Bank Math

PrepBank · পাতা ৬৯ / ১৬১ · ৬,৮০১–৬,৯০০ / ১৬,১২৪

৬,৮০১.

A man can do a work in 20 days and a woman in 15 days. If they work on it together for 5 days, then the fraction of the work that is left is-

ক
1/12
খ
1/10
গ
5/12
ঘ
7/15

ব্যাখ্যা

Question: A man can do a work in 20 days and a woman in 15 days. If they work on it together for 5 days, then the fraction of the work that is left is-

Solution:
Man’s 1 day’s work = 1/20
Woman’s 1 day’s work = 1/15

(Man + woman)’s 1 day’s work = (1/20 + 1/15) = 7/60
(Man + woman)’s 5 day’s work = (7/60 × 5) = 7/12

Thus, Remaining work = 1 - 7/12 = 5/12

৬,৮০২.

AB and CD are two parallel chords on the opposite sides of the center of the circle. If AB = 10 cm , CD = 24 cm and the radius of the circle is 13 cm, the distance between the chords is-

ক
17 cm
খ
10 cm
গ
16 cm
ঘ
18 cm
ঙ
24 cm

ব্যাখ্যা

Question: AB and CD are two parallel chords on the opposite sides of the center of the circle. If AB = 10 cm , CD = 24 cm and the radius of the circle is 13 cm, the distance between the chords is-

Solution:

From O draw OL ⊥ AB and OM ⊥ CD. Join OA and OC.
AL = AB/2 = 5cm , OA = 13 cm.
OL² = OA² - AL² = (13)² - 5² = (169 - 25) = 144
⇒ OL = √144 = 12 cm.

Now,
CM = CD/2 = 12 cm and OC = 13c m.
∴ OM² = OC² - CM² = (13)² - (12)² = (169 - 144) = 25
⇒ OM =√25 = 5 cm.

∴ ML = OM + OL = (5 +12 ) cm = 17cm.

৬,৮০৩.

The number of subsets of a set with 6 elements is:

ক
24
খ
64
গ
32
ঘ
48

ব্যাখ্যা

Question: The number of subsets of a set with 6 elements is:

Solution:
- কোনো সেট থেকে যতগুলো সেট গঠন করা যায়, এদের প্রত্যেকটি সেটকে ঐ সেটের উপসেট (subset) বলা হয়।
কোনো সেটের উপাদানের সংখ্যা, n = 6
ঐ সেটের উপসেট (subset) সংখ্যা = 2ⁿ
=2⁶
= 64

৬,৮০৪.

A shirt is sold for Tk. 1500 at a profit of 20%. What would have been the actual profit or loss if it had been sold for Tk. 1200?

ক
profit 5%
খ
loss of 8%
গ
profit 4.5%
ঘ
None of these

ব্যাখ্যা

Question: A shirt is sold for Tk. 1500 at a profit of 20%. What would have been the actual profit or loss if it had been sold for Tk. 1200?

Solution:
Firstly let us find the cost price of the same. C.P. = 1500 × (100/120) = 1250.
New selling price = 1200
Loss = 1250 - 1200 = 50

∴ Loss percentage = 100 × (50/1250)
= 4%.

If sold at Tk. 1200, there would be a loss of 4%.

৬,৮০৫.

A milkman claims to sell milk at its cost price, still, he is making a profit of 30% since he has mixed some amount of water in the milk. What is the % of milk in the mixture?

ক
ক) 71.02%
খ
খ) 76.92%
গ
গ) 63.22%
ঘ
ঘ) 86.42%

ব্যাখ্যা

Let the milk he bought is 1000 ml
Let C.P of 1000 ml is Tk. 100

Here let he is mixing K ml of water
He is getting 30% profit

⇒ Now, the selling price is also Tk. 100 for 1000 ml
⇒ 100 : K%
⇒ 100 : 30
10 : 3 is the ratio of milk to water

Percentage of milk = 10 x 100/13
= 1000/13
= 76.92%

৬,৮০৬.

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

ক
5 hours
খ
6 hours
গ
4 hours
ঘ
7 hours
ঙ
10 hours

ব্যাখ্যা

We are given that,
3 pumps, working 4 hours a day, can empty a tank in 2 days.
Therefore, it means that:
3 pumps take a total of 8 hours to empty the tank.
Hence, 1 pump will take 8 x 3 = 24 hours

As the number of pumps decreases, the time required increases.
So, if 4 pumps work, the time required decreases.
∴ 24/4 = 6hrs. needed to empty the tank in 1 day.

৬,৮০৭.

tanA + sinA = m and tanA - sinA = n, then (m² - n²)/4 = ?

ক
√m
খ
mn
গ
√mn
ঘ
√mn/2

ব্যাখ্যা

প্রশ্ন: tanA + sinA = m and tanA - sinA = n, then (m² - n²)/4 = ?

সমাধান:
(m² - n²)/4
= {(tanA + sinA)² - (tanA - sinA)²}/4
= (4tanA . sinA)/4 [(a + b)² - (a - b)² = 4ab]
= √(tan²A . sin²A)
= √{tan²A (1- cos²A)}
= √(tan²A - tan²A . cos²A)
= √(tan²A - (sin²A/cos²A) . cos²A)
= √(tan²A - sin²A)
= √{(tanA + sinA)(tanA - sinA)}
= √mn

৬,৮০৮.

A manufacturer sells a pair of shoes to a wholesale dealer at a profit of 20%. Wholesalers sell them to retailers at a profit of 25 %. The shoes are again sold to the customer for Tk. 50.50, thereby earning a profit of 30 %. Find the cost price of the manufacturer.

ক
Tk. 20.36
খ
Tk. 22.90
গ
Tk. 25.89
ঘ
Tk. 30.50

ব্যাখ্যা

Profit earned by manufacturer = 20%
Profit earned by wholesaler = 25%
Profit earned by retailer = 30%
S.P. of shoes = Tk. 50

Therefore, 130% of 125% of 120% of x = 50.50
⇒ 120/100 × 125/100 × 130/100 × x = 5050/100
⇒ (195/100) x = 5050/100
⇒ x = (5050 × 100)/(195 × 100)
⇒ x = 25.89
Cost price of shoes = Tk. 25.89

৬,৮০৯.

