Bank Math

PrepBank · পাতা ৩৮ / ১৬১ · ৩,৭০১–৩,৮০০ / ১৬,১২৪

৩,৭০১.

M is a girl and has the same number of brothers as sisters. N is a boy and has twice as many sisters as brothers. M and N are the children of Q. How many children does Q have?

ক
3
খ
4
গ
7
ঘ
9

ব্যাখ্যা

Question: M is a girl and has the same number of brothers as sisters. N is a boy and has twice as many sisters as brothers. M and N are the children of Q. How many children does Q have?

Solution:
Let d and s represent the number of daughters and sons respectively.
Then, we have :
d - 1 = s
and
2(s - 1) = d
2s - 2 = s + 1
2s - s = 1 + 2
s = 3

d = 3 + 1 = 4

Total children = 4 + 3 = 7

৩,৭০২.

The perimeter of a circle measures 28π cm. What is the area of the circle in sq. cm?

ক
49π sq. cm
খ
144π sq. cm
গ
196π sq. cm
ঘ
256π sq. cm

ব্যাখ্যা

Question: The perimeter of a circle measures 28π cm. What is the area of the circle in sq. cm?

Solution:
মনে করি
বৃত্তের ব্যাসার্ধ = r
∴ বৃত্তের পরিধি = 2πr
এবং বৃত্তের ক্ষেত্রফল = πr²

প্রশ্নমতে,
2πr = 28π
⇒ 2r = 28
⇒ r = 14

∴ বৃত্তের ক্ষেত্রফল = πr²
= π(14)²
= 196π sq. cm

৩,৭০৩.

If 9x² - px + 16 is a square number, then p/2 = ?

ক
9
খ
12
গ
16
ঘ
24

ব্যাখ্যা

Question: If 9x² - px + 16 is a square number, then p/2 = ?

Solution:
9x² - px + 16
= (3a)² - 2 · 3a · 4 + 4²
= (3a)² - 24a + 4²

Here, p= 24
∴ p/2 = 24/2 = 12

৩,৭০৪.

What will come at the place of the question mark?
5, 6, 8, 12, 20, ?

ক
36
খ
22
গ
24
ঘ
28

ব্যাখ্যা

Question: What will come at the place of the question mark?
5, 6, 8, 12, 20, ?

Solution:
5 + 1 =6
6 + 2 = 8
8 + 4 = 12
12 + 8 = 20
20 + 16 = 36

৩,৭০৫.

The difference between two numbers is 1365. When the larger number is divided by the smaller one, the quotient is 6 and the remainder is 15. What is the smaller number?

ক
ক) 240
খ
খ) 270
গ
গ) 295
ঘ
ঘ) 360

ব্যাখ্যা

Let the smaller number be x.
Then, a large number
= x + 1365.
∴ x + 1365 = 6x + 15
5x = 1350
x = 270.

৩,৭০৬.

6 men and 8 women can complete a work in 10 days. 26 men and 48 women can finish the same work in 2 days. 15 men and 20 women can do the same work in _____ days.

ক
6 days
খ
4 days
গ
8 days
ঘ
2 days

ব্যাখ্যা

Let work done by 1 man in 1 day = m and work done by 1 woman in 1 day = b
Work is done by 6 men and 8 women in 1 day = 1/10
⇒ 6m + 8b = 1/10
⇒ 60m + 80b = 1 --- (1)
Work done by 26 men and 48 women in 1 day = 1/2
⇒ 26m + 48b = 1/2
⇒ 52m + 96b = 1--- (2)
Solving equation 1 and equation 2.
We get m = 1/100 and b = 1/200
Work is done by 15 men and 20 women in 1 day
= 15/100 + 20/200 =1/4
⇒ Time is taken by 15 men and 20 women in doing the work = 4 days.

৩,৭০৭.

The profit earned by selling a chair for Tk. 750 is 1.5 times the loss occurred when the same chair was sold for Tk. 300. What is the cost price of the chair?

ক
ক) Tk. 360
খ
খ) Tk. 420
গ
গ) Tk. 425
ঘ
ঘ) Tk. 480

ব্যাখ্যা

Question: The profit earned by selling a chair for Tk. 750 is 1.5 times the loss occurred when the same chair was sold for Tk. 300. What is the cost price of the chair?

Solution:
Let the cost price of the chair be x
ATQ,
750 - x = 1.5 (x - 300)
⇒ 7500 - 10x = 15x - 4500
⇒ 15x + 10x = 7500 + 4500
⇒ 25x = 12000
⇒ x = 12000/25
∴ x = 480

∴ The price of the chair is Tk. 480

৩,৭০৮.

A group of students decided to collect as many paise from each member of group as is the number of members. If the total collection amounts to Tk. 59.29, the number of the member is the group is:

ক
ক) 73
খ
খ) 75
গ
গ) 77
ঘ
ঘ) 79

ব্যাখ্যা

Money collected = (59.29 x 100) paise = 5929 paise.
Number of members = √5929 = 77.

৩,৭০৯.

Three bells ring at intervals of 12 sec, 18 sec, and 30 sec. If they start together, how many times will all three ring together in 2 hours?

ক
101
খ
76
গ
61
ঘ
41
ঙ
31

ব্যাখ্যা

Question: Three bells ring at intervals of 12 sec, 18 sec, and 30 sec. If they start together, how many times will all three ring together in 2 hours?

Solution:
Prime factorization:
12 = 2² × 3
18 = 2 × 3²
30 = 2 × 3 × 5

LCM = 2² × 3² × 5
= 4 × 9 × 5
= 180 sec
So, all three bells ring together every 180 seconds.

2 hours = 2 × 60 × 60
= 7200 sec

Number of times = 7200/180 + 1 [As they are starting together, they will ring at 0 sec. So, +1]
= 40 + 1 = 41

৩,৭১০.

Which of the following is a common factor of both x³ + 7x²- x -7 and 2x⁴- x² - 1 polynomials?

ক
(x - 2)(x + 1)
খ
(x-1)(x - 2)
গ
(x+1)(x -1)
ঘ
(x + 1)(x + 2)
ঙ
(x - 2)(x - 3)

ব্যাখ্যা

ধরি, P(x) = x³ + 7x² -x -7
এবং Q(x) = 2x⁴ - x² - 1
এখন, P(-1) = (-1)³+ 7(-1)²-(-1) - 7 = 0
∴ (x + 1), P(x) এর উৎপদক।
Q(-1) = 2(-1)⁴-(-1)² - 1 = 0
∴ (x + 1), Q(x) এর উৎপদক।

আবার, P(1) = (1)³+7(1)²-1-7 = 0
∴ (x-1), P(x) এর উৎপদক।
Q(1) = 2(1)⁴-(1)²-1 = 0
∴ (x - 1), Q(x) এর উৎপদক।

∴ (x + 1) এবং (x - 1) উভয়েই x³ + 7x² - x - 7 এবং 2x4 - x² - 1 বহুপদীদ্বয়ের উৎপাদক।

৩,৭১১.

A number reduced by 25% becomes 300. What percent should it be increased so that it becomes 460?

ক
ক) 15%
খ
খ) 20%
গ
গ) 25%
ঘ
ঘ) 30%

ব্যাখ্যা

Question: A number reduced by 25% becomes 300. What percent should it be increased so that it becomes 460?

Solution:
Le the number be x
Then,
x - 25% of x = 300
⇒ x - (25x/100) = 300
⇒ 3x/4 = 300
⇒ 3x = 1200
∴ x = 400

∴ Increase = (460 - 400) = 60

The percent should be increased = {(60 × 100)/400}%
= 15%

৩,৭১২.

An individual is cycling at a speed of 30 km per hour. He catches his predecessor who had started earlier in three hours. What is the speed of his predecessor who had started 3 hours earlier?

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

ব্যাখ্যা

Question: An individual is cycling at a speed of 30 km per hour. He catches his predecessor who had started earlier in three hours. What is the speed of his predecessor who had started 3 hours earlier?

Solution:
The distance covered in three hour = 3 × 30
= 90 km

Time taken by first individual = (3h + 3h) = 6h

Then, the speed of predecessor, = 90/6 kmph
= 15 kmph.

৩,৭১৩.

A trader mixes 26 kg of rice at Tk. 20 per kg with 30 kg of rice of other variety at Tk. 36 per kg and sells the mixture at Tk. 28 per kg. His profit or loss percent is-

ক
2% loss
খ
5% profit
গ
2% profit
ঘ
3% loss

ব্যাখ্যা

Question: A trader mixes 26 kg of rice at Tk. 20 per kg with 30 kg of rice of other variety at Tk. 36 per kg and sells the mixture at Tk. 28 per kg. His profit or loss percent is-

Solution:
C.P. of 56 kg rice = (26 × 20 + 30 × 36)
= (520 + 1080)
=1600

S.P. of 56 kg rice = (56 × 28)
= 1568

∴ Loss = 1600 - 1568
= 32

∴ Loss percentage = (32/1600) × 100% = 2%

৩,৭১৪.

