Bank Math

PrepBank · পাতা ১২৭ / ১৬১ · ১২,৬০১–১২,৭০০ / ১৬,১২৪

১২,৬০১.

An integer n between 1 and 100, inclusive, is to be chosen at random. What is the probability that n(n + 1) will be divisible by 5?

ক
ক) 2/3
খ
খ) 2/5
গ
গ) 1/5
ঘ
ঘ) None of these

সঠিক উত্তর: খ

খ) 2/5

উত্তর

সঠিক উত্তর: খ

খ) 2/5

খ

ব্যাখ্যা

Question: An integer n between 1 and 100, inclusive, is to be chosen at random. What is the probability that n(n+1) will be divisible by 5?

Solution:
total number = 100
n(n+1) will be divisible by 5 if n or n + 1 is divisible by 5

when n is divisible by 5, there are 20 such number (5, 10, 15, 20, 25,.....,100)
when n + 1 is divisible by 5, there are 20 such number (4, 9, 14,.....,99)

∴ proability = (20 + 20)/100
= 40/100
= 2/5

১২,৬০২.

The average age of all the students in a driving class is 22 years. The average age of the boys is 24 years and that of the girls is 18 years. If there are 15 girls in the class, find the number of boys.

ক
30
খ
25
গ
24
ঘ
20

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: The average age of all the students in a driving class is 22 years. The average age of the boys is 24 years and that of the girls is 18 years. If there are 15 girls in the class, find the number of boys.

Solution:
Let the number of boys in the class be x.

According to the question,
22 × (x + 15) = 24x + (18 × 15)
⇒ 22x + 330 = 24x + 270
⇒ 24x + 270 = 22x + 330
⇒ 24x - 22x = 330 - 270
⇒ 2x = 60
⇒ x = 30

∴ The number of boys in the class is 30.

১২,৬০৩.

Simplify the expression,

ক
14
খ
12
গ
8
ঘ
16

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: Simplify the expression,

Solution:

১২,৬০৪.

x³ + 5x + k = 0 solution is - 3, but what is the value of k?

ক
27
খ
- 32
গ
42
ঘ
- 18

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: x³ + 5x + k = 0 solution is - 3, but what is the value of k?

Solution:
∴ একটি সমাধান = - 3
x = - 3 বসিয়ে পাই,
⇒ (- 3)³ + 5(- 3) + k = 0
⇒ - 27 - 15 + k = 0
⇒ k - 42 = 0
∴ k = 42

১২,৬০৫.

The value of log₃2⋅log₄3⋅log₅4⋅log₆5⋅log₇6⋅log₈7 is-

ক
1/4
খ
1/2
গ
1/3
ঘ
None of these

সঠিক উত্তর: গ

1/3

উত্তর

সঠিক উত্তর: গ

1/3

গ

ব্যাখ্যা

Question: The value of log₃2⋅log₄3⋅log₅4⋅log₆5⋅log₇6⋅log₈7 is-

Solution:
log₃2⋅log₄3⋅log₅4⋅log₆5⋅log₇6⋅log₈7
⇒ (log₃2 ⋅ log₄3) (log₅4 ⋅ log₆5) (log₇6 ⋅ log₈7)
⇒ log₄2 .log₆4. log₈6 [log_bM × log_ab = log_aM]
⇒ (log₄2 .log₆4) log₈6
⇒ log₆2 ⋅ log₈6
⇒ log₈2
⇒ 1/log₂8
⇒ 1/log₂2³
= 1/(3log₂2)
= 1/(3 × 1) [∵ log₂2 = 1]
= 1/3

∴ log₃2⋅log₄3⋅log₅4⋅log₆5⋅log₇6⋅log₈7 = 1/3

১২,৬০৬.

In a group of 6 boys and 4 girls, four children are to be selected. In how many different ways can they be selected such that at least one girl should be there?

ক
ক) 210 ways
খ
খ) 209 ways
গ
গ) 195 ways
ঘ
ঘ) 192 ways

সঠিক উত্তর: গ

গ) 195 ways

উত্তর

সঠিক উত্তর: গ

গ) 195 ways

গ

ব্যাখ্যা

Question: In a group of 6 boys and 4 girls, four children are to be selected. In how many different ways can they be selected such that at least one girl should be there?

Solution:
There are total 10 children

Selecting 4 children out of 10 = ¹⁰C₄
= 210 ways

selecting 4 boys out of 6 boys = ⁶C₄
= 15

∴ different ways can they be selected such that at least one girl should be there is = 210 - 15 ways
= 195 ways

১২,৬০৭.

Kamal and Rajon enter into a partnership with their capitals in the ratio 5 : 6. At the end of 6 months Kamal withdraws her capital. If they receive the profit in the ratio of 5 : 9, find how long was Rajon's capital used?

ক
11 months
খ
10 months
গ
8 months
ঘ
None of these

সঠিক উত্তর: ঘ

None of these

উত্তর

সঠিক উত্তর: ঘ

None of these

ঘ

ব্যাখ্যা

Question: Kamal and Rajon enter into a partnership with their capitals in the ratio 5 : 6. At the end of 6 months Kamal withdraws her capital. If they receive the profit in the ratio of 5 : 9, find how long was Rajon's capital used?

Solution:
Let,
Kamal invested Tk. 5x for 6 months
And
Rajon invested Tk. 6x for y months
Then,
(5x × 6)/(6x × y) = 5/9
⇒ 30x/6xy = 5/9
⇒ 5/y = 5/9
∴ y = 9

So Rajon capital was used for 9 months

১২,৬০৮.

Kalam buys an article at a discount of 25%. At what percentage above the cost price should he sell it to make a profit of 25% over the original list price?

ক
20%
খ
25%
গ
40%
ঘ
66.67%

সঠিক উত্তর: ঘ

66.67%

উত্তর

সঠিক উত্তর: ঘ

66.67%

ঘ

ব্যাখ্যা

Question: Kalam buys an article at a discount of 25%. At what percentage above the cost price should he sell it to make a profit of 25% over the original list price?

Solution:
Let, the original price is 100 taka
Buying price = 100 - 25
= 75 taka

To make 25% over original price = 100 + 25
= 125 taka

percentage above the cost price = {(125 - 75)/75} × 100%
= (50/75) × 100%
= (2/3) × 100%
= 66.67%

১২,৬০৯.

In a school, there are 55% students play Cricket, 50% students play Hockey and 25% students play both games. The difference between the number of students who play both games and the number of students who do not play any games is 40. Find the number of students who play only one game.

ক
490
খ
400
গ
440
ঘ
390

সঠিক উত্তর: গ

440

উত্তর

সঠিক উত্তর: গ

440

গ

ব্যাখ্যা

Question: In a school, there are 55% students play Cricket, 50% students play Hockey and 25% students play both games. The difference between the number of students who play both games and the number of students who do not play any games is 40. Find the number of students who play only one game.

Solution:
Total number of students in a school be N.
Total number of students who play only Cricket = (55/100 - 25/100)N
= 30N/100

Total number of students who play only Hockey = (50/100 - 25/100)N
= 25N/100

Total number of students who do not play any Game = N - 30N/100 - 25N/100 - 25N/100
= 20N/100

ATQ,
25N/100 - 20N/100 = 40
⇒ 5N/100 = 40
⇒ N = (40 × 100)/5
∴ N = 800

Total number of students who play only one game = 30N/100 + 25N/100
= 55N/100
= (55 × 800)/100
= 440

১২,৬১০.

A canteen requires 420 bananas for a week. Totally, how many bananas will it require for the months of January, February, and March in 2024?

ক
ক) 4860
খ
খ) 5000
গ
গ) 5220
ঘ
ঘ) 5460

সঠিক উত্তর: ঘ

ঘ) 5460

উত্তর

সঠিক উত্তর: ঘ

ঘ) 5460

ঘ

ব্যাখ্যা

Question: A canteen requires 420 bananas for a week. Totally, how many bananas will it require for the months of January, February, and March in 2024?

Solution:
January = 31 days
February = 29 days
March = 31 days

Total = 31 + 29 +31 = 91 days

In 7 days the canteen requires 420 bananas
In 1 day the canteen requires 420/7 bananas
In 91 days the canteen requires (420 × 91)/7 bananas
= 5460 bananas

১২,৬১১.

A man can row upstream at 24km/hr and downstream at 16km/hr. The man rowing speed in still water is?

ক
20 km/hr
খ
18 km/hr
গ
15 km/hr
ঘ
10 km/hr

সঠিক উত্তর: ক

20 km/hr

উত্তর

সঠিক উত্তর: ক

20 km/hr

ক

ব্যাখ্যা

Question: A man can row upstream at 24km/hr and downstream at 16km/hr. The man rowing speed in still water is?

Solution:
Let, x = downstream speed and y = upstream speed
We know,
Speed of boat in still water = (x + y)/2
∴ Boat's speed = (24 + 16)/2 = 40/2 = 20 km/hr.

১২,৬১২.

A man invested Tk. 1552 in a stock at 97 to obtain an income of Tk. 128. The dividend from the stock is:

ক
ক) 7%
খ
খ) 8%
গ
গ) 9.7%
ঘ
ঘ) None of these

সঠিক উত্তর: খ

খ) 8%

উত্তর

সঠিক উত্তর: খ

খ) 8%

খ

ব্যাখ্যা

By investing Tk.1552, income = Tk 128.
By investing Tk.97, income = Tk.(128 / 1552×97) = Tk 8
∴Dividend = 8%

১২,৬১৩.

In what time will a train 100 metres long cross an electric pole, if its speed be 144 km/hr?

ক
2.5 second
খ
4.25 second
গ
5 second
ঘ
12.5 second

সঠিক উত্তর: ক

2.5 second

উত্তর

সঠিক উত্তর: ক

2.5 second

ক

ব্যাখ্যা

Question: In what time will a train 100 metres long cross an electric pole, if its speed be 144 km/hr?

Solution:
Speed = 144 km/hr
= (144 × 1000)/3600 m/sec
= 40 m/sec

∴Required Time = (Distance/speed) sec
= (100/40) sec
= 2.5 sec.

১২,৬১৪.

The surface area of a sphere is 144π cm². Find the radius of the sphere.