Nine times a whole number is equal to five less than twice the square of the number. Find the number?

ক
3
খ
9
গ
5
ঘ
10

ব্যাখ্যা

Question: Nine times a whole number is equal to five less than twice the square of the number. Find the number?

Solution: Let the required whole number be x.

According to the question,
9x = 2x² - 5
⇒ 2x² - 9x - 5 = 0
⇒(x - 5)(2x + 1) = 0
⇒ x - 5 = 0 or 2x + 1 = 0
⇒ x = 5 or x = - 1/2

Since x is supposed to be a whole number, the answer, i.e., the required whole number is 5.

৬,৮১০.

If 8 men and 3 boys working together can do five times as much work per hour as a man and a boy together, working capacities of a man and a boy are in the ratio-

ক
3 : 2
খ
2 : 3
গ
3 : 4
ঘ
4 : 3

ব্যাখ্যা

Question: If 8 men and 3 boys working together can do five times as much work per hour as a man and a boy together, working capacities of a man and a boy are in the ratio-

Solution:
Let,
1 man 1 day work = p
1 boy 1 day work = q

Now,
8p + 3q = 5(p + q)
or, 8p + 3q = 5p + 5q
or, 8p - 5p = 5q - 3q
or, 3p = 2q
or, p/q = 2/3
∴ p : q = 2 : 3

৬,৮১১.

The average of 7 consecutive numbers is n. If the next two numbers are included, the average will-

ক
increased by 2.5
খ
remains the same
গ
increased by 1
ঘ
increased by 2
ঙ
None of these

ব্যাখ্যা

Question: The average of 7 consecutive numbers is n. If the next two numbers are included, the average will-

Solution:

The average of 7 consecutive numbers is n implies that the 4th term is equal to n.

Now if we include next two terms then the average of 9 terms will be the 5th term. Now as the terms are consecutive, so the 5th term will be n + 1.

(1 + 2 + 3 + 4 + 5 + 6 + 7)/7 = 28/7 = 4

(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9)/9 = 45/9 = 5

৬,৮১২.

Two people start jogging at the same point and time but in opposite directions. If the rate of one jogger is 2 miles per hour faster than the other and after 3 hours, they are 30 miles apart. What is the rate of the faster jogger?

ক
ক) 3
খ
খ) 4
গ
গ) 5
ঘ
ঘ) 6

ব্যাখ্যা

মনেকরি,
প্রথম জনের গতিবেগ x মাইল/ঘণ্টা
দ্বিতীয় জনের গতিবেগ x + 2 মাইল/ঘণ্টা

প্রশ্নমতে,
3x + 3(x + 2) = 30
3x + 3x + 6 = 30
6x + 6 = 30
6x = 30 - 6
6x = 24
x = 4

দ্বিতীয় জনের গতিবেগ = 4 + 2 মাইল/ঘণ্টা
= 6 মাইল/ঘণ্টা

৬,৮১৩.

One of the factors of the expression: a³ - 6a² + 12a - 9

ক
ক) a - 1
খ
খ) a + 1
গ
গ) a - 3
ঘ
ঘ) a + 3

ব্যাখ্যা

Question: One of the factors of the expression: a³ - 6a² + 12a - 9

Solution:
Given that
a³ - 6a² + 12a - 9

Let
f(a) = a³ - 6a² + 12a - 9
f(3) = 3³ - 6 × 3² + 12 × 3 - 9
= 27 - 54 + 36 - 9
= 63 - 63
= 0
a - 3 one of the factors of the expression a³ - 6a² + 12a - 9

৬,৮১৪.

If x and y are positive integers, each greater than 1, and if 13(x - 1) = 17(y - 1), what is the least possible value of (x + y)?

ক
32
খ
30
গ
26
ঘ
25

ব্যাখ্যা

Question: If x and y are positive integers, each greater than 1, and if 13(x - 1) = 17(y - 1), what is the least possible value of (x + y)?

Solution:
13(x - 1) = 17(y - 1) .....(1)
⇒ 13x - 13 = 17y - 17
⇒ 13x + 4 = 17y
⇒ 4 = 17y - 13x
⇒ 4 - 4y = 13y - 13x
⇒ 4(1 - y) = 13(y - x)

Since x and y are integers, we know that (y - x) is an integer, which means 13(y - x) is a multiple of 13. From this, we can conclude that 4(1 - y) is a multiple of 13
What is the smallest value of y (given that y is a positive integer greater than 1) such that 4(1 - y) is a multiple of 13?

If y = 14, then 4(1 - y) = 4(1 - 14) = 4(- 13) = - 52
So, y = 14 is the smallest value of y to meet the given conditions.

Now, putting the value of y = 14 in equation (1) we get,
13(x - 1) = 17(y - 1)
⇒ 13x - 13 = 17(14 - 1)
⇒ 13x = 17 × 13 + 13
⇒ 13x = 18 × 13
∴ x = 18

∴ x + y = 18 + 14 = 32

৬,৮১৫.

A man buys a chair and table for Tk. 6000. He sells the chair at a loss of 10% and the table at gain of 10%. He still gains Tk. 100 on the whole. Cost price of chair is:

ক
2500
খ
2850
গ
3050
ঘ
2100

ব্যাখ্যা

Question: A man buys a chair and table for Tk. 6000. He sells the chair at a loss of 10% and the table at gain of 10%. He still gains Tk. 100 on the whole. Cost price of chair is:

Solution:
Given,
Total cost price (CP) of chair and table = Tk. 6000
Total profit = Tk. 100

Let, the Cost Price (CP) of the chair be x Tk.
So, the Cost Price of the table is = (6000 - x) Tk.

