Bank Math

PrepBank · পাতা ১৩৮ / ১৬১ · ১৩,৭০১–১৩,৮০০ / ১৬,১২৪

১৩,৭০১.

The difference of the areas of two squares drawn on two line segments of different lengths is 32 sq. cm. Find the length of the greater line segment if one is longer than the other by 2 cm.

ক
7 cm
খ
9 cm
গ
11 cm
ঘ
16 cm

ব্যাখ্যা

Question: The difference of the areas of two squares drawn on two line segments of different lengths is 32 sq. cm. Find the length of the greater line segment if one is longer than the other by 2 cm.

Solution:
Let the lengths of the line segments be x and (x + 2) cm.

ATQ,
(x + 2)² - x² = 32
⇒ x² + 4x + 4 - x² = 32
⇒ 4x = 28
∴ x = 7 cm

Hence, the greater line should be, x + 2 = 7 + 2 = 9 cm

১৩,৭০২.

If n(U) = 50, n(A) = 28, n(B) = 26 and n(A ∩ B) = 12 then what is n(A ∪ B)′ ?

ক
9
খ
12
গ
8
ঘ
10
ঙ
None of these

ব্যাখ্যা

Question: If n(U) = 50, n(A) = 28, n(B) = 26 and n(A ∩ B) = 12 then what is n(A ∪ B)′ ?

Solution:
We know that,
n(A ∪ B)= n(A) + n(B) - (A ∩ B)
= 28 + 26 - 12
= 42

Now,
n(A ∪ B)′= n(U) - n(A ∪ B)
= 50 - 42
= 8

So, n(A ∪ B)′ = 8

১৩,৭০৩.

Find the average of the numbers 32, 18, 95, 52 and 86.

ক
57.5
খ
65.2
গ
63.5
ঘ
56.6

ব্যাখ্যা

Question: Find the average of the numbers 32, 18, 95, 52 and 86.

Solution:
Average = Sum of all observations/ Number of observations
∴ Average = (32 + 18 + 95 + 52 + 86)/5
= 283/5
= 56.6
∴ The correct answer is 56.6.

১৩,৭০৪.

A boat takes 6 hours to travel 30 km upstream and 3 hours to travel the same distance downstream. Find the distance travelled by the boat in 7 hours in still water.

ক
52.5 km
খ
42.6 km
গ
62.5 km
ঘ
45.5 km

ব্যাখ্যা

Question: A boat takes 6 hours to travel 30 km upstream and 3 hours to travel the same distance downstream. Find the distance travelled by the boat in 7 hours in still water.

Solution:
Let the speed of the boat in still water be b km/h
and the speed of the current (stream) be c km/h.
Then we get,
Upstream speed = b - c
Downstream speed = b + c

Now, upstream speed = 30/6 = 5 km/h
∴ b - c = 5 ……… (1)
And downstream speed = 30/3 = 10 km/h
∴ b + c = 10 ……… (2)

Add equations (1) and (2) then we get,
(b - c) + (b + c) = 5 + 10
⇒ 2b = 15
⇒ b = 15/2
∴ b = 7.5 km/h
So, the speed of the boat in still water is 7.5 km/h.

∴ Distance travel in still water in 7 hours = speed × time
= 7.5 × 7
= 52.5 km

So the boat will travel 52.5 km in 7 hours in still water.

১৩,৭০৫.

Neha completes 75% of a work in 15 days. Then she calls in Farin, and together they finish the remaining work in 4 days. How long would Farin alone take to complete the whole work?

ক
80 days
খ
75 days
গ
70 days
ঘ
68 days

ব্যাখ্যা

Question: Neha completes 75% of a work in 15 days. Then she calls in Farin, and together they finish the remaining work in 4 days. How long would Farin alone take to complete the whole work?

Solution:
Here,
Neha does 75% or 3/4 work in 15 days
in one day, she does = (3/4 ÷ 15) = 1/20 part
in 4 days, she does 4/20 = 1/5 part

work remaining = 1/4 - 1/5
= 1/20 part

Farin does 1/20 parts in 4 days
she will complete the work {4 ÷ (1/20)} days
= 80 days

১৩,৭০৬.

If both 5² and 3² are factors of m where m = z × 2⁵ × 6² × 7³ , what is the smallest positive value of z?

ক
9
খ
16
গ
25
ঘ
32
ঙ
36

ব্যাখ্যা

Question: If both 5² and 3² are factors of m where m = z × 2⁵ × 6² × 7³ , what is the smallest positive value of z?

Solution:
Given, m = z × 2⁵ × 6² × 7³

We simplify this by expressing all terms in their prime factorizations except z,
∴ 6² = (2 × 3)² = 2² × 3²
∴ m = z × 2⁵ × 2² × 3² × 7³
= z × 2⁷ × 3² × 7³

If both 5² and 3² are factors, then they must be present in the number.
From the simplified expression of m, we can see that the factor 3² is already present but the required factor 5² is missing.
To make 5² a factor of m, it must come from z.

∴ The smallest positive value of z = 5² = 25.

১৩,৭০৭.

The average temperature for Wednesday, Thursday, and Friday was 38°C. The average for Thursday, Friday, and Saturday was 40° C. If the temperature on Saturday was 41° C, what was the temperature on Wednesday?

ক
35° C
খ
36° C
গ
37° C
ঘ
38° C

ব্যাখ্যা

Question: The average temperature for Wednesday, Thursday, and Friday was 38°C. The average for Thursday, Friday, and Saturday was 40° C. If the temperature on Saturday was 41° C, what was the temperature on Wednesday?

Solution:
Average temperature for Wednesday, Thursday, and Friday = 38° C
∴ Total temperature = 3 × 38 = 114° C

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

Temperature on Saturday = 41° C

Now,
(Thursday + Friday + Saturday) - (Wednesday + Thursday + Friday) = 120 - 114
= 106° C

Hence, Saturday - Wednesday = 6
∴ Wednesday = 41 - 6 = 35° C

১৩,৭০৮.

A train 'B' speeding with 100 kmph crosses another train C, running in the same direction, in 2 minutes. If the length of the train B and C be 150 metre and 250 metre respectively, what is the speed of the train C (in kmph)?

ক
90 kmph
খ
88 kmph
গ
80 kmph
ঘ
72 kmph

ব্যাখ্যা

Question: A train 'B' speeding with 100 kmph crosses another train C, running in the same direction, in 2 minutes. If the length of the train B and C be 150 metre and 250 metre respectively, what is the speed of the train C (in kmph)?

Solution:
Let,
the speed of train C be = x kmph.
Relative speed of B = (100 - x ) kmph.

Time taken in crossing = Length of both trains/Relative speed
2/60 = [{(150 + 250)/1000}/(100 - x)]
⇒ 1/30 = 2/{5(100 - x)}
⇒ 1/6 = 2/(100 - x)
⇒ 100 - x = 12
⇒ x = 100 – 12
∴ x = 88 kmph

১৩,৭০৯.

A container contains 40 litres of milk. From this container 4 litres of milk was taken out and replaced by water. This process was repeated further two times. How much milk is now contained by the container?

ক
26.34 litres
খ
27.36 litres
গ
28 litres
ঘ
29.16 litres

ব্যাখ্যা

Question: A container contains 40 litres of milk. From this container 4 litres of milk was taken out and replaced by water. This process was repeated further two times. How much milk is now contained by the container?

Solution:
Amount of milk left after 3 operations = 40(1 - 4/40)³ litres
= 40 × 36/40 × 36/40 × 36/40
= 40 × 9/10 × 9/10 × 9/10
= 29.16 litres

১৩,৭১০.

When 52416 is divided by 312, the quotient is 168. What will be the quotient when 52.416 is divided by 0.0168?

ক
ক) 3.120
খ
খ) 31.20
গ
গ) 3120
ঘ
ঘ) None of these

ব্যাখ্যা

Given,
52416/312=168
⇔ 52416/ 168 =312
Now, 52.416/ 0.0168
= 524160/ 168
= (52416/ 168 )×10
=312×10
=3120

১৩,৭১১.

If |x - 1| = 2x, what is the value of x?

ক
ক) -1
খ
খ) 1/3
গ
গ) 2
ঘ
ঘ) 4/3
ঙ
ঙ) None

ব্যাখ্যা

এখানে, |x - 1| = 2x
যদি x এর মান ধনাত্মক হয়, x - 1 = 2x
বা, 2x - x = -1
বা, x = -1 [গ্রহণযোগ্য নয়, কারণ এতে করে পরম মান ঋণাত্মক হয়, যা অসম্ভব]
যদি x এর মান ঋণাত্মক হয়, -(x - 1) = 2x
বা, - x + 1 = 2x
বা, 3x = 1
বা, x = 1/3

১৩,৭১২.

The angle of elevation at the top of a tower at a point on the ground is 30° at a distance of 75 metre from the foot. Find the height of the tower.

ক
5/√3
খ
25√3
গ
√3/25
ঘ
25/√3

ব্যাখ্যা

Question: The angle of elevation at the top of a tower at a point on the ground is 30° at a distance of 75 metre from the foot. Find the height of the tower.