ক
12 cm
খ
10 cm
গ
6 cm
ঘ
11 cm

সঠিক উত্তর: গ

6 cm

উত্তর

সঠিক উত্তর: গ

6 cm

গ

ব্যাখ্যা

Question: The surface area of a sphere is 144π cm². Find the radius of the sphere.

Solution:
Let, r is radius of the sphere

A.T.Q,
4πr² = 144π
r² = 36
∴ r = 6

The radius of the sphere is 6 cm.

১২,৬১৫.

The sum of three consecutive odd integers is 48 more than the last of the numbers. What is the middle number?

ক
21
খ
23
গ
25
ঘ
27
ঙ
31

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: The sum of three consecutive odd integers is 48 more than the last of the numbers. What is the middle number?

Solution:
Let the first odd number be x
∴ 2nd odd number is x + 2
∴ 3rd odd number is x + 4

According to the question,
Sum of the odd numbers = (x + 4) + 48

The equation,
x + (x + 2) + (x + 4) = (x + 4) + 48
⇒ 3x + 6 = x + 4 + 48
⇒ 3x + 6 = x + 52
⇒ 3x - x = 52 - 6
⇒ 2x = 46
⇒ x = 46/2
⇒ x = 23

First number is 23
The second number is 25
The third number is 27

∴ The middle number is 25

১২,৬১৬.

The sum of first 55 natural numbers is -

ক
1610
খ
1590
গ
1580
ঘ
1540

সঠিক উত্তর: ঘ

1540

উত্তর

সঠিক উত্তর: ঘ

1540

ঘ

ব্যাখ্যা

Question: The sum of first 55 natural numbers is -

Solution:
We know that,
The sum of the first n natural numbers = n(n+1)/2

Sum of first 55 natural numbers is = 55(55 + 1)/2
= (55 × 56)/2
= 1540

১২,৬১৭.

Alal sold an article at 6% loss. Had he sold it for Tk. 64 more, he would have made a profit of 10%. Then the cost of the article is =?

ক
Tk. 320
খ
Tk. 400
গ
Tk. 450
ঘ
Tk. 480

সঠিক উত্তর: খ

Tk. 400

উত্তর

সঠিক উত্তর: খ

Tk. 400

খ

ব্যাখ্যা

Question: Alal sold an article at 6% loss. Had he sold it for Tk. 64 more, he would have made a profit of 10%. Then the cost of the article is =?

Solution:
Let, the cost of the article is x taka

Selling price = 0.94x

ATQ,
0.94x + 64 = 1.1x
⇒ 1.1x - 0.94x = 64
⇒ 0.16x = 64
⇒ x = 64/0.16 = 400 taka

১২,৬১৮.

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 4 days. B can do the work alone in-

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

সঠিক উত্তর: ক

12 days

উত্তর

সঠিক উত্তর: ক

12 days

ক

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 4 days. B can do the work alone in-

Solution:
Suppose A, B and C take x, x/2 and x/3 days respectively to finish the work.
Then,
1/x + 2/x + 3/x = 1/4
⇒ 6/x = 1/4
⇒ x/6 = 2
∴ x = 24

So, B takes (24/2) = 12 days to finish the work.

১২,৬১৯.

What is the difference between the biggest and the smallest fraction of 2/3, 3/5, 4/9, 6/11?

ক
ক) 1/6
খ
খ) 2/9
গ
গ) 3/8
ঘ
ঘ) 4/5

সঠিক উত্তর: খ

খ) 2/9

উত্তর

সঠিক উত্তর: খ

খ) 2/9

খ

ব্যাখ্যা

Question: What is the difference between the biggest and the smallest fraction of 2/3, 3/5, 4/9, 6/11?

Solution:
Here,
2/3 = 0.667
3/5 = 0.6
4/9 = 0.444
6/11 = 0.545

Biggest fraction is 2/3
and the smallest fraction is 4/9

∴ The difference between them,
= (2/3) - (4/9)
= (6 - 4)/9
= 2/9

১২,৬২০.

If the price of sugar increases by 10%, by what percentage a housewife should reduce the consumption of sugar so that expense is not increased?

ক
1%
খ
7%
গ
11%
ঘ
None of the above

সঠিক উত্তর: ঘ

None of the above

উত্তর

সঠিক উত্তর: ঘ

None of the above

ঘ

ব্যাখ্যা

Question: If the price of sugar increases by 10%, by what percentage a housewife should reduce the consumption of sugar so that expense is not increased?

Solution:
Let the price be Tk. x per kg
Consumption be y kg

Expenditure = price × consumption
⇒ Expenditure = x × y

Price of sugar is increased by 10%
New price of sugar = Tk. 1.1x per kg

Let new consumption be z kg
New Expenditure = (1.1x) × z

New Expenditure = old Expenditure
⇒ (1.1x) × z = x × y
⇒ z = y/1.1

Reduction in consumption = (y - z) = y - y/1.1 = y/11

Percentage reduction in consumption = (reduction in consumption/original consumption)× 100
Percentage reduction in consumption = [(y/11)/y] × 100 = 100/11 = 9.09% ≈ 9.1%

১২,৬২১.

220% of a number X is 44. What is 44% of X.

ক
8.8
খ
8.9
গ
6.6
ঘ
7.7
ঙ
None of these

সঠিক উত্তর: ক

8.8

উত্তর

সঠিক উত্তর: ক

8.8

ক

ব্যাখ্যা

Question: 220% of a number X is 44. What is 44% of X.

Solution:
Given,
220% of X = 44
⇒ (220X)/100 = 44
⇒ X = (44 × 100)/220
Or, X = 20

Thus,
44% of 20
= (44 × 20)/100
= 8.8

১২,৬২২.

A man takes 50 minutes to cover a certain distance at a speed of 6 km/hr. If he walks with a speed of 10 km/hr, he covers the same distance in-

ক
ক) 10 min
খ
খ) 20 min
গ
গ) 30 min
ঘ
ঘ) 40 min

সঠিক উত্তর: গ

গ) 30 min

উত্তর

সঠিক উত্তর: গ

গ) 30 min

গ

ব্যাখ্যা

Distance = Speed×Time
Distance = (6× 50)/60)km
Distance = 5 km
∴ Required time
= Distance/ Speed
= 5/10 hrs
= 1/2 hrs
= 30 min

১২,৬২৩.

A cylindrical rod of iron, whose height is equal to its radius, is melted and cast into spherical balls whose radius is 1/4 the radius of the rod. Find the number of balls?

ক
6
খ
48
গ
24
ঘ
36

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: A cylindrical rod of iron, whose height is equal to its radius, is melted and cast into spherical balls whose radius is 1/4 the radius of the rod. Find the number of balls?

Solution:
ধরি, সিলিন্ডার আকৃতির রডের ব্যাসার্ধ r এবং উচ্চতা h
দেওয়া আছে, রডের ব্যাসার্ধ এবং উচ্চতা সমান ।
∴ রডের আয়তন = πr²h = πr² × r = πr³ ঘন একক

আবার ধরি,
গোলকের ব্যাসার্ধ, r' = r/4

∴ প্রতিটি গোলকের আয়তন = (4/3) × πr'³ = (4/3) × π × (r/4)³
= πr³/48 ঘন একক

∴ গোলকের সংখ্যা = রডের আয়তন/প্রতিটি গোলকের আয়তন
= πr³/(πr³/48)
= (πr³/πr³) × 48
= 48

১২,৬২৪.

Out of two numbers, 40% of the greater number is equal to 60% of the smaller. If the sum of the numbers is 150, then the greater number is-

ক
50
খ
76
গ
80
ঘ
90

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: Out of two numbers, 40% of the greater number is equal to 60% of the smaller. If the sum of the numbers is 150, then the greater number is-

Solution:
Let, greater number be = x.
Smaller number = 150 - x

ATQ,
(40 × x)/100 = {60(150 - x)}/100
⇒ 40x/100 = (9000 - 60x)/100
⇒ 100x = 9000
⇒ x = 9000/100
∴ x = 90

১২,৬২৫.

At what rate percent of simple interest will a sum of money triples itself in 6 years?

ক
13.33%
খ
20.50%
গ
33.33%
ঘ
29.23%

সঠিক উত্তর: গ

33.33%

উত্তর

সঠিক উত্তর: গ

33.33%

গ

ব্যাখ্যা

Question: At what rate percent of simple interest will a sum of money triples itself in 6 years?

Solution:
Let,
Total = 100 tk

Now,
After 6 years capital with interest = 100 × 3 tk
= 300 tk
∴ Total interest will be = 300 - 100 tk
= 200 tk

If the interest in 6 years is tk 200, then the interest in 1 year will be = 200/6
= 33.33%

১২,৬২৬.

A container is 1/2 full. When 8 gallons is removed the container is 1/10 full. What is the capacity of the container in gallon?

ক
20
খ
24
গ
16
ঘ
22

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: A container is 1/2 full. When 8 gallons is removed the container is 1/10 full. What is the capacity of the container in gallon?

Solution:
Let,
The capacity of the container in gallon is x gallons

ATQ,
(x/2) - 8 = x/10
⇒ (x - 16)/2 = x/10
⇒ 10(x - 16) = 2x
⇒ 10x - 160 = 2x
⇒ 10x - 2x = 160
⇒ 8x = 160
∴ x = 160/8 = 20 gallons

১২,৬২৭.

If 1st October is Monday, then 1st November will be -

ক
ক) Wednesday
খ
খ) Tuesday
গ
গ) Thursday
ঘ
ঘ) Friday

সঠিক উত্তর: গ

গ) Thursday

উত্তর

সঠিক উত্তর: গ

গ) Thursday

গ

ব্যাখ্যা

Question: If 1st October is Monday, then 1st November will be -

Solution:
Given that 1st October is Monday.
Number of days in October = 31
31 days = 3 odd days. [As we can reduce multiples of 7 from odd days which will not change anything]

∴ 1st November = (Monday + 3 odd days) = Thursday.

১২,৬২৮.

Which of the numbers below is not equivalent to 2%?

ক
ক) 1/50
খ
খ) 1/10
গ
গ) 0.02
ঘ
ঘ) 2/100

সঠিক উত্তর: খ

খ) 1/10

উত্তর

সঠিক উত্তর: খ

খ) 1/10

খ

ব্যাখ্যা

2% = 2/100 = 1/50 = 0.02

১২,৬২৯.