At 10% loss,
Selling Price of chair = x - 10% of x
= x - (10x/100)
= 90x/100

At 10% gain,
Selling Price of table = (6000 - x) + 10% of (6000 - x)
= (6000 - x) + {(10/100) × (6000 - x)}
= 110(6000 - x)/100

So, Total Selling Price of chair and table,
= 90x/100 + {110(6000 - x)/100}
= {90x + 110(6000 - x)}/100

Now,
Total SP = Total CP + Profit
⇒ {90x + 110(6000 - x)}/100 = 6000 + 100
⇒ 90x + 110(6000 - x) = 100 × 6100
⇒ 90x + 660000 - 110x = 610000
⇒ - 20x = - 50000
⇒ x = 2500
∴ Cost price of the chair = Tk. 2500

৬,৮১৬.

A is younger than B by 7 years. If their ages are in the respective ratio of 7 : 9, how old is A?

ক
ক) 18
খ
খ) 22
গ
গ) 24
ঘ
ঘ) None of these

ব্যাখ্যা

Let B's age be x years
Therefore, A's age = (x - 7) years
(x - 7)/x = 7/9
⇒ 7x = 9x - 63
⇒ 2x = 63
⇒ x = 31.5
A's age = 31.5 - 7 = 24.5 years

৬,৮১৭.

A cake is divided into 18 pieces. If Mohiuddin takes 1/3 of the cake and Morshed takes 1/3 of the rest that are left, how many pieces are still left?

ক
4
খ
8
গ
6
ঘ
10
ঙ
None of these

ব্যাখ্যা

Question: A cake is divided into 18 pieces. If Mohiuddin takes 1/3 of the cake and Morshed takes 1/3 of the rest that are left, how many pieces are still left?

Solution:
Mohiuddin takes 1/3 of cake
Left after Mohiuddin takes = (1 - 1/3) of cake
= 2/3 of cake

Morshed takes (1/3) × (2/3) = 2/9 of cake

Mohiuddin and Morshed takes = (1/3 + 2/9) = (3 + 2)/9 = 5/9 of cake
Left after both take = (1 - 5/9) = 4/9 of cake

full cake divided into 18 pieces
∴ 4/9 of cake divided into (18 × 4)/9 pieces = 8 pieces

৬,৮১৮.

Tickets numbered 1 to 20 are mixed up and then a ticket is drawn at random. What is the probability that the ticket drawn has a number which is a multiple of 3 or 5?

ক
6/20
খ
11/20
গ
7/20
ঘ
9/20

ব্যাখ্যা

Question: Tickets numbered 1 to 20 are mixed up and then a ticket is drawn at random. What is the probability that the ticket drawn has a number which is a multiple of 3 or 5?

Solution:
Total number of tickets = 20

The numbers which are multiple of 3 or 5 are {3, 5, 6, 9, 10, 12, 15, 18, 20}
∴ Total expected events = 9

∴ The probability = 9/20

৬,৮১৯.

A college has 10 basketball players. A 5-member team and a captain will be selected out of these 10 players. How many different selections can be made?

ক
210
খ
620
গ
960
ঘ
1260

ব্যাখ্যা

Question: A college has 10 basketball players. A 5-member team and a captain will be selected out of these 10 players. How many different selections can be made?

Solution:
A team of 6 members (5 players and 1 captain) has to be selected from 10 players.
this can be done in = ¹⁰C₆ = 210 ways.
A captain can be selected from these 6 in 6 ways.
∴ total number of ways = 210 × 6 = 1260 ways

৬,৮২০.

In a 50-liter mixture of milk and water, the ratio of milk to water is 4 : 1. How much more water must be added to change the ratio to 2 : 3?

ক
45 liters
খ
50 liters
গ
60 liters
ঘ
40 liters

ব্যাখ্যা

Question: In a 50-liter mixture of milk and water, the ratio of milk to water is 4 : 1. How much more water must be added to change the ratio to 2 : 3?

Solution:
Given that,
Milk : Water = 4 : 1
Total mixture = 50 liters

Milk = (50 of 4/5) = 40 liters
Water = (50 of 1/5) = 10 liters
Let x = additional water to be added.
New water = 10 + x liters

ATQ,
⇒ 40 : 10 + x = 2 : 3
⇒ 40/(10 + x) = 2/3
⇒ 20 + 2x = 120
⇒ 2x = 120 - 20
⇒ 2x = 100
⇒ x = 100/2
∴ x = 50
∴ 50 liters of water must be added to achieve the 2 : 3 ratio.

৬,৮২১.

If x² is odd, what will x² - x be?

ক
Even
খ
Odd
গ
Prime
ঘ
A perfect square

ব্যাখ্যা

Question: If x² is odd, what will x² - x be?

Solution:
যেহেতু x² বিজোড় তাই x ও বিজোড় হবে।

এখন,
x² - x
= x(x - 1)
= (x - 1)x
∴ (x - 1) এবং x দুইটি ক্রমিক সংখ্যা।

x বিজোড় সংখ্যা হলে (x - 1) অবশ্যই জোড় সংখ্যা হবে।
কারণ দুইটি ক্রমিক সংখ্যার মধ্যে একটি বিজোড় হলে অন্যটি জোড় হবে।

সুতরাং, x ও (x - 1) এর গুনফল,
= x(x - 1)
= x² - x, একটি জোড় সংখ্যা। [জোড় × বিজোড় = জোড়]

৬,৮২২.

A man is walking at a speed of 10 kmph. After every km, he takes rest for 5 minutes. How much time will he take to cover a distance of 5 km?

ক
ক) 60 min
খ
খ) 50 min
গ
গ) 55 min
ঘ
ঘ) 70 min

ব্যাখ্যা

Time taken by man if he did not stop
= 5 km/10 kmph
= 1/2 hr
= 30 min
∵Man takes rest for 5 minutes on each km
Total rest time= 5×4= 20 min
Total travelling time:
= 30 min+20 min
= 50 min

৬,৮২৩.

How much water should be added to 80 liters of pure milk to gain extra 20% profit when selling the mixture at the price of pure milk?

ক
6 liters
খ
12 liters
গ
8 liters
ঘ
16 liters

ব্যাখ্যা

Question: How much water should be added to 80 liters of pure milk to gain extra 20% profit when selling the mixture at the price of pure milk?

