratio of A's and C's income is 2 : 3 and the ratio of B's and D's income is 1 : 2, then the incomes of A, B, C and D are respectively

ক
ক) Tk. 60, Tk.120, Tk. 320 and Tk. 240
খ
খ) Tk.300, Tk.160, Tk. 600 and Tk. 520
গ
গ) Tk.400, Tk.260, Tk. 600 and Tk. 520
ঘ
ঘ) Tk.320, Tk.180, Tk. 480 and Tk. 360

ব্যাখ্যা

A : C = 2 : 3 or 2x : 3x
B : D = 1 : 2 or y : 2y
According to question,
2x - 140 = y .... (i)
3x - 80 = 2y .... (ii)

multiply equation (i) by 2 and solved
4x - 280 - 3x + 80 = 2y - 2y
⇒ x - 200 = 0
⇒ x = 200

Now put value of x in equation (i)
(2 × 200) - 140 = y
⇒ 400 - 140 = y
⇒ y = 260

A's salary = 2x = Tk. 400
B's salary = y = Tk. 260
C's salary = 3x = Tk. 600
D's Salary = 2y = Tk. 520

৩,৯৩৩.

A fruit vendor claims to sell oranges at cost price, but he secretly adds lower quality oranges to his premium ones and gains 25%. Find the percentage of lower quality oranges in the mixture.

ক
18%
খ
20%
গ
22%
ঘ
None of the above

ব্যাখ্যা

Question: A fruit vendor claims to sell oranges at cost price, but he secretly adds lower quality oranges to his premium ones and gains 25%. Find the percentage of lower quality oranges in the mixture.

Solution:
let,
cost price = 100
sell price = 125
∴ the amount of premium oranges = 100/125
= 4/5
∴ the amount of lower quality oranges = {1 - (4/5)} × 100%
= 20%

৩,৯৩৪.

If for non zero x, x² - 4x - 1 = 0, then the value of x² + (1/x²) is?

ক
ক) 14
খ
খ) 16
গ
গ) 18
ঘ
ঘ) 20

ব্যাখ্যা

Question: If for non zero x, x² - 4x - 1 = 0, then the value of x² + (1/x²) is?

Solution:
x² − 4x − 1 = 0
⇒ x² − 1 = 4x
⇒ x − (1/x) = 4 (divide x both sides)
⇒ {x − (1/x)}² = 4²
⇒ x² + (1/x²) − 2.x.(1/x) = 16
∴ x² + (1/x²) = 18

৩,৯৩৫.

In the figure at the right end, PS is perpendicular to QR. If PQ = PR = 26 and PS= 24, then QR =?

ক
20
খ
18
গ
16
ঘ
14

ব্যাখ্যা

Question: In the figure at the right end, PS is perpendicular to QR. If PQ = PR = 26 and PS= 24, then QR =?

Solution:
In the figure,
PS is perpendicular to QR. PQ = PR = 26 and PS= 24,

In ΔPQS,
PQ² = PS² + QS²
⇒ QS² = PQ² - PS²
∴ QS = √(PQ² - PS²)
= √(26² - 24²)
= √(676 - 576)
= √100
= 10

Similarly,
In ΔPRS,
PR² = PS² + RS²
⇒ RS² = PR² - PS²
∴ RS = √(PQ² - PS²)
= √(26² - 24²)
= √(676 - 576)
= √100
= 10

∴ QR = QS + RS = 10 + 10 = 20

৩,৯৩৬.

Everybody in a room shakes hands with everybody else. The total number of handshakes is 66. Then the total number of persons in the room is-

ক
ক) 11
খ
খ) 12
গ
গ) 16
ঘ
ঘ) 15

ব্যাখ্যা

Question: Everybody in a room shakes hands with everybody else. The total number of handshakes is 66. Then the total number of persons in the room is-

Solution:
⇒n(n - 1)/2 = 66
⇒n(n - 1) = 66 × 2
n² - n = 132
n² - n - 132 = 0
n² - 12n + 11n - 132 = 0
n(n - 12) + 11(n - 12) = 0
(n - 12)(n + 11) = 0
n = 12, - 11

৩,৯৩৭.

If 1 + sinθ = x cosθ , then tanθ is -

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

ব্যাখ্যা

দেয়া আছে,
1 + sinθ = x cosθ
⇒ (1 + sinθ)/cosθ = x cosθ/cosθ [উভয়পক্ষে cosθ দ্বারা ভাগ করে পাই]
⇒ (1/cosθ) + sinθ/cosθ = x
⇒ secθ + tanθ = x ..........(i)
আমরা জানি,
sec²θ - tan²θ = 1
⇒ (secθ + tanθ)(secθ - tanθ) = 1
⇒ x(secθ - tanθ) = 1
⇒ secθ - tanθ = 1/x .........(ii)
সমীকরণ (i) থেকে (ii) বিয়োগ করে পাই,
(secθ + tanθ) - (secθ - tanθ) = x - (1/x)
2 tanθ = (x² - 1)/x
tanθ = (x² - 1)/2x.

৩,৯৩৮.

Five machines working together can process 1200 items in 4 days. If one machine breaks down, how many items can the remaining machines process in 6 days?

ক
1080 items
খ
1200 items
গ
1440 items
ঘ
1620 items

ব্যাখ্যা

Question: Five machines working together can process 1200 items in 4 days. If one machine breaks down, how many items can the remaining machines process in 6 days?

Solution:
Total items processed by 5 machines in 4 days = 1200
Items processed by 1 machine in 4 days = (1200/5) items
Items processed by 1 machine in 1 day = {1200/(5 × 4)} items
= 60 items

Therefore,
Items processed by 4 machines in 6 days = 60 × 6 × 4
= 1440 items

৩,৯৩৯.

A sum of money amounts to Tk. 21000 in 2 years at 20% simple interest per annum. Find the sum.

ক
Tk. 15000
খ
Tk. 18000
গ
Tk. 14000
ঘ
Tk. 16000

ব্যাখ্যা

Question: A sum of money amounts to Tk. 21000 in 2 years at 20% simple interest per annum. Find the sum.

Solution:
Here,
A = 21000, T = 2, R = 20%
⇒ A = P + SI
⇒ A = P + (P × R × T/100)
⇒ A = P [1 + (R × T/100)]
⇒ 21000 = P [1 + 0.4]
⇒ P = 21000/1.4
⇒ P = 15000
Thus, the required sum is Tk. 15000

৩,৯৪০.

How many different words can we make using the letters A, B, E and L without repeating the letters?