Which of the following are not a factor of p³ - 7p - 6

ক
(p + 1)
খ
(p + 2)
গ
(p + 3)
ঘ
(p - 3)

সঠিক উত্তর: গ

(p + 3)

উত্তর

সঠিক উত্তর: গ

(p + 3)

গ

ব্যাখ্যা

Question: Which of the following are not a factor of p³ - 7p - 6

Solution:
p³ - 7p - 6
= p³+ p² - p² - p - 6p - 6
= p²(p + 1) - p(p + 1) - 6(p + 1)
= (p + 1)(p² - p - 6)
= (p + 1)(p² - 3p + 2p - 6)
= (p + 1){p(p - 3) + 2(p - 3)}
= (p + 1)(p + 2)(p - 3)

১২,৬৩০.

What number will replace the '?' mark?
8 : 63 :: 10 : ?

ক
101
খ
98
গ
99
ঘ
100

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: What number will replace the '?' mark?
8 : 63 :: 10 : ?

Solution:
এখানে
8 ⇒ 8² - 1 = 63
অর্থাৎ
সম্পর্কটি হলো
x : x² - 1
10 ⇒ 10² - 1 = 99

১২,৬৩১.

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 38 years now, the son's age six years back was:

ক
13 years
খ
24 years
গ
34 years
ঘ
None of these

সঠিক উত্তর: ক

13 years

উত্তর

সঠিক উত্তর: ক

13 years

ক

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 38 years now, the son's age five years back was:

Solution:

Let, at present son is x years old

so, at the time of his birth, the father was x years old

∴ at present the age of the father is = (x + x) year

= 2x years

Now,

2x = 38

⇒ x = 38/2

∴ x = 19 years

∴ The son's age five years back was = 19 - 6

= 13 years

১২,৬৩২.

25% of Rasel's yearly income is equal to 75% of Aman's monthly income. If Aman's yearly income is 240000 taka. What is Rasel's monthly income?

ক
Tk. 4500
খ
Tk. 6000
গ
Tk. 5000
ঘ
Tk.2300

সঠিক উত্তর: গ

Tk. 5000

উত্তর

সঠিক উত্তর: গ

Tk. 5000

গ

ব্যাখ্যা

Question: 25% of Rasel's yearly income is equal to 75% of Aman's monthly income. If Aman's yearly income is 240000 taka. What is Rasel's monthly income?

Solution:
Aman's yearly income is 240000 taka
Aman's monthly income is 240000/12 taka
= 20000 taka

75% of Aman's monthly income = 0.75 × 20000 = 15000 taka

25% of Rasel's yearly income = 15000
Rasel's yearly income = 15000 × 4 = 60000 taka

∴ Rasel's monthly income = 60000/12 = Tk. 5000

১২,৬৩৩.

A, B and C are the three contestants in one km race. If A can give B a start of 40 metres and A can give C a start of 64 metres, how many metres start can B give C?

ক
35 m
খ
25 m
গ
20 m
ঘ
22 m

সঠিক উত্তর: খ

25 m

উত্তর

সঠিক উত্তর: খ

25 m

খ

ব্যাখ্যা

While A covers 1000 m, B covers (1000 - 40) m = 960 m
And C covers (1000 - 64) m = 936 m.

When B covers 960 m, C covers 936 m.
When B covers 1000 m, C covers
= (936/960) × 1000 = 975 m

∴ B can give C a start of
(1000 - 975) = 25 m.

১২,৬৩৪.

If 8265P is divisible by 9, what is the value of P?

ক
6
খ
5
গ
8
ঘ
9

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: If 8265P is divisible by 9, what is the value of P?

Solution:
একটি সংখ্যা 9 দ্বারা বিভাজ্য হবে যদি সংখ্যাটির অঙ্কগুলোর সমষ্টি 9 দ্বারা বিভাজ্য হয়।

8 + 2 + 6 + 5 = 21; এর সাথে P যোগ করলে (21 + P) হবে, যা 9 দ্বারা বিভাজ্য হতে হবে।

21 + P = 27 (যা 9 দ্বারা বিভাজ্য নিকটতম সংখ্যা)
⇒ P = 27 - 21 = 6
∴ P = 6

১২,৬৩৫.

If p and q are positive integers and (p - q) is an even number, then (p² - q²) will be always divisible by-

ক
3
খ
4
গ
6
ঘ
9
ঙ
12

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: If p and q are positive integers and (p - q) is an even number, then (p²- q²) will be always divisible by-

Solution:
When, p = 3 and q = 1 then, p - q = 3 - 1 = 2 (even)
∴ p² - q² = 3² - 1²
= 9 - 1
= 8
When, p = 4 and q = 2 then, p - q = 4 - 2 = 2 (even)
p² - q² = 4² - 2²
= 16 - 4
= 12
When, p = 5 and q = 3 then, p - q = 5 - 3 = 2 (even)
p² - q² = 5² - 3² = 25 - 9
= 16

H.C.F. of 8, 12, 16 = 4
So, the number is divisible by 4.

১২,৬৩৬.

A man spent 1/2 of his money and then lost 1/4 of the remainder. He was left with Tk. 4.236. How much did he start with?

ক
ক) 9600
খ
খ) 10400
গ
গ) 11,296
ঘ
ঘ) 12800

সঠিক উত্তর: গ

গ) 11,296

উত্তর

সঠিক উত্তর: গ

গ) 11,296

গ

ব্যাখ্যা

Let,
He has TK. x
ATQ, x - x/2 - x/2×1/4 = 4236
or, x/2 - x/8 = 4236
or, (4x - x)/8 = 4236
∴ x = (4236×8)/3 = 11,296

১২,৬৩৭.

All possible three-digit numbers are formed by 1, 3, 5. If one number is chosen randomly, the probability that it would be divisible by 5 is –

ক
1/2
খ
1/3
গ
1
ঘ
0

সঠিক উত্তর: খ

1/3

উত্তর

সঠিক উত্তর: খ

1/3

খ

ব্যাখ্যা

Question: All possible three-digit numbers are formed by 1, 3, 5. If one number is chosen randomly, the probability that it would be divisible by 5 is –

Solution:
1, 3, 5 এই তিনটি অংক দ্বারা 3! = 6 উপায়ে সংখ্যা গঠন করা যায়
5 দ্বারা বিভাজ্য হতে হলে শেষের অংক 5 রেখে সংখ্যার প্রথম দুটি স্থান বাকি দুইটি অংক দিয়ে 2! = 2 উপায়ে গঠন করা যায়।
∴ সংখ্যাটি 5 দ্বারা বিভাজ্য হবার সম্ভাবনা = 2/6 = 1/3

১২,৬৩৮.

মুসতাকিম 35 কিলোমিটার দূরুত্বের কিছু অংশ 4 kmph বেগে এবং কিছু অংশ 5 kmph বেগে অতিক্রম করে। সে যদি পুরো দূরুত্ব 5 kmph বেগে যেত তাহলে ঐ একই সময়ে আরো 2 কিলোমিটার বেশি দূরুত্ব অতিক্রম করতে পারতো। মোট ভ্রমণে তাঁর কত সময় লেগেছিল?

ক
7 ঘণ্টা 40 মিনিট
খ
7 ঘণ্টা 24 মিনিট
গ
6 ঘণ্টা 24 মিনিট
ঘ
8 ঘণ্টা 24 মিনিট
ঙ
কোনটিই নয়

সঠিক উত্তর: খ

7 ঘণ্টা 24 মিনিট

উত্তর

সঠিক উত্তর: খ

7 ঘণ্টা 24 মিনিট

খ

ব্যাখ্যা

প্রশ্ন: মুসতাকিম 35 কিলোমিটার দূরুত্বের কিছু অংশ 4 kmph বেগে এবং কিছু অংশ 5 kmph বেগে অতিক্রম করে। সে যদি পুরো দূরুত্ব 5 kmph বেগে যেত তাহলে ঐ একই সময়ে আরো 2 কিলোমিটার বেশি দূরুত্ব অতিক্রম করতে পারতো। মোট ভ্রমণে তাঁর কত সময় লেগেছিল?

সমাধান:
ধরি,
4 kmph বেগে যায় x কি.মি.
5 kmph বেগে যায় (35 - x) কি.মি.

প্রশ্নমতে,
(x/4) + (35 - x)/5 = (35 + 2)/5
⇒ (5x + 140 - 4x)/20 = 37/5
⇒ x + 140 = 148
∴ x = 8

মোট সময় লাগবে = (8/4) + (35 - 8)/5
= 2 + 5.4
= 7.4
= 7 ঘণ্টা 24 মিনিট।

১২,৬৩৯.

The ratio between the speeds of a bus and bike is 5 : 8. If the bus travels 120 km in 2 hours, find the speed of the bike.

ক
88 km/h
খ
86 km/h
গ
96 km/h
ঘ
98 km/h

সঠিক উত্তর: গ

96 km/h

উত্তর

সঠিক উত্তর: গ

96 km/h

গ

ব্যাখ্যা

Question: The ratio between the speeds of a bus and bike is 5 : 8. If the bus travels 120 km in 2 hours, find the speed of the bike.

Solution:
Let,
the speed of bus and bike is 5x and 8x.

Speed of bus = 120/2
= 60 km/h

ATQ,
5x = 60
⇒ x= 60/5
∴ x = 12

Speed of bike = 8x = (8 × 12) = 96 km/h

১২,৬৪০.

The average of a, b, c is 8 and a - b = 4, ab = 60, what is the value of c?

ক
ক) 7
খ
খ) 8
গ
গ) 9
ঘ
ঘ) 10

সঠিক উত্তর: খ

খ) 8

উত্তর

সঠিক উত্তর: খ

খ) 8

খ

ব্যাখ্যা

দেয়া আছে,
(a + b + c)/3 = 8
a + b + c = 24 ............ (1)

a - b = 4,
ab = 60

আমরা জানি
(a + b)² = (a - b)² + 4ab
(a + b)² = (4)² + 4 × 60
(a + b)²= 16 + 240
(a + b)² = 256
a + b = 16
(1) নং এ a + b এর মান বসিয়ে পাই,
a + b + c = 24
16 + c = 24
c = 24 - 16
c = 8

১২,৬৪১.