Solution:

Let the height of the tower is AB = h meter. The angle of elevation at C from the foot of the tower BC = 75 metre of A on the ground is ∠ACB = 30°

From triangle ABC
∴ tan∠ACB = AB/BC
⇒ tan30° = AB/75
⇒ 1/√3 = h/75
⇒ h = 75/√3
⇒ h = 75√3/3
∴ h = 25√3

১৩,৭১৩.

Abir invested Tk 333000 in 5(1/2) % stocks at 110 .If brokerage is Tk. 1, what is his annual income from his investment.

ক
ক) Tk. 16500
খ
খ) Tk. 12000
গ
গ) Tk. 12500
ঘ
ঘ) Tk. 18000

ব্যাখ্যা

Investment = Tk. 333000
Since face value is not given, we can take it as Tk.100
and dividend per share = Tk. 11/2
Market Value = 110 + 1 = 111
Number of shares purchased = 333000/111 = 3000
Total income = 3000 × (11/2)
= Tk. 16500.

১৩,৭১৪.

A pipe can fill a tank in x hours and another pipe can empty it in y (y > x) hours. If both pipes are open, in how many hours will the tank is filled?

ক
xy/(x - y) hours
খ
(x - y) hours
গ
xy/(y - x) hours
ঘ
None of the above

ব্যাখ্যা

Question: A pipe can fill a tank in x hours and another pipe can empty it in y (y > x) hours. If both pipes are open, in how many hours will the tank is filled?

Solution:
Net part filled in 1 hour
= (1/x) - (1/y)
= (y - x)/xy hours

∴ The tank will be filled in = xy/(y - x) hours.

১৩,৭১৫.

At a certain restaurant, 1/8 pizzas sold in one week were mushroom and 1/3 of the remaining pizzas sold were pepperoni. If 3n/7 of the pizzas sold were mushroom, how many were pepperoni?

ক
3n
খ
n
গ
n/3
ঘ
(n + 5)/7

ব্যাখ্যা

Question: At a certain restaurant, 1/8 pizzas sold in one week were mushroom and 1/3 of the remaining pizzas sold were pepperoni. If 3n/7 of the pizzas sold were mushroom, how many were pepperoni?

Solution:
mushroom pizza = 1/8 of total pizza = 3n/7
total pizza = 24n/7

remaining pizza = (24n/7) - (3n/7)
= 21n/7
= 3n

pepperoni pizza = 3n/3
= n

১৩,৭১৬.

Which of the following is the smallest given that x > y > 1 ?

ক
x/y
খ
(x + 2)/(y + 2)
গ
(x + 1)/(y + 1)
ঘ
(x + 1)/y
ঙ
xy

ব্যাখ্যা

Question: Which of the following is the smallest given that x > y > 1 ?

Solution:
Given that, x > y > 1,
Consider the expressions,
∴ (x/y) >1 (since x > y)

Adding the same positive number to numerator and denominator reduces the fraction when numerator > denominator.
Thus,
(x/y) > (x+1)/(y+1) > (x + 2)/(y + 2)
Also, (x + 1)/y > (x+1)/(y+1) (larger denominator)
and xy > x > y > 1

∴ The smallest is (x + 2)/(y + 2)

১৩,৭১৭.

A train 120 meters long takes 30 seconds to cross a 480-meter-long bridge. How much time will the train take to cross a 400-meter-long platform?

ক
24 seconds
খ
26 seconds
গ
30 seconds
ঘ
32 seconds

ব্যাখ্যা

Question: A train 120 meters long takes 30 seconds to cross a 480-meter-long bridge. How much time will the train take to cross a 400-meter-long platform?

Solution:
Length of train = 120 m
Length of bridge = 480 m
∴ Total distance to cross bridge = 120 + 480 = 600 m
Time taken = 30 seconds
∴ Speed of train = Total distance/Time
= 600/30 = 20 m/s

Length of platform = 400 m
∴ Total distance to cross platform = 120 + 400 = 520 m

∴ Time taken = Total distance/Speed
= 520/20 seconds
= 26 seconds

১৩,৭১৮.

What number has to be added to the terms of 3 : 5 to make the ratio 5 : 6?

ক
ক) 5
খ
খ) 6
গ
গ) 7
ঘ
ঘ) 8

ব্যাখ্যা

Question: What number has to be added to the terms of 3 : 5 to make the ratio 5 : 6?

Solution:
Let the number to be added is X.
Then,
(3 + X)/(5 + X) = 5/6
6(3 + X) = 5(5 + X)
18 + 6X = 25 + 5X
X = 7

১৩,৭১৯.

The smallest prime factor of (2⁴)² − 1 is -

ক
ক) 3
খ
খ) 5
গ
গ) 7
ঘ
ঘ) 9

ব্যাখ্যা

(2⁴)² − 1
= 2⁸ - 1
= 256 - 1
= 255
= 3 × 5 × 17

So, the smallest prime factor is 3

১৩,৭২০.

A boat covers a certain distance downstream in 1 hour, while it comes back in 7/3 hours. If the speed of the stream be 6 kmph, what is the speed of the boat in still water?

ক
ক) 9 kmph
খ
খ) 12 kmph
গ
গ) 15 kmph
ঘ
ঘ) 18 kmph

ব্যাখ্যা

Question: A boat covers a certain distance downstream in 1 hour, while it comes back in 7/3 hours. If the speed of the stream be 6 kmph, what is the speed of the boat in still water?

Solution:
Let the speed of the boat in still water be x kmph.
Then,
Speed downstream = (x + 6) kmph,
Speed upstream = (x - 6) kmph.

Now
(x + 6) x 1 =(x - 6)(7/3)
3(x + 6) = 7(x - 6)
3x + 18 = 7x - 42
7x - 3x = 42 + 18
4x = 60
x = 15 kmph

১৩,৭২১.

A group of 7 members having a majority of boys is to be formed out of 7 boys and 4 girls. The number of ways can be formed is the group-

ক
80
খ
100
গ
294
ঘ
110

ব্যাখ্যা

Question: A group of 7 members having a majority of boys is to be formed out of 7 boys and 4 girls. The number of ways can be formed is the group-

Solution:
Boys Girls Ways
-------------------------------------------------------
6 1 ⁷C₆ × ⁴C₁ = 7 × 4 = 28
5 2 ⁷C₅ × ⁴C₂ = 21 × 6 = 126
4 3 ⁷C₄ × ⁴C₃ = 35 × 4 = 140

∴ Total number of ways = 28 + 126 + 140 = 294

১৩,৭২২.

A and B can do a work in 12 days. B and C can do it in 15 days. A and C can do it in 20 days. If all of them work together, in how many days can they finish the work?

ক
ক) 9
খ
খ) 10
গ
গ) 12
ঘ
ঘ) 13

ব্যাখ্যা

(A + B) 1 দিনে করে কাজটির 1/12 অংশ
(B + C ) 1 দিনে করে কাজটির 1/15 অংশ
(C + A) 1 দিনে করে কাজটির 1/20 অংশ

(A + B + B + C + C + A) 1 দিনে করে কাজটির = (1/12) + (1/15) + (1/20) অংশ
(2A + 2B + 2C) 1 দিনে করে কাজটির = (5 + 4 + 3)/60 অংশ
2(A + B + C) 1 দিনে করে কাজটির = 1/5 অংশ
(A + B + C) 1 দিনে করে কাজটির = 1/10 অংশ

(A + B + C) 1/10 অংশ কাজ করে 1 দিনে
(A + B + C) 1 বা সম্পূর্ণ অংশ কাজ করে = (10 × 1)/1 দিনে
= 10 দিনে

১৩,৭২৩.

If a photocopier makes 3 copies in 1/4 seconds, at the same rate how many copies does it make in 5 minutes?

ক
3200 copies
খ
3400 copies
গ
3600 copies
ঘ
3800 copies

ব্যাখ্যা

Question: If a photocopier makes 3 copies in 1/4 seconds, at the same rate how many copies does it make in 5 minutes?

Solution:
In 1/4 seconds he can make 3 copies
In 1 second he can make 3/(1/4) copies = 12 copies
∴ In 300 seconds he can make (3 × 4 × 300) copies = 3600 copies

১৩,৭২৪.

A boat running downstream covers a distance of 30 km in 3 hours, while it takes 5 hours to cover the same distance upstream. What is the speed of the boat in still water?

ক
9 km/h
খ
5 km/h
গ
8 km/h
ঘ
10 km/h

ব্যাখ্যা

Question: A boat running downstream covers a distance of 30 km in 3 hours, while it takes 5 hours to cover the same distance upstream. What is the speed of the boat in still water?

Solution:
Rate while running downstream = (30/3) km/h
= 10 km/h

Rate while running upstream = (30/5) km/h
= 6 km/h

∴ Speed of the boat in still water
= (10 + 6)/2 km/h
= 8 km/h

So, the speed of the boat in still water is 8 km/h.

১৩,৭২৫.

Ten years ago, the average age of A, B, C and D was 40 yr. with E joining them now, the average of all the five is 49 yr. How old is E?

ক
ক) 44 years
খ
খ) 45 years
গ
গ) 46 years
ঘ
ঘ) 47 years

ব্যাখ্যা

Question: Ten years ago, the average age of A, B, C and D was 40 yr. with E joining them now, the average of all the five is 49 yr. How old is E?