ক
24
খ
16
গ
256
ঘ
none

ব্যাখ্যা

Question: How many different words can we make using the letters A, B, E and L without repeating the letters?

Solution:
We have 4 choices for the first letter, 3 choices for the second letter, 2 choices for the third letter and 1 choice for the fourth letter. Hence the number of words is given by 4 × 3 × 2 × 1 = 4! = 24

৩,৯৪১.

There are 9 non-collinear points. How many triangles can be drawn by joining these points?

ক
66
খ
72
গ
84
ঘ
108

ব্যাখ্যা

Question: There are 9 non-collinear points. How many triangles can be drawn by joining these points?

Solution:
We know,
A triangle is formed by joining any three non-collinear points in pairs.

Given.
There are 9 non-collinear points. অর্থাৎ নয়টি বিন্দু সমরেখ নয়।

∴ The number of triangles formed = ⁹C₃
= (9 × 8 × 7 × 6!)/(9 - 3)! × 3!
= (9 × 8 × 7 × 6!)/(6! × 3 × 2)
= 84

৩,৯৪২.

The average weight of 20 students is 50 kg, and the average weight of another 30 students is 60 kg. What is the average weight of all 50 students combined?

ক
54.5 kg
খ
55 kg
গ
56 kg
ঘ
57.5 kg

ব্যাখ্যা

Question: The average weight of 20 students is 50 kg, and the average weight of another 30 students is 60 kg. What is the average weight of all 50 students combined?

Solution:
20 জন ছাত্রের গড় ওজন = 50 কেজি
∴ 20 জন ছাত্রের ওজনের সমষ্টি = 20 × 50 = 1000 কেজি

30 জন ছাত্রের গড় ওজন = 60 কেজি
∴ 30 জন ছাত্রের ওজনের সমষ্টি = 30 × 60 = 1800 কেজি

মোট ওজনের সমষ্টি = 1000 + 1800 = 2800 কেজি
মোট ছাত্র সংখ্যা = 20 + 30 = 50 জন

সুতরাং, সম্মিলিত গড় ওজন = 2800/50 = 56 কেজি

৩,৯৪৩.

There are 8 multiple-choice questions in an examination, and each question has 4 options. In how many ways can these questions be answered?

ক
32 ways
খ
4⁸ ways
গ
8⁴ ways
ঘ
12 ways

ব্যাখ্যা

Question: There are 8 multiple-choice questions in an examination, and each question has 4 options. In how many ways can these questions be answered?

Solution:
একটি বহু নির্বাচনী প্রশ্নে উত্তর দেওয়ার সম্ভাব্য উপায় সংখ্যা হলো 4টি (কারণ প্রতিটি প্রশ্নে 4টি করে বিকল্প আছে)।

যেহেতু প্রশ্ন সংখ্যা 8টি এবং প্রতিটি প্রশ্নের উত্তর একে অপরের উপর নির্ভরশীল নয়, তাই মোট সম্ভাব্য উপায় সংখ্যা হবে প্রতিটি প্রশ্নের উত্তরের উপায়গুলোর গুণফল।

প্রথম প্রশ্নের সম্ভাব্য উপায় = 4
দ্বিতীয় প্রশ্নের সম্ভাব্য উপায় = 4
............
অষ্টম প্রশ্নের সম্ভাব্য উপায় = 4

∴ নির্ণেয় মোট উপায় = 4 × 4 × 4 × … (8 বার) = 4⁸
অতএব, 8টি বহু নির্বাচনী প্রশ্নের উত্তর 4⁸ উপায়ে দেওয়া যেতে পারে।

৩,৯৪৪.

If a and b are positive real numbers, then (a⁰ - 3b⁰)⁵ =

ক
ক) 0
খ
খ) 1
গ
গ) -1
ঘ
ঘ) -32

ব্যাখ্যা

(a⁰ - 3b⁰)⁵
= (1 - 3)5
= -2⁵
= -32

৩,৯৪৫.

The average expenditure of a man for the first five months is TK.1200 and for the next seven months is TK.1300. If he saves TK.2900 in that year, his monthly average income is

ক
TK. 1400
খ
TK. 1500
গ
TK. 1600
ঘ
TK. 1800

ব্যাখ্যা

প্রশ্ন: The average expenditure of a man for the first five months is TK.1200 and for the next seven months is TK.1300. If he saves TK.2900 in that year, his monthly average income is

Solution:
Total annual expenditure = TK.(5 × 1200) + (7 × 1300)
= TK.(6000 + 9100)
= TK. 15100
His total annual income = Total expenditure + Total savings
= (15100 + 2900)
= TK. 18000

∴ Average monthly income = 18000/12 = TK. 1500

৩,৯৪৬.

An employee's annual salary was increased to Tk. 20,000. If her new annual salary now equals Tk. 86,000 what was the percent increase?

ক
ক) 20%
খ
খ) 22%
গ
গ) 25%
ঘ
ঘ) 30%

ব্যাখ্যা

New annual salary = 86,000
Salary increase = 20,000.
Original salary = 86,000 - 20,000.
= 66,000

% Increase = ($20,000/ $66,000 )×100
= 30%

৩,৯৪৭.

A group of 20 friends formed an investment club, with each member contributing an equal amount to the general fund. The club then invested the entire fund, which amounted to 1 taka, in stock X. The value of the stock subsequently increased 40 percent, at that point the stock was sold and the proceeds divided evenly among the members. In terms of t, how much money did each member of the club receive from the sale?

ক
800t
খ
(t/2)
গ
(7t/5)
ঘ
(7t/100)

ব্যাখ্যা

Question: A group of 20 friends formed an investment club, with each member contributing an equal amount to the general fund. The club then invested the entire fund, which amounted to t taka, in stock X. The value of the stock subsequently increased 40 percent, at that point the stock was sold and the proceeds divided evenly among the members. In terms of t, how much money did each member of the club receive from the sale?

Solution:
After 40% increase, amount = t + 40% of t
= t + 0.4t
= 1.4t

each member of the club received = 1.4t/20
= 14t/(20 × 10)
= 7t/100

৩,৯৪৮.

You have saved 104 Taka by purchasing a blanket with 13% discount on it. what is the quoted price of the blanket in Taka?

ক
ক) 820
খ
খ) 800
গ
গ) 840
ঘ
ঘ) 900

ব্যাখ্যা

13% ছাড়ে
13 টাকা ছাড় পাওয়া যায় যখন তালিকামূল্য 100 টাকা
1 টাকা ছাড় পাওয়া যায় যখন তালিকামূল্য 100/13 টাকা
104 টাকা ছাড় পাওয়া যায় যখন তালিকামূল্য (100 × 104)/13 টাকা
= 800 টাকা

৩,৯৪৯.