A ladder 26 m long is placed against a wall of height 13 m such that it just touches the top of the wall. Find the angle of elevation made by the ladder with the ground.

ক
45°
খ
90°
গ
60°
ঘ
30°

ব্যাখ্যা

Question: A ladder 26 m long is placed against a wall of height 13 m such that it just touches the top of the wall. Find the angle of elevation made by the ladder with the ground.

Solution:

AC = 26 meter
AB = 13 meter
∠ACB = θ

∴ sin θ = AB/AC = 13/26 = 1/2
⇒ sin θ = sin 30°

∴ θ = 30°

৩,৭১৫.

Shefali has a rectangular piece of cloth with dimensions 20 m and 15 m. She wants to paint a border of breadth 4 m inside the four sides of the rectangle. The paint would cost her Tk. 6 per sq m. Find the cost of painting the complete border.

ক
ক) Tk. 1083
খ
খ) Tk. 1296
গ
গ) Tk. 1500
ঘ
ঘ) Tk. 1548

ব্যাখ্যা

Paint Area = Total area - Non-Paint area
Subtracting width of the border from all sides we get,
Length = 20 - 4 - 4 = 12
Breadth = 15 - 4 - 4 = 7

∴ Paint Area = (20 x 15) - (12 x 7) = 300 - 84 = 216

Total Cost of painting the border = Rate x Area
= 6 x 216
= Tk. 1296

৩,৭১৬.

If 1 is added to the numerator of a fraction, the fraction becomes 1. If 1 is added to the denominator, the fraction becomes 1/2. Find the fraction.

ক
1/4
খ
1/2
গ
2/3
ঘ
3/8

ব্যাখ্যা

Question: If 1 is added to the numerator of a fraction, the fraction becomes 1. If 1 is added to the denominator, the fraction becomes 1/2. Find the fraction.

Solution:
ধরি, ভগ্নাংশটি = x/y

শর্তমতে,
(x + 1) / y = 1
⇒ x + 1 = y ............(1)

আবার,
x / (y + 1) = 1/2
⇒ 2x = y + 1
⇒ 2x - 1 = y ............(2)

(1) ও (2) থেকে,
2x - 1 = x + 1
⇒ x = 2

x = 2 হলে,
y = x + 1 = 2 + 1 = 3

∴ ভগ্নাংশটি = 2/3

৩,৭১৭.

If x² + 1/x² = 34, then x +1/x =?

ক
3
খ
4
গ
5
ঘ
None of these

ব্যাখ্যা

Question: If x² + 1/x² = 34, then x +1/x =?

Solution:
x² + 1/x² = 34
⇒ (x + 1/x)² - 2.x.(1/x) = 34
⇒ (x + 1/x)² = 36
∴ (x +1/x) = ± 6

৩,৭১৮.

Mijan,Jewel, and Tasif rent a pasture. Mijan puts 10 oxen for 7 months, Jewel puts 12 oxen for 5 months and Tasif puts 15 oxen for 3 months for grazing. If the rent of the pasture is Tk. 1750, how much must Tasif pay as his share of rent?

ক
ক) Tk. 700
খ
খ) Tk. 600
গ
গ) Tk. 450
ঘ
ঘ) Tk. 900

ব্যাখ্যা

প্রশ্ন: Mijan,Jewel, and Tasif rent a pasture. Mijan puts 10 oxen for 7 months, Jewel puts 12 oxen for 5 months and Tasif puts 15 oxen for 3 months for grazing. If the rent of the pasture is Tk. 1750, how much must Tasif pay as his share of rent?

সমাধান:
Mijan : Jewel : Tasif = (10 × 7) : (12 × 5) : (15 × 3)
= 70 : 60 : 45
= 14 : 12 : 9

∴ Tasif's share of rent = 1750 × (9/35) = 450.

৩,৭১৯.

The difference of the ages of Raju and Manik is 21 years and the product of their age is 72 years. Find the ratio of the ages of Raju and Manik.?

ক
8:1
খ
6:5
গ
7:4
ঘ
2:3

ব্যাখ্যা

Let a be the age of Raju and b be the age of manik.
According to the question,
a - b = 21 ...(1) and ab = 72
⇒ b = 72/a
Then (1) becomes, a - 72/a = 21
⇒ a² - 72 = 21a
⇒ a² - 21a - 72 = 0
⇒ a² - 24a + 3a -72 = 0
⇒ a(a - 24) + 3(a - 24) = 0
⇒ (a - 24)(a + 3) = 0
⇒ a = 24 or a = -3
Since age cannot be a negative number, the age of Raju will be 24 years
Therefore b = 3.
Hence the age of manik is 3 years
Then the required ratio = Raju/manik = 24/3 = 8/1
Hence the answer is 8:1

৩,৭২০.

If a sum of money trebles itself in 40 years, what is the rate of interest?

ক
7%
খ
4%
গ
9%
ঘ
3%
ঙ
5%

ব্যাখ্যা

Let the sum of money be x.
x becomes 3x in 40 years.
Simple interest = 3x - x
= 2x
R = (100 × 2x)/(x × 40)
= 5%

৩,৭২১.

A barge, when empty, travels twice as fast as when it is full. If it travel 20 miles north with a cargo, spends 30 minutes for unloading and returns to its original port empty to complete the entire trip taking 8 hours, what is the speeds of the barge when it is empty?

ক
12mph
খ
8mph
গ
10mph
ঘ
15mph
ঙ
.

ব্যাখ্যা

Question: A barge, when empty, travels twice as fast as when it is full. If it travel 20 miles north with a cargo, spends 30 minutes for unloading and returns to its original port empty to complete the entire trip taking 8 hours, what is the speeds of the barge when it is empty?

Solution:
ধরি,
Barge এর মাল বোঝাই অবস্থায় গতি বেগ x মাইল /ঘণ্টা
Barge এর খালি অবস্থায় গতি বেগ 2x মাইল/ঘণ্টা

মাল বোঝাই অবস্থায়
Barge এর 20 মাইল যেতে সময় লাগে = 20/x ঘণ্টা

খালি অবস্থায়
Barge এর 20 মাইল যেতে সময় লাগে =20/2x = 10/x ঘণ্টা

প্রশ্নমতে,
20/x + 10/x + 30/60 = 8
20/x + 10/x + 1/2 = 8
(40 + 20 + x)/2x = 8
16x = 60 + x
16x - x = 60
15x = 60
x = 4

Barge এর খালি অবস্থায় গতি বেগ = 2x মাইল/ঘণ্টা
= 2 × 4 মাইল/ঘণ্টা
= 8 মাইল/ঘণ্টা

৩,৭২২.

If the average of p numbers is q² and the average of q numbers is p², find the average of all (p + q) numbers.

ক
p + q
খ
pq
গ
p² + q²
ঘ
p²q²

ব্যাখ্যা

Question: If the average of p numbers is q² and the average of q numbers is p², find the average of all (p + q) numbers.

Solution:
Sum of p numbers = p × q²Sum of q numbers = q × p²Total sum = pq² + qp² = pq(p + q)

Total numbers = p + q

∴ Average of all (p + q) numbers = Total sum/Total numbers
= [pq(p + q)]/(p + q)
= pq

৩,৭২৩.

Find the missing number.

ক
ক) 10
খ
খ) 25
গ
গ) 50
ঘ
ঘ) 100

ব্যাখ্যা

Find the missing number.

Solution:
১ম চিত্রে, ১৬ = ৮ × ২
৫০ = ২৫ × ২
২৬ = ১৩ × ২

দ্বিতীয় চিত্রে,
৮ = ৮ × ১
২৫ = ২৫ × ১
১৩ = ১৩ × ১

তৃতীয় চিত্রে,
৩২ = ৮ × ৪
৫২ = ১৩ × ৪
তাহলে খালি ঘরে ২৫ × ৪ = ১০০ বসবে।

৩,৭২৪.

A train 900 meter long is running at a speed of 90 km/h. If it crosses a tunnel in 1 min, then the length of the tunnel is-

ক
500 meter
খ
600 meter
গ
700 meter
ঘ
800 meter

ব্যাখ্যা

Question: A train 900 meter long is running at a speed of 90 km/h. If it crosses a tunnel in 1 min, then the length of the tunnel is-

Solution:
Let
the length of the tunnel = x meter
Then, distance = (900 + x) meter
Time = 1 minute = 60 seconds

∴ Speed = 90 km/h
= {90 × (5/18)} m/s
= 25 m/s

ATQ,
900 + x = 60 × 25
⇒ 900 + x = 1500
∴ x = 600

So, the length of the tunnel is 600 meter

৩,৭২৫.