Solution:
Let’s assume,
Price of pure milk per liter = 100 Taka
So, the price of 50 liters of pure milk = 100 × 80 = 8000 Taka

Now assume,
Water added to the milk = x liters
Then the total quantity of the milk-water mixture = (80 + x) liters

Since the mixture is sold at the price of pure milk,
The selling price of (80 + x) liters = 100(80 + x) Taka

According to the question,
100(80 + x) = 8000 + 8000 of 20%
⇒ 8000 + 100x = 8000 + {8000 × (20/100)}
⇒ 8000 + 100x = 8000 + 1600
⇒ 8000 - 8000 + 100x = 1600
⇒ 100x = 1600
⇒ x = 1600/100
⇒ x = 16

∴ Amount of water to be added = 16 liters

৬,৮২৪.

Working 10 hours a day, Polas can Complete a work in 4 days. Working 3 hours a day, Mahin can complete the same work in 20 days. Working 4 hours a day, they can jointly complete the work in

ক
7 days
খ
5 days
গ
3 days
ঘ
4 days

ব্যাখ্যা

Questin: Working 10 hours a day, Polas can Complete a work in 4 days. Working 3 hours a day, Mahin can complete the same work in 20 days. Working 8 hours a day, they can jointly complete the work in

Solution:
Given,
Working 10 hours a day, Polash can complete the work in 4 days
Polash's total working hour = (10 × 4) hours
= 40 hours

∴ Polas's 1 hour's work = 1/40 part

Working 3 hours a day, Mahin can complete the work in 20 days
Mahin's total working hour = (3 × 20) hours
= 60 hours

∴ Mahin's 1 hour's work = 1/60 part

( Polas + Mahin )'s 1 hour's work = (1/40 + 1/60) part
= ( 3 + 2)/120 part
= 5/120
= 1/24 part

So (Polas + Mahin) complete whole part = 24 hours

Hence, they work 8 hours a day
i.e. they require 3 days to complete the work.

৬,৮২৫.

Two ships are sailing in the sea on the two sides of a lighthouse. The angle of elevation of the top of the lighthouse is observed from the ships are 30° and 45° respectively. If the lighthouse is 100 m high, the distance between the two ships is-

ক
173 m
খ
200 m
গ
273 m
ঘ
300 m

ব্যাখ্যা

Question: Two ships are sailing in the sea on the two sides of a lighthouse. The angle of elevation of the top of the lighthouse is observed from the ships are 30° and 45° respectively. If the lighthouse is 100 m high, the distance between the two ships is-

Solution:

Let AB be the lighthouse and C and D be the positions of the ships.

Then, AB = 100 m, ACB = 30° and ADB = 45°.

AB/AC = tan 30° = 1/√3
⇒ AC = AB × √3 = 100√3 m.

AB/AD = tan 45° = 1
⇒ AD = AB = 100 m.

∴ CD = (AC + AD) = (100√3 + 100) m
= 100(√3 + 1) m
= 100 × 2.73 m
= 273 m

৬,৮২৬.

A tank is 25 m long, 12 m wide and 6 m deep. The cost of plastering its walls and bottom at 75 paise per sq. m, is

ক
ক) 456
খ
খ) 458
গ
গ) 558
ঘ
ঘ) 568

ব্যাখ্যা

Area to be plastered= [2(l + b) x h] + (l x b)
= {[2(25 + 12) x 6] + (25 x 12)} m²
= (444 + 300) m²= 744 m².
Cost of plastering = Rs.744 x (75/100)
= Rs. 558

৬,৮২৭.

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
20
খ
22
গ
16
ঘ
18

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
⇒ 247x² = 2223
⇒ x² = 9
∴ x = 3

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

৬,৮২৮.

The present age of Jamal and Kamal are in the ratio of 6 : 4. Five years ago, their ages were in the ratio of 5 : 3. How old is Jamal now?

ক
ক) 30 years
খ
খ) 35 years
গ
গ) 40 years
ঘ
ঘ) 45 years

ব্যাখ্যা

Let, the present ages of Jamal and Kamal are 6x and 4x respectively.
(6x - 5)/(4x - 5) = 5/3
20x - 25 = 18x - 15
2x = 10
x = 5

Habib's age = 6 × 5 = 30 years

৬,৮২৯.

The angle of elevation of the top of a tower of height x metre from a point on the ground is found to be 60°. By going y metre away from that point, it becomes 30°. Which one of the following relations is correct?

ক
x = y
খ
2x = √3y
গ
2x = 3y
ঘ
None of the above

ব্যাখ্যা

Question: The angle of elevation of the top of a tower of height x metre from a point on the ground is found to be 60°. By going y metre away from that point, it becomes 30°. Which one of the following relations is correct?

Solution:
Given that,
The angle of elevation of the top of a tower of height x meter from a point on the ground is found to be 60°.
By going y metre away from that point, it becomes 30°.

According to the question,
tan60° = AB/BC
⇒ √3 = x/BC
⇒ BC = x/√3 .........(1)

Again,
tan30° = AB/BD
⇒ 1/√3 = AB/BD
⇒ 1/√3 = x/BD
⇒ BD = √3x .........(2)

Now,
BD = BC + CD
⇒ √3x = (x/√3) + y [From equation (1 and 2)]
⇒ √3x = (x + √3y)/√3
⇒ 3x = x + √3y
⇒ 3x - x = √3y
⇒ 2x = √3y

৬,৮৩০.

The area of an isosceles triangle is 25√3 cm², and the measure of each of the equal sides is 10 cm, what is the angle between the equal sides?

ক
30°
খ
45°
গ
50°
ঘ
60°

ব্যাখ্যা

Question: The area of an isosceles triangle is 25√3 cm², and the measure of each of the equal sides is 10 cm, what is the angle between the equal sides?

Solution:
Area of the traingle :
⇒ 1/2absinθ = 25√3
⇒ 10 × 10 sinθ = 50√3
⇒ sinθ = √3/2
∴ θ = 60°

৬,৮৩১.