In a simultaneous throw of two dice what is the probability of getting a doublet?

ক
ক) 1/6
খ
খ) 1/4
গ
গ) 2/3
ঘ
ঘ) 1/3

ব্যাখ্যা

Total number of possibilities = 36
Event of getting a doublet = [(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)] = 6
Thus, probability of getting a doublet = 6/36 = 1/6

৩,৯৫০.

The marked price of a shirt and trousers are in the ratio 1 : 2. The shopkeeper gives 40% discount on the shirt. If the total discount in the set of the shirt and trousers is 30%, the discount offered on the trousers is:

ক
15%
খ
20%
গ
25%
ঘ
30%

ব্যাখ্যা

Question: The marked price of a shirt and trousers are in the ratio 1 : 2. The shopkeeper gives 40% discount on the shirt. If the total discount in the set of the shirt and trousers is 30%, the discount offered on the trousers is:

Solution:
Let,
the price of shirt and trouser be Tk. 100 and Tk. 200 respectively.
Then, price of set of shirt and trouser = Tk. 300.

After giving 30% discount on the set,
Selling Price = 300 - 30% of 300
= 210.

∴ Total Discount on Set = Original price × Discount percentage
= 300 × (30/100)
= 300 × 0.30
= Tk. 90

And Discount on shirt is 20% alone,
SP of shirt alone = 100 - 40% of 100 = 60.
Tk. 40 is the discount on shirt then Tk. 50 must be the discount on the trouser.
So, discount on trouser = (50 × 100)/200 = 25%

৩,৯৫১.

If a = b = 2c and abc = 256, then a = ?

ক
2
খ
4
গ
6
ঘ
8

ব্যাখ্যা

Question: If a = b = 2c and abc = 256, then a = ?

Solution:
abc = 256
⇒ (2c) (2c) c = 256
⇒ 4c³ = 256
⇒ c³ = 64
⇒ c = 4

∴ a = 2c = (2 × 4) = 8

৩,৯৫২.

ক
ক) - 13
খ
খ) - 7
গ
গ) 5
ঘ
ঘ) 4

ব্যাখ্যা

Question:

Solution:

৩,৯৫৩.

If a sum on compound interest becomes three times in 4 years, then with the same interest rate, the sum will become 81 times in -

ক
ক) 18 years
খ
খ) 12 years
গ
গ) 16 years
ঘ
ঘ) 14 years

ব্যাখ্যা

Let the sum be P
The sum P becomes 3P in 4 years on compound interest
3P = P{(1 + R/100)}⁴
3 = {1 + R/100)}⁴
Let the sum P becomes 81P in n years
81P = P{1 + (R/100)}ⁿ
81 = {1 + (R/100)}ⁿ
3⁴ = {1 + (R/100)}ⁿ
[{1 + (R/100)}⁴]⁴ = {1 + (R/100)}ⁿ
{1 + (R/100)}¹⁶ = {1 + (R/100)}ⁿ
n = 16.

৩,৯৫৪.

Square ABCD is inscribed in a circle whose radius is 4cm. Calculate the area of the square.

ক
36 sq. cm
খ
32 sq. cm
গ
16 sq. cm
ঘ
24 sq. cm

ব্যাখ্যা

Question: Square ABCD is inscribed in a circle whose radius is 4cm. Calculate the area of the square.

Solution:
Diameter = 2 × 4 = 8 cm = diagonal of the square

let, the side of square is x

√2x = 8
⇒ x = 8/√2

∴ Area of the square = x²
= (8/√2)²
= 64/2
= 32 sq. cm

৩,৯৫৫.

A cricket team has won 40 games out of 60 played. It has 30 more games to play. How many of these must the team win to make a record 70% win for them?

ক
27
খ
23
গ
21
ঘ
29

ব্যাখ্যা

Team won 40 games out of 60 and the remaining games were 30.
Total games = 60 + 30
= 90
70% of 90 = 63
Team has to win 63 games in total.
Team has already won 40.
∴ Games to win = 63 - 40
= 23

৩,৯৫৬.

A fort had provision of food for 150 men for 45 days. After 10 days, 25 men left the fort. The number of days for which the remaining food will last, is:

ক
ক) 29
খ
খ) 37
গ
গ) 42
ঘ
ঘ) 54
ঙ
ঙ) 60

ব্যাখ্যা

After 10 days : 150 men had food for 35 days.
Suppose 125 men had food for x days.
Now, Less men, More days (Indirect Proportion)
∴ 125 : 150 :: 35 : x
⇒ 125 × x = 150 × 35
⇒ x = (150 × 35)/125
= 42

৩,৯৫৭.

After replacing an old member with a new member it was found that the average age of five members of a club is the same as it was 3 years ago. What is the difference between the ages of the replaced and the new member?

ক
ক) 15
খ
খ) 20
গ
গ) 25
ঘ
ঘ) None of these

ব্যাখ্যা

Question: After replacing an old member with a new member it was found that the average age of five members of a club is the same as it was 3 years ago. What is the difference between the ages of the replaced and the new member?

Solution:
(পাঁচ সদস্যের একটি গ্রুপে পুরাতন একজন সদস্যের পরিবর্তে নতুন একজন সদস্য নেওয়া হলে তাদের গড় বয়স ৩ বছর আগে যত ছিল তত হয়। পরিবর্তিত সদস্য এবং নতুন সদস্যের বয়সের পার্থক্য কত?)

এখানে
⇒ ৫ জন সদস্যের মোট বয়স ৩ বছর আগে যতই থাকুক ৩ বছর পর (৫ × ৩ ) = ১৫ বেড়ে যাওয়ার কথা।
⇒ কিন্তু নতুন সদস্য আসার কারণে ১৫ বছর কমে গিয়েছে, অর্থাৎ ৩ বছর আগের গড় বয়সের সমান হয়ে যায়।
⇒ অর্থাৎ নতুন সদস্যের বয়স পরিবর্তিত সদস্যের বয়স থেকে ১৫ বছর বেশি।

৩,৯৫৮.

If the price of sugar decreases by 20%, by what percentage must sugar consumption increase to keep the total expenditure the same?

ক
20%
খ
25%
গ
30%
ঘ
10%

ব্যাখ্যা

Question: If the price of sugar decreases by 20%, by what percentage must sugar consumption increase to keep the total expenditure the same?