Rice worth Tk. 126 per kg and Tk. 134 per kg are mixed with a third variety in the ratio 1 : 1 : 2. If the mixture is worth Tk. 177 per kg, the price of the third variety per kg will be:

ক
ক) Tk 224
খ
খ) Tk 200
গ
গ) Tk 250
ঘ
ঘ) Tk 150
ঙ
ঙ) Tk 75

ব্যাখ্যা

Let, the price of third variety = x
ATQ,
Or, 126×1 + 134×1 + x×2 = 177×4
Or, 260 + 2x = 708
Or, 2x = 448
Or, x = 224 [Answer.]

৩,৭২৬.

A vessel is filled with liquid, 3 parts of which are water and 5 parts syrup. How much of the mixture must be drawn off and replaced with water so that the mixture may be half water and half syrup?

ক
1/3
খ
1/4
গ
1/5
ঘ
1/7

ব্যাখ্যা

Question: A vessel is filled with liquid, 3 parts of which are water and 5 parts syrup. How much of the mixture must be drawn off and replaced with water so that the mixture may be half water and half syrup?

Solution:
Suppose the vessel initially contains 8 litres of liquid.
Let x litres of this liquid be replaced with water.

Quantity of water in new mixture = {3 - (3x/8) + x} litres

Quantity of syrup in new mixture = (5 - 5x/8) litres

ATQ,
{3 - (3x/8) + x} = (5 - 5x/8)
⇒ 5x + 24 = 40 - 5x
⇒ 10x = 16
∴ x = 8/5

So, part of the mixture replaced = (8/5) × (1/8) = 1/5

৩,৭২৭.

6⁴ⁿ - 5⁴ⁿ, where n is an integer > 0, is divisible by-

ক
11
খ
15
গ
25
ঘ
31

ব্যাখ্যা

Question: 6⁴ⁿ - 5⁴ⁿ, where n is an integer > 0, is divisible by-

Solution:
6⁴ⁿ - 5⁴ⁿ

Assume that n = 1
= 6⁴ - 5⁴
= (6²)² - (5²)²
= (6² - 5²)(6² + 5²)
= (36 - 25) × (36 + 25)
= 11 × 61
The expression is divisible by 11 (and also 61 when n = 1)

৩,৭২৮.

Of three numbers, the average of the first and second numbers is 12 more than the average of the second and third numbers. What is the difference between the first and third numbers?

ক
30
খ
26
গ
24
ঘ
22

ব্যাখ্যা

Question: Of three numbers, the average of the first and second numbers is 12 more than the average of the second and third numbers. What is the difference between the first and third numbers?

Solution:
Let,
these numbers are x, y and z respectively.

ATQ,
{(x + y)/2} - {(y + z)/2} = 12
⇒ {(x + y) - (y + z)}/2 = 12
⇒ (x + y - y - z)/2 = 12
∴ x - z = 24

∴ the difference between the first and the third number is = 24.

৩,৭২৯.

A circular garden with a diameter of 20 meters is surrounded by a walkway of width 1 meter. What is the area of the walkway?

ক
41πm²
খ
41m²
গ
21πm²
ঘ
21m²

ব্যাখ্যা

Question: A circular garden with a diameter of 20 meters is surrounded by a walkway of width 1 meter. What is the area of the walkway?

Solution:
Given that,
Radius of the garden = 20/2 = 10 m
Width of walkway = 1 m
So, radius of the outer circle (garden + walkway) = 10 + 1 = 11 m

We know,
Area of circle = πr²

Now,
Area of outer circle = π × (11²) = π × 121 = 121π m²And,
Area of inner circle (garden only) = π × (10²)= π × 100 = 100π m²

Now,
Area of walkway = Area of outer circle - Area of inner circle = (121π - 100π) m² = 21π m²

So the area of the walkway = 21π m²

৩,৭৩০.

The average of 50 numbers is 30. If two numbers, 35 and 40 are discarded, then the average of the remaining numbers is nearly:

ক
19.68
খ
20
গ
29.68
ঘ
23

ব্যাখ্যা

Question: The average of 50 numbers is 30. If two numbers, 35 and 40 are discarded, then the average of the remaining numbers is nearly:

Solution:

The total sum of 48 numbers,

= (50 × 30) - (35 +40)

= 1500 - 75

= 1425

Average = 1425/48 = 29.68

৩,৭৩১.

3 pumps working 8 hours a day, can empty a tank in 2 days. How many hours a day must 4 pumps work to empty the tank in 1 day?

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

ব্যাখ্যা

Question: 3 pumps working 8 hours a day, can empty a tank in 2 days. How many hours a day must 4 pumps work to empty the tank in 1 day?

Solution:
3 pumps need 2 × 8 hours = 16 hours
1 pump needs 16 × 3 hours
4 pumps need (16 × 3)/4 hours
= 12 hours

৩,৭৩২.

A card is drawn from a pack of 52 cards. The probability of getting a queen of club or a king of heart is?

ক
ক) 1/52
খ
খ) 3/26
গ
গ) 3/52
ঘ
ঘ) 1/26

ব্যাখ্যা

Here,
n(S) = 52
Let E = event of getting a queen of club or a king of heart.
Then, n(E)= 2

P(E)=n(E)/n(S)
= 2/52
= 1/26

৩,৭৩৩.

Product of present age of Rahim and Latif is 2223 years and their present age ratio is 19 : 13 find the difference age of Rahim and Latif.

ক
18 years
খ
21 years
গ
16 years
ঘ
23 years

ব্যাখ্যা

Question: Product of present age of Rahim and Latif is 2223 years and their present age ratio is 19 : 13 find the difference age of Rahim and Latif.

Solution:
Let, age of Rahim be 19x and age of Latif be 13x
Then, product of their ages = 19x × 13x = 247x²

Now,
247x² = 2223
⇒ x² = 9
∴ x = 3

Hence, required difference = 19x - 13x = 6x = 6 × 3 = 18 years

৩,৭৩৪.

How many different ways can 4 students line up to purchase a new laptop?

ক
ক) 12
খ
খ) 20
গ
গ) 24
ঘ
ঘ) 30

ব্যাখ্যা

Question: How many different ways can 4 students line up to purchase a new laptop?

Solution:
4 students line up to purchase a new laptop
= 4!
= 24

৩,৭৩৫.

Shejan deposited Tk. 10000 for 4 year at a rate of 6% p.a. Find the amount Shejan got after 4 years.

ক
Tk. 12600
খ
Tk. 12350
গ
Tk. 12000
ঘ
Tk. 12400

ব্যাখ্যা

Question: Shejan deposited Tk. 10000 for 4 year at a rate of 6% p.a. Find the amount Shejan got after 4 years.

Solution:
Principal (P) = 10000,
Time (n) = 4 years,
Rate (r) = 6% p.a.

Simple Interest (SI) = {Principal (P) × Rate (r) × Time (n)}/100
= (10000 × 6 × 4)/100
= 2400

Amount (A) = Principal (P) + Interest (I)
= 10000 + 2400
= 12400

৩,৭৩৬.

5^{3x - 2} = 625, find the value of x.

ক
0
খ
3
গ
1
ঘ
2

ব্যাখ্যা

Question: 5^{3x - 2} = 625, find the value of x.

Solution:
5^{3x - 2} = 625
⇒ 5^{3x - 2} = 5⁴
⇒ 3x - 2 = 4
⇒ 3x = 6
∴ x = 2

৩,৭৩৭.

In a lottery, there are 10 prizes and 25 blanks. A lottery is drawn at random. What is the probability of not getting a prize?

ক
2/7
খ
5/7
গ
1/5
ঘ
1/2
ঙ
None of the above

ব্যাখ্যা

Total number of outcomes possible, n(S) = 10 + 25 = 35
Total number of prizes, n(E) = 10
Probability of getting prize P(E) = n(E)/n(S)
= 10/35
= 2/7
Probability of not getting prize = 1 - (2/7)
= 5/7

৩,৭৩৮.

If the average (arithmetic mean) of x and y is 60 and the average (arithmetic mean) of y and z is 80, what is the value of (z-x)?

ক
ক) 70
খ
খ) 40
গ
গ) 20
ঘ
ঘ) None

ব্যাখ্যা

এখানে (x + y)/2 = 60 বা, x + y = 120…..(1)
এবং (y + z)/2 = 80 বা, y + z = 160…..(2)
(2) নং সমীকরণ থেকে (1) নং বিয়োগ করে পাই,
z - x = 40

৩,৭৩৯.