Father is aged three times more than his son Roni. After 8 years, he would be two and a half times of Roni's age. After another 8 years, how many times would he be of Ronit's age?

ক
ক) 2 times
খ
খ) 2.5 times
গ
গ) 2.75 times
ঘ
ঘ) 3 times
ঙ
ঙ) 3.5 times

ব্যাখ্যা

Let Roni's present age be x years. Then, father's present age =(x + 3x) years = 4x years.
there4 (4x + 8) = (5/2)(x + 8)
=> 8x + 16 = 5x + 40
=> 3x = 24
=> x = 8.

Hence, required ratio = (4x + 16)/(x + 16)
= 48/24
= 2

৬,৮৩২.

How many cases do you need if you have to pack 119 pairs of shoes into cases that each hold 34 shoes?

ক
ক) 5
খ
খ) 6
গ
গ) 7
ঘ
ঘ) 8

ব্যাখ্যা

প্রশ্ন : How many cases do you need if you have to pack 119 pairs of shoes into cases that each hold 34 shoes?

এখানে,
34 টি জুতা = 17 জোড়া জুতা
∴ আবরণ (case) প্রয়োজন হবে = 119/17 = 7টি

৬,৮৩৩.

The least number which when divided by 4, 6, 8 and 9 leave zero remainder in each case and when divided by 19 leaves a remainder of 15 = ?

ক
72
খ
75
গ
78
ঘ
79
ঙ
88

ব্যাখ্যা

L.C.M. of (4, 6, 8, 9)
= 2 × 2 × 3 × 2 × 3
= 72
∴ Required result should be = 72
If we divide 72 by 19, we get 15 as a remainder.
Therefore, the required least number is 72.

৬,৮৩৪.

Find the value of cosec(- π/3)

ক
- 2/√3
খ
√3/2
গ
1
ঘ
1/√2

ব্যাখ্যা

Question: Find the value of cosec(- π/3)

Solution:
cosec(- π/3)
= - cosec(π/3)
= - 1/sin(π/3)
= - 1/sin60°
= - 1/(√3/2)
= - 2/√3

৬,৮৩৫.

Two trains of equal length, running in opposite directions, pass a pole in 24 and 12 seconds. The train will cross each other in -

ক
ক) 16 sec
খ
খ) 18 sec
গ
গ) 14 sec
ঘ
ঘ) 20 sec

ব্যাখ্যা

Question: Two trains of equal length, running in opposite directions, pass a pole in 24 and 12 seconds. The train will cross each other in -

Solution:
Let the length of both train be x meters
Speed of the first train = x/24 m/s
and speed of the second train = x/12 m/s

When running in opposite directions, relative speed = x/24 + x/12
= 3x/24
= x/8

To cross each other, distance to be covered = x + x = 2x meters

Taken taken = (2x)/(x/8) = 16 sec

৬,৮৩৬.

Shanto was asked to find the value of 7/12 of a sum of money. Instead of multiplying the same by 7/12, he divided it by 7/12 and his answer exceeded the correct answer by 95. The correct answer is

ক
ক) 48
খ
খ) 89
গ
গ) 84
ঘ
ঘ) 69
ঙ
ঙ) 49

ব্যাখ্যা

Question: Shanto was asked to find the value of 7/12 of a sum of money. Instead of multiplying the same by 7/12, he divided it by 7/12 and his answer exceeded the correct answer by 95. The correct answer is-

Solution:
Let,
sum of money = x

ATQ,
x/(7/12) - x × (7/12) = 95
⇒ (12x)/7 - (7x)/12 = 95
⇒ (12x × 12 - 7x × 7)/84 = 95
⇒ 144x - 49x = 95 × 84
⇒ 95x = 95 × 84
∴ x = 84

The correct answer is 84 × (7/12) = 49

৬,৮৩৭.

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

ক
50 metres
খ
100 metres
গ
200 metres
ঘ
250 metres

ব্যাখ্যা

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

Solution:
Area of the floor = (14 × 9) m² = 126 m²∴ Length of the carpet = {(126/63) × 100}m = 200 metres

৬,৮৩৮.

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

ক
7 hours
খ
8 hours
গ
6 hours
ঘ
5 hours
ঙ
None of these

ব্যাখ্যা

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

Solution:
As number of pumps increase, the time required decreases and when working hours increase, fewer days are required to complete the work. Hence, this is a problem related to indirect proportion.

More pumps (↑),Less working hours (↓)
More working hours (↑),Less days (↓)

⇒ 4 × 3 × 2 = 4 × 1 × x
⇒ 24 = 4x
∴ x = 6

৬,৮৩৯.

If 0.05 is a% of 0.5, then the value of a is-

ক
ক) 100
খ
খ) 1
গ
গ) 10
ঘ
ঘ) 1/10

ব্যাখ্যা

a% of 0.5 = 0.05
⇒ 0.5 × (a/100) = 0.05
⇒ a/100 =0.05/0.5
⇒ a/100 = 0.1
∴ a = 10

৬,৮৪০.

Runa is shorter than Shila but taller than Tuli. Fahim is taller than Runa. Shila is the second-tallest person among them. Akash is shorter than Tuli. Who is the third-tallest person among them?

ক
Tuli
খ
Runa
গ
Fahim
ঘ
Shila

ব্যাখ্যা

Question: Runa is shorter than Shila but taller than Tuli. Fahim is taller than Runa. Shila is the second-tallest person among them. Akash is shorter than Tuli. Who is the third-tallest person among them?

Solution:
First statement: Shila > Runa > Tuli

Second statement: Fahim > Runa

Third statement: Shila is the second-tallest, meaning one person is taller than Shila. Since Fahim is taller than Runa and Shila is taller than Runa, Fahim must be taller than Shila.
Therefore, Fahim > Shila > Runa

Fourth statement: Tuli > Akash

Putting everyone together: Fahim > Shila > Runa > Tuli > Akash

∴ The third-tallest person is Runa.

৬,৮৪১.