Solution:
মনে করি,
চিনির পূর্বমূল্য = 100 টাকা

চিনির মূল্য 20% হ্রাস পেলে,
নতুন মূল্য হয়= 100 - 100 এর 20% = 100 - 20 = 80 টাকা

চিনির মূল্য হ্রাস পায় = 100 - 80 = 20 টাকা

তাহলে, খরচ একই রাখতে হলে 20 টাকার সমপরিমান চিনি বেশি ব্যাবহার করতে হবে।

এখন,
চিনির ব্যাবহার 80 টাকায় বৃদ্ধি করতে হবে = 20 টাকা
∴ 1 টাকায় বৃদ্ধি করতে হবে = 20/80 টাকা
∴ 100 টাকায় বৃদ্ধি করতে হবে = (20× 100) / 80 = 25 টাকা

সুতরাং চিনির ব্যাবহার শতকরা 25% বৃদ্ধি করতে হবে।

৩,৯৫৯.

The ratio of third proportional to 12 and 30 and the mean proportional between 9 and 25 is-

ক
5 : 2
খ
3 : 2
গ
5 : 3
ঘ
5 : 1

ব্যাখ্যা

Question: The ratio of third proportional to 12 and 30 and the mean proportional between 9 and 25 is-

Solution:
Let, the third proportional to 12 and 30 be x.

ATQ,
12 : 30 : : 30 : x
⇒ 12/30 = 30/x
⇒ 12x = 30 × 30
⇒ x = (30 × 30)/12
∴ x = 75

∴ Third proportional to 12 and 30 = 75.
Mean proportional between 9 and 25 = √(9 × 25) = 15
∴ Required ratio = 75/15 = 5 : 1 .

৩,৯৬০.

If x and y are prime integers and x < y, which of the following cannot be true?

ক
xy is even
খ
y + xy is even
গ
2x + y is even
ঘ
x + y is odd

ব্যাখ্যা

Question: If x and y are prime integers and x < y, which of the following cannot be true? x is even y + xy is even x + y is odd 2x + y is even.

Solution:
2x is always even.
y is always odd

We know, odd + even = odd
So 2x + y must be odd

৩,৯৬১.

If log_x625 = 4 then x = ?

ক
ক) 15
খ
খ) 5
গ
গ) 4
ঘ
ঘ) 25

ব্যাখ্যা

log_x625 = 4
Or, x⁴ = 625 = 5⁴
Or, x = 5

৩,৯৬২.

Two trains start from station A and B and travels towards each other at speed of 48km/hr and 72km/hr respectively. At the time of their meeting, the second train has traveled 144 km more than the first. The distance between A and B is-

ক
680 km
খ
720 km
গ
760 km
ঘ
800 km

ব্যাখ্যা

Question: Two trains start from station A and B and travels towards each other at speed of 48km/hr and 72km/hr respectively. At the time of their meeting, the second train has traveled 144 km more than the first. The distance between A and B is-

Solution:
The second train has traveled 144 km more than the first train because the speed of second train is 24 km/hr more than first.
Time taken by second train to cover 144 km with surplus 24km/hr = 144/24 = 6 hours.
then, time taken by both train before meeting is 6 hours.
So, their relative speed = 48 + 72 = 120
Total distance travel by both = 120 × 6 = 720 km
Distance between A and B = 720 km

৩,৯৬৩.

Of the three consecutive even numbers, the sum of 1st and 2nd is 166 and the sum of the 2nd and 3rd is 170 and the sum of 3rd and twice of 1st is 250. The second number is-

ক
ক) 78
খ
খ) 82
গ
গ) 86
ঘ
ঘ) 80
ঙ
ঙ) 84

ব্যাখ্যা

x + y = 166,
y + z = 170,
z + 2x = 250.

Solving the equation, y = 84.

৩,৯৬৪.

If the sum of 12 numbers is 756, the average of the first 6 is 54 and the average of the last 5 is 72, what is the 7^th number?

ক
ক) 72
খ
খ) 70
গ
গ) 62
ঘ
ঘ) 74

ব্যাখ্যা

Question: If the sum of 12 numbers is 756, the average of the first 6 is 54 and the average of the last 5 is 72, what is the 7^th number?

Solution:
প্রথম ১২টি সংখ্যার সমষ্টি = ৭৫৬
প্রথম ৬টি সংখ্যার সমষ্টি = (৫৪ × ৬) = ৩২৪

আবার,
শেষ ৫টি সংখ্যার গড় = ৭২
শেষ ৫টি সংখ্যার সমষ্টি = (৭২ × ৫) = ৩৬০

∴ ৭ম সংখ্যাটি = {৭৫৬ - (৩২৪ + ৩৬০)}
= ৭২

৩,৯৬৫.

A jar contains a mixture of oil and water in the ratio 7 : 5. If 9 liters of the mixture is removed and replaced with the same amount of water, the new ratio becomes 7 : 9 . What was the initial quantity of oil in the jar?

ক
10 liters
খ
20 liters
গ
21 liters
ঘ
25 liters

ব্যাখ্যা

Question: A jar contains a mixture of oil and water in the ratio 7 : 5. If 9 liters of the mixture is removed and replaced with the same amount of water, the new ratio becomes 7 : 9 . What was the initial quantity of oil in the jar?

Solution:
ধরি,
শুরুতে জারের মধ্যে তেলের পরিমাণ = 7x লিটার
পানির পরিমাণ = 5x লিটার
∴ মোট অংশ = 12x

9 লিটার মিশ্রণ ফেলে দিলে,
ফেলে দেওয়া মিশ্রণে তেলের পরিমাণ = 9 এর (7x/12x) = 21/4 লিটার
এবং পানির পরিমাণ = 9 এর (5x/12x) = 15/4 লিটার

বাকি মিশ্রণে,
তেলের পরিমাণ = 7x - (21/4) = (28x - 21)/4
পানির পরিমাণ = 5x - (15/4) = (20x - 15)/4

মিশ্রণে 9 লিটার পানি যোগ করা হলে পানির নতুন পরিমাণ = {(20x - 15)/4} + 9 = (20x - 15 + 36)/4 = (20x + 21)/4

প্রশ্নমতে,
{(28x - 21)/4}/{(20x + 21)/4} = 7/9
⇒ (28x - 21)/(20x + 21) = 7/9
⇒ 9(28x - 21) = 7(20x + 21)
⇒ 252x - 189 = 140x + 147
⇒ 252x - 140x = 189 + 147
⇒ 112x = 336
⇒ x = 336/112
⇒ x = 3

∴ শুরুতে মিশ্রণে তেলের পরিমাণ ছিলো = (7 × 3) লিটার = 21 লিটার

৩,৯৬৬.