The LCM and HCF of two numbers are 90 and 15 respectively. If one number is 45, the other number is -

ক
ক) 30
খ
খ) 60
গ
গ) 15
ঘ
ঘ) 75

ব্যাখ্যা

Let, the other number be x

We know, HCF×LCM = product of two numbers
Or, 90×15 = 45×x
Or, x = 1350/45 = 30

So, 30 is the other number

৩,৭৪০.

If 2 kg of metal, of which 1/3 is zinc and the rest is copper be mixed with 3 kg of metal of which 1/4 is zinc and the rest is copper, What is the ratio of zinc to copper in the mixture?

ক
23 : 39
খ
5 : 9
গ
17 : 43
ঘ
3 : 7

ব্যাখ্যা

Question: If 2 kg of metal, of which 1/3 is zinc and the rest is copper be mixed with 3 kg of metal of which 1/4 is zinc and the rest is copper, What is the ratio of zinc to copper in the mixture?

Solution:
Zinc = {2 × (1/3)} + {3 × (1/4)} = 17/12
Copper = 5 - (17/12) = 43/12

∴ Zinc : Copper = (17/12) : (43/12)
= 17 : 43

৩,৭৪১.

An ingoing pipe was used to fill a tank. But while using an outgoing pipe that can pour water in 10 hours at the same time, it took 8 hours to fill the tank. The ingoing pipe can fill the tank alone in -

ক
30/9 hours
খ
40/11 hours
গ
20/3 hours
ঘ
40/9 hours

ব্যাখ্যা

Question: An ingoing pipe was used to fill a tank. But while using an outgoing pipe that can pour water in 10 hours at the same time, it took 8 hours to fill the tank. The ingoing pipe can fill the tank alone in -

Solution:
Let,
the ingoing pipe can fill the tank in X hours.
so, total fill-up in one hour = 1/X - 1/10
= (10 - X)/10X
ATQ,
10X/(10 - X) = 8
10X = 80 - 8X
18X = 80
X = 80/18 = 40/9 hours

৩,৭৪২.

Two-fifth of one-third of three-seventh of a number is 15. What is 40 percent of that number?

ক
72
খ
84
গ
136
ঘ
None of these

ব্যাখ্যা

Question: Two-fifth of one-third of three-seventh of a number is 15. What is 40 percent of that number?

Solution:
let the number be x

(2/5) × (1/3) × (3/7) x = 15
⇒ x = (15 × 35)/2

40% of x = {(15 × 35)/2 } × .4
= {(15 × 35)/2} × (4/10)
= 105

৩,৭৪৩.

Three dice are thrown together. Find the probability of getting a total of at least 6?

ক
ক) 103/108
খ
খ) 103/208
গ
গ) 103/216
ঘ
ঘ) 36/103

ব্যাখ্যা

Total number of events occure when the dice are thrown = 6 × 6 × 6 = 216
Let A be the event of getting a total of at least 6 and B denoted event of getting a total of less than 6 i.e.,3,4,5.

So, B = {(1,1,1), (1,1,2), (1,2,1), (2,1,1), (1,1,3), (1,3,1), (3,1,1), (1,2,2), (2,1,2), (2,2,1)}
Favorable number of cases = 10
So, P(B) = 10/216 = 5/108
We know, P(A) + P(B) = 1
∴ P(A) = 1 – P(B)
= 1 – (5/108)
= 103/108

৩,৭৪৪.

A tap can fill a cistern in 8 hours. Due to a drain pipe in the bottom, it takes 10 hours to fill the same cistern. If the cistern is full, how much time will the drain pipe take to empty it?

ক
21 hours
খ
27 hours
গ
35 hours
ঘ
40 hours

ব্যাখ্যা

Question: A tap can fill a cistern in 8 hours. Due to a drain pipe in the bottom, it takes 10 hours to fill the same cistern. If the cistern is full, how much time will the drain pipe take to empty it?

সমাধান:
ট্যাপ দ্বারা 1 ঘন্টায় পূর্ণ হয় 1/8 অংশ।
নল ও ছিদ্র দ্বারা একত্রে 1 ঘন্টায় পূর্ণ হয় 1/10 অংশ।

∴ ছিদ্র দ্বারা 1 ঘন্টায় খালি হয় = (ট্যাপের কাজ - যৌথ কাজ)
= (1/8 - 1/10) অংশ
= (5 - 4)/40 অংশ
= 1/40 অংশ।

∴ ছিদ্রটি সম্পূর্ণ চৌবাচ্চাটি খালি করতে 40 ঘন্টা সময় নেবে।

৩,৭৪৫.

ক
1
খ
1/2
গ
0
ঘ
∞

ব্যাখ্যা

Question:

Solution:

৩,৭৪৬.

A, B, and C can do a piece of work in 20, 30 and 60 days respectively. In how many days can A do the work if he is assisted by B and C on every third day?

ক
12 days
খ
15 days
গ
16 days
ঘ
18 days

ব্যাখ্যা

A's 2 day's work = (1/20) × 2
= 1/10
(A + B + C)'s 1 day's work = (1/20) + (1/30) + (1/60)
= 6/60
= 1/10
(A + B + C) work in 3 days = (1/10) + (1/10)
= 1/5.
Now, 1/5 work is done in 3 days.
∴ Whole work is done in (3 × 5)
= 15 days.

৩,৭৪৭.

Out of 10 persons working on a project, 4 are graduated. If 3 are selected, what is the probability that there is at least one graduate among them?

ক
ক) 1/6
খ
খ) 5/8
গ
গ) 3/8
ঘ
ঘ) 5/6

ব্যাখ্যা

Question: Out of 10 persons working on a project, 4 are graduated. If 3 are selected, what is the probability that there is at least one graduate among them?

Solution:
মোট কর্মীর সংখ্যা 10 জন
গ্র্যাজুয়েট কর্মীর সংখ্যা = 4 জন

কমপক্ষে 1 জন গ্র্যাজুয়েট হওয়ার সম্ভাবনা = 1 - নন গ্র্যাজুয়েটহওয়ার সম্ভাবনা
= 1 - (⁶C₃/¹⁰C₃)
= 1 - (20/120)
= (6 - 1)/6
= 5/6

৩,৭৪৮.

A takes twice as much time as B or thrice as much time as C to finish a piece of work. Working together, they can finish the work in 2 days. B can do the work alone in:

ক
4 days
খ
6 days
গ
8 days
ঘ
12 days

ব্যাখ্যা

Question: A takes twice as much time as B or thrice as much time as C to finish a piece of work. Working together, they can finish the work in 2 days. B can do the work alone in:

Solution:
Let
A, B and C take x, x/2 and x/3 days respectively to finish the work.

Now
(1/x) + (2/x) + (3/x) = 1/2
(1 + 2 + 3)/x = 1/2
⇒ 6/x = 1/2
⇒ x = 12.

So, B takes to finish the work = 12/2 days

৩,৭৪৯.

How many times are the short hand and the long hand of a clock of a right angles during their motion from 1:00 pm to 10:00 pm?

ক
10
খ
14
গ
18
ঘ
22

ব্যাখ্যা

Question: How many times are the short hand and the long hand of a clock of a right angles during their motion from 1:00 pm to 10:00 pm?

Solution:

The duration from 1:00 pm to 10:00 pm is 9 hours.
During each of these 9 hours, the hands of the clock are at right angles twice.

So, required number = 9 × 2 = 18

৩,৭৫০.

If √a = √4 - √5 then the value of a² - 18a + 81 is?

ক
160
খ
120
গ
80
ঘ
81

ব্যাখ্যা

Question: If √a = √4 - √5 then the value of a² - 18a + 81 is?

Solution:
Given,
√a = √4 - √5
⇒ (√a)²= (√4 - √5)²
⇒ a = (√4)² - 2 . √4 .√5 + (√5)²
⇒ a = 4 - 2√20 + 5
⇒ a = 9 - 2√20
⇒ a - 9 = - 2√20
⇒ (a - 9)² = (- 2√20)²⇒ a² - 2 . a . 9 + 9² = 80
∴ a² - 18a + 81 = 80

৩,৭৫১.

A salesman receives daily wage of Tk. 450 and earns a commission of 10% on all sales he makes. How much taka worth of sales does he need to make in order to bring his total daily income of Tk. 1200?

ক
Tk. 9000
খ
Tk. 8500
গ
Tk. 7500
ঘ
Tk. 6500

ব্যাখ্যা

Question: A salesman receives daily wage of Tk. 450 and earns a commission of 10% on all sales he makes. How much taka worth of sales does he need to make in order to bring his total daily income of Tk. 1200?