A man and a boy can finish a job working together for 4 days. The man can work as much as two boys can do. Then in how many days can the man do it alone?

ক
6 days
খ
8 days
গ
10 days
ঘ
12 days

সঠিক উত্তর: ক

6 days

উত্তর

সঠিক উত্তর: ক

6 days

ক

ব্যাখ্যা

Question: A man and a boy can finish a job working together for 4 days. The man can work as much as two boys can do. Then in how many days can the man do it alone?

Solution:
1 man and 1 boy can do in 1 day 1/4 part
2 man and 2 boy can do in 1 day 2/4 part = 1/2 part

Now 1 man = 2 boys
(2 + 1) man can do in 1 day is 1/2 part
3 man can do in 1 day is 1/2 part
1 man can do in 1 day is 1/(2 × 3) part = 1/6 part

∴ Man alone can do the work in 6 days.

১২,৬৪২.

The ratio of expenditure and savings is 3 : 2 . If the income increases by 15% and the savings increase by 6%, then by how much percent should his expenditure increases?

ক
20%
খ
21%
গ
23%
ঘ
25%

সঠিক উত্তর: খ

21%

উত্তর

সঠিক উত্তর: খ

21%

খ

ব্যাখ্যা

Question: The ratio of expenditure and savings is 3 : 2 . If the income increases by 15% and the savings increase by 6%, then by how much percent should his expenditure increases?

Solution:
Let, expenditure 300 and savings be 200
Income = 300 + 200
= 500

New income = 500 + 500 × 15/100
= 500 + 75
= 575

New savings = 200 + 200 × 6/100
= 212

New expenditure = 575 - 212
= 363

% increase = {(363 - 300)/300} × 100%
= 21%

১২,৬৪৩.

The captain of a cricket team of 11 members is 26 years old, and the wicket-keeper is three years older than the captain. If the ages of captain and wicketkeeper are excluded, the average age of the remaining players of the team is one year less than the average age of the whole team. What is the average age of the team?

ক
19
খ
21
গ
17
ঘ
23

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: The captain of a cricket team of 11 members is 26 years old, and the wicket-keeper is three years older than the captain. If the ages of captain and wicketkeeper are excluded, the average age of the remaining players of the team is one year less than the average age of the whole team. What is the average age of the team?

Solution:
Let the average age of the whole team by x years.
Total age of the whole team 11x years
Age of the captain 26 years
Age of the wiket-keeper 26 + 3 = 29 years

The average age of the remaining players after excluding the ages of captain and wicketkeeper = x - 1
∴ Total age of the players without the ages of captain and wicketkeeper = 9(x - 1)

We can say,
11x - (26 + 29) = 9(x - 1)
⇒ 11x - 55 = 9x - 9
⇒ 11x - 9x = - 9 + 55
⇒ 2x = 46
∴ x = 23 Years.

১২,৬৪৪.

A tank is filled in 5 hours by three pipes A, B and C. The pipe C is twice as fast as B and B is twice as fast as A. How much time will pipe A alone take to fill the tank?

ক
45 hours
খ
40 hours
গ
30 hours
ঘ
35 hours

সঠিক উত্তর: ঘ

35 hours

উত্তর

সঠিক উত্তর: ঘ

35 hours

ঘ

ব্যাখ্যা

Question: A tank is filled in 5 hours by three pipes A, B and C. The pipe C is twice as fast as B and B is twice as fast as A. How much time will pipe A alone take to fill the tank?

Solution:
Let,
pipe A alone takes x hours to fill the tank.
Then, pipe B take x/2 hours to fill the tank.
and pipe C will take x/4 hours to fill the tank.

ATQ,
(1/x) + (2/x) + (4/x) = 1/5
⇒ 7/x = 1/5
∴ x = 35 hours.

১২,৬৪৫.

A man buys a TV priced at Tk. 16000. He pays Tk. 4000 at once and the rest after 15 months on which he charges a simple interest at the rate of 12% per year. The total amount he pays for TV is -

ক
ক) Tk. 18200
খ
খ) Tk. 17800
গ
গ) Tk. 17200
ঘ
ঘ) Tk. 16800

সঠিক উত্তর: খ

খ) Tk. 17800

উত্তর

সঠিক উত্তর: খ

খ) Tk. 17800

খ

ব্যাখ্যা

Total price of TV = Tk. 16000
Initial payment = Tk. 4000
Remaining amount = Tk. 12000
Simple interest in 15 months for Tk. 12000

S.I. = (P × R × T)/100
= ( 12000 × 12 × 15)/(100 × 12)
= Tk. 1800

⇒ With S.I. total amount to be paid for principal amount Tk. 12000
= Tk. (12000 + 1800)
= Tk. 13800

Therefore,
total amount he pays for the TV is
= 4000 + 13800
= Tk. 17800

১২,৬৪৬.

A man walking at a speed of 3 km/hr. crosses a square field diagonally in 5 minutes. What is the area of the field (in m²) ?

ক
ক) 32550 m²
খ
খ) 32150 m²
গ
গ) 31520 m²
ঘ
ঘ) 31250 m²

সঠিক উত্তর: ঘ

ঘ) 31250 m²

উত্তর

সঠিক উত্তর: ঘ

ঘ) 31250 m²

ঘ

ব্যাখ্যা

Question:
A man walking at a speed of 3 km/hr. crosses a square field diagonally in 5 minutes. What is the area of the field (in m²) ?

Solution:
Man walks in 5 mins = 3 × (5/60) = 1/4 km = 250m

So, the length of the diagonal of the square field = 250m
Let the measure of each side of the square field be X.

According to the question,
X√2 = 250
⇒ X = 250√2
⇒ X² = 31250

∴ The area of the square field is 31250 m².

১২,৬৪৭.

A train crossed a platform in 45 seconds travelling with a speed of 36 km/h. If the length of train is 200 meters, then what will be the length (in meters) of the platform?

ক
250
খ
200
গ
300
ঘ
350

সঠিক উত্তর: ক

250

উত্তর

সঠিক উত্তর: ক

250

ক

ব্যাখ্যা

Question: A train crossed a platform in 45 seconds travelling with a speed of 36 km/h. If the length of train is 200 meters, then what will be the length (in meters) of the platform?

Solution:
Speed = 36 km/hr
= (36 × 1000)/3600 m/s
= 10 m/s

In 45 seconds, the train travelled = 45 × 10 m
= 450 m

length (in meters) of the platform = 450 - 200
= 250 m

১২,৬৪৮.

Which of the following is not a prime number?

ক
ক) 241
খ
খ) 337
গ
গ) 391
ঘ
ঘ) 571

সঠিক উত্তর: গ

গ) 391

উত্তর

সঠিক উত্তর: গ

গ) 391

গ

ব্যাখ্যা

Clearly, 16² > 241.
∴ 16 > √241
Now,
Prime numbers less than 16 are 2, 3, 5, 7, 11, 13.
241 is not divisible by any of them.
∴ 241 is a prime number
In that same fashion,
337 and 571 are also prime numbers.

Now, 391
clearly, 20² > 391
∴ 20 > √391
Now, Prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, 19
Here, 391 is divisible by 17.
∴ 391 is not a prime number.

১২,৬৪৯.

The equation 2x² + kx + 8 = 0 has two equal roots, then the value of k is-

ক
± 10
খ
± 8
গ
± 2√2
ঘ
± √3

সঠিক উত্তর: খ

± 8

উত্তর

সঠিক উত্তর: খ

± 8

খ

ব্যাখ্যা

Question: The equation 2x² + kx + 8 = 0 has two equal roots, then the value of k is-

Solution:
Here
a = 2, b = k and c = 8
Since the equation has two equal roots
: b² - 4ac = 0
⇒ (k)² - 4 × 2 × 8 = 0
⇒ k² = 64
⇒ k = ± √(64)
k = ± 8

১২,৬৫০.

Two pipes, P and Q can fill a cistern in 12 and 15 minutes respectively. Both are opened together, but at the end of 3 minutes, P is turned off. In how many more minutes will Q fill the cistern?

ক
ক) 5 minutes 15 seconds
খ
খ) 6 minutes 15 seconds
গ
গ) 8 minutes 15 seconds
ঘ
ঘ) 12 minutes

সঠিক উত্তর: গ

গ) 8 minutes 15 seconds

উত্তর

সঠিক উত্তর: গ

গ) 8 minutes 15 seconds

গ

ব্যাখ্যা

P can fill cistern in 12 minutes
P fills cistern in 1 minute = 1/12 part

Q can fill cistern in 15 minutes
Q fills cistern in 1 minute = 1/15 part
P and Q together can fill cistern in 1 minute,
=1/12 + 1/15 = 9/60 part
So, they can together fill cistern in 3 minute,
= 3 × 9/60 = 9/20 part
Rest Cistern
= 1− 9/20
= 11/20 part
cistern must be filled by Q in (11/20) ÷ (1/15)
= 8 minutes 15 seconds

১২,৬৫১.

If CAT is coded as 3120, how will DOG be coded?

ক
4157
খ
5147
গ
4175
ঘ
1574

সঠিক উত্তর: ক

4157

উত্তর

সঠিক উত্তর: ক

4157

ক

ব্যাখ্যা

Question: If CAT is coded as 3120, how will DOG be coded?

Solution:

Convert letters to their positions in the alphabet
C = 3
A = 1
T = 20
So, CAT = 3 1 20 = 3120

Now,
apply the same logic to DOG
D = 4
O = 15
G = 7
So, DOG = 4 15 7 = 4157

∴ DOG = 4157

১২,৬৫২.

Two men earn a yearly salary in the ratio 10 : 13. If their spending is in the ratio of 4 : 5 and the man spending lesser of the two saves Tk. 6000 while the other one saves Tk. 8000, then find the salary of the person who is higher paid.

ক
Tk. 10000
খ
Tk. 13000
গ
Tk. 15000
ঘ
Tk. 17000

সঠিক উত্তর: খ

Tk. 13000

উত্তর

সঠিক উত্তর: খ

Tk. 13000

খ

ব্যাখ্যা

Question: Two men earn a yearly salary in the ratio 10 : 13. If their spending is in the ratio of 4 : 5 and the man spending lesser of the two saves Tk. 6000 while the other one saves Tk. 8000, then find the salary of the person who is higher paid.