Out of three positive numbers, the ratio of the first and the second numbers is 3 : 4 that of the second and the third numbers is 5 : 6 if the product of the second and the third numbers is 4320. What is the sum of three numbers?

ক
ক) 177
খ
খ) 165
গ
গ) 185
ঘ
ঘ) 160

ব্যাখ্যা

The ratio of 1st and 2nd numbers = 3 : 4
The ratio of 2nd and 3rd numbers = 5 : 6
Let,
the 2nd number = 5x, third number = 6x
Product of 2nd and 3rd numbers = 4320
5x × 6x = 4320
x²= 144
x = 12
2nd number = 60, 3rd number = 72,
1st number = (60/4) × 3 = 45
Sum of three numbers = 60 + 72 + 45 = 177

৩,৯৬৭.

Six bells start ringing together and ring at intervals of 4, 8, 10, 12, 15, and 20 seconds respectively. How many times will they ring together in 60 minutes?

ক
ক) 31
খ
খ) 15
গ
গ) 30
ঘ
ঘ) 16

ব্যাখ্যা

LCM of 4, 8, 10, 12, 15 and 20 = 120
120 seconds = 2 minutes
Hence all the six bells will ring together in every 2 minutes
Hence, number of times they will ring together in 60 minutes = 1 + (60/2)
= (2+ 60)/2
= 31.

The HCF of a group of numbers will always be a factor of their LCM.
HCF is the product of all common prime factors using the least power of each common prime factor.
LCM is the product of the highest powers of all prime factors.

৩,৯৬৮.

A man covers half of his journey by train at 60 km/hr, half of the remaining by bus at 30 km/hr and the rest by cycle at 10 km/hr. Find his average speed during the entire journey?

ক
32 kmph
খ
20 kmph
গ
18 kmph
ঘ
24 kmph

ব্যাখ্যা

Question: A man covers half of his journey by train at 60 km/hr, half of the remaining by bus at 30 km/hr and the rest by cycle at 10 km/hr. Find his average speed during the entire journey?

Solution:
Let the total distance be 120 km
⇒Time taken to cover distance at 60 km/h = 60/60 = 1 hr
⇒ Time taken to cover distance at 30 km/h = 30/30 = 1 hr
⇒ Time taken to cover distance at 10 km/h = 30/10 = 3 hr

∴ Average speed = 120/(1 + 1 + 3) = 24 km/h

৩,৯৬৯.

A rectangular water tank is 6 m high, 4m long and 2.5 m high wide. How many liters of water can it hold?

ক
ক) 40000 litre
খ
খ) 50000 litre
গ
গ) 60000 litre
ঘ
ঘ) 70000 litre

ব্যাখ্যা

Question: A rectangular water tank is 6 m high, 4m long and 2.5 m high wide. How many liters of water can it hold?

Solution:
Volume = length × width × height
= 6 × 2.5 × 4 m³
= 60 m³

1 m³ = 1000 litre
60 m³ = 60 × 1000 litre
= 60000 litre

৩,৯৭০.

Parimal got two and a half time as many marks in English as in History. If his total marks in the two subjects are 140, the marks obtained by him in English are:

ক
40
খ
75
গ
90
ঘ
100

ব্যাখ্যা

Question: Parimal got two and a half time as many marks in English as in History. If his total marks in the two subjects are 140, the marks obtained by him in English are:

Solution:
ধরি,
পরিমল ইতিহাসে পায় x নম্বর
∴ পরিমল ইংরেজিতে পায় 2.5x নম্বর

প্রশ্নমতে,
x + 2.5x = 140
⇒ 3.5x = 140
⇒ x = 140/3.5
∴ x = 40

∴ পরিমল ইংরেজিতে পায় 2.5x = 2.5 × 40 = 100 নম্বর

৩,৯৭১.

A number when divided by 627 leaves a remainder 43. By dividing the same number by 19 , the remainder will be__

ক
ক) 32
খ
খ) 43
গ
গ) 13
ঘ
ঘ) 5

ব্যাখ্যা

Let the number be = 627 + 43
= 670
So, the remainder is,

19) 670 (35
57
-------
100
95
--------
5

৩,৯৭২.

Find the value of x if log_x324= 4

ক
ক) 2√3
খ
খ) 3√2
গ
গ) √6
ঘ
ঘ) 6

ব্যাখ্যা

log_x324= 4
⇒ x⁴ = 324
⇒ x⁴ = (3√2)⁴
∴ x = 3√2

৩,৯৭৩.

January 1, 2008, is Monday. What day of the week lies on Jan 1, 2009?

ক
Monday
খ
Tuesday
গ
Wednesday
ঘ
Friday

ব্যাখ্যা

Question: January 1, 2008, is Monday. What day of the week lies on Jan 1, 2009?

Solution:
কোন বছরের প্রথম দিন যদি শনিবার হয় তাহলে পরবর্তী বছরে প্রথম দিন একদিন বেশি বা বরিবার হবে।

কিন্তু ২০০৮ সাল লিপ ইয়ার হওয়ার কারনে এখানে অতিরিক্ত ১ দিন যোগ হবে। অর্থাৎ ২ দিন বেশি হবে।
তাই, ২০০৮ সালের প্রথম দিন সোমবার হওয়ায় ২০০৯ সালের প্রথম দিন বুধবার হবে।

৩,৯৭৪.

How many line segments can be drawn inside a pentagon that connects non-adjacent vertices?

ক
9
খ
2
গ
3
ঘ
5

ব্যাখ্যা

Question: How many line segments can be drawn inside a pentagon that connects non-adjacent vertices?

Solution:
A pentagon has 5 sides. We obtain the diagonals by joining the vertices in pairs.
Total number of sides and diagonals,
= ⁵C₂
= 10
This includes its 5 sides also.

∴ Diagonals = 10 – 5 = 5

৩,৯৭৫.

What is the square root of (8 + 2√15)?

ক
√5 + √3
খ
2√2 + 2√6
গ
2√5 + 2√3
ঘ
√2 + √6

ব্যাখ্যা

Question: What is the square root of (8 + 2√15)?

Solution:

৩,৯৭৬.

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

ব্যাখ্যা

প্রশ্ন: 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 -

সমাধান:
১, ২, ৩ দ্বারা গঠিত তিন অংকের সংখ্যা হবে ৬টি
সংখ্যাগুলো হবে,
১২৩
১৩২
২১৩
২৩১
৩১২
৩২১

এই ৬টি সংখ্যার একটিও ১১১ দ্বারা বিভাজ্য নয়।

সম্ভাবনা = ০/৬ = ০

৩,৯৭৭.