Soultion:
Total present age of A, B, C and D
= (40 + 10) × 4
= 200 years
Total age present age of A, B, C, D and E
= 49 × 5
= 245 years
So, Age of E = 45 years

১৩,৭২৬.

What is the probability of getting 53 Mondays in a leap year?

ক
3/5
খ
2/7
গ
4/9
ঘ
5/8

ব্যাখ্যা

Question: What is the probability of getting 53 Mondays in a leap year?

Solution:
1 year = 365 days . A leap year has 366 days
A year has 52 weeks. Hence there will be 52 Sundays for sure.
52 weeks = 52 × 7 = 364days
366 - 364 = 2 days

In a leap year there will be 52 Sundays and 2 days will be left.
These 2 days can be:
1. Sunday, Monday
2. Monday, Tuesday
3. Tuesday, Wednesday
4. Wednesday, Thursday
5. Thursday, Friday
6. Friday, Saturday
7. Saturday, Sunday
Of these total 7 outcomes, the favourable outcomes are 2.

Hence the probability of getting 53 days = 2/7

১৩,৭২৭.

How many prime numbers are there between 56 and 100?

ক
ক) 8
খ
খ) 9
গ
গ) 10
ঘ
ঘ) 11
ঙ
ঙ) 12

ব্যাখ্যা

Prime numbers between 56 and 100
59, 61, 67, 71, 73, 79, 83, 89, 97 are the prime numbers between 56 and 100.

১৩,৭২৮.

If |x - 2| < 5, what are the values of a and b for which a < 2x + 9 < b?

ক
a = 1 and b = 13
খ
a = 5 and b = 27
গ
a = 7 and b = 19
ঘ
a = 3 and b = 23
ঙ
None

ব্যাখ্যা

Question: If |x - 2| < 5, what are the values of a and b for which a < 2x + 9 < b?

Solution:
|x - 2| < 5
⇒ - 5 < x - 2 < 5
⇒ - 5 + 2 < x - 2 + 2 < 5 + 2
⇒ - 3 < x < 7
⇒ - 6 < 2x < 14
⇒ - 6 + 9 < 2x + 9 < 14 + 9
∴ 3 < 2x + 9 < 23

Compare with a < 2x + 9 < b.
We get, a = 3 and b = 23

১৩,৭২৯.

A rectangular park 60 m long and 40 m wide has two concrete crossroads running in the middle of the park and rest of the park has been used as a lawn. If the area of the lawn is 2109 sq. m, then what is the width of the road?

ক
ক) 2.91 m
খ
খ) 3 m
গ
গ) 5.82 m
ঘ
ঘ) None of these

ব্যাখ্যা

Area of the park = (60 x 40) m² = 2400 m².
Area of the lawn = 2109 m².
Area of the crossroads = (2400 - 2109) m² = 291 m^2.Let the width of the road be x metres. Then,
60x + 40x - x² = 291
x² - 100x + 291 = 0
(x - 97)(x - 3) = 0
x = 3.

১৩,৭৩০.

A dishonest milkman profess to sell his milk at cost price but he mixes it with water and thereby gains 25%. The percentage of water in the mixture is-

ক
ক) 20%
খ
খ) 25%
গ
গ) 15%
ঘ
ঘ) 10%

ব্যাখ্যা

Question: A dishonest milkman profess to sell his milk at cost price but he mixes it with water and thereby gains 25%. The percentage of water in the mixture is-

Solution:
Let, the cost price of 1 liter of milk be = 100 Tk
So, the selling price of 1 liter mixture is also = 100 Tk

Here, in 100 Tk, SP gain = 25%
So, cost price of the mixture = (100 × 100)/125 = 80 Tk

So, water in the mixture = 100 - 80 = Tk 20

১৩,৭৩১.

When three-fifths of a number are subtracted from two-thirds of the number, the result is 18. What's the number?

ক
260
খ
250
গ
280
ঘ
270

ব্যাখ্যা

Question: When three-fifths of a number are subtracted from two-thirds of the number, the result is 18. What's the number?

Solution:
Let,
The number be n

ATQ,
(2/3) × n - (3/5) × n = 18
⇒ (2n/3) - (3n/5) = 18
⇒ (10n - 9n)/15 = 18
⇒ n/15 = 18
∴ n = 270

১৩,৭৩২.

ক
144
খ
14.4
গ
1.44
ঘ
0.0144

ব্যাখ্যা

Question:

Solution:

১৩,৭৩৩.

The four digit smallest positive number which when divided by 4, 5, 6 or 7, it leaves always the remainder as 3:

ক
1000
খ
1257
গ
1263
ঘ
1683

ব্যাখ্যা

Question: The four digit smallest positive number which when divided by 4, 5, 6 or 7, it leaves always the remainder as 3:

Solution:
The least possible number = (LCM of 4, 5, 6 and 7) + 3 = 420 + 3 = 423
The next higher number is, (420m + 3) now we put a least value of m such that (420m + 3) ≥ 1000
value = 420 × 3 + 3 = 1263
∴ m = 3

১৩,৭৩৪.

16. Value of (-1) ^-13 is-

ক
1
খ
-1
গ
1/13
ঘ
-13
ঙ
None of the above

ব্যাখ্যা

Since base one and power is negative so answer in -1.

১৩,৭৩৫.

ক
0
খ
1
গ
2
ঘ
3

ব্যাখ্যা

Question:

Solution:

১৩,৭৩৬.

Two pipes can fill a tank in 10 and 14 minutes respectively and a waste pipe can empty 4 gallons per minute. If all the pipes working together can fill the tank in 6 minutes, what is the capacity of the tank?

ক
ক) 120 gallons
খ
খ) 240 gallons
গ
গ) 450 gallons
ঘ
ঘ) 840 gallons

ব্যাখ্যা

Question: Two pipes can fill a tank in 10 and 14 minutes respectively and a waste pipe can empty 4 gallons per minute. If all the pipes working together can fill the tank in 6 minutes, what is the capacity of the tank?

Solution:
Work done by the waste pipe in 1 minute
= (1/6) - {(1/10 + 1/14 )}
= (1/6) - (24/140)
= -1/210 [-ve sign means emptying]

∴ Volume of 1/210 part = 4 gallons
∴ Volume of whole = (4 × 210 ) gallons = 840 gallons

১৩,৭৩৭.

What number should come next in the series:
5, 7, 13, 31, 85,.......?

ক
217
খ
247
গ
271
ঘ
284

ব্যাখ্যা

Question: What number should come next in the series:
5, 7, 13, 31, 85,.......?

Solution:
দেওয়া আছে,
সিরিজটি হলো: 5, 7, 13, 31, 85,.......

পার্থক্যগুলোর মধ্যে একটি প্যাটার্ন রয়েছে। প্রতিটি পার্থক্য আগের পার্থক্যের 3 গুণ।

5 থেকে 7 পর্যন্ত পার্থক্য: 2
7 থেকে 13 পর্যন্ত পার্থক্য: 6 (2 × 3)
13 থেকে 31 পর্যন্ত পার্থক্য: 18 (6 × 3)
31 থেকে 85 পর্যন্ত পার্থক্য: 54 (18 × 3)

সুতরাং, পরবর্তী পার্থক্যটি হবে:
54 × 3 = 162

পরবর্তী সংখ্যাটি হবে শেষ সংখ্যা এবং এই পার্থক্যের যোগফল:
85 + 162 = 247
∴ পরবর্তী সংখ্যাটি হলো 247

১৩,৭৩৮.

A train covers a distance of 600 meters in 25 seconds, whereas a car covers a distance of 43.2 km in 36 minutes. What is the ratio of the speed of the train to the speed of the car?

ক
3 : 7
খ
6 : 5
গ
8 : 3
ঘ
7 : 4

ব্যাখ্যা

Question: A train covers a distance of 600 meters in 25 seconds, whereas a car covers a distance of 43.2 km in 36 minutes. What is the ratio of the speed of the train to the speed of the car?

Solution:
ট্রেনের গতিবেগ নির্ণয়:
দূরত্ব = 600 মিটার, সময় = 25 সেকেন্ড।
∴ ট্রেনের গতিবেগ = 600/25 মিটার/সেকেন্ড
= 24 মিটার/সেকেন্ড।

গাড়ির গতিবেগ নির্ণয়:
দূরত্ব = 43.2 কিমি = 43.2 × 1000 = 43200 মিটার।
সময় = 36 মিনিট = 36 × 60 = 2160 সেকেন্ড।
∴ গাড়ির গতিবেগ = 43200/2160 মিটার/সেকেন্ড
= 20 মিটার/সেকেন্ড।

গতিবেগের অনুপাত = ট্রেনের গতিবেগ : গাড়ির গতিবেগ
= 24 : 20
= 6 : 5
∴ তাদের গতিবেগের অনুপাত হলো 6 : 5

১৩,৭৩৯.

After paying a 10 percent tax on all income over Tk. 3,000, a person had a net income of Tk. 12,000. What was the income before taxes?

ক
Tk. 13,000
খ
Tk. 12900
গ
Tk. 10000
ঘ
Tk. 9000

ব্যাখ্যা

Question: After paying a 10 percent tax on all income over Tk. 3,000, a person had a net income of Tk. 12,000. What was the income before taxes?