Eight years back, Ashik's age was (1/8)th of Zaber's age. Ten years from now, Zaber's age will be double of Ashik's age. How old is Ashik now?

ক
11 years
খ
8 years
গ
15 years
ঘ
13 years

ব্যাখ্যা

Question: Eight years back, Ashik's age was (1/8)th of Zaher's age. Ten years from now, Zaher's age will be double of Ashik's age. How old is Ashik now?

Solution:
ধরি,
জহিরের বর্তমান বয়স = ক বছর
৮ বছর পূর্বে জহিরের বয়স = (ক - ৮) বছর
∴ ৮ বছর পূর্বে আশিকের বয়স = (ক - ৮)/৮ বছর

বর্তমানে,
আশিকের বয়স = {(ক - ৮)/৮} + ৮ বছর
= (ক - ৮ + ৬৪)/৮ বছর
= (ক + ৫৬)/৮ বছর

∴ ১০ বছর পর জহিরের বয়স হবে = ক + ১০ বছর
∴ ১০ বছর পর আশিকের বয়স হবে = {(ক + ৫৬)/৮} + ১০ বছর
= (ক + ৫৬ + ৮০)/৮ বছর
= (ক + ১৩৬)/৮ বছর

প্রশ্নমতে,
ক + ১০ = ২ × {(ক + ১৩৬)/৮}
⇒ ক + ১০ = (ক + ১৩৬)/৪
⇒ ৪(ক + ১০) = ক + ১৩৬
⇒ ৪ক + ৪০ = ক + ১৩৬
⇒ ৪ক - ক = ১৩৬ - ৪০
⇒ ৩ক = ৯৬
⇒ ক = ৯৬/৩
∴ ক = ৩২

∴ আশিকের বর্তমান বয়স = (৩২ + ৫৬)/৮ বছর
= ৮৮/৮ বছর
= ১১ বছর

৬,৮৪২.

What will be the least number which when doubled will be exactly divisible by 18, 24, 28, and 36?

ক
504
খ
320
গ
252
ঘ
222

ব্যাখ্যা

Question: What will be the least number which when doubled will be exactly divisible by 18, 24, 28, and 36?

Solution:
LCM of 18, 24, 28, and 36 is = 504
So, the number will be half of 504 = 504/2
= 252

৬,৮৪৩.

The sum of the perfect squares between 110 and 300 is -

ক
ক) 1144
খ
খ) 1204
গ
গ) 2311
ঘ
ঘ) 1400

ব্যাখ্যা

Question: The sum of the perfect squares between 110 and 300 is -

Solution:
Perfect squares between 110 and 300 = 121, 144, 169, 196, 225, 256, 289
Sum = 121 + 144 + 169 + 196 + 225 + 256 + 289 = 1400

৬,৮৪৪.

x^-3 - 0.001 = 0 হলে x² = ?

ক
ক) 100
খ
খ) 1/100
গ
গ) 10
ঘ
ঘ) 1/10

ব্যাখ্যা

x^-3 - 0.0001 = 0
বা, 1/x³ = 0.001
বা, 1/x³ = 1/10³
বা, x³ = 10³
বা, x = 10
∴ x² = 10² = 101

৬,৮৪৫.

If log_x2 = a and log_x5 = b, then log_x50 =

ক
ক) a + b
খ
খ) a + b²
গ
গ) ab²
ঘ
ঘ) a + 2b

ব্যাখ্যা

Here, log_x50 = log_x(2×25)
= log_x2 + log_x5²
= log_x2 + 2log_x5
= a + 2b [As, log_x2 = a; log_x5 = b]

৬,৮৪৬.

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

ক
ক) 98
খ
খ) 88
গ
গ) 99
ঘ
ঘ) 89

ব্যাখ্যা

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

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

৬,৮৪৭.

5a² - 4a - 3 - 3(a² + a + 4) = 0. What is the value of a?

ক
ক) 5, 3/2
খ
খ) 5, -3/2
গ
গ) 2, -3/2
ঘ
ঘ) - 3, 3/2

ব্যাখ্যা

Question: 5a² - 4a - 3 - 3(a² + a + 4) = 0. What is the value of a?

Solution:
5a² - 4a - 3 - 3(a² + a + 4) = 0
⇒ 5a² - 4a - 3 - 3a² - 3a - 12 = 0
⇒ 2a² - 7a - 15 = 0
⇒ 2a² - 10a + 3a - 15 = 0
⇒ 2a(a - 5) + 3(a - 5) = 0
∴ (a - 5)(2a + 3) = 0

হয়,
a - 5 = 0
a = 5

অথবা
2a + 3 = 0
2a = - 3
a = - 3/2

৬,৮৪৮.

In a box, there are 7 red, 8 blue and 9 green balls. One ball is picked up randomly. What is the probability that it is neither red nor green?

ক
1/2
খ
2/3
গ
1/3
ঘ
1/4

ব্যাখ্যা

Question: In a box, there are 7 red, 8 blue and 9 green balls. One ball is picked up randomly. What is the probability that it is neither red nor green?

Solution:
Total number of balls, n(S) = (8 + 7 + 6) = 24

Let,
E = event that the ball drawn in neither red nor green = even that the ball drawn in blue.
∴ n(E) = 8

∴ P(E) = n(E)/n(S)
= 8/24
= 1/3

৬,৮৪৯.

If cos⁴θ - sin⁴θ = 2/3 then the value of (1 - 2sin²θ) is -

ক
ক) 1/3
খ
খ) 1/2
গ
গ) 2/3
ঘ
ঘ) 1

ব্যাখ্যা

Question: If cos⁴θ - sin⁴θ = 2/3 then the value of (1 - 2sin²θ) is -

Solution:
cos⁴θ - sin⁴θ = 2/3
⇒ (cos²θ - sin²θ) (cos²θ + sin²θ) = 2/3
⇒ cos²θ - sin²θ = 2/3
⇒ 1 - sin²θ - sin²θ = 2/3
∴ 1 - 2sin²θ = 2/3

৬,৮৫০.