The present age of three persons are in the ratio of 4 : 7 : 9. Eight years ago, their total age was 56 years. In five years what will be the age of the oldest person?

ক
31 years
খ
21 years
গ
39 years
ঘ
41 years

ব্যাখ্যা

Question: The present age of three persons are in the ratio of 4 : 7 : 9. Eight years ago, their total age was 56 years. In five years what will be the age of the oldest person?

Solution:
Present age ratio of three persons is 4 : 7 : 9

Let,
Their age is 4X, 7X and 9X respectively.

ATQ,
4X - 8 + 7X - 8 + 9X - 8 = 56
Or, 20X - 24 = 56
Or, 20X = 80
Or, X = 4

∴ The present age of the oldest person is = 9X = 9 x 4 = 36 years.
In five years, his age will be = (36 + 5 ) = 41 years.

৩,৯৭৮.

A storm breaks a tree. The broken part of tree bends so that the top of the tree touches the ground and makes an angle of 60° with the horizontal plane. If the distance between the base of the tree and the point where top of tree touches the ground is 10 m, find the height of the tree?

ক
37.3 m
খ
17.3 m
গ
27.3 m
ঘ
20.3 m

ব্যাখ্যা

Question: A storm breaks a tree. The broken part of tree bends so that the top of the tree touches the ground and makes an angle of 60° with the horizontal plane. If the distance between the base of the tree and the point where top of tree touches the ground is 10 m, find the height of the tree?

Solution:

PQ = 10 and let RQ be X.

RQ/PQ = tan60°
⇒ X/10 = √3
∴ X = 10√3

Now,
PR² = X² + (10)²
PR² = (10√3)² + (10)² = 300 + 100
PR² = 400
∴ PR = 20

Height of tree = RQ + PR
= X + 20
= 10√3 + 20
= 10 × 1.73 + 20
= 17.3 + 20
= 37.3 meter

৩,৯৭৯.

Find the number of triangles in the given figure.

ক
14
খ
16
গ
13
ঘ
15

ব্যাখ্যা

Question: Find the number of triangles in the given figure.

Solution:

Triangles with one empty cell each: 1, 2, 3, 4, 5, 6 = 6 triangles
Triangles with two empty cells each: 23, 45, 16 = 3 triangles
Triangles with three empty cells each: 123, 234, 345, 456, 126, 156 = 6 triangles
Triangle with all cells filled: 123456 = 1 triangle

Therefore, total number of triangles = 6 + 6 + 3 + 1 = 16

৩,৯৮০.

Pipes A and B can fill a tank in 15 hours and 20 hours respectively and pipe C can empty the full tank in 30 hours. If all the pipes are opened together, how much time will be needed to make the tank full?

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

ব্যাখ্যা

Time taken by pipe A to fill the tank =15 hours
Portion of tank filled by pipe A in 1 hour =1/15

Time taken by pipe B to fill the tank =20 hours
Portion of Tank filled by pipe B in 1 hour 1/20

Time taken by pipe C to empty the tank =30 hours
Portion of tank emptied by pipe C in 1 hour 1/30

Now, Portion of tank filled by all three pipes together in 1 hour
=1/15 + 1/20 - 1/30
= (4 + 3 - 2)/60
= 5/60
= 1/12

Time taken to filled the tank when all three pipes are opened together = 12 hours

৩,৯৮১.

The H.C.F. of two numbers is 11 and their L.C.M. is 7700. If one of the numbers is 308, then the other is:

ক
242
খ
275
গ
228
ঘ
280
ঙ
295

ব্যাখ্যা

Question: The H.C.F. of two numbers is 11 and their L.C.M. is 7700. If one of the numbers is 308, then the other is:

Solution:
We know that,
L.C.M × H.C.F. = Product of two numbers
⇒ 7700 × 11 = 308 × other number
⇒ Other number = (7700 × 11)/308
∴ Other number = 275

৩,৯৮২.

A 5-digit identification code is to be created as a sequence that contains each integer in the set {1, 2, 3, 4, 5} exactly once. If all identification codes are possible except those containing even integers next to each other, how many different identification codes are possible?

ক
60
খ
68
গ
72
ঘ
102

ব্যাখ্যা

Question: A 5-digit identification code is to be created as a sequence that contains each integer in the set {1, 2, 3, 4, 5} exactly once. If all identification codes are possible except those containing even integers next to each other, how many different identification codes are possible?

Solution:
Total ways to create 5-digit identification code = 5!

Both even number cannot be together.
When both the even digits are taken together as one digit, we have 4 digits, that is 1, 3, 5 and (24) or, 4! to arrange 4 digits
and 2 & 4 can be arranged in 2! ways within themselves. Hence, 4! × 2

Ways we are looking at = 5! - 4 × 2! = 120 - 48 = 72

৩,৯৮৩.

If log₇14 + log₇(5x + 1) - 1 = log₇(x + 5), then what is the value of x?

ক
2/3
খ
1/3
গ
3/5
ঘ
3/7

ব্যাখ্যা

Question: If log₇14 + log₇(5x + 1) - 1 = log₇(x + 5), then what is the value of x?

Solution:
log₇14 + log₇(5x + 1) - 1 = log₇(x + 5)
⇒ log₇14 + log₇(5x + 1) - log₇7 = log₇(x + 5)
⇒ log₇[{14 × (5x + 1)}/7] = log₇(x + 5)
⇒ 2(5x + 1) = x + 5
⇒ 10x + 2 = x + 5
⇒ 10x - x = 5 - 2
⇒ 9x = 3
⇒ x = 3/9
∴ x = 1/3

৩,৯৮৪.

A and B together can paint a wall in 8 days. A can do it alone in 12 days. How many days would it take B to do this job alone?

ক
24 days
খ
20 days
গ
15 days
ঘ
14 days
ঙ
None

ব্যাখ্যা

Question: A and B together can paint a wall in 8 days. A can do it alone in 12 days. How many days would it take B to do this job alone?

Solution:
A's 1 day's work 1/12
A and B's 1 day's work 1/8
∴ B's 1 day's work = 1/8 - 1/12
= (6 - 4)/48
= 2/48
= 1/24

1/24 part of job done by B in 1 day
∴ Full job done by B in (24/1) days
= 24 days

৩,৯৮৫.

A train is moving at the rate 8 mi/h along a piece of circular track of radius 2,500 ft. Through what angle does it turn in 1 min?