Solution:
প্রশ্নে ৩০০০ টাকা পর্যন্ত আয়ের উপর কোন কর দেয়া লাগে না এবং ৩০০০ টাকার উপর আয় করলে বাকি টাকার উপর কর দিতে হয়।

ধরি,
আয়ের করযুক্ত অংশ ক টাকা

∴ ক - ক এর ১০% + ৩০০০ = ১২০০০
বা, ক - (১০ক)/১০০ + ৩০০০ = ১২০০০
বা, ক - ক/১০ = ৯০০০
বা, ১০ক - ক = ৯০০০০
বা, ৯ক = ৯০০০০
∴ ক = ১০০০০

∴ করযুক্ত টাকা ১০০০০ এবং করমুক্ত টাকা ৩০০০
∴ মোট টাকা (১০০০০ + ৩০০০) = ১৩০০০ টাকা

∴ যদি কর না দিত তাহলে তাঁর আয় ১৩০০০ টাকা হত। কিন্তু ১০০০ টাকা কর দিয়ে দেয়ায় তার আয় থাকে ১২০০০ টাকা

১৩,৭৪০.

A retailer bought a glass at wholesale and marked it up 80% to its initial retail price of Tk.45. By how many more taka does he need to increase the price to achieve a 100% markup?

ক
Tk. 9
খ
Tk. 6
গ
Tk. 5
ঘ
Tk. 4

ব্যাখ্যা

Question: A retailer bought a glass at wholesale and marked it up 80% to its initial retail price of Tk.45. By how many more taka does he need to increase the price to achieve a 100% markup?

Solution:
let, wholesale price x

1.8x = 45
⇒ x = 45/1.8
∴ x = 25 taka

to achieve a 100% markup retail price = 25 + 25 = 50 taka

he needs to increase = 50 - 45 taka
= Tk. 5

১৩,৭৪১.

In traveling from a dormitory to a certain city, a student went 1/5 of the way by foot, 2/3 of the way by bus, and the remaining 8 kilometers by car. What is the distance, in kilometers, from the dormitory to the city?

ক
30
খ
45
গ
60
ঘ
90

ব্যাখ্যা

Question: In traveling from a dormitory to a certain city, a student went 1/5 of the way by foot, 2/3 of the way by bus, and the remaining 8 kilometers by car. What is the distance, in kilometers, from the dormitory to the city?

Solution:
Let,
total distance x kilometers

ATQ,
x/5 + 2x/3 + 8 = x
⇒ x - x/5 - 2x/3 = 8
⇒ (15x - 3x - 10x)/15 = 8
⇒ 2x = 120
∴ x = 60

১৩,৭৪২.

The average of the two-digit numbers which remain the same when the digits interchange their positions is-

ক
ক) 33
খ
খ) 44
গ
গ) 55
ঘ
ঘ) 66

ব্যাখ্যা

প্রশ্ন: The average of the two-digit numbers which remain the same when the digits interchange their positions is-

সমাধান:
দুই অঙ্কের একটি সংখ্যার অঙ্কদ্বয় স্থান পরিবর্তন করলে, সংখ্যাটি একই থাকে।
অর্থাৎ, একক ও দশক স্থানীয় অঙ্ক একই হবে।

এমন সংখ্যাগুলো হল ১১, ২২, ৩৩, ৪৪, ৫৫, ৬৬, ৭৭, ৮৮, ৯৯

∴ সংখ্যাগুলোর গড় = (১১ + ২২ + ৩৩ + ৪৪ + ৫৫ + ৬৬ + ৭৭ + ৮৮ + ৯৯)/ ৯
= ৪৯৫/৯
= ৫৫

১৩,৭৪৩.

In a class, 25 students play football, 15 students play cricket, and 5 students play both. 10 students play neither football nor cricket. What is the total number of students in the class?

ক
45
খ
60
গ
50
ঘ
72

ব্যাখ্যা

Question: In a class, 25 students play football, 15 students play cricket, and 5 students play both. 10 students play neither football nor cricket. What is the total number of students in the class?

Solution:
Number of students who play football, n(F) = 25
Number of students who play cricket, n(C) = 15
Number of students who play both football and cricket, n(F ∩ C) = 5
Number of students who play neither = 10

n(F ∪ C) = n(F) + n(C) - n(F ∩ C)
= 25 + 15 - 5
= 35

Total students in the class = students who play football or cricket + students who play neither
= 35 + 10
= 45

∴ There are 45 students in the class.

১৩,৭৪৪.

A person saves Tk. 5000 in his bank account. If the bank gives 10% annual profit, how much will be his balance after 6 years?

ক
ক) Tk. 3000
খ
খ) Tk. 5000
গ
গ) Tk. 7000
ঘ
ঘ) Tk. 8000

ব্যাখ্যা

প্রশ্ন: A person saves Tk. 5000 in his bank account. If the bank gives 10% annual profit, how much will be his balance after 6 years?

সমাধান:
দেওয়া আছে,
আসল, P = 5000 টাকা
মুনাফার হার, r = 10%
সময়, n = 6 বছর

আমরা জানি,
I = Pnr
⇒ I = 5000 × 6 × 10%
⇒ I = 5000 × 6 × 10/100
⇒ I = 3000 টাকা
∴ 6 বছর পর মোট টাকা = 5000 + 3000 = 8000 টাকা

১৩,৭৪৫.

Protik rolled a dice twice and he saw that the addition of two numbers that appeared on the top face was 8. Find the probability of getting a 4 on the top face of the dice in the first throw.

ক
ক) 1/36
খ
খ) 2/36
গ
গ) 1/6
ঘ
ঘ) 1/5

ব্যাখ্যা

A dice has 6 faces.
So there are 6 possible outcomes
Dice are rolled once AND then again.
So total possibilities = 6 x 6 = 36

The sum should be 8 of the 2 throws.
So which combination of numbers from 1 to 6 will yield us a sum of 8?
They are - (2,6); (6,2); (3,5); (5,3); (4,4)

So there are a total of 5 possibilities where the addition is 8.
But only 1 possibility where the first throw of dice is 4.

So, the Probability for the first throw to be 4 and sum to be 8 = 1/36.

১৩,৭৪৬.

The least number by which 320 must be multiplied to make it perfect square, is-

ক
5
খ
25
গ
10
ঘ
15

ব্যাখ্যা

Question: The least number by which 320 must be multiplied to make it perfect square, is-

Solution:
Here,
320 = 2 × 2 × 2 × 2 × 2 × 2 × 5 = 2⁶ × 5
To make it perfect squre, it must be multiplied by 5

Therefore, the least number by which 320 must be multiplied to make it a perfect square is 5.

১৩,৭৪৭.

Ratul is four times as old as Tarek. 12 years ago, Ratul was 10 times as old as Tarek. What will be the sum of their ages after 5 years?

ক
75
খ
90
গ
85
ঘ
100

ব্যাখ্যা

Question: Ratul is four times as old as Tarek. 12 years ago, Ratul was 10 times as old as Tarek. What will be the sum of their ages after 5 years?

Solution:
ধরি, বর্তমানে Tarek-এর বয়স = x বছর।
তাহলে Ratul-এর বয়স = 4x বছর।

12 বছর আগে তাদের বয়স ছিল:
Tarek = x - 12 বছর
Ratul = 4x - 12 বছর

প্রশ্নমতে,
4x - 12 = 10(x - 12)
⇒ 4x - 12 = 10x - 120
⇒ 120 - 12 = 10x - 4x
⇒ 108 = 6x
⇒ x = 108/6
∴ x = 18

∴ বর্তমানে Tarek এর বয়স = 18 বছর,
Ratul এর বয়স = 4 × 18 = 72 বছর

5 বছর পর তাদের বয়স হবে:
Tarek = 18 + 5 = 23 বছর
Ratul = 72 + 5 = 77 বছর

∴ 5 বছর পর তাদের বয়সের যোগফল = 23 + 77 = 100 বছর

১৩,৭৪৮.

Solve for x if 16x-4 = 68+7x.

ক
ক) x = 16
খ
খ) x = -16
গ
গ) x = 8
ঘ
ঘ) x = 5

ব্যাখ্যা

16x - 4 = 68 + 7x
বা, 9x = 72
বা, x = 8.

১৩,৭৪৯.

Akash can do a piece of work in 30 days. He works at it for 6 days and then Rakib finishes it in 18 days. In what time can Akash and Rakib together finish the work?

ক
90/7 days
খ
90/4 days
গ
90/11 days
ঘ
90 days

ব্যাখ্যা

Question: Akash can do a piece of work in 30 days. He works at it for 6 days and then Rakib finishes it in 18 days. In what time can Akash and Rakib together finish the work?

Solution:
আকাশ ৩০ দিনে করে সম্পূর্ণ অংশ
১ দিনে করে ১/৩০ অংশ
৬ দিনে করে ৬/৩০ অংশ
= ১/৫ অংশ

বাকি থাকে ১ - ১/৫ অংশ
= ৪/৫ অংশ

রাকিব ১৮ দিনে করে ৪/৫ অংশ
১ দিনে করে ২/৪৫ অংশ

রাকিব ও আকাশ ১ দিনে করে ১/৩০ + ২/৪৫ অংশ
= ৩ + ৪/৯০
= ৭/৯০

সম্পুর্ণ কাজ করতে সময় লাগে = ৯০/৭ দিন

১৩,৭৫০.