Solution:
He has to earn only with commission (1200 - 450) Taka = 750 Taka

He earns 10 taka when sells 100 taka
∴ He earns 1 taka when sells 100/10 taka = 10 taka
∴ He earns 750 taka when sells 10 × 750 taka = 7500 taka

৩,৭৫২.

If x² − 8x +15 < 0 then solve the inequality-

ক
3 < x < 5
খ
- 3 < x > 5
গ
1 > x > 6
ঘ
- 2 < x > 9

ব্যাখ্যা

Question: If x² − 8x +15 < 0 then solve the inequality-

Solution:
Given,
x² − 8x +15 < 0
⇒ x² - 3x - 5x + 15 < 0
⇒ x(x - 3) - 5(x - 3) < 0
⇒ (x - 3)(x - 5) < 0

The inequality will be true if x - 3 > 0 and x - 5 < 0 .
x - 3 > 0
or, x > 3

x - 5 < 0
or, x < 5
The inequality will be true if 3 < x < 5

∴ The solution of the inequality is 3 < x < 5

৩,৭৫৩.

If 7x - 3y = 31 and 9x - 5y = 41, then (x, y) =?

ক
(- 7, - 3)
খ
(- 4, 3)
গ
(5, 1)
ঘ
(4, - 1)
ঙ
(- 5, 3)

ব্যাখ্যা

Question: If 7x - 3y = 31 and 9x - 5y = 41, then (x, y) =?

Solution:
7x - 3y = 31 ................(1)
9x - 5y = 41 ...............(2)

(1) × 5 - (2) × 3 ⇒
35x - 15y - 27x + 15y = 155 - 123
8x = 32
x = 32/8
x = 4

(1) ⇒
7x - 3y = 31
28 - 3y = 31
3y = - 3
y = - 1

(x, y) = (4, - 1)

৩,৭৫৪.

Which one of the following is the minimum value of sum of two integers whose product is 64?

ক
ক) 16
খ
খ) 18
গ
গ) 34
ঘ
ঘ) 20

ব্যাখ্যা

এখানে,
4 + 16 = 18 এবং 4 × 16= 64
2 + 32= 34 এবং 2 × 32= 64
8 + 8 = 16 এবং 8 × 8 = 64

৩,৭৫৫.

The ratio of ages of Mary and Maria is 4 : 5. After 12 years their ratio becomes 5 : 6. What will be the age of Mary after 2 years?

ক
ক) 45 years
খ
খ) 48 years
গ
গ) 50 years
ঘ
ঘ) 56 years

ব্যাখ্যা

Since the ages of Mary and Maria are in the ratio 4 : 5,
the actual ages can be taken as 4x years and 5x years respectively.
After 12 years, their ages will be 4x + 12 years and 5x + 12 years.
Therefore, as per the problem (4x + 12)/(5x + 12) = 5/6 => x = 12
Mary's age now is 4x years = 48 years.
So after 2 years Mary will be 50 years.

৩,৭৫৬.

The ratio between the perimeter and the breadth of a rectangle is 5 : 1. If the area of the rectangle is 216 sq. cm, what is the length of the rectangle?

ক
16 cm
খ
18 cm
গ
24 cm
ঘ
20 cm

ব্যাখ্যা

Question: The ratio between the perimeter and the breadth of a rectangle is 5 : 1. If the area of the rectangle is 216 sq. cm, what is the length of the rectangle?

Solution:
Let the length of the rectangle = L cm
Let the breadth of the rectangle = B cm

Given that,
Ratio of perimeter to breadth = 5 : 1
Area of the rectangle = 216 cm²

Perimeter of rectangle = 2(L + B)
So, according to the ratio,
⇒ 2(L + B)/B = 5/1
⇒ 2(L + B) = 5B
⇒ 2L + 2B = 5B
⇒ 2L = 5B − 2B
⇒ 2L = 3B
∴ L = (3/2)B .........(1)

Also given that,
Area = L × B = 216
∴ L × B = 216 ......... (2)

Substitute L from equation (1) into equation (2). Then we get,
⇒ (3/2)B × B = 216
⇒ (3/2)B² = 216
⇒ B² = 216 × (2/3)
⇒ B² = 144 = 12²
∴ B = 12 cm
Now find length, L = (3/2) × 12 = 18 cm

So, the length of the rectangle is 18 cm.

৩,৭৫৭.

যদি a² - √11a + 1 = 0 হয়, তবে a² + a^{- 2} এর মান কত?

ক
3
খ
6
গ
7
ঘ
9
ঙ
0

ব্যাখ্যা

প্রশ্ন: যদি a² - √11a + 1 = 0 হয়, তবে a² + a^{- 2} এর মান কত?

সমাধান:
a² - √11a + 1= 0
⇒ a² + 1 = √11a
⇒ (a²/a) + (1/a) = √11a/a
⇒ a + (1/a) = √11

প্রদত্ত রাশি = a² + a^{- 2}
= a² + (1/a²)
= {a + (1/a)}²- 2 ⋅ a ⋅ (1/a)
= (√11)² - 2
= 11 - 2
= 9

৩,৭৫৮.

The difference between two numbers is 1550. If 8 % of one number is 10 % of the other number, then find the two numbers.

ক
ক) 4973, 6523
খ
খ) 5450, 7000
গ
গ) 6200, 7750
ঘ
ঘ) 6500, 4950

ব্যাখ্যা

Question: The difference between two numbers is 1550. If 8 % of one number is 10 % of the other number, then find the two numbers.

Solution:
Let two numbers be x and y.
It is given that, 8 % of x = 10 % of y
Therefore,
(8x)/100 = (10y)/100
⇒ 8x = 10y
⇒ x = (10y)/8
∴ x = (5y)/4

Difference between two numbers (x - y) = 1550
Substituting the value of x, we get,
(5y)/4 - y = 1550
⇒ y/4 = 1550
⇒ y =1550 × 4
∴ y = 6200

And,
x = (5 × 6200)/4 = 7750

৩,৭৫৯.

Two trains running in opposite directions cross a man standing on the platform in 27 seconds and 17 seconds respectively and they cross each other in 23 seconds. The ratio of their speeds is-

ক
ক) 1:3
খ
খ) 3:2
গ
গ) 3:4
ঘ
ঘ) None of these

ব্যাখ্যা

Let the speeds of the two trains be x m/sec and y m/sec respectively.
Then, length of the first train = 27x metres
and length of the second train = 17y metres
∴ 27x+17y/(x+y) = 23
⇒ 27x+17y = 23x+23y
⇒ 4x=6y
⇒ x/y= 3/2

৩,৭৬০.

If the diagonals of a rhombus are 24 cm an 10 cm. What is the perimeter of the rhombus ?

ক
ক) 48 cm
খ
খ) 50 cm
গ
গ) 52 cm
ঘ
ঘ) 54 cm

ব্যাখ্যা

OA = (1/2) × 24 = 12
OB = (1/2) × 10 = 5

ΔAOB
AB = √{12² + 5²}
= √(144 + 25)
= √169
=13
The perimeter of the rhombus = 13 × 4 = 52

৩,৭৬১.

If the price of the commodity is increased by 50% by what fraction must its consumption be reduced so as to keep the same expenditure on its consumption?

ক
1/4
খ
1/3
গ
1/2
ঘ
2/3
ঙ
None of these

ব্যাখ্যা

Question: If the price of the commodity is increased by 50% by what fraction must its consumption be reduced so as to keep the same expenditure on its consumption?

Solution:
Let, x kg of the commodity costs 100 taka

Increased by 50%,
∴ x kg costs = 100 + 100 × .5 = 150 taka

In 100 taka, new quantity of commodity = (100 × x)/150 = 2x/3 kg

Fraction must its consumption be reduced = {x - (2x/3)}/x
= (x/3)/x
= 1/3

৩,৭৬২.

A train of length 150m takes 10 seconds to cross another train 100m long coming from the opposite direction. If the speed of first train is 30 km/hr, what is the speed of second train ?

ক
ক) 48 km/hr
খ
খ) 54 km/hr
গ
গ) 60 km/hr
ঘ
ঘ) 72 km/hr

ব্যাখ্যা

ধরি
২য় ট্রেনের গতিবেগ x

দেয়া আছে,
১ম ট্রেনের দৈর্ঘ্য = 150 মিটার
২য় ট্রেনের দৈর্ঘ্য = 100মিটার

১ম ট্রেনের গতিবেগ = 30 কি.মি./ঘণ্টা
এখন
10 = (150 + 100)/{(30 + x)(5/18)}
⇒18 × 250 = 10(150 + 5x)
⇒ 4500 = 1500 + 50x
⇒ 50x = 4500 - 1500
⇒ 50x = 3000
⇒ x = 60km/hr

৩,৭৬৩.