Solution:
Let, their salary 10x, 13x taka

expenditure 4y, 5y taka

10x - 4y = 6000
⇒ 50x - 20y = 30000

13x - 5y = 8000
⇒ 52x - 20y = 32000

52x - 20y - 50x + 20y = 32000 - 30000
⇒ 2x = 2000
∴ x = 1000

the salary of the person who is higher paid is 13000 taka.

১২,৬৫৩.

The sum of the first five prime number is:

ক
11
খ
18
গ
26
ঘ
28

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: The sum of the first five prime number is:

Solution:
The first five prime number are 2, 3, 5, 7, 11
Required sum =(2 + 3 + 5 + 7 + 11) = 28

১২,৬৫৪.

Solve the inequality (7x + 5)/(3x - 1) < 5

ক
(1/3, 0)
খ
(- ∞, 1/3) ∪ (5/4, ∞)
গ
(1/3, 5/4)
ঘ
(- ∞, 1/3) ∪ (5/4, 7)
ঙ
None of these

সঠিক উত্তর: খ

(- ∞, 1/3) ∪ (5/4, ∞)

উত্তর

সঠিক উত্তর: খ

(- ∞, 1/3) ∪ (5/4, ∞)

খ

ব্যাখ্যা

Question: Solve the inequality (7x + 5)/(3x - 1) < 5

Solution:

The critical points are x = 1/3, 5/4
Plot these points on the number line,

The given inequality is positive, the solution is x ∈ (- ∞, 1/3) ∪ (5/4, ∞).

১২,৬৫৫.

A fruit seller had some bananas. He sells 40% bananas and still has 420 bananas. Originally, he had how many bananas?

ক
200
খ
500
গ
650
ঘ
700

সঠিক উত্তর: ঘ

700

উত্তর

সঠিক উত্তর: ঘ

700

ঘ

ব্যাখ্যা

Question: A fruit seller had some bananas. He sells 40% bananas and still has 420 bananas. Originally, he had how many bananas?

Solution:
Suppose originally he had x bananas.

ATQ,
(100 - 40)% of x = 420
⇒ 60% of x = 420
⇒ 60x/100 = 420
⇒ 60x = 42000
⇒ x = 42000/60
∴ x = 700

Originally he had 700 pcs bananas.

১২,৬৫৬.

If the clock in the mirror shows 5:25, what is the time on the real clock?

ক
8:35
খ
5:35
গ
7:35
ঘ
6:35

সঠিক উত্তর: ঘ

6:35

উত্তর

সঠিক উত্তর: ঘ

6:35

ঘ

ব্যাখ্যা

Question: If the clock in the mirror shows 5:25, what is the time on the real clock?

Solution:
∴ Actual time = 11:60 - time in the mirror
= 11:60 - 5:25
= 6:35

১২,৬৫৭.

An investor earns 3% return on (1/4)^thof his capital, 5% on (2/3)^rdand 11% on the reminder. What is the average rate of return he earns on his total capital?

ক
5%
খ
5.5%
গ
10%
ঘ
10.5%

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: An investor earns 3% return on (1/4)^thof his capital, 5% on (2/3)^rdand 11% on the reminder. What is the average rate of return he earns on his total capital?

Solution:
Let, the capital be x taka

Earns on 3% rate = (x/4) × 3/100 = 3x/400
Earns on 5% interest = (2x/3) × 5/100 = x/30

Remainder = x - (x/4) - (2x/3)
= x{1 - (11/12)}
= x/12

Earns on 11% interest = (x/12) × (11/100)
= 11x/1200

Total return = (3x/400) + (x/30) + (11x/1200)
= (9x + 40x + 11x)/1200
= 60x/1200
= x/20

The average rate of return he earns on his total capital = {(x/20)/x} × 100%
= 5%

১২,৬৫৮.

According to the following symbol substitutions: '+' means '×', '×' means '-', '-' means '÷', and '÷' means '+', what is the result of 20 + 5 - 10 × 4 ÷ 2?

ক
10
খ
8
গ
6
ঘ
12

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: According to the following symbol substitutions: '+' means '×', '×' means '-', '-' means '÷', and '÷' means '+', what is the result of 20 + 5 - 10 × 4 ÷ 2?

Solution:
If '+' means '×', '×' means '-', '-' means '÷', and '÷' means '+',

Now,
20 + 5 - 10 × 4 ÷ 2

The transformed expression will be:
20 × 5 ÷ 10 - 4 + 2
= 100 ÷ 10 - 4 + 2
= 10 - 4 + 2
= 6 + 2
= 8

১২,৬৫৯.

The milk and water in a mixture are in the ratio 7 : 5. When 15 litres of water are added to it, the ratio of milk and water in the new mixture becomes 7 : 8. The total quantity of water in the new mixture is-

ক
35 litres
খ
40 litres
গ
45 litres
ঘ
32 litres
ঙ
None of these

সঠিক উত্তর: খ

40 litres

উত্তর

সঠিক উত্তর: খ

40 litres

খ

ব্যাখ্যা

Question: The milk and water in a mixture are in the ratio 7 : 5. When 15 litres of water are added to it, the ratio of milk and water in the new mixture becomes 7 : 8. The total quantity of water in the new mixture is-

Solution:
Let the quantity of milk in the mixture = 7x litres and that of water = 5x litres.

ATQ,
⇒ 7x/5x+15 = 7/8
⇒ 56x = 35x + 105
⇒ 56x - 35x = 105
⇒ 21x = 105
⇒ x = 105/21
∴ x = 5

∴ Required quantity of water = (5x + 15) litres
= (5 × 5) + 15 = 40 litres

১২,৬৬০.

Five consecutive integers are given. If the sum of the three integers is 24, what is the sum of the last three?

ক
27
খ
28
গ
29
ঘ
30
ঙ
None of these

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

X + X + 1 + X + 2 = 24
or 3X + 3 = 24
or, X = 7
last three (7 + 2) + (7 + 3) + (7 + 4) = 30.

১২,৬৬১.

The sum of the ages of a man and his son is 70 years. After 10 years, the ratio of their ages will be 2 : 1. What is the son's present age?

ক
16
খ
22
গ
20
ঘ
None of these

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: The sum of the ages of a man and his son is 70 years. After 10 years, the ratio of their ages will be 2 : 1. What is the son's present age?

Solution:
ধরি,
বাবার বর্তমান বয়স x বছর
ছেলের বর্তমান বয়স y বছর

প্রশ্নমতে,
x + y = 70 ............ (1)

আবার, 10 বছর পর,
(x + 10)/(y + 10) = 2/1
⇒ x + 10 = 2(y + 10)
⇒ x + 10 = 2y + 20
⇒ x - 2y = 10 ............. (2)

Equation, (1) - (2)
x + y - x + 2y = 70 - 10
⇒ 3y = 60
⇒ y = 20

∴ পুত্রের বর্তমান বয়স = 20 বছর

১২,৬৬২.

ক : খ = ৪ : ৫, খ : গ = ৬ : ৭ হলে ক : খ : গ = কত?

ক
ক) ২৪ : ৩৫ : ৪৫
খ
খ) ২৪ : ৩০ : ৩৫
গ
গ) ২৪ : ৩০ : ৪০
ঘ
ঘ) ২১ : ৩০ : ৩৫

সঠিক উত্তর: খ

খ) ২৪ : ৩০ : ৩৫

উত্তর

সঠিক উত্তর: খ

খ) ২৪ : ৩০ : ৩৫

খ

ব্যাখ্যা

দেয়া আছে,
ক : খ = ৪ : ৫ = ৪ × ৬ : ৫ × ৬ = ২৪ : ৩০
খ : গ = ৬ : ৭ = ৬ × ৫ : ৭ × ৫ = ৩০ : ৩৫
ক : খ : গ = ২৪ : ৩০ : ৩৫

১২,৬৬৩.

How much water must be added to a bucket which contains 40 liters of milk at the cost price of Tk 3.50 per liter so that the cost of milk reduces to Tk 2 per liter?

ক
ক) 25 liters
খ
খ) 28 litres
গ
গ) 30 liters
ঘ
ঘ) 35 liters

সঠিক উত্তর: গ

গ) 30 liters

উত্তর

সঠিক উত্তর: গ

গ) 30 liters

গ

ব্যাখ্যা

Total cost price = Tk (40 x 7/2) = Tk 140
Cost per litre = Tk n2,
Total quantity = 140/2 = 70 Litres.
Water to be added =(70-40) =30 Litres.

১২,৬৬৪.

In how many different ways can the letters of the word "OFFICES" be arranged?

ক
5040
খ
2520
গ
2050
ঘ
1880

সঠিক উত্তর: খ

2520

উত্তর

সঠিক উত্তর: খ

2520

খ

ব্যাখ্যা

Question: In how many different ways can the letters of the word "OFFICES" be arranged?

Solution:
Here,
OFFICES word has 7 letters

Where,
F appears 2 times
All other letters appear once

∴ Total arrangements = 7!/2! = 5040/2 = 2520

১২,৬৬৫.

A radio sold a loss of 2.5%. Had he sold it for Tk. 100 more, he would have gained 7.5%. In order to gain 12.5%, he should sell it for:

ক
ক) 850
খ
খ) 925
গ
গ) 1080
ঘ
ঘ) 1125
ঙ
ঙ) 1200

সঠিক উত্তর: ঘ

ঘ) 1125

উত্তর

সঠিক উত্তর: ঘ

ঘ) 1125

ঘ

ব্যাখ্যা

Cost Price = 100
Selling price = 100 - 2.5 = 97.5
New selling price at 7.5% = 107.5
difference between two selling price = 107.5 - 97.5 = 10
10% price is equal to Tk. 100
1% price is equal to Tk. 100/10%
100% price is equal to Tk. (100/10%) × 100
= 1000
so Cost price = 1000
SP at 12.5% = (100+12.5/100) × 1000
= 1125

১২,৬৬৬.

If 1 < p < 4 and 2 < q < 6, which of the following best describes p - q?