Simple interest on a certain sum is 16/25 of the sum. Find the rate percent & time, if both are equal.

ক
ক) 5% and 5 years.
খ
খ) 6% and 6 years.
গ
গ) 8% and 8 years.
ঘ
ঘ) 9% and 9 years.

ব্যাখ্যা

Question: Simple interest on a certain sum is 16/25 of the sum. What is the rate percent & time, if both are equal.

Solution:
Let.
The sum, P = Tk. x
Then, Simple interest, I = 16x/25
Rate of interest = r%
Since, rate percent = time
So r = n

We know,
I = Pnr
⇒ 16x/25 = x × r × r%
⇒ xr²/100 = 16x/25
⇒ r²/100 = 16/25
⇒ r² = 1600/25
⇒ r = 40/5
∴ r = 8

∴ Rate = 8% and Time = 8 years.

১৩,৭৫১.

A father said to his son, I was as old as you are at the present at the time of your birth. If the father's age is 42 years now, the son's age four years back was -

ক
ক) 16
খ
খ) 17
গ
গ) 19
ঘ
ঘ) 21

ব্যাখ্যা

Question: A father said to his son, I was as old as you are at the present at the time of your birth. If the father's age is 42 years now, the son's age four years back was -

Solution:
বর্তমানে পুত্রের যে বয়স, পুত্রের জন্মের সময় পিতার বয়স তত ছিল।
অর্থাৎ, বর্তমানে পিতার বয়স পুত্রের দ্বিগুণ।
∴ পুত্রের বর্তমান বয়স = ৪২/২ = ২১ বছর।

৪ বছর পূর্বে পুত্রের বয়স ছিল = ২১ - ৪ = ১৭ বছর।

১৩,৭৫২.

What will replace the '?' mark?

ক
60
খ
70
গ
85
ঘ
100

ব্যাখ্যা

Question: What will replace the '?' mark?

Solution:
এখানে, উপরের সংখ্যাটি নিচের দুটি সংখ্যার যোগফলের দ্বিগুণ।

১ম চিত্রে,
(15 + 10) × 2 = 25 × 2 = 50

২য় চিত্রে,
(30 + 15) × 2 = 45 × 2 = 90

একইভাবে, ৩য় চিত্রে,
(25 + 10) × 2 = 35 × 2 = 70

সুতরাং, সঠিক উত্তরটি হবে 70।

১৩,৭৫৩.

A room 5m × 8m is to be carpeted leaving a margin of 10 cm from each wall. If the cost of the carpet is Tk. 18 per sq. meter, the cost of carpeting the room will be :

ক
ক) Tk. 673.92
খ
খ) Tk. 682.46
গ
গ) Tk. 691.80
ঘ
ঘ) Tk. 702.60

ব্যাখ্যা

Area of the carpet :
= [(5 - 0.20) × (8 - 0.20)] m²
= (4.8 × 7.8) m²
= 37.44 m²
∴ Cost of carpeting :
= Tk. (37.44 × 18)
= Tk. 673.92

১৩,৭৫৪.

If selling a chair for 720 Taka causes a 10% loss, determine the selling price required to earn a 20% profit on the same item.

ক
Tk. 760
খ
Tk. 800
গ
Tk. 920
ঘ
Tk. 960

ব্যাখ্যা

Question: If selling a chair for 720 Taka causes a 10% loss, determine the selling price required to earn a 20% profit on the same item.

Solution:

১৩,৭৫৫.

What is the greatest number of four digits, which, when divided by 6, 12, and 18, leaves a remainder of 3 in each case?

ক
9990
খ
9985
গ
9979
ঘ
9975

ব্যাখ্যা

Question: What is the greatest number of four digits, which, when divided by 6, 12, and 18, leaves a remainder of 3 in each case?

Solution:
the largest four-digit number is = 9999
the L.C.M of 6, 12, 18 is = 36
dividing 9999 by 36 we get the remainder of 27

so, the number is = (9999 - 27) + 3 = 9975

১৩,৭৫৬.

Find the rate of discount when the marked price is Tk.1880 and selling price is Tk. 1598.

ক
ক) 12%
খ
খ) 18%
গ
গ) 8%
ঘ
ঘ) 15%

ব্যাখ্যা

Question: Find the rate of discount when the marked price is Tk.1880 and selling price is Tk. 1598.

Solution:

বইয়ের লিখিত মূল্য, x = Tk. 1880
বইয়ের বিক্রয় মূল্য, y = Tk. 1598

Formula:
If Tk. x be MP of the article and Tk. y be S.P of the article, then

ছাড় = (x - y)
ছাড় % = (Discount/MP) × 100

ছাড় = 1880 - 1598 = Tk. 282
∴ ছাড়ের শতাংশ = (282/1880) × 100 = 15%

১৩,৭৫৭.

A rectangular swimming pool has dimensions 45 m by 20 m. A concrete path 3.5 m wide is laid round it. Find the cost of laying the path at a rate of Tk. 25 per meter square.

ক
Tk. 12600
খ
Tk. 14600
গ
Tk. 9800
ঘ
Tk. 13650

ব্যাখ্যা

Question: A rectangular swimming pool has dimensions 45 m by 20 m. A concrete path 3.5 m wide is laid round it. Find the cost of laying the path at a rate of Tk. 25 per meter square.

Solution:
রাস্তাসহ সুইমিং পুলের দৈর্ঘ্য = 45 + 2 × 3.5 = 45 + 7 = 52 m
রাস্তাসহ সুইমিং পুলের প্রস্থ = 20 + 2 × 3.5 = 20 + 7 = 27 m

রাস্তাসহ সুইমিং পুলের ক্ষেত্রফল = 52 × 27 = 1404 m²
রাস্তাছাড়া সুইমিং পুলের ক্ষেত্রফল = 45 × 20 = 900 m²

∴ রাস্তার ক্ষেত্রফল = 1404 - 900 m²
= 504 m²

মোট খরচ = (504 × 25) টাকা
= 12600 টাকা

১৩,৭৫৮.

Sina and Sium are preparing a report. Sina takes 8 hours to write 48 pages on a computer, while Sium takes 7 hours to write 42 pages. How much time will it take, working together on two different computers, to write a report of 132 pages?

ক
9 hours
খ
10 hours
গ
11 hours
ঘ
12 hours

ব্যাখ্যা

Question: Sina and Sium are preparing a report. Sina takes 8 hours to write 48 pages on a computer, while Sium takes 7 hours to write 42 pages. How much time will it take, working together on two different computers, to write a report of 132 pages?

Solution:
Sina takes 8 hours to write 48 pages.
So, Sina writes in 1 hour = 48/8 = 6 pages

Sium takes 7 hours to write 42 pages.
So, Sium writes in 1 hour = 42/7 = 6 pages.

Together, Sina and Sium write in 1 hour = 6 + 6 = 12 pages

Now, Time taken to write 1 page = 1/12 hours
Time taken to write 132 pages = (1/12) × 132 hours
= 11

১৩,৭৫৯.

What will be the equation form for the following statement?
The sum of 3 times x and 11 is 32.

ক
x + 3 × 11 = 32
খ
(x/3) + 11 = 32
গ
3(x + 11) = 32
ঘ
3x + 11 = 32

ব্যাখ্যা

Question: What will be the equation form for the following statement?
The sum of 3 times x and 11 is 32.

Solution:
3 times x = 3x
The sum of 3 times x and 11 is 32.
3x + 11 = 32

১৩,৭৬০.

ক
194
খ
210
গ
180
ঘ
164

ব্যাখ্যা

Question:

Solution:

১৩,৭৬১.

If a, b, and c are three consecutive odd integers such that 10 < a < b < c < 20 and if b and c are prime numbers, what is the value of a + b?

ক
32
খ
30
গ
28
ঘ
36

ব্যাখ্যা

Question: If a, b, and c are three consecutive odd integers such that 10 < a < b < c < 20 and if b and c are prime numbers, what is the value of a + b?

Solution:
Consecutive odd integers between 10 and 20 are 11,13,15,17,19
b and c are prime numbers.
Since a is at least 11, then b and c must be 17 and 19.

Hence a = 15, b = 17, c = 19.
∴ a + b = 32

১৩,৭৬২.

In a business A and C invested amounts in the ratio 2 : 1 whereas A and B invested amounts in the ratio 3 : 2 . If their annual profit be Tk. 157300, then B's share in the profit is -

ক
ক) Tk. 24200
খ
খ) Tk. 24200
গ
গ) Tk. 48000
ঘ
ঘ) Tk. 48400

ব্যাখ্যা

A:B = 3:2 = 6:4
A:C = 2:1 = 6:3
A:B:C = 6:4:3
∴ B's share = 4/13 × 157300 = 48400

১৩,৭৬৩.