ক
ক) 0.3816 rad or 18.13°
খ
খ) 0.2816 rad or 16.13°
গ
গ) 0.2844 rad or 16.30°
ঘ
ঘ) 0.2916 rad or 17.13°

ব্যাখ্যা

A train is moving at the rate 8 mi/h along a piece of circular track of radius 2,500 ft. Through what angle does it turn in 1 min?

সমাধান:
৮ মাইল/ঘণ্টা = (৮ × ৫২৮০)/৬০ ফুট/মিনিট [১ মাইল = ৫২৮০ ফুট]
= ৭০৪ ফুট/মিনিট

২৫০০ ফুট ব্যাসার্ধ বিশিষ্ট পথের পরিধি = ২ × π × ২৫০০ ফুট
= ৫০০০π ফুট

মোট সময় = ৫০০০π ÷ ৭০৪ মিনিট
= ৬২৫π/৮৮ মিনিট

একবার ঘুরলে ট্রেনটি অতিক্রম করে ৩৬০° = ২π রেডিয়ান

৬২৫π/৮৮ মিনিটে উৎপন্ন করে ২π রেডিয়ান
∴ ১ মিনিটে উৎপন্ন করে (২π × ৮৮)/৬২৫π রেডিয়ান
= ০.২৮১৬ রেডিয়ান

এখন,
০.২৮১৬ রেডিয়ান
= (০.২৮১৬ × ১৮০)/π
= ১৬.১৩°

৩,৯৮৬.

Which amount to be received after 4 years at the rate of 7% p.a. of simple interest on a sum of Tk. 2600?

ক
Tk. 3328
খ
Tk. 2238
গ
Tk. 3388
ঘ
Tk. 4428

ব্যাখ্যা

Question: Which amount to be received after 4 years at the rate of 7% p.a. of simple interest on a sum of Tk. 2600?

Soution:
Given,
Rate of interest, r = 7%
Pricipal, P = 2600 Tk.
Time, n = 4 years.

∴ Total interest, I = Pnr
= (2600 × 4 × 7)/100
= 728 Tk.

So, amount = Principal + Simple interest
= (2600 + 728) Tk.
= 3328 Tk.

৩,৯৮৭.

7 years ago, the ages of A and B were in the ratio 4 : 5 and 7 years hence, they will be in the ratio 5 : 6. The present age of B is?

ক
73 years
খ
69 years
গ
81 years
ঘ
77 years

ব্যাখ্যা

Question: 7 years ago, the ages of A and B were in the ratio 4 : 5 and 7 years hence, they will be in the ratio 5 : 6. The present age of B is?

Solution:
Let 7 years ago, ages of A and B were 4k years and 5k years, respectively.
Then, the present age of A = 4k + 7
and present age of B = 5k + 7

ATQ,
∴ (4k + 7 + 7)/(5k + 7 + 7) = 5/6
⇒ 24k + 84 = 25k + 70
⇒ k = 14
Hence, B's present age = 5 × 14 + 7 = 77 years

৩,৯৮৮.

In a class average age of 15 boys is 11. If 5 boys each of age 9 years are added, what would be the new average?

ক
20 years
খ
10 years
গ
10.5 years
ঘ
23 years

ব্যাখ্যা

Question: In a class average age of 15 boys is 11. If 5 boys each of age 9 years are added, what would be the new average?

Solution:
Sum of ages of 15 boys = 15 × 11= 165
Sum of ages of 5 boys = 5 × 9 = 45
Total age of 20 boys = 165 + 45 = 210
∴ Average of ages of 20 boys = 210/20 = 10.5 years

৩,৯৮৯.

The maximum number of students among whom 1001 pens and 910 pencils can be distributed in such a way that each student gets the same number of pens and same number of pencils is-

ক
101
খ
91
গ
1001
ঘ
910

ব্যাখ্যা

Question: The maximum number of students among whom 1001 pens and 910 pencils can be distributed in such a way that each student gets the same number of pens and same number of pencils is-

Solution:
We use prime factorization,
1001 = 7 × 11 × 13
910 = 2 × 5 × 7 × 13
∴ Common factors are 7 and 13

∴ HCF = 7 × 13 = 91

∴ Maximum number of students = HCF of 1001 and 910 = 91

৩,৯৯০.

The three sides of a triangle are 2x, 3x + 1, and 4x − 1 respectively, and the perimeter is 36 cm. What is the length of the longest side?

ক
18 cm
খ
21 cm
গ
15 cm
ঘ
20 cm

ব্যাখ্যা

Question: The three sides of a triangle are 2x, 3x + 1, and 4x − 1 respectively, and the perimeter is 36 cm. What is the length of the longest side?

Solution:
প্রশ্নমতে, ত্রিভুজের তিনটি বাহুর দৈর্ঘ্যের যোগফল তার পরিসীমার সমান।
2x + (3x + 1) + (4x − 1) = 36

সমীকরণটি সমাধান করে পাই,
(2x + 3x + 4x) + (1 − 1) = 36
9x = 36
x = 36 / 9
x = 4

এখন, x এর মান বসিয়ে বাহুগুলোর দৈর্ঘ্য নির্ণয় করি:
প্রথম বাহু = 2x = 2 × 4 = 8 সেমি
দ্বিতীয় বাহু = 3x + 1 = 3 × 4 + 1 = 13 সেমি
তৃতীয় বাহু = 4x − 1 = 4 × 4 − 1 = 15 সেমি

সুতরাং, সবচেয়ে বড় বাহুটি হলো 15 সেমি।

৩,৯৯১.

A 280 metre long train crosses a platform thrice its length in 50 seconds. What is the speed of the train in km/hr?

ক
40.64 km/h
খ
80.64 km/h
গ
50.64 km/h
ঘ
70.64 km/h

ব্যাখ্যা

Question: A 280 metre long train crosses a platform thrice its length in 50 seconds. What is the speed of the train in km/hr?

Solution:
Given that,
Length of the train = 280 m
Length of the platform = 3 × 280 = 840 m

∴ Total distance to be covered
= Train length + Platform length
= 280 + 840
= 1120m
And, Time taken = 50 seconds

∴ Speed = Distance/Time
= 1120/50
= 22.4 m/s
= 22.4 × 3.6 ;[1 m/s = 3.6 km/h]
= 80.64 km/h

∴ The speed of the train is 80.64 km/h.

৩,৯৯২.

A cube-shaped box with a side length of 9 meters contains another cube-shaped box inside it. The empty space accommodates 217,000 liters of water. What is the surface area of the smaller box?