ক
- 5 < p - q < 2
খ
- 4 < p - q < 5
গ
- 1 < p - q < 4
ঘ
0 < p - q < 5

সঠিক উত্তর: ক

- 5 < p - q < 2

উত্তর

সঠিক উত্তর: ক

- 5 < p - q < 2

ক

ব্যাখ্যা

Question: If 1 < p < 4 and 2 < q < 6, which of the following best describes p - q?

Solution: দেয়া আছে:
1 < p < 4 -------------(1)
2 < q < 6 -------------(2)

এখন, আমরা p - q এর সীমা বের করতে চাই।

(2)⇒
2 < q < 6
⇒ - 2 > -q > - 6 (যদি -1 দ্বারা গুণ করি)
⇒ - 6 < -q < - 2 ----------(3)

(1) এবং (3) যোগ করি,
⇒ (1 + (- 6 )) < p - q < (4 + (- 2))
⇒ - 5 < p - q < 2

১২,৬৬৭.

Find

ক
3
খ
6
গ
10
ঘ
12
ঙ
14

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: Find

Solution:

১২,৬৬৮.

A person borrows Rs. 5000 for 2 years at 4% p.a. simple interest. He immediately lends it to another person at 6% p.a compound interest for 2 years. Find his total gain in the transaction.

ক
Tk. 258
খ
Tk. 180
গ
Tk. 150
ঘ
Tk. 218

সঠিক উত্তর: ঘ

Tk. 218

উত্তর

সঠিক উত্তর: ঘ

Tk. 218

ঘ

ব্যাখ্যা

Question: A person borrows Rs. 5000 for 2 years at 4% p.a. simple interest. He immediately lends it to another person at 6% p.a compound interest for 2 years. Find his total gain in the transaction.

Solution:
Given,
P = Tk. 5000
n = 2 years
r = 4% = 4/100 = 1/25

So the simple interest = 5000 × (1/25) × 2
= Tk. 400

Again,
compound interest rate r = 6% = 6/100 = 3/50

So,
The compound interest = P(1 + r)ⁿ - P
= 5000(1 + 3/50)² - 5000
= {5000 × (53/50)²} - 5000
= (5000 × 2809/2500) - 5000
= 5618 - 5000
= 618

So the total gain in the transaction = (618 - 400)
= Tk. 218

১২,৬৬৯.

If 4x² - px + 25 is a square number, then p =?

ক
9
খ
16
গ
25
ঘ
20

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: If 4x² - px + 25 is a square number, then p =?

Solution:
4x² - px + 25
= (2x)² - 2. 2x. 5 + 5² - px + 20x
= (2x - 5)² - px + 20x

Now,
- px + 20x = 0 [The expression is a square number]
⇒ px = 20x
∴ p = 20

১২,৬৭০.

A tank can be filled with water by two pipes A and B together in 36 minutes. If the pipe B was stopped after 30 minutes, the tank is filled in 40 minutes. The pipe B can along fill the tank in:

ক
45 minutes
খ
60 minutes
গ
75 minutes
ঘ
90 minutes

সঠিক উত্তর: ঘ

90 minutes

উত্তর

সঠিক উত্তর: ঘ

90 minutes

ঘ

ব্যাখ্যা

Question: A tank can be filled with water by two pipes A and B together in 36 minutes. If the pipe B was stopped after 30 minutes, the tank is filled in 40 minutes. The pipe B can along fill the tank in:

Solution:
Let the pipes fill the tank in x minutes.
Part of tank filled by pipes A and B in one minute:
1/36 Part of the tank filled by pipe A in 1 minute = 1/36 − 1/x
According to the question,
30×1/x + 40(1/36 − 1/x) = 1
⇒ 30/x + 40/36 − 40/x = 1
⇒ 30/x + 10/9 − 40/x = 1
⇒ 10/x = 1/9
Hence, x = 90 minutes.

১২,৬৭১.

In how many ways can 8 Bangladeshi and, 4 American and 4 Englishmen can be seated in a row so that all person of the same nationality sit together?

ক
ক) 3! 8! 4! 4!
খ
খ) 8! 4! 4!/3!
গ
গ) 8! 4! 4!
ঘ
ঘ) 3! 8! 4! 4!/3

সঠিক উত্তর: ক

ক) 3! 8! 4! 4!

উত্তর

সঠিক উত্তর: ক

ক) 3! 8! 4! 4!

ক

ব্যাখ্যা

Taking all person of same nationality as one person, then we will have only three people.
These three person can be arranged themselves in 3! Ways.
8 Bangladeshi can be arranged themselves in 8! Way.
4 American can be arranged themselves in 4! Ways.
4 Englishman can be arranged themselves in 4! Ways.
Hence, required number of ways = 3! 8! 4! 4! Ways.
----------------------------------------------------------
Alternative way:

১২,৬৭২.

If (y - 1) (y + 2) = (y + 4) (y - 2) then what is the value of y?

ক
ক) 12
খ
খ) 9
গ
গ) 6
ঘ
ঘ) 3

সঠিক উত্তর: গ

গ) 6

উত্তর

সঠিক উত্তর: গ

গ) 6

গ

ব্যাখ্যা

Question: If (y - 1) (y + 2) = (y + 4) (y - 2) then what is the value of y?

Solution:
(y - 1) (y + 2) = (y + 4) (y - 2)
⇒ y² - y + 2y - 2 = y² + 4y - 2y - 8
⇒ y - 2 = 2y - 8
⇒ y - 2y = - 8 + 2
∴ y = 6

১২,৬৭৩.

The product of two numbers is 9375 and the quotient, when the larger one is divided by the smaller, is 15. The sum of the numbers is:

ক
ক) 420
খ
খ) 400
গ
গ) 450
ঘ
ঘ) 600

সঠিক উত্তর: খ

খ) 400

উত্তর

সঠিক উত্তর: খ

খ) 400

খ

ব্যাখ্যা

Let
the larger numbers be x and
the smaller numbers be y.

Then, xy = 9375 and
x/y = 15
now
xy/(x/y) = 9375/15
y² = 625
y² = 25²
y = 25

x = 25 × 15 = 375

x + y = 375 + 25 = 400

১২,৬৭৪.

The average attendance of a college for the first three days of a week is 327 and the first four days is 321. How many were present on the fourth day?

ক
ক) 310
খ
খ) 320
গ
গ) 303
ঘ
ঘ) 307

সঠিক উত্তর: গ

গ) 303

উত্তর

সঠিক উত্তর: গ

গ) 303

গ

ব্যাখ্যা

প্রশ্ন : The average attendance of a college for the first three days of a week is 327 and the first four days is 321. How many were present on the fourth day?
সমাধান :
Total attendance for first 3 days = 327×3 = 981
Total attendance for first 4 days = 321×4 = 1284
∴ Present on the 4th day = 1284 - 981 = 303

১২,৬৭৫.

125, 127, 130, 135, 142, 153, 165

ক
ক) 130
খ
খ) 142
গ
গ) 153
ঘ
ঘ) 165

সঠিক উত্তর: ঘ

ঘ) 165

উত্তর

সঠিক উত্তর: ঘ

ঘ) 165

ঘ

ব্যাখ্যা

Prime numbers 2, 3, 5, 7, 11, 13 are to be added successively. So, 165 is wrong.

১২,৬৭৬.

4 men and 6 women can complete a work in 8 days, while 3 men and 7 women can complete it in 10 days. In how many days will 10 women complete it?

ক
ক) 40 days
খ
খ) 36 days
গ
গ) 32 days
ঘ
ঘ) 34 days

সঠিক উত্তর: ক

ক) 40 days

উত্তর

সঠিক উত্তর: ক

ক) 40 days

ক

ব্যাখ্যা

Let 1 man's 1 day's work = x and 1 woman's 1 day's work = y
Then, 4x + 6y = 1/8 and 3x + 7y = 1/10
Solving the two equations, we get
x = 11/400, y = 1/400
∴ 1 woman's 1 day's work = 1/400
⇒ 10 women's 1 day's work = ((1/400)×10) = 1/40
Hence, 10 women will complete the work in 40 days

১২,৬৭৭.

If 75% of the students in a school are boys and the number of girls is 520, the number of boys is:

ক
1450
খ
1560
গ
1230
ঘ
1280

সঠিক উত্তর: খ

1560

উত্তর

সঠিক উত্তর: খ

1560

খ

ব্যাখ্যা

Question: If 75% of the students in a school are boys and the number of girls is 520, the number of boys is:

Solution:
Percentage of boys = 75%
Percentage of girls = 25%

ATQ,
∴ Number of Girls = 25% = 520
⇒ 25% = 520
⇒ 1% = 520/25
∴ 75% = (520 × 75)/25
= 1560

১২,৬৭৮.

Today is Sunday. What time will be after 353 days?

ক
Tuesday
খ
Wednesday
গ
Sunday
ঘ
Monday

সঠিক উত্তর: খ

Wednesday

উত্তর

সঠিক উত্তর: খ

Wednesday

খ

ব্যাখ্যা

Question: Today is Friday. What time will be after 353 days?

Solution:
এখানে,
৩৫৩ কে ৭ দিয়ে ভাগ করলে ভাগশেষ থাকে ৩।
৩৫৩ দিন থেকে ৩ দিন বাদ দিলে হয় ৩৫০ দিন।

আমরা জানি,
যে কোনো তারিখ হতে ৭ দিন পর পর (৮ম দিনে) একই বার পাওয়া যায়।
অর্থ্যাৎ, শুক্রবারের ৭ দিন পর বা ৮ম দিনে গিয়ে আবার শুক্রবার পাওয়া যাবে।

অর্থাৎ,
৩৫১ তম দিন রবিবার।
৩৫২ তম দিন সোমবার।
৩৫৩ তম দিন মঙ্গলবার ।
৩৫৪ তম দিন বুধবার।

১২,৬৭৯.

What is the least number of digits (including repetitions) needed to express 10¹⁰⁰ in decimal notation?

ক
100 digits
খ
1000 digits
গ
1001 digits
ঘ
101 digits
ঙ
None of the above

সঠিক উত্তর: ঘ

101 digits

উত্তর

সঠিক উত্তর: ঘ

101 digits

ঘ

ব্যাখ্যা

Question: What is the least number of digits (including repetitions) needed to express 10¹⁰⁰ in decimal notation?