An apple costs 7 taka each. An orange costs 5 taka each. Rasel spends 38 taka on these fruits. The number apple purchased is-

ক
2
খ
3
গ
4
ঘ
Can't be determined

ব্যাখ্যা

Question: An apple costs 7 taka each. An orange costs 5 taka each. Rasel spends 38 taka on these fruits. The number apple purchased is-

Solution:
ধরি, x টি আপেল ও y টি কমলা কিনেছে।

প্রশ্নমতে,
7x + 5y = 38
⇒ 5y = (38 - 7x)
∴ y = (38 - 7x)/5

x = 2, y= (38 - 14)/5 = 24/5; যা পূর্ণসংখ্যা নয়।
x = 3, y= (38 - 21)/5 = 17/5; যা পূর্ণসংখ্যা নয়।
x = 4, y= (38 - 28)/5 = 10/5 = 2; যা পূর্ণসংখ্যা।

১৩,৭৬৪.

What is the simple average of 3³⁰, 3³¹ and 3³²?

ক
3³¹
খ
13(3²⁸)
গ
16(3³⁰)
ঘ
13(3²⁹)

ব্যাখ্যা

Question: What is the simple average of 3³⁰, 3³¹ and 3³²?

Solution:
the simple average of 3³⁰, 3³¹ and 3³² = (3³⁰ + 3³¹ + 3³²)/3
= (3³⁰/3) + (3³¹/3) + (3³²/3)
= 3²⁹ + 3³⁰ + 3³¹
= 3²⁹ (1 + 3 + 3²)
= 3²⁹ (1 + 3 + 9)
= 13(3²⁹)

১৩,৭৬৫.

The perimeter of a rectangular field is 104 meters. If the length of the field is 10 meters more than twice the width, what is the area of that field in square meters?

ক
530
খ
532
গ
580
ঘ
588

ব্যাখ্যা

Question: The perimeter of a rectangular field is 104 meters. If the length of the field is 10 meters more than twice the width, what is the area of that field in square meters?

Solution:
Let,
The width of the rectangular field is x meter
∴ The length of the rectangular field is 2x + 10 meter

ATQ,
2(2x + 10 + x) = 104
⇒ 3x + 10 = 52
⇒ 3x = 42
∴ x = 14

∴ The area of that field is = (2x + 10) × x = (2 × 14 + 10) × 14 = (28 + 10) × 14 square meters
= 38 × 14 square meters
= 532 square meters

১৩,৭৬৬.

Sum one of the interior angles of parallelogram is _____ degree.

ক
ক) 240
খ
খ) 180
গ
গ) 360
ঘ
ঘ) 270

ব্যাখ্যা

সামান্তরিক একটি চতুর্ভুজ
যেকোনো চতুর্ভুজের 4টি কোণের সমষ্টি 360 ডিগ্রি

১৩,৭৬৭.

For 9 innings, Roman has an average of 65 runs. In the tenth inning, he scores 200 runs, thus increasing his average. His average increased by-

ক
78.5
খ
72
গ
13.5
ঘ
77.5

ব্যাখ্যা

Question: For 9 innings, Roman has an average of 65 runs. In the tenth inning, he scores 200 runs, thus increasing his average. His average increased by-

Solution:

Total score for 9 innings = 65 × 9 = 585

Total score after 10^th innings = 585 + 200 = 785

So, the new average is 785/10 = 78.5

So, the increment is 78.5 - 65 = 13.5

১৩,৭৬৮.

Determine the full perimeter of the figure illustrated above.

ক
280
খ
420
গ
260
ঘ
380

ব্যাখ্যা

Question:

Determine the full perimeter of the figure illustrated above.

Solution:

Apply the Pythagorean theorem,
We get: 60²+ x² = 100²

⇒ 3600 + x²= 10000
⇒ x²= 6400
∴ x = 80

∴ Perimeter = 60 + 70 + 100 + 80 + 70 = 380

১৩,৭৬৯.

A trader expects a gain of 15% on his cost price. If in a week his sale is of Tk. 580, then what is his profit?

ক
ক) 75.65
খ
খ) 73.26
গ
গ) 72.50
ঘ
ঘ) 70.78

ব্যাখ্যা

We are given selling price = Tk. 580 and expected profit of 15%
Therefore, we can easily solve this numerical, considering basic formulae of profit and loss.
Let cost price = x
Selling price = C.P. + Profit
S.P. = C.P. + (15% of C.P.) [We know that profit is gained on cost price]
580 = x + (0.15 x)
580 = 1.15 x

Therefore,
x = 504.347
Cost Price = Tk. 504.35
Now,
we have the cost price and hence,
Profit = S.P. – C.P. = 580 – 504.35
= Tk. 75.65
The trader gets a profit of Tk. 75.65

১৩,৭৭০.

If m is the average of the first 7 positive multiples of 4 and if M is the median of the first 7 positive multiples of 4, what is the value of M - m?

ক
0
খ
8
গ
16
ঘ
10

ব্যাখ্যা

Question: If m is the average of the first 7 positive multiples of 4 and if M is the median of the first 7 positive multiples of 4, what is the value of M - m?

Solution:
4-এর প্রথম 7টি ধনাত্মক গুণিতক হলো= {4, 8, 12, 16, 20, 24, 28}
(এখানে পদের সংখ্যা, n = 7)

ধাপ 1: গড় (m) নির্ণয়:
যেহেতু এটি একটি সমান্তর ধারা (Arithmetic Progression), গড় হলো প্রথম ও শেষ পদের গড়।
m = (প্রথম পদ + শেষ পদ)/2
∴ m = (4 + 28)/2 = 32/2 = 16

ধাপ 2: মধ্যমা (M) নির্ণয়:
যেহেতু পদসংখ্যা বিজোড় (n = 7),
∴ মধ্যমা হবে (7 + 1)/2 বা 4র্থ পদ।
4র্থ পদ = 16
∴ M = 16

ধাপ 3: M - m এর মান নির্ণয়:
M - m = 16 - 16
∴ M - m = 0

অতএব, M - m এর মান হলো 0।

Shortcut:
যেহেতু 4-এর গুণিতকগুলো একটি সমান্তর ধারা (Arithmetic Progression) তৈরি করে, তাই যেকোনো সমান্তর ধারার ক্ষেত্রে গড় (m) এবং মধ্যমা (M) সর্বদা সমান হয়। অতএব, M - m = 0 হবে।

১৩,৭৭১.

A shopkeeper expects a gain of 22.5% on his cost price. If in a week, his sale was of Tk. 392, what was his profit?

ক
ক) Tk. 56
খ
খ) Tk. 65
গ
গ) Tk. 72
ঘ
ঘ) Tk. 84

ব্যাখ্যা

C.P. = Tk. {(100/122.5) x 392}
= Tk. {(1000/1225) x 392}
= 320

Profit = Tk. (392 - 320)
= Tk. 72

১৩,৭৭২.

A person 1.8 meters tall sees the top of a tree in a small mirror placed on the ground. The mirror is 1 meter away from the person's feet and 150 meters away from the base of the tree. What is the height of the tree?

ক
250 meters
খ
180 meters
গ
270 meters
ঘ
300 meters

ব্যাখ্যা

Question: A person 1.8 meters tall sees the top of a tree in a small mirror placed on the ground. The mirror is 1 meter away from the person's feet and 150 meters away from the base of the tree. What is the height of the tree?

Solution:

ধরি, মানুষের উচ্চতা, AB = 1.8 m
মানুষ এবং আয়নার দূরত্ব, BC = 1 m
গাছের উচ্চতা, ED = h
গাছ এবং আয়নার দূরত্ব, CD = 150 m

আলোর প্রতিফলনের সূত্র অনুসারে, ∠ACB = ∠ECD (আপতন কোণ = প্রতিফলন কোণ)।
∴ ΔABC এবং ΔEDC সদৃশ।

সদৃশ ত্রিভুজের ধর্ম অনুসারে:
AB/ED = BC/CD
⇒ 1.8/h = 1/150
⇒ h × 1 = 1.8 × 150
⇒ h = 270 m

অতএব, গাছটির উচ্চতা = 270 meters।

১৩,৭৭৩.

If P = 5 + √3, then find the value of P² ?

ক
20 + 10√3
খ
15 + 10√3
গ
25 + 10√3
ঘ
28 + 10√3

ব্যাখ্যা

Question: If P = 5 + √3, then find the value of P²?

Solution:
Given that,
P = 5 + √3
∴ P²= (5 + √3)²
= 5² + 2 × 5 × √3 + (√3)²
= 25 + 10√3 + 3
= 28 + 10√3

১৩,৭৭৪.

A mixture contains two liquids A and B are in the ratio 2 : 1. If 6 litres of mixture is withdrawn and replaced with 6 litres of B, then the ratio becomes 3 : 2. What was the initial quantity of A?

ক
30 litres
খ
40 litres
গ
50 litres
ঘ
60 litres

ব্যাখ্যা

Question: A mixture contains two liquids A and B are in the ratio 2 : 1. If 6 litres of mixture is withdrawn and replaced with 6 litres of B, then the ratio becomes 3 : 2. What was the initial quantity of A?

Solution:
মনে করি,
মিশ্রণের প্রাথমিক পরিমাণ = 3X লিটার

A এর পরিমাণ = 2X লিটার
B এর পরিমাণ = X লিটার

∴ 6 লিটার মিশ্রণ তুলে নেওয়ার পর,
A এর পরিমাণ = 2X - (2/3) × 6 = (2X - 4) লিটার
B এর পরিমাণ = X - (1/3) × 6 = (X - 2) লিটার

আবার,
B তে 6 লিটার যোগ করার পর,
B এর পরিমাণ = X - 2 + 6 = (X + 4) লিটার

প্রদত্ত অনুপাত,
(2X - 4) /(X + 4) = 3/2
বা, 4X - 8 = 3X + 12
বা, X = 12 + 8
∴ X = 20

তাহলে, A এর পরিমাণ = 2 × 20 = 40 লিটার

১৩,৭৭৫.