What comes next?
0.12, 0.12, 0.24, 0.72, ___

ক
ক) 1
খ
খ) 2.48
গ
গ) 2.88
ঘ
ঘ) 4.88

ব্যাখ্যা

Question: What comes next?
0.12, 0.12, 0.24, 0.72, ?

Solution:
0.12 × 1 = 0.12
0.12 × 2 = 0.24
0.24 × 3 = 0.72
next number, 0.72 × 4
= 2.88

৬,৮৫১.

The sum of the present ages of a father and his son is 60 years. five years ago, father's age was four times the age of the son. So now the son's age will be-

ক
25
খ
10
গ
15
ঘ
20

ব্যাখ্যা

Question: The sum of the present ages of a father and his son is 60 years. five years ago, father's age was four times the age of the son. So now the son's age will be-

Solution:
Let the present ages of son and father be x and (60 - x) years respectively.

Then,
(60 - x) - 5 = 4(x - 5)
⇒ 55 - x = 4x - 20
⇒ 5x = 75
∴ x = 15

৬,৮৫২.

Two trains start from the same starting points and move towards the same destination. The first one starts half an hour earlier than the second one. The first one runs at a speed of 90 km/hr while the second one runs at 30 km/hr faster. At what distance from the starting point will the two trains meet?

ক
ক) 150 km
খ
খ) 180 km
গ
গ) 360 km
ঘ
ঘ) 450 km

ব্যাখ্যা

We know,
Distance(D) = Speed(S) × Time(T)
⇒ D = S × T
∴ S = D/T; T = D/S

Since the second train is 30 km/hr faster, it is moving at 120 km/hr
Now, let train A travel T hours before meeting train B.
Time for which train B travels = (T - 1/2) hrs.
This is because it starts half an hour late
Distance travelled is same.

∴ D = D
∴ 90 km/hr × T hrs = 120 km/hr × {T - (1/2} hrs
∴ 90T = 120T - 60
∴ T = 2 hours

Distance travelled by train A = 90 km/hr × 2 hours = 180km.
Thus they meet 180 km from starting point.

৬,৮৫৩.

A printer was sold at the loss of 6% If it were sold at Tk. 2600 more, there would be a profit of 7%. What was the cost price of the printer?

ক
10,000 tk.
খ
15,000 tk.
গ
20,000 tk.
ঘ
25,000 tk.

ব্যাখ্যা

Question: A printer was sold at the loss of 6% If it were sold at Tk. 2600 more, there would be a profit of 7%. What was the cost price of the printer?

Solution:
মনে করি,
ক্রয়মূল্য = 100 টাকা
∴ 6% ক্ষতিতে বিক্রয়মূল্য = 100 - 6 = 94 টাকা

এবং
7% লাভে বিক্রয়মূল্য = 100 + 7 = 107 টাকা
∴ বিক্রয়মূল্য (107 - 94) = 13 টাকা বেশি হলে, 7% লাভে হতো।

এখন,
বিক্রয়মূল্য 13 টাকা বেশি হলে ক্রয়মূল্য 100 টাকা
∴ বিক্রয়মূল্য 1 টাকা বেশি হলে ক্রয়মূল্য (100/13) টাকা
∴ বিক্রয়মূল্য 2600 টাকা বেশি ক্রয়মূল্য (100 × 2600/13) টাকা
= 20000 টাকা।

সুতরাং, ক্রয়মূল্য 20000 টাকা।

৬,৮৫৪.

A question paper has two parts, A and B, each containing 5 questions. If a student has to choose 3 from part A and 2 from part B, in how many ways can he choose the questions?

ক
ক) 50
খ
খ) 70
গ
গ) 85
ঘ
ঘ) 100

ব্যাখ্যা

Question: A question paper has two parts, A and B, each containing 5 questions. If a student has to choose 3 from part A and 2 from part B, in how many ways can he choose the questions?

Solution:
ways to choose 3 from part A = ⁵C₃
ways to choose 2 from part B = ⁵C₂

choose 3 from part A and 2 from part B = ⁵C₃ × ⁵C₂
= {5!/(3! 2!)} × {5!/(2! 3!)}
= 10 × 10
= 100

৬,৮৫৫.

Akib can do 1/6 of a work in 7 days. In how many days will he complete the work?

ক
36 days
খ
42 days
গ
46 days
ঘ
50 days

ব্যাখ্যা

Question: Akib can do 1/6 of a work in 7 days. In how many days will he complete the work?

Solution:
Akib can do 1/6 of a work in 7 days
∴ he will complete the work in (6 × 7) days
= 42 days

∴ Akib will complete the work in 42 days.

৬,৮৫৬.

A hall measures 40 m in length, 25 m in width, and 20 m in height. If each person needs 200 cubic meters of space, how many people can the hall accommodate?

ক
80
খ
90
গ
100
ঘ
110

ব্যাখ্যা

Question: A hall measures 40 m in length, 25 m in width, and 20 m in height. If each person needs 200 cubic meters of space, how many people can the hall accommodate?

Solution:
Length of the hall = 40 m
Width of hall = 25 m
Height of hall = 20 m

∴ Volume of the hall
= 40 × 25 × 20
= 20000 m³
∴ Space occupied by each person = 200 m³
∴ Number of people that can be accommodated in the hall
= 20000/200
= 100

৬,৮৫৭.

Find the smallest number of five digits exactly divisible by 16, 24, 36, and 54.

ক
10688
খ
10638
গ
12368
ঘ
10368

ব্যাখ্যা

Question: Find the smallest number of five digits exactly divisible by 16, 24, 36, and 54.

Solution:
The smallest number of five digits is = 10000
The number must be divisible by the LCM of 16, 24, 36, and 54 = 432
On dividing 10000 by 432, we get 64 as the remainder.

So, required number is = 10000 + (432 - 64)
= 10000 + 368
= 10368

৬,৮৫৮.

ক
1
খ
a
গ
a^1/3
ঘ
a³
ঙ
None of these

ব্যাখ্যা

Question:

Solution:

৬,৮৫৯.