In a group of 52 persons, 16 drink tea but not coffee, and 33 drink tea. How many drink coffee but not tea?

ক
ক) 3
খ
খ) 7
গ
গ) 19
ঘ
ঘ) 17

ব্যাখ্যা

Question: In a group of 52 persons, 16 drink tea but not coffee, and 33 drink tea. How many drink coffee but not tea?

Solution:
N(T) = 33
n(T) = no. of people who drink only tea = 16

So, n(C ∩ T) = N(T) - n(T)
= 33 - 16
= 17

∴ N (T ∪ C) = 52

So, no. of people who drink only coffee = 52 - 33 = 19

৩,৭৬৪.

ক
ক) f (0) = ∝
খ
খ) f (1) = - 1
গ
গ) f (1) = 0
ঘ
ঘ) f (- 1) = - 1/2

ব্যাখ্যা

Solution:
f(- 1) = (- 1)² + {1/(- 1 - 1)} - 1
= 1 - 1/2 - 1
= - 1/2

৩,৭৬৫.

When the speed is increased to 4 kmph it takes 4 hours less to cover a distance of 32 km. Find the previous speed.

ক
2 kmph
খ
4 kmph
গ
8 kmph
ঘ
2 kmph
ঙ
None of the above

ব্যাখ্যা

Previous speed = x
A/Q,
32/x - 32/(x + 4) = 4
Or, 32x + 128 - 32x/x(x + 4) = 4
Or, (x + 4)x = 32
Or, x² + 4x - 32 = 0
Or, x² + 8x - 4x - 32 = 0
Or, (x + 8) (x - 4) = 0
So, previous speed is 4 kmph.
And, Present speed will be 4 + 4 = 8 kmph.

৩,৭৬৬.

The banker's gain of a certain sum due 2 years hence at 10% per annum is Tk. 28. Find the present worth.

ক
600
খ
670
গ
700
ঘ
740

ব্যাখ্যা

Question: The banker's gain of a certain sum due 2 years hence at 10% per annum is Tk. 28. Find the present worth.

Solution:
Banker Gain = 28
True Discount = (Bnakers gain × 100)/(rate × time) = (28 × 100)/(10 × 2) = 140

Banker's Gain = Banker's Discount - True discount
⇒ 28 = Banker's discount - 140
∴ Banker's discount = 140 + 28 = 168

Now, banker's discount is the interest on the face value
Let the face value = x
So,
x × (10/100) × 2 = 168
⇒ x × 0.1 × 2 = 168
⇒ x = 168/0.2 = 840

Now,
Present Worth = Face value - True discount
= 840 - 140
= 700

৩,৭৬৭.

If 10 persons can do a job in 20 days, then 20 persons with twice the efficiency can do the same job in -

ক
ক) 10 days
খ
খ) 8 days
গ
গ) 5 days
ঘ
ঘ) 4 days

ব্যাখ্যা

Question: If 10 persons can do a job in 20 days, then 20 persons with twice the efficiency can do the same job in -

Solution:
10 persons in one day can do = 1/20
20 persons in one day can do = 2 × 1/20 = 1/10

with double efficiency, in one day they can do = 2 × 1/10 = 1/5

so, the work will be finished in just 5 days

৩,৭৬৮.

A person having bought goods for Tk 400 sells half of it at a gain of 5%, at what gain % must he sell the remainder so as to gain 20% on the whole?

ক
ক) 32%
খ
খ) 33%
গ
গ) 34%
ঘ
ঘ) 35%

ব্যাখ্যা

To gain 20% on whole he must sell all good for,
400 + 20% of 400 = 480
As he get 5% gain on half of the goods i.e.
200 + 5% of 200 = 210
So required balance = 480 - 210 = 270
He must gain Tk. 70 on rest Tk. 200
% gain on remainder goods =(70×100)/200 = 35%

৩,৭৬৯.

Which one is the smallest?

ক
ক) 0.02
খ
খ) 1 ÷ 100
গ
গ) 10 ^{- 3}
ঘ
ঘ) None

ব্যাখ্যা

Question: Which one is the smallest?

Solution:
1 ÷ 100 = 0.01
10^{- 3} = 1/10³ = 1/1000 = 0.001

৩,৭৭০.

How many 3-digit numbers can be formed from the digits 3, 5, 6, 7 and 9, which are divisible by 5 and none of the digits is repeated?

ক
ক) 60 ways
খ
খ) 20 ways
গ
গ) 16 ways
ঘ
ঘ) 12 ways

ব্যাখ্যা

Question: How many 3-digit numbers can be formed from the digits 3, 5, 6, 7 and 9, which are divisible by 5 and none of the digits is repeated?

Solution:
Number will be divisible by 5 if the last number is 5.
So, first number can be chosen in ⁴C₁ ways
= 4 ways

As the digit is not repeated
second number can be chosen in ³C₁
= 3 ways

∴ Total ways = 4 × 3 ways
= 12 ways

৩,৭৭১.

A boat having a length 3.5 m and breadth 2 m is floating on a lake. The boat sinks by 1 cm when a man gets on it. The mass of the man is:

ক
ক) 60 kg
খ
খ) 70 kg
গ
গ) 75 kg
ঘ
ঘ) 80 kg

ব্যাখ্যা

Volume of water displaced
= (3.5 × 2 × 0.01) m³
= 0.07 m³.

Mass of man
= Volume of water displaced × Density of water
= (0.07 × 1000) kg
= 70 kg.

৩,৭৭২.

The population of a town increases every year by 4%. If its present population is 50,000, then after 2 years it will be-

ক
54280
খ
50500
গ
60250
ঘ
54080

ব্যাখ্যা

Question: The population of a town increases every year by 4%. If its present population is 50,000, then after 2 years it will be-
Solution:
Here we can use the compound interest based formula,
Population after n years
= P × [1 + (r/100)]ⁿ

∴ Population after 2 years,
= 50000 × [1 + (4/100)]²
= 50000 × [1 + 0.04]²
= 50000 × (1.04)²
= 54080

∴ Population after 2 years = 54080

৩,৭৭৩.

The ratio of girls to boys in a class is 8 : 7. Which of the following can't be the number of the total students-

ক
30
খ
45
গ
50
ঘ
60

ব্যাখ্যা

Question: The ratio of girls to boys in a class is 8 : 7. Which of the following can't be the number of the total students-

Solution:
ছাত্রী ও ছাত্রের অনুপাত = ৮ : ৭

যোগফল = ৮ + ৭ = ১৫
অতএব, মোট শিক্ষার্থী সংখ্যা ১৫ দ্বারা নি:শেষে বিভাজ্য হতে হবে।

৫০, ১৫ দ্বারা নি:শেষে বিভাজ্য নয়।

৩,৭৭৪.

Four gardeners with four grass mowers mow 400 sq. m of ground in 4 hours. How long would it take for eight gardeners with eight grassmowers to mow 800 sq.m of the ground?

ক
2 hours
খ
4 hours
গ
6 hours
ঘ
8 hours

ব্যাখ্যা

Question: Four gardeners with four grassmowers mow 400 sq. m of ground in 4 hours. How long would it take for eight gardeners with eight grass mowers to mow 800 sq. m of the ground?

Solution:
Since each gardener would work with one grassmower,
So, we shall consider 1 gardener and 1 grass mower as one unit.
Let the required time be x hours

ATQ,
8 × 400 × x = 4 × 800 × 4
⇒ x = (4 × 800 × 4)/(8 × 400)
∴ x = 4 hours

৩,৭৭৫.

Find the value of (1 + tanA - secA) (1 + cotA + cosecA).

ক
2
খ
1
গ
0
ঘ
Undefined

ব্যাখ্যা

Question: Find the value of (1 + tanA - secA) (1 + cotA + cosecA).

Solution:

৩,৭৭৬.

If 15 people attend a conference and each person shakes hands with every other person exactly once, find the total number of handshakes.

ক
105
খ
65
গ
125
ঘ
85

ব্যাখ্যা

Question: If 15 people attend a conference and each person shakes hands with every other person exactly once, find the total number of handshakes.

Solution:
Total handshakes = ¹⁵C₂ = 15!/2!(15 - 2)!
= (15 × 14 × 13!)/(2 × 13!)
= 105

Thus, the total number of handshakes is 105.

৩,৭৭৭.