Two boys start running at the same time in the same direction at a speed of 10 km/hr and 12 km/hr respectively. In what time they will be 8 km apart?

ক
3 hours
খ
6 hours
গ
4 hours
ঘ
5 hours

ব্যাখ্যা

Question: Two boys start running at the same time in the same direction at a speed of 10 km/hr and 12 km/hr respectively. In what time they will be 8 km apart?

Solution:
Both the boys are in motion so we will consider relative speed to find the time.
Relative speed = 12 - 10 = 2 km/hr

Distance = 8 km

Time = 8/2 = 4 hours

১৩,৭৭৬.

A square and a circle have the same perimeter. The side of the length of square is 44 cm, what is the area of the circle?

ক
1656 sq. cm.
খ
2464 sq. cm.
গ
1000 sq. cm.
ঘ
1884 sq. cm.

ব্যাখ্যা

Question: A square and a circle have the same perimeter. The side of the length of square is 44 cm, what is the area of the circle?

Solution:
Perimeter of the square = 4 × side length
= 4 × 44 cm
= 176 cm

As per the question, the square and circle have the same perimeter.
∴ Circumference of the circle = 176 cm
We know that, Circumference of the circle = 2πr
∴ 2πr = 176
⇒ r = 176/(2π)
⇒ r = 88/π
⇒ r = 88/(22/7)
⇒ r = 88 × 7/22
⇒ r = 4 × 7
⇒ r = 28 cm

Area of the circle = πr²
= (22/7) × 28²
= (22/7) × (28 × 28)
= 22 × 4 × 28
= 2464 sq. cm

∴ The area of the circle is 2464 sq. cm.

১৩,৭৭৭.

A right triangle has sides 6 cm, 8 cm, and 10 cm. What is its area?

ক
24 cm²
খ
30 cm²
গ
48 cm²
ঘ
60 cm²

ব্যাখ্যা

Question: A right triangle has sides 6 cm, 8 cm, and 10 cm. What is its area?

Solution:
Given that,
A right triangle with sides are 6 cm, 8 cm, and 10 cm.

We know,
Area = (1/2) × base × height
= (1/2) × 6 × 8
= 24 cm²

So the area of the right triangle is 24cm².

১৩,৭৭৮.

40 workers can build 40 engines working 6 hours a day. How many workers need to be appointed extra to boost the production to double if they work 8 hours a days?

ক
15 workers
খ
10 workers
গ
20 workers
ঘ
25 workers

ব্যাখ্যা

Question: 40 workers can build 40 engines working 6 hours a day. How many workers need to be appointed extra to boost the production to double if they work 8 hours a days?

Solution:
6 hours to build 40 engines by 50 workers
1 hour to build 1 engine by = (40 × 6)/40 workers
8 hours to build 80 engine by = (6 × 80)/8 workers
= 60 workers

∴ extra workers = 60 - 40 = 20 workers

১৩,৭৭৯.

ক
- 2
খ
- 1/2
গ
1/2
ঘ
2
ঙ
3

ব্যাখ্যা

Question:

Solution:

১৩,৭৮০.

A solid metal sphere of radius 10 cm is melted and recast into solid cones of radius 5 cm and height 12 cm. How many cones can be made?

ক
7
খ
10
গ
11
ঘ
13

ব্যাখ্যা

Question: A solid metal sphere of radius 10 cm is melted and recast into solid cones of radius 5 cm and height 12 cm. How many cones can be made?

Solution:
The volume of a sphere = (4/3)πr³
= (4/3)π(10)³
= (4000π/3) cm³

The volume of a cone = (1/3)πr²h
= (1/3)π(5)²(12)
= 100π cm³

Number of cones = (4000π/3)/100π
= 40/3
= 13. 33
= 13 (approx)

১৩,৭৮১.

Given a circle with a 14 cm diameter, what would be the circumference?

ক
4 m
খ
0.92 m
গ
8 m
ঘ
0.44 m

ব্যাখ্যা

Question: Given a circle with a 14 cm diameter, what would be the circumference?

Solution:
Radius of the circle r = 14/2 = 7

The circumference of the circle = 2πr
= 2 × (22/7) × 7
= 44 cm
= 44/100 m
= 0.44 m

১৩,৭৮২.

If a man covers 15.2 km in 4 hours, the distance covered by him in 5 hours is:

ক
ক) 15 km
খ
খ) 17 km
গ
গ) 19 km
ঘ
ঘ) 20 km

ব্যাখ্যা

Question: If a man covers 15.2 km in 4 hours, the distance covered by him in 5 hours is:

Solution:
Speed = (15.2/4) km/hr
= 3.8 km/hr.

Distance covered in 5 hours = ( 3.8 × 5) km
= 19 km.

১৩,৭৮৩.

If sin3A = x, then value of x is-

ক
3sinA + 4sin³A
খ
4sin³A - 3sinA
গ
4sin³A + sinA
ঘ
3sinA - 4sin³A

ব্যাখ্যা

Question: If sin3A = x, then value of x is-

Solution:
sin3A = sin(2A + A)
⇒ sin3A = sin2A.cosA + cos2A.sinA
⇒ sin3A = 2sinA.cosA.cosA + (1 - 2sin²A).sin A
⇒ sin3A = 2sinA(1 - sin²A) + sinA - 2sin³A
⇒ sin3A = 2sinA - 2sin³A + sinA - 2sin³A
∴ sin3A = 3sinA - 4sin³A

১৩,৭৮৪.

If SINGLE = 66, Then DOUBLE =?

ক
59
খ
33
গ
60
ঘ
132

ব্যাখ্যা

Question: If SINGLE = 66, Then DOUBLE =?

Solution:

S + I + N + G + L + E
19 + 9 + 14 + 7 + 12 + 5
= 66

Now,
D + O + U + B + L + E
4 + 15 + 21 + 2 + 12 + 5
= 59

১৩,৭৮৫.

State the equation of the vertical line that goes through the point (-3, 5).

ক
x = 5
খ
y = 5
গ
y = - 3
ঘ
x = - 3

ব্যাখ্যা

Question: State the equation of the vertical line that goes through the point (-3, 5).

Solution:

একটি উল্লম্ব রেখা (vertical line) হলো এমন একটি সরলরেখা যা Y-অক্ষের সমান্তরাল। এই ধরনের রেখার একটি বিশেষ বৈশিষ্ট্য হলো, রেখার উপর অবস্থিত প্রতিটি বিন্দুর x-স্থানাঙ্ক (x-coordinate) একই থাকে, কিন্তু y-স্থানাঙ্ক (y-coordinate) পরিবর্তিত হতে পারে।

উল্লম্ব রেখার সাধারণ সমীকরণ হলো: x = a, যেখানে a একটি ধ্রুবক সংখ্যা এবং রেখার প্রতিটি বিন্দুর x এর মান একই থাকে।

প্রশ্নে বলা হয়েছে রেখাটি (- 3, 5) বিন্দুর মধ্য দিয়ে যায়।
যেহেতু এই বিন্দুর x-স্থানাঙ্ক হলো - 3,
সুতরাং রেখাটির সমীকরণ হবে: x = - 3

১৩,৭৮৬.

Last year, Company X made p dollars in profit. Half of the profit went to the company's founder. The rest was split evenly among his four other partners. In terms of p, how much did each of the other partners receive?

ক
p/4
খ
p/6
গ
2p/5
ঘ
p/8

ব্যাখ্যা

Question: Last year, Company X made p dollars in profit. Half of the profit went to the company's founder. The rest was split evenly among his four other partners. In terms of p, how much did each of the other partners receive?

Solution:
Here,
Company X's profit is p dollars.

Half of the profit went to the founder, so the founder received,
1/2 of p = p/2

Remaining Profit = p - p/2 = p/2

The remaining profit was split among 4 partners.
∴ Each partner gets = (p/2) ÷ 4
= (p/2) × (1/4)
= p/8

১৩,৭৮৭.

The sum of three consecutive even integers is 54. Find the least integer among them.

ক
16
খ
14
গ
12
ঘ
18

ব্যাখ্যা

Question: The sum of three consecutive even integers is 54. Find the least integer among them.

Solution:
Let us consider three consecutive even integers be x, x + 2 and x + 4

According to the question,
x + (x + 2) + (x + 4) = 54
⇒ 3x + 6 = 54
⇒ 3x = 48
⇒ x = 16

Hence, the least integer = 16

১৩,৭৮৮.

Two numbers are in the ratio 2 : 3. If 4 is subtracted from the first number, the ratio becomes 1 : 2. What are the numbers?

ক
14, 21
খ
16, 24
গ
18, 27
ঘ
20, 30

ব্যাখ্যা

Question: Two numbers are in the ratio 2 : 3. If 4 is subtracted from the first number, the ratio becomes 1 : 2. What are the numbers?