ক
364m²
খ
334m²
গ
328m²
ঘ
384m²
ঙ
None of the above

ব্যাখ্যা

Question: A cube-shaped box with a side length of 9 meters contains another cube-shaped box inside it. The empty space accommodates 217,000 liters of water. What is the surface area of the smaller box?

Solution:
আমরা জানি,
১০০০ লিটার = ১ ঘন মি.
∴ ২১৭০০০ লিটার = (২১৭০০০/১০০০) = ২১৭ ঘন মি.

অর্থাৎ, খালি অংশের আয়তন = ২১৭ ঘন মি.

বড় বক্সের আয়তন = (৯)^৩ ঘন মি. = ৭২৯ ঘন মি.

∴ ছোট বক্সের আয়তন = ৭২৯ - ২১৭ = ৫১২ ঘন মি.

ছোট বক্সের এক বাহুর দৈর্ঘ্য = ^৩√৫১২ = ৮ মি.

∴ ছোট বক্সের সমগ্রতলের ক্ষেত্রফল = ৬ × (৮)^২ বর্গ মি.
= ৩৮৪ বর্গ মি.

৩,৯৯৩.

A retail fruit vendor buys pineapples at a score for Tk. 200, and retails them at a dozen for Tk 156. Did he gain or lose in the transaction and what % was his gain or loss?

ক
30% loss
খ
22% loss
গ
22% gain
ঘ
30% gain

ব্যাখ্যা

Question: A retail fruit vendor buys pineapples at a score for Tk. 200, and retails them at a dozen for Tk 156. Did he gain or lose in the transaction and what % was his gain or loss?

Solution:
C.P of 1 Pineapple = Tk. 200/score = 200/20 = 10 (Note: 1 score = 20)

S.P of 1 Pineapple = Tk.156/dozen = 156/12 = 13 (Note: 1 dozen = 12)

∴ Profit of 1 Pineapple = 13 - 10 = 3

∴ % Profit = (3/10) × 100 % = 30%

৩,৯৯৪.

If x is an integer, then which of the following statements about x² - x - 1 is true?

ক
It is always odd
খ
It is always even.
গ
It is always positive.
ঘ
It is even when x is even and odd when x is odd

ব্যাখ্যা

Question: If x is an integer, then which of the following statements about x² - x - 1 is true?

Solution:
If x = 1,
x² - x - 1 = 1² - 1 - 1 = - 1
so, the value can be negative.

x = even,
x² will be even,
x² - x will also be even,
x² - x - 1 will be odd.

x = odd,
x² will be odd,
x² - x will also be even,
x² - x - 1 will be odd.

So, It is always odd.

৩,৯৯৫.

A right triangle with sides 6 cm, 8 cm, and 10 cm is rotated on the side of 6 cm to form a cone. The volume of the cone so formed is -

ক
ক) 48π cm³
খ
খ) 64π cm³
গ
গ) 72π cm³
ঘ
ঘ) 96π cm³

ব্যাখ্যা

Question: A right triangle with sides 6 cm, 8 cm, and 10 cm is rotated on the side of 6 cm to form a cone. The volume of the cone so formed is -

Solution:

We have,
r = 6cm
h = 8cm

∴ Volume = (1/3)πr²h
= (1/3) × π × 6² × 8 cm³
= 96π cm³

৩,৯৯৬.

A moving train, 66 metres long, overtakes another train of 88 metres long, moving in the same direction in 0.168 minutes. If the second train is moving at 40 km/hr, at what speed is the first train moving ?

ক
ক) 95 km/hr
খ
খ) 85 km/hr
গ
গ) 75 km/hr
ঘ
ঘ) 79 km/hr

ব্যাখ্যা

Let the speed of the first train be x km/hr.
Then,
Sum of lengths of trains = (66 + 88)m = 154 m.

Relative speed of two trains = (x - 40) km/hr
= {(x - 40) × (5/18)} m/s

∴ 154/{(x - 40) × (5/18)} = 0.168 × 60
⇒ 5(x - 40) = (154 × 18)/10.08
⇒ 5(x - 40) = 275
⇒ x - 40 = 55
⇒ x = 95 km/hr.

৩,৯৯৭.

What is the compound interest on a sum of tk 40, 000 for 3 years at the rate of 11% per annum?

ক
15704.82
খ
14705.24
গ
12095.24
ঘ
13234.42
ঙ
11987.35

ব্যাখ্যা

Amount after 3 years = 40000 {(1 + 11)/100}³
= 40000 ( 111 /100 )³
= 54705.24
Compound interest = 54705.24 − 40000
= 14705.24

৩,৯৯৮.

Five years ago the average of A, B, C, D was 45 years. By including X the present average of all the five is 49 years . Then the present age of X is-

ক
ক) 40 years
খ
খ) 45 years
গ
গ) 42 years
ঘ
ঘ) 49 years

ব্যাখ্যা

Five years ago, the average age of A,B,C and D was 45 years

Sum of present age of A, B, C and D = 45 × 4 + 5 × 4 = 200 years
Present average age of A, B, C, D and X is 49
⇒ Sum of present age of A, B, C, D and X is (49 × 5) = 245 years
Present age of X is (245 - 200) = 45 years

∴ The present age of X is 45 years.

৩,৯৯৯.

A started a business with Tk. 2100 and is joined afterwards by B with Tk. 3600. After how many months did B join if the profits at the end of the year are divided equally?

ক
ক) 4 months.
খ
খ) 5 months.
গ
গ) 6 months.
ঘ
ঘ) 7 months.

ব্যাখ্যা

Question: A started a business with Tk. 2100 and is joined afterwards by B with Tk. 3600. After how many months did B join if the profits at the end of the year are divided equally?

Solution:
Suppose,
B joined after x months.

Then,
2100 × 12 = 3600 × (12 - x)
⇒ 252 = 432 - 36x
⇒ 36x = 180
∴ x = 5

∴ B joined after 5 months.

৪,০০০.

If then 2A : B : 2C = ?

ক
ক) 2 : 3 : 4
খ
খ) 3 : 2 : 4
গ
গ) 6 : 3 : 8
ঘ
ঘ) 4 : 3 : 8

ব্যাখ্যা

Question: If then 2A : B : 2C = ?

Solution:
দেওয়া আছে,
A/2 = B/3
A : B = 2 : 3
এবং,
B/3 = C/4
B : C = 3 : 4

এখন,
A : B = 2 : 3
B : C = 3 : 4
A : B : C = 2 : 3 : 4

∴ 2A : B : 2C = (2 × 2) : 3 : (4 × 2)
= 4 : 3 : 8