The sum of all even numbers between 1 and 11 is-

ক
20
খ
22
গ
26
ঘ
30

ব্যাখ্যা

Question: The sum of all even numbers between 1 and 11 is-

Solution:
The even numbers between 1 and 11 are:
2, 4, 6, 8, 10

∴ The sum of these numbers = (2 + 4 + 6 + 8 + 10) = 30

৩,৭৭৮.

A 40-meter cable is attached from the top of a vertical pole down to the ground. If the cable makes an angle of 30 degrees with the ground, find the height of the pole.

ক
20 meters
খ
20√3 meters
গ
24 meters
ঘ
30 meters

ব্যাখ্যা

Question: A 40-meter cable is attached from the top of a vertical pole down to the ground. If the cable makes an angle of 30 degrees with the ground, find the height of the pole.

Solution:

ধরি, উচ্চতা(Height), AB = h
দেয়া আছে, AC = 40m
∠ACB = 30°

∴ sin30°= AB/AC
⇒ 1/2 = h/40
⇒ h = 40 × 1/2
∴ h = 20 m

৩,৭৭৯.

A man can row at 5 kmph in still water. If the velocity of the current is 1 kmph and it takes him 1 hour to row to a place and comes back, how far is the place?

ক
ক) 2 km
খ
খ) 2.4 km
গ
গ) 3 km
ঘ
ঘ) 3.6 km

ব্যাখ্যা

Question: A man can row at 5 kmph in still water. If the velocity of the current is 1 kmph and it takes him 1 hour to row to a place and comes back, how far is the place?

Solution
Speed downstream = (5 + 1) kmph = 6 kmph
Speed upstream = (5 - 1) kmph = 4 kmph
Let the required distance be x km

Then,
x/6 + x/4 = 1
⇒ (2x + 3x)/12 = 1
⇒ 2x + 3x = 12
⇒ 5x = 12
⇒ x = 2.4 km

৩,৭৮০.

Three men, four women, and six children can complete a work in seven days. A woman does double the work a man does and a child does half the work a man does. How many women alone can complete this work in 7 days?

ক
4 days
খ
5 days
গ
6 days
ঘ
7 days

ব্যাখ্যা

Question: Three men, four women, and six children can complete a work in seven days. A woman does double the work a man does and a child does half the work a man does. How many women alone can complete this work in 7 days?

Solution:
Let, women's 1 day's work = x
1 man's 1 day's work = x/2
1 child's 1 day's work = x/4

ATQ,
3x/2 + 4x + 6x/4 = 1/7
⇒ (6x + 16x + 6x)/4 = 1/7
⇒ 28x/4 = 1/7
⇒ 7x = 1/7
∴ x = 1/49

1 woman alone can complete 1/49 of the work in 1 day.
So, 1 woman alone can complete the work in 49 days.
To complete the work in 7 days, the number of women required = 49/7 = 7 days.

৩,৭৮১.

If A : B = 1 : 2, B : C = 4 : 3 and A + B + C = 630, what is the value of C?

ক
210
খ
70
গ
140
ঘ
270

ব্যাখ্যা

Question: If A : B = 1 : 2, B : C = 4 : 3 and A + B + C = 630, what is the value of C?

Solution:
Given,
A : B = 1 : 2 = 2 : 4
B : C = 4 : 3

∴ A : B : C = 2 : 4 : 3

∴ Value of C = 630 × 3/9
= 210

৩,৭৮২.

Two cards are drawn from a pack of 52 cards. What is the probability that both are black cards?

ক
2/103
খ
3/108
গ
25/102
ঘ
15/26

ব্যাখ্যা

Question: Two cards are drawn from a pack of 52 cards. What is the probability that both are black cards?

Solution:
number of black cards = 26

probability both card black = ²⁶C₂/⁵²C₂
= 325/1326
= 25/102

৩,৭৮৩.

How far apart are the centers of two circles with diameters of 20 cm and radii of 8 cm, when they touch each other externally?

ক
22 cm
খ
10 cm
গ
12 cm
ঘ
18 cm

ব্যাখ্যা

প্রশ্ন: How far apart are the centers of two circles with diameters of 20 cm and radii of 8 cm, when they touch each other externally?
(20 সে.মি. ব্যাস ও 8 সে. মি. ব্যাসার্ধ বিশিষ্ট দুইটি বৃত্ত পরস্পরকে বহিঃস্পর্শ করলে বৃত্ত দুটির কেন্দ্রদ্বয়ের মধ্যবর্তী দূরত্ব কত?)

সমাধান:
আমরা জানি,
দুইটি বৃত্ত পরস্পরকে বহিঃস্পর্শ করলে কেন্দ্রদ্বয়ের মধ্যবর্তী দূরত্ব বৃত্ত দুইটির ব্যাসার্ধের যোগফলের সমান।
এখানে,
১ম বৃত্তের ব্যাসার্ধ = 20/2 = 10 সে.মি.
২য় বৃত্তের ব্যাসার্ধ = 8 সে. মি.

∴ কেন্দ্রদ্বয়ের মধ্যবর্তী দূরত্ব = (10 + 8) সে.মি.
= 18 সে.মি.

৩,৭৮৪.

A sum of tk. 558 is to be divided among A, B and C such that 2 times A's share, 3 times B's share and 5 times C's share are all equal. The share of C is

ক
135
খ
122
গ
108
ঘ
102

ব্যাখ্যা

Question: A sum of tk. 558 is to be divided among A, B and C such that 2 times A's share, 3 times B's share and 5 times C's share are all equal. The share of C is

Solution:
Let,
2A = 3B = 5C = x
∴ A = x/2 , B = x/3 , C = x/5

A : B : C = x/2 : x/3 : x/5
To eliminate the fractions, find the LCM of 2, 3, and 5, which is 30.

Multiply each term by 30 to get:
A : B : C = 30x/2 : 30x/3 : 30x/5
= 15x : 10x : 6x

So, the ratio of A : B : C is = 15 : 10 : 6.

The total parts = (15 + 10 + 6) = 31 parts.

The share of C is = (558 × 6)/31
= 108

৩,৭৮৫.

The speed of a boat along the stream is 18 km/hr and against the stream is 12 km/hr. What is the time taken by the boat to sail 45 km in still water?

ক
1 hours
খ
2 hours
গ
3 hours
ঘ
4 hours
ঙ
4.5 hours

ব্যাখ্যা

Speed of boat in still water = (18 + 12)/2 = 15 km/hr [ speed = distance/time ]
Time taken by the boat in still water = 45/15 = 3 hours

৩,৭৮৬.

At what rate percent per annum will the simple interest on a sum of money 3/5 of the amount in 5 years?

ক
12%
খ
10%
গ
14%
ঘ
8%

ব্যাখ্যা

Question: At what rate percent per annum will the simple interest on a sum of money 3/5 of the amount in 5 years?

Solution:
Let the principal be p Tk
So, interest = 3p/5 Tk
Time = 5 years
rate of interest, r =?

Now,
I = pnr
r = I/pn
= (3p/5)/(p × 5)
= (3/25) × 100%
= 12%

৩,৭৮৭.

In a certain town, the ratio of the number of rented residences to the number of owned residences is 7 : 4. Those are the only two kinds of residences in this town. If there are 150 more rented than owned residences in the town?

ক
ক) 200
খ
খ) 350
গ
গ) 550
ঘ
ঘ) 700

ব্যাখ্যা

ধরি,
বাড়িওয়ালার সংখ্যা x জন।
ভাড়াটের সংখ্যা (x + 150) জন।

প্রশ্নমতে,
(x + 150) : x = 7 : 4
(x + 150)/x = 7/4
7x = 4x + 600
3x = 600
x = 200

অতএব
বাড়িওয়ালার সংখ্যা 200 জন।
ভাড়াটের সংখ্যা = 200 + 150 = 350 জন।
মোট অধিবাসীর সংখ্যা = 200 + 350 = 550 জন।

৩,৭৮৮.

If 12 people attend a meeting and each person shakes hands with every other person exactly once, find the total number of handshakes.

ক
55
খ
66
গ
40
ঘ
60

ব্যাখ্যা

Question: If 12 people attend a meeting and each person shakes hands with every other person exactly once, find the total number of handshakes.

Solution:

Total handshakes = ¹²C₂ = 12!/2!(12 - 2)!
= (12 × 11 × 10!)/(2 × 10!)
= 66

∴ The total number of handshakes is 66.

৩,৭৮৯.

The ratio between the perimeter and the breadth of a rectangle is 3 : 1. If the area of the rectangle is 288 sq. cm, what is the length and breadth of the rectangle?

ক
12 cm & 24 cm
খ
16 cm & 32 cm
গ
25 cm & 50 cm
ঘ
None of the above

ব্যাখ্যা

Question: The ratio between the perimeter and the breadth of a rectangle is 3 : 1. If the area of the rectangle is 288 sq. cm, what is the length and breadth of the rectangle?