Solution:
Let the two numbers be: 2x and 3x

According to the question,
(2x - 4)/3x = 1/2
⇒ 2(2x - 4) = 3x
⇒ 4x - 8 = 3x
⇒ x = 8

∴ First number = 2 × 8 = 16
∴ Second number = 3 × 8 = 24

১৩,৭৮৯.

একটি বর্গের বাহুর দৈর্ঘ্য দ্বিগুণ বৃদ্ধি হলে তার ক্ষেত্রফল কতগুণ বৃদ্ধি পাবে?

ক
ক) ২ গুণ
খ
খ) ৪ গুণ
গ
গ) ৬ গুণ
ঘ
ঘ) ৮ গুণ

ব্যাখ্যা

প্রশ্ন: একটি বর্গের বাহুর দৈর্ঘ্য দ্বিগুণ বৃদ্ধি হলে তার ক্ষেত্রফল কতগুণ বৃদ্ধি পাবে?

সমাধান:
ধরি,
বর্গের বাহুর দৈর্ঘ্য ক একক
∴ বর্গের ক্ষেত্রফল ক^২ বর্গএকক

এখন,
বাহুর দৈর্ঘ্য দ্বিগুণ হলে বাহু হবে ২ক একক
∴ বর্গের ক্ষেত্রফল হবে (২ক)^২ বর্গএকক
= ৪ক^২ বর্গএকক

ক্ষেত্রফল পূর্বের চেয়ে ৪ গুণ বৃদ্ধি পায়।

১৩,৭৯০.

A number increased by 37.5% gives 33. The number is -

ক
ক) 22
খ
খ) 24
গ
গ) 25
ঘ
ঘ) 27

ব্যাখ্যা

Let,
the number be x.
Then, (100 + 37.5)% of x
⇒ 137.5% of x = 33
⇒ 137.5 × (1/100) × x = 33
⇒ x = (33 × 100)/137.5
⇒ x = 24.

১৩,৭৯১.

A dealer purchased a washing machine for Tk.7660. He allows a discount of 12% on its marked price and still gains 10%. Find the marked price of the machine.

ক
ক) 9600 Taka
খ
খ) 8700 Taka
গ
গ) 9500 Taka
ঘ
ঘ) 9575 Taka

ব্যাখ্যা

প্রশ্ন: A dealer purchased a washing machine for Tk.7660. He allows a discount of 12% on its marked price and still gains 10%. Find the marked price of the machine.

সমাধান:
১০% লাভে,
ক্রয়মূল্য ১০০ টাকা হলে বিক্রয়মূল্য ১১০ টাকা
∴ ক্রয়মূল্য ৭৬৬০ টাকা হলে বিক্রয়মূল্য (১১০ × ৭৬৬০)/১০০ টাকা
= ৮৪২৬ টাকা

১২% কমিশনে,
বিক্রয়মূল্য ৮৮ টাকা হলে গায়ে লিখা মূল্য ১০০ টাকা
∴ বিক্রয়মূল্য ৮৪২৬ টাকা হলে গায়ে লিখা মূল্য (১০০ × ৮৪২৬)/৮৮ টাকা
= ৯৫৭৫ টাকা

১৩,৭৯২.

100 liter solution contains 30% salt. How much water should be evaporated to make the solution 40% salt?

ক
20 liter
খ
25 liter
গ
40 liter
ঘ
30 liter
ঙ
None of these

ব্যাখ্যা

Question: 100 liter solution contains 30% salt. How much water should be evaporated to make the solution 40% salt?

Solution:
100 লিটারের 30% লবণ রয়েছে, অর্থাৎ,
(30/100) × 100 = 30 লিটার লবণ

∴ পানির পরিমাণ = 100 - 30 = 70 লিটার

ধরি,
x লিটার পানি বাষ্পীভূত করতে হবে ।
তাহলে নতুন দ্রবণের পরিমাণ হবে, 100 - x লিটার

প্রশ্নমতে,
⇒ 30/(100 - x) = 40/100
⇒ 40(100 - x) = 30 × 100
⇒ 4000 - 40x = 3000
⇒ 40x = 4000 - 3000
⇒ 40x = 1000
⇒ x = 1000/40
⇒ x = 25

∴ 25 লিটার পানি বাষ্পীভূত করতে হবে যাতে লবণের পরিমাণ ৪০% হয়।

১৩,৭৯৩.

If a + b + c = 15 and a² + b² + c² = 75, then find ab + bc + ca.

ক
25
খ
50
গ
75
ঘ
100

ব্যাখ্যা

Question: If a + b + c = 15 and a² + b² + c² = 75, then find ab + bc + ca.

Solution:
Given,
a + b + c = 15
a² + b² + c² = 75

We know,
(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
⇒ 2(ab + bc + ca) = (a + b + c)² - (a² + b² + c²)
⇒ 2(ab + bc + ca) = (15)² - 75
⇒ 2(ab + bc + ca) = 225 - 75
⇒ 2(ab + bc + ca) = 150
∴ ab + bc + ca = 75

১৩,৭৯৪.

In how many ways can 4 people from a group of 7 people be seated around a circular table?

ক
840
খ
210
গ
120
ঘ
280

ব্যাখ্যা

Question: In how many ways can 4 people from a group of 7 people be seated around a circular table?

Solution:
4 people out of 7 = ⁷C₄
= 7!/4!(7 - 4)!
= 7!/(4! × 3!)
= 35

And 4 people around a circular table = (4 - 1)! = 3! = 6

∴ Total ways = 6 × 35 = 210

১৩,৭৯৫.

A pipe can fill a cistern in 6 hour. After adding another pipe the whole process took only 4 hour. The second pipe alone can do it in-

ক
9 hours
খ
10 hours
গ
11 hours
ঘ
12 hours

ব্যাখ্যা

Question: A pipe can fill a cistern in 6 hour. After adding another pipe the whole process took only 4 hour. The second pipe alone can do it in-

Solution:
Let the socond pipe can do the work in X hours
so in one hour it can fill = 1/X of the cistern

the first pipe can do in one hour = 1/6 of the cistern

ATQ,
1/X + 1/6 = 1/4
(6 + X)/6X = 1/4
6X = 24 + 4X
x = 12
∴ the second pipe can fill the cistern in 12 hours.

১৩,৭৯৬.

A committee of 3 men and 2 women is to be formed from 5 men and 4 women. In how many ways can the committee be formed?

ক
120 ways
খ
160 ways
গ
90 ways
ঘ
60 ways

ব্যাখ্যা

Question: A committee of 3 men and 2 women is to be formed from 5 men and 4 women. In how many ways can the committee be formed?

Solution:
We have 5 men and 4 women.
We need to choose 3 men from 5 and 2 women from 4.

∴ Number of ways = ⁵C₃ × ⁴C₂
= {5!/3!(5 - 3)!} × {4!/2!(4 - 2)!}
= {(5 × 4)/2} × {(4 × 3)/2}
= 10 × 6
= 60 ways

১৩,৭৯৭.

Find a number such that when 15 is subtracted from 7 times the number, the result is 10 more than twice the number.

ক
ক) 5
খ
খ) 10
গ
গ) 15
ঘ
ঘ) 20

ব্যাখ্যা

Let, the number be z,
Then, 7z – 15 = 2z + 10
⇒ 5z = 25
⇒ z = 5.
Hence, the required number is 5.

১৩,৭৯৮.

A rectangular field has to be fenced on three sides leaving a side of 20 feet uncovered. If the area of the field is 680 sq. feet, how many feet of fencing will be required?

ক
88
খ
83
গ
79
ঘ
76
ঙ
89

ব্যাখ্যা

Given that,
The area of the field = 680 sq. feet
⇒ lb = 680 sq. feet
Length(l) = 20 feet
⇒ 20 × b = 680
⇒ b = 680/20
= 34 feet

∴ Required length of the fencing = l + 2b
= 20 + (2 × 34)
= 88 feet

১৩,৭৯৯.

If the total surface area of the hemisphere be 48π sq. cm, then its radius is -

ক
ক) 3√2 cm
খ
খ) 4 cm
গ
গ) 6 cm
ঘ
ঘ) 2√3 cm

ব্যাখ্যা

We know surface area of hemisphere = 3πr²
ATQ, 3πr²= 48π
⇒ r²= 16
⇒ r = 4

১৩,৮০০.

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

ক
ক) 4 hrs 15 min
খ
খ) 3 hrs 45 min
গ
গ) 3 hrs 24 min
ঘ
ঘ) 4 hrs 51 min

ব্যাখ্যা

Time taken by one tap to fill half the tank = 3 hrs.
Part filled by one tap in 1 hour = 1/6
Part filled by four taps in 1 hour = (4×1/6) = 2/3
Remaining part = (1−1/2) = 1/2
2/3 of the tank is filled by four taps in 1 hour.
So, 1/2 of the tank is filled in = 3/2×1/2=3/4 hours
3/4 hours = 3/4 × 60 = 45 min
So, the total time taken = 3 hrs + 45 min = 3 hrs 45 min or 225 min
-------------------------------------------------
Alternative way:
A tap can fill a tank in 6 hours.
A tap can fill half of a tank in 6/2 or 3 hours.
4 tap can fill half of a tank in 3/4 hours = 45 min
Total time taken = 3 hours 45 min