Solution:
10¹ = 2 digits
10² = 3 digits
10³ = 4digits
10⁴ = 5digits
...........
...........
10ⁿ= n + 1 digits
Therefore, 10¹⁰⁰ = 100 + 1 = 101 digits

১২,৬৮০.

What is the least number that, when tripled, is exactly divisible by 9, 12, 15, and 18?

ক
45
খ
60
গ
84
ঘ
120

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: What is the least number that, when tripled, is exactly divisible by 9, 12, 15, and 18?

Solution:
Let the number be x.
Tripled the number is 3x.

9 = 3 × 3
12 = 2 × 2 × 3
15 = 3 × 5
18 = 2 × 3 × 3

LCM = 2 × 2 × 3 × 3 × 5 = 180

∴ 3x = 180
∴ x = 180/3 = 60

১২,৬৮১.

The salaries of P, Q, and R are in the ratio 5 : 8 : 6 and their expenses are in the ratio 4 : 7 : 5. If P saves 1/3rd of his salary then the savings of P, Q, and R are in the ratio of-

ক
11 : 15 : 17
খ
7 : 15 : 12
গ
13 : 19 : 9
ঘ
10 : 13 : 11

সঠিক উত্তর: ঘ

10 : 13 : 11

উত্তর

সঠিক উত্তর: ঘ

10 : 13 : 11

ঘ

ব্যাখ্যা

Question: The salaries of P, Q, and R are in the ratio 5 : 8 : 6 and their expenses are in the ratio 4 : 7 : 5. If P saves 1/3rd of his salary then the savings of P, Q, and R are in the ratio of-

Solution:
Let the salaries of P, Q and R are 5x, 8x, and 6x respectively
and expenses of P, Q and R are 4y, 7y and 5y respectively
ATQ,
5x - 4y = 5x × 1/3
⇒ 15x - 12y = 5x
⇒ 10x = 12y
⇒ x : y = 12 : 10

∴ The ratio of savings of P, Q, and R
= (5x - 4y) : (8x - 7y) : (6x - 5y)
= (5 × 12 - 4 × 10) : (8 × 12 - 7 × 10) : (6 × 12 - 5 × 10)
= (60 - 40) : (96 - 70) : (72 - 50)
= 20 : 26 : 22
= 10 : 13 : 11
Therefore, the savings of P, Q, and R are in the ratio 10 : 13 : 11

১২,৬৮২.

There are 3 white ball, 4 green ball and 5 orange colour balls in a bag. If a ball is picked at random, what is the probability of having either a green or a orange balls?

ক
3/4
খ
2/3
গ
1/3
ঘ
3/5

সঠিক উত্তর: ক

3/4

উত্তর

সঠিক উত্তর: ক

3/4

ক

ব্যাখ্যা

Question: There are 3 white ball, 4 green ball and 5 orange colour balls in a bag. If a ball is picked at random, what is the probability of having either a green or a orange balls?

Solution:
Here, total number of balls = (3 + 4 + 5) = 12

The probability of getting a green ball = 4/12
The probability of getting a orange ball = 5/12

∴ The probability of getting either green or a orange ball = (4/12) + (5/12)
= (4 + 5)/12
= 9/12
= 3/4

১২,৬৮৩.

If P = {multiples of 3 between 1 and 20} and Q = {even natural numbers up to 15}, then find P - Q.

ক
{3, 6, 9, 15, 18}
খ
{6, 12}
গ
{3, 9, 15}
ঘ
{3, 9, 15, 18}

সঠিক উত্তর: ঘ

{3, 9, 15, 18}

উত্তর

সঠিক উত্তর: ঘ

{3, 9, 15, 18}

ঘ

ব্যাখ্যা

Question: If P = {multiples of 3 between 1 and 20} and Q = {even natural numbers up to 15}, then find P - Q.

Solution:
P = {multiples of 3 between 1 and 20}
∴ P = {3, 6, 9, 12, 15, 18}

Q = {even natural numbers upto 15}
∴ Q = {2, 4, 6, 8, 10, 12, 14}

Hence,
P - Q = {3, 6, 9, 12, 15, 18} - {2, 4, 6, 8, 10, 12, 14}
= {3, 9, 15, 18}

১২,৬৮৪.

ABCD is a rhombus. AB = 3x - 2, AC = 4x + 4, and BD = 2x. Find x.

ক
5
খ
3
গ
2
ঘ
1.5

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: ABCD is a rhombus. AB = 3x - 2, AC = 4x + 4, and BD = 2x. Find x.

Solution:
A rhombus is a quadrilateral with four sides of equal length. Rhombuses have diagonals that bisect each other at right angles.

Thus, we can consider the right triangle AED and use the Pythagorean Theorem to solve for x. From the problem:
AD = 3x - 2
BD = 2x
AC = 4x + 4

Because the diagonals bisect each other, we know:
ED = x
AE = 2x + 2

Using the Pythagorean Theorem,
a² + b² = c²
∴ AE² + ED² = AD²
⇒ (2x + 2)² + (x)² = (3x - 2)²
⇒ 4x² + 8x + 4 + x² = 9x² - 12x + 4
⇒ 0 = 4x² - 20x

Factoring,
0 = x and 0 = 4x - 20
The first solution is nonsensical for this problem.
0 = 4x - 20
∴ x = 5

১২,৬৮৫.

Due to a 25% reduction in sugar price, a housewife can buy 5 extra kilograms for 600 Taka. Find the sugar’s original price per kg.

ক
Tk. 32
খ
Tk. 40
গ
Tk. 28
ঘ
Tk. 48

সঠিক উত্তর: খ

Tk. 40

উত্তর

সঠিক উত্তর: খ

Tk. 40

খ

ব্যাখ্যা

Question: Due to a 25% reduction in sugar price, a housewife can buy 5 extra kilograms for 600 Taka. Find the sugar’s original price per kg.

Solution:
Let original price per kg = p Taka
Reduced price = p - p of 25% = 0.75p
Original quantity for 600 Taka
Q₁ = 600/p kg

And
New quantity for 600 Taka
Q₂ = 600/0.75p = 800/p kg

Now, quantity difference,
Q₂ - Q₁ = 5
(800/p) - (600/p) = 5
200/p = 5
p = 200/5
p = 40

∴ Original price per kg = Tk. 40

১২,৬৮৬.

A man complete a journey in 5 hours. He travels first half of the journey at the rate of 16 km/hr and second half at the rate of 24 km/hr. Find the total journey in km.

ক
ক) 48km
খ
খ) 72km
গ
গ) 96km
ঘ
ঘ) 108km

সঠিক উত্তর: গ

গ) 96km

উত্তর

সঠিক উত্তর: গ

গ) 96km

গ

ব্যাখ্যা

Question: A man complete a journey in 5 hours. He travels first half of the journey at the rate of 16 km/hr and second half at the rate of 24 km/hr. Find the total journey in km.

Solution:
Let
The total journey x km
Now
{(1/2)x/16} + {(1/2)x/24} = 5
⇒ (x/16) + (x/24) = 10
⇒ (3x + 2x)/48 = 10
⇒ 5x/48 = 10
⇒ x = (10 × 48)/5
x = 96km

১২,৬৮৭.

Find the highest common factor of 36 and 84.

ক
4
খ
6
গ
12
ঘ
18

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: Find the highest common factor of 36 and 84.

Solution:
36 = 2 × 2 × 3 × 3
84 = 2 × 2 × 3 × 7

H.C.F. of 36 and 84 = 2 × 2 × 3 = 12.

১২,৬৮৮.

A man bought 40 shares of Tk. 60 at 5 discount, the rate of dividend being 12(1/2)%. The rate of interest obtained is -

ক
ক) 15.5%
খ
খ) 14%
গ
গ) 13.64%
ঘ
ঘ) 14.25%

সঠিক উত্তর: গ

গ) 13.64%

উত্তর

সঠিক উত্তর: গ

গ) 13.64%

গ

ব্যাখ্যা

Face Value of a share = Tk. 60
He bought each share at Tk. 60 - Tk. 5 = Tk. 55
Number of shares = 40
Dividend = 12(1/2)%
= (25/2)%

Dividend per share = (60 × 25)/(2 × 100)
= Tk. 7.5

Total dividend = (40 × 7.5)
∴ He got a dividend of (40 × 7.5) for an investment of Tk. (40 × 55)

Interest obtained
= (40 × 7.5 × 100)/(40 × 55)
= 13.64%

১২,৬৮৯.

The top and bottom of a flag on a building subtend angles of 60° and 30° respectively at a point B which is 48 meter away from the building. Find the height of the flag?

ক
32 m
খ
32√3 m
গ
18.49 m
ঘ
16 m

সঠিক উত্তর: খ

32√3 m

উত্তর

সঠিক উত্তর: খ

32√3 m

খ

ব্যাখ্যা

Question: The top and bottom of a flag on a building subtend angles of 60° and 30° respectively at a point B which is 48 meter away from the building. Find the height of the flag?

Solution:

Let height of building be AC = X and height of flag be CD = h.

In ΔDAB
tan60° = (X + h)/48
⇒ √3 = (X + h)/48
⇒ X + h = 48√3
∴ h = 48√3 - X ..................(1)

In ΔCAB
tan30° = X/48
⇒ 1/√3 = X/48
∴ X = 48/√3

From (1) we get,
h = 48√3 - 48/√3
= (48 × 3 - 48)/√3
= (144 - 48)/√3
= 96/√3
= (32 × 3)/√3
= 32√3

১২,৬৯০.

In how many ways can the letters of the word 'ARRANGE' be arranged?

ক
360
খ
1530
গ
720
ঘ
1260

সঠিক উত্তর: ঘ

1260

উত্তর

সঠিক উত্তর: ঘ

1260

ঘ

ব্যাখ্যা

Question: In how many ways can the letters of the word 'ARRANGE' be arranged?

Solution:
'ARRANGE' শব্দটিতে মোট 7 টি বর্ণ আছে।

এই বর্ণগুলোর মধ্যে
'A' আছে 2 বার,
'R' আছে 2 বার,
এবং বাকি বর্ণগুলো ভিন্ন ভিন্ন।

সুতরাং, মোট সাজানোর সংখ্যা = 7!/(2! × 2!)
= 5040/(2 × 2)
= 5040/4
= 1260

∴ মোট বিন্যাস সংখ্যা = 1260

১২,৬৯১.