Solution:
Let, Length = l
And, breadth = b
∴ Perimeter of a rectangle = 2(l + b)

Now,
2(l + b)/b = 3/1
⇒ 2l + 2b = 3b
⇒ b = 2l

∴ Area, b × l = 288
⇒ 2l × l = 288
⇒ l² = 144
⇒ l = 12
∴ b = 2 × 12 = 24

∴ The length & breadth of the rectangle is 12 cm & 24 cm respectively.

৩,৭৯০.

Two alarm clocks ring their alarms at regular intervals of 50 seconds and 48 seconds. If they first beep together at 2.30 PM, at what time will they beep again for first time?

ক
2 : 50 PM
খ
3 : 10 PM
গ
2 : 55 PM
ঘ
3 : 00 PM

ব্যাখ্যা

Question: Two alarm clocks ring their alarms at regular intervals of 50 seconds and 48 seconds. If they first beep together at 2 : 30 PM, at what time will they beep again for first time?

Solution:
They will ring together after,
LCM of 48 and 50 secs.
48 = 2 × 2 × 2 × 2 × 3
50 = 2 × 5 × 5

∴ LCM = 2 × 2 × 2 × 2 × 3 × 5 × 5 = 1200 secs
= 1200/60 = 20 min. [1 min = 60 sec]

∴ They will beep together at 2 : 30 + 20 = 2 : 50 PM

৩,৭৯১.

A worker’s regular pay is Tk. 15 per hour up to 40 hours. Overtime is paid at twice the regular rate. If he was paid Tk. 1,050, how many hours of overtime did he work?

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

ব্যাখ্যা

Question: A worker’s regular pay is Tk. 15 per hour up to 40 hours. Overtime is paid at twice the regular rate. If he was paid Tk. 1,050, how many hours of overtime did he work?

Solution:

ধরি,
তিনি x ঘণ্টা overtime কাজ করেছেন।

সাধারণ বেতন (40 ঘণ্টার জন্য) = 15 × 40 = 600 টাকা
Overtime বেতন = 15 × 2 × x = 30x টাকা

মোট বেতন = সাধারণ বেতন + overtime বেতন

অর্থাৎ,
600 + 30x = 1,050
⇒ 30x = 1,050 - 600
⇒ 30x = 450
⇒ x = 450/30
∴ x = 15

∴ তিনি 15 ঘন্টা overtime করেছেন।

৩,৭৯২.

Nobody went there, ______?

ক
did he
খ
did they
গ
didn't he
ঘ
did you

ব্যাখ্যা

• Complete Sentence: Nobody went there, did they?

• Tag question করার নিয়ম:
- Tag question ব্যবহার করা হয় উক্তিটি সত্য না মিথ্যা তা নিশ্চিত করার জন্য।
- Statement positive হলে tag question টি negative হবে।
- Statement negative হলে tag question টি positive হবে।
- Subject tense অনুসারে auxiliary verb দ্বারা tag question হয়।
- Anybody, anyone, no one, nobody, none, neither যদি কোনো sentence এর subject রূপে ব্যবহৃত হয় তাহলে tag এর subject হবে they (you, he , she নয়)।

Source: A Passage to the English Language, S.M. Zakir Hussain.

৩,৭৯৩.

In how many ways 4 boys and 3 girls can be seated in a row so that they are alternate.

ক
144
খ
160
গ
120
ঘ
220

ব্যাখ্যা

Question: In how many ways 4 boys and 3 girls can be seated in a row so that they are alternate.

Solution:
Let the Arrangement be,
B G B G B G B
4 boys can be seated in 4! Ways
Girl can be seated in 3! Ways

∴ Required number of ways,
= 4! × 3!
= 24 × 6
= 144

৩,৭৯৪.

A tap can fill a tank in 4 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) 1 hour 30 minutes
খ
খ) 2 hour 30 minutes
গ
গ) 3 hour 30 minutes
ঘ
ঘ) 4 hour 30 minutes

ব্যাখ্যা

A tap can fill a tank in 4 hours.
Therefore the tap can fill half the tank in 2 hours.

Remaining = 1/2

After half the tank is filled, three more similar taps are opened.
Hence, the total number of taps becomes 4.

Part filled by one tap in 1 hour = 1/4
Part filled by four taps in 1 hour = 4 × (1/4) = 1
i.e., 4 taps can fill the remaining half in 30 minutes.

Total time taken
= 2 hour + 30 minute = 2 hour 30 minutes.

৩,৭৯৫.

The present age of the daughter is half of the present age of her mother. Ten years ago, her mother's age was thrice the age of her daughter. What is the present age of the daughter?

ক
35 years
খ
30 years
গ
25 years
ঘ
20 years

ব্যাখ্যা

Question: The present age of the daughter is half of the present age of her mother. Ten years ago, her mother's age was thrice the age of her daughter. What is the present age of the daughter?

Solution:

Let the mother's age be 2x years
Then, the daughter's age = x years

According to the question,

2x - 10 = 3(x - 10)

⇒ 2x - 10 = 3x - 30

⇒ x = 20

∴ The present age of the daughter is 20 years.

৩,৭৯৬.

To attract more visitors, the zoo authorities announced a 20% discount on each ticket, the original price of which was 25 paisa. As a result, the total ticket-sale revenue increased by 36%. What is the percentage increase in the number of visitors?

ক
40%
খ
60%
গ
70%
ঘ
No change

ব্যাখ্যা

Question: To attract more visitors, the zoo authorities announced a 20% discount on each ticket, the original price of which was 25 paisa. As a result, the total ticket-sale revenue increased by 36%. What is the percentage increase in the number of visitors?

Solution:
Let the number of visitors in the beginning = 100

Original ticket price = 25 paisa = 0.25 Tk
∴ Total original revenue = 0.25 × 100
= 25 Tk

At 20% discount on every ticket,
New ticket price = 25 × (80/100)
= 20 paisa
= 0.20 Tk

Sale increased by 36%
∴ New revenue = 25 × (136/100)
= 34 Tk

Number of visitors now = New revenue/New ticket price
= 34/0.20
= 34/(20/100)
= 34 × 5
= 170

∴ Percentage increase in number of visitors = [(170 − 100)/100] × 100
= (70 × 100)/100
= 70%

৩,৭৯৭.

Hamed's average on 4 tests is 80. Assuming he can earn no more than 100 on any test, what is the least he can earn on his 5^th test and still have a chance for an 85 average after seven tests?

ক
ক) 60
খ
খ) 70
গ
গ) 75
ঘ
ঘ) 85
ঙ
ঙ) None of these

ব্যাখ্যা

Question: Hamed's average on 4 tests is 80. Assuming he can earn no more than 100 on any test, what is the least he can earn on his 5^th test and still have a chance for an 85 average after seven tests?

Solution:
হামিদ 4টি পরীক্ষায় মোট পায় = 4 × 80 = 320 নম্বর
হামিদ 7টি পরীক্ষায় মোট পায় = 7 × 85 = 595 নম্বর

হামিদ 3টি পরীক্ষায় মোট পায় = 595 - 320 = 275 নম্বর

হামিদ ৬ষ্ঠ ও ৭ম পরীক্ষায় সর্বোচ্চ নম্বর পায় = 100 + 100 = 200
হামিদ ৫ম পরীক্ষায় সর্বনিম্ন নম্বর পায় = 275 - 200 = 75

৩,৭৯৮.

If the list price of a mobile phone is Tk. 10000, and a Tk. 1500 discount is offered on the mobile phone, then what is the discount percentage?

ক
10%
খ
12.5%
গ
15%
ঘ
18.5%

ব্যাখ্যা

Question: If the list price of a mobile phone is Tk. 10000, and a Tk. 1500 discount is offered on the mobile phone, then what is the discount percentage?

Solution:
Marked Price = Tk. 10000
Discount = Tk. 1500

Discount % = (Discount/marked Price) × 100%
Discount (%) = (1500/10000) × 100%
= 15%

∴ So the discount percentage is 15%.

৩,৭৯৯.

ক
1
খ
0
গ
1/2
ঘ
2/3

ব্যাখ্যা

Question:

Solution:

৩,৮০০.

Which of the following is closest to 1/3 + 0.4 + 65%?

ক
1
খ
1.4
গ
2.5
ঘ
2.8

ব্যাখ্যা

Question: Which of the following is closest to 1/3 + 0.4 + 65%?

Solution:
1/3 + 0.4 + 65%
= (1/3) + (4/10) + (65/100)
= (1/3) + (2/5) + (13/20)
= 1.4