Find the area of a triangle with side lengths of 5 feet, 12 feet, and 13 feet.

ক
17 square feet
খ
27 square feet
গ
30 square feet
ঘ
37 square feet

সঠিক উত্তর: গ

30 square feet

উত্তর

সঠিক উত্তর: গ

30 square feet

গ

ব্যাখ্যা

Question: Find the area of a triangle with side lengths of 5 feet, 12 feet, and 13 feet.

Solution:
ধরি,
তিনবাহুর দৈর্ঘ্য a = 5 ফুট, b = 12 ফুট ও c = 13 ফুট

∴ ত্রিভুজটির অর্ধ-পরিসীমা, s = (5 + 12 + 13)/2 = 15 ফুট

∴ ক্ষেত্রফল = √{s(s - a) (s - b) (s - c)} বর্গ ফুট
= √{15(15 - 5) (15 - 12) (15 - 13)} বর্গ ফুট
= √(15 × 10 × 3 × 2)
= √900 বর্গ ফুট
= 30 বর্গ ফুট

১২,৬৯২.

Age of Kamal will be 4 times the age of Rina in 6 years from today. If ages of Kamal and Mahi are 7 times and 6 times the age of Rina respectively, what is present age of Kamal?

ক
38 years
খ
52 years
গ
48 years
ঘ
36 years
ঙ
42 years

সঠিক উত্তর: ঙ

42 years

উত্তর

সঠিক উত্তর: ঙ

42 years

ঙ

ব্যাখ্যা

Question: Age of Kamal will be 4 times the age of Rina in 6 years from today. If ages of Kamal and Mahi are 7 times and 6 times the age of Rina respectively, what is present age of Kamal?

Solution:
Let,
Rina's present age = R years
Then,
Kamal's present age = 7 × R = 7R
Mahi's present age = 6 × R = 6R

Age of Kamal will be 4 times the age of Rina in 6 years.
Kamal’s age after 6 years = 7R +6
Rina’s age after 6 years = R + 6

ATQ,
7R + 6 = 4(R + 6)
⇒ 7R + 6 = 4R + 24
⇒ 7R - 4R = 18
⇒ R = 18/3
∴ R = 6

∴ Kamal's present age = 7 × R = 7R = 7 × 6 = 42 years

১২,৬৯৩.

Hasan rides his bike at a speed of 24 km/h and arrives at his office 5 minutes late. If he had traveled 25% faster, he would have arrived 4 minutes before the scheduled time. What is the distance between his house and his office (in km)?

ক
18 km
খ
24 km
গ
36 km
ঘ
40 km
ঙ
none of these

সঠিক উত্তর: ক

18 km

উত্তর

সঠিক উত্তর: ক

18 km

ক

ব্যাখ্যা

Question: Hasan rides his bike at a speed of 24 km/h and arrives at his office 5 minutes late. If he had traveled 25% faster, he would have arrived 4 minutes before the scheduled time. What is the distance between his house and his office (in km)?

Solution:
দেওয়া আছে, হাসানের প্রাথমিক গতিবেগ = 24 কিমি/ঘন্টা

25% গতিবেগ বৃদ্ধি পেলে নতুন গতিবেগ হবে = 24 + (24 × 25%)
= 24 + 6 = 30 কিমি/ঘন্টা

মনে করি, বাড়ি থেকে অফিসের দূরত্ব x কিমি।
∴ প্রথম ক্ষেত্রে সময় লাগে = x/24 ঘন্টা
দ্বিতীয় ক্ষেত্রে সময় লাগে = x/30 ঘন্টা

সময়ের পার্থক্য = 5 মিনিট (দেরি) + 4 মিনিট (আগে) = 9 মিনিট
= 9/60 ঘণ্টা

প্রশ্নমতে,
(x/24) - (x/30) = 9/60
⇒ (5x - 4x) /120 = 3/20
⇒ x/120 = 3/20
⇒ x = (3 × 120)/20
⇒ x = 3 × 6
∴ x = 18

সুতরাং, তার বাড়ি থেকে অফিসের দূরত্ব 18 কিমি।

১২,৬৯৪.

When x is divided by 7, the remainder is 6. Which of the following must be an even number?

ক
x + 6
খ
x² + x
গ
x - 4
ঘ
None

সঠিক উত্তর: খ

x² + x

উত্তর

সঠিক উত্তর: খ

x² + x

খ

ব্যাখ্যা

Question: When x is divided by 7, the remainder is 6. Which of the following must be an even number?

Solution:
Given that when x is divided by 7, the remainder is 6, we can express x as:
x = 7k + 6 where k is an integer.

Now, let's evaluate each option to see which, if any, must be an even number.

Option ক)
x + 6
x + 6 = (7k + 6) + 6 = 7k +12
Since 7k is odd or even depending on k, adding 12 (which is even) does not guarantee that x + 6 will be even.
Therefore, x + 6 is not necessarily even.

Option খ)
x² + x
x² + x = (7k+6)² + (7k+6) Let's expand and simplify:
x² + x = (49k² + 84k + 36) + (7k + 6) = 49k² + 91k + 42
All terms are multiples of 7, and 42 is an even number, making the entire expression even.
Thus, x² + x must be an even number.

Option গ)
x - 4
x - 4 = (7k + 6) - 4 = 7k + 2
This is not necessarily even because 7k + 2 depends on the parity of k.
Therefore, x - 4 is not necessarily even.

Conclusion:
The correct answer is খ) x² + x

১২,৬৯৫.

A merchant has three different types of milk- 324 litres, 351 litres and 459 litres. Find the minimum number of casks of equal size that can store the milk without mixing.

ক
61
খ
45
গ
51
ঘ
42
ঙ
None of the above

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: A merchant has three different types of milk- 324 litres, 351 litres and 459 litres. Find the minimum number of casks of equal size that can store the milk without mixing.

Solution:
Since a minimum number of casks are required, the size of the cask is greatest.
Also the cask in three cases is of equal size. The size of the cask is the H.C.F. of 324 litres, 351 litres and 459 litres which is 27.

Now, the number of casks required for storing the milk = (324 + 351+ 459)/27 = 42.

১২,৬৯৬.

Nirmal and Kapil started a business investing Tk 9000 and Tk 12000 respectively. After 6 months, Kapil withdrew half of his investment. If after a year, the total profit was Tk 4600, what was Kapil’s share initially?

ক
ক) Tk 2300
খ
খ) Tk 2400
গ
গ) Tk 2500
ঘ
ঘ) None of above

সঠিক উত্তর: ক

ক) Tk 2300

উত্তর

সঠিক উত্তর: ক

ক) Tk 2300

ক

ব্যাখ্যা

Nirmal:Kapil = 9000×12:(12000×6 + 6000×6)= 1:1
Kapil's share = Tk [4600 ×(1/2)) = Tk. 2300

১২,৬৯৭.

An investor earns 3% return on 1/4 of his capital, 5% on 2/3 and 11% on the remainder. What is the average rate of return he earns on his total capital?

ক
ক) 4%
খ
খ) 3%
গ
গ) 5%
ঘ
ঘ) 6%

সঠিক উত্তর: গ

গ) 5%

উত্তর

সঠিক উত্তর: গ

গ) 5%

গ

ব্যাখ্যা

Let
The investor total capital is x

Total return = 3% of (x/4) + 5% of (2x/3) + 11% of {x - (x/4) - (2x/3)}
= (3x/400) + (10x/300) + (11/100) of {(12x - 3x - 8x)/12}
= (3x/400) + (10x/300) + (11x/1200)
= (9x + 40x + 11x)/1200
= 60x/1200
= x/20

the average rate of return =[{(x/20)/x} × 100]%
= 5%

১২,৬৯৮.

Find the least number which is when divided by 12 and 16 will have remainders of 5 and 9 respectively?

ক
ক) 42
খ
খ) 41
গ
গ) 43
ঘ
ঘ) 44

সঠিক উত্তর: খ

খ) 41

উত্তর

সঠিক উত্তর: খ

খ) 41

খ

ব্যাখ্যা

Question: Find the least number which is when divided by 12 and 16 will have remainders of 5 and 9 respectively?
Solution:
১২ - ৫ = ৭; ১৬ - ৯ = ৭
এখন,
১২ = ২ × ২ × ৩
১৬ = ২ × ২ × ২ × ২
∴ ১২, ১৬ এর লসাগু = ২ × ২ × ৩ × ২ × ২ = ৪৮
সুতরাং, নির্ণেয় সংখ্যা = ৪৮ - ৭ = ৪১

১২,৬৯৯.

A man sells a watch for Tk. 960 at a loss of 20%. What was the original cost price of the watch?

ক
1500 Tk.
খ
1220 Tk.
গ
100 Tk.
ঘ
1200 Tk.

সঠিক উত্তর: ঘ

1200 Tk.

উত্তর

সঠিক উত্তর: ঘ

1200 Tk.

ঘ

ব্যাখ্যা

Question: A man sells a watch for Tk. 960 at a loss of 20%. What was the original cost price of the watch?

Answer:
Here,
Selling Price = 960 Tk.
and Loss 20 %,
Now we can write,
The Selling Price is 80 Tk. and the cost price is = 100 Tk.
∴ The Selling Price is 1 Tk. then the cost price is = 100/80 Tk.
∴ The Selling Price is 960 Tk. then the cost price is = (100×960)/80 Tk.
= 1200 Tk.

১২,৭০০.

If 7x + 3y = 17 and 3x + 7y = 19, what is the average of x and y?

ক
1.55
খ
2.05
গ
1.8
ঘ
3.6
ঙ
None

সঠিক উত্তর: গ

1.8

উত্তর

সঠিক উত্তর: গ

1.8

গ

ব্যাখ্যা

প্রশ্ন: If 7x + 3y = 17 and 3x + 7y = 19, what is the average of x and y?

সমাধান:
দেওয়া আছে,
7x + 3y = 17 .......... (1)
3x + 7y = 19 ......... (2)

(1) নং + (2) নং ⇒
7x + 3y + 3x + 7y = 17 + 19
⇒ 10x + 10y = 36
⇒ 10(x + y) = 36
⇒ x + y = 36/10
⇒ x + y = 3.6

অতএব, গড় = (x + y)/2
= 3.6/2
= 1.8