If log₁₀5+ log₁₀(5x + 1) = log₁₀(x + 5) + 1, then what is the value of x ?

ক
1
খ
3
গ
5
ঘ
10
ঙ
15

ব্যাখ্যা

Question: If log₁₀5+ log₁₀(5x + 1) = log₁₀(x + 5) + 1, then what is the value of x ?

Solution:
log₁₀5+ log₁₀(5x + 1) = log₁₀(x + 5) + 1
⇒ log₁₀5+ log₁₀(5x + 1) = log₁₀(x + 5) + log₁₀10
⇒ log₁₀[5(5x + 1)] = log₁₀[10(x + 5)
⇒ 5(5x + 1) = 10(x + 5)
⇒ 5x + 1 = 2x + 10
⇒ 3x = 9
∴ x = 3

৬,৮৬০.

Asif, Sami and Riad started a shop by investing Tk. 2700, Tk. 9000 and Tk. 6300 respectively. At the end of one year, the profit was distributed. If Riad's share was Tk. 2100, what was their total profit?

ক
Tk. 5000
খ
Tk. 4000
গ
Tk. 7000
ঘ
Tk. 6000

ব্যাখ্যা

Question: Asif, Sami and Riad started a shop by investing Tk. 2700, Tk. 9000 and Tk. 6300 respectively. At the end of one year, the profit was distributed. If Riad's share was Tk. 2100, what was their total profit?

Solution:
Let the total profit is = x

Here, Asif : Sami : Riad = 2700 : 9000 : 6300
= 3 : 10 : 7

then, Riad's share = (7/20) × x = 7x/20
ATQ,
7x/20 = 2100
⇒ x = (2100 × 20)/7
∴ x = 6000

∴ The total profit = Tk. 6000

৬,৮৬১.

Find the highest three-digit number which, when divided by 14, 21, 35 and 42, leaves 5 as remainder in each case.

ক
ক) 835
খ
খ) 840
গ
গ) 855
ঘ
ঘ) 845

ব্যাখ্যা

Finding the L.C.M of 14, 21, 35 and 42
14 = 2 × 7
21 = 3 × 7
35 = 5 × 7
42 = 2 × 3 × 7

∴ L.C.M = 2 × 3 × 5 × 7 = 210

Now, 210 × 4 = 840 is the highest three digits number divisible by 14, 21, 35 and 42
∴ Required number is = 840 + 5 = 845

৬,৮৬২.

The ratio of principal to interest for a sum of money is 5 : 2 when invested at 8% per annum. Find the time period for which the money was invested.

ক
3 years
খ
3.5 years
গ
4.5 years
ঘ
5 years
ঙ
6 years

ব্যাখ্যা

Question: The ratio of principal to interest for a sum of money is 5 : 2 when invested at 8% per annum. Find the time period for which the money was invested.

Solution:
Given
Rate (R) = 8% per annum
Principal : Interest = 5 : 2

This means P/I = 5/2
⇒ P/(P × 8 × T/100) = 5/2
⇒ 100/(8 × T) = 5/2
⇒ 100 = 8 × T × 5/2
⇒ 100 = 20T
∴ T = 5
Therefore, the money was invested for 5 years.

[I = P × R × T/100; where I = Interest, P = Principal, R = Rate, T = Time in years]

৬,৮৬৩.

Question:

ক
0
খ
1
গ
a³
ঘ
x
ঙ
None of these

ব্যাখ্যা

Question:

Solution:

৬,৮৬৪.

If a and b are roots of P² - 4P - 21 = 0, where a > b, find the equation whose roots are a/b and b/a.

ক
21P² + 58P + 21 = 0
খ
25P² + 58P + 21 = 0
গ
21P² + 40P + 21 = 0
ঘ
21P² + 56P + 25 = 0
ঙ
None of these

ব্যাখ্যা

Question: If a and b are roots of P² - 4P - 21 = 0, where a > b, find the equation whose roots are a/b and b/a.

Solution:
Given the equation as P² - 4P - 21 = 0.
As a and b are its roots,
∴ a + b = - {(- 4)/1} = 4
ab = - 21

Given new roots are (a/b) and (b/a)
Hence sum of roots = (a/b) + (b/a) = (a² + b²)/ab
= {(a + b)² - 2ab}/ab
= (16 + 42)/(- 21)
= - 58/21

So new equation = P² - (- 58/21)P + 1 = 0
⇒ 21P² + 58P + 21 = 0.

৬,৮৬৫.

If the n^th term of an arithmetic progression is 5n + 2, then what is the common difference?

ক
4
খ
5
গ
6
ঘ
9

ব্যাখ্যা

Question: If the n^th term of an arithmetic progression is 5n + 2, then what is the common difference?

Solution:
The n^th term of an arithmetic progression is T_n = 5n + 2
n = 1 then, T₁ = 5 × 1 + 2 = 7
n = 2 then, T₂ = 5 × 2 + 2 = 12
n = 3 then, T₃ = 5 × 3 + 2 = 17
n = 4 then, T₄ = 5 × 4 + 2 = 22
............................

Common difference,
T₂ - T₁ = 12 - 7 = 5
T₄ - T₃ = 22 - 17 = 5

∴ The common difference is 5.

৬,৮৬৬.

A man's speed with the current is 18 km/h and the speed of the current is 3 km/h. The man's speed against the current is-

ক
10 km/h
খ
12 km/h
গ
14 km/h
ঘ
15 km/h

ব্যাখ্যা

Question: A man's speed with the current is 18 km/h and the speed of the current is 3 km/h. The man's speed against the current is-

Solution:
Given,
man's speed with the current is 18 km/h
the speed of the current is 3 km/h

∴ Man's rate in still water = (Man's speed with the current - The speed of the current)
= (18 - 3) km/h
= 15 km/h

∴ Man's rate against the current = (Man's rate in still water - The speed of the current)
= (15 - 3) km/h
= 12 km/h

৬,৮৬৭.

What are the solutions to the equation 2x²+ 9x + 9 = 0

ক
ক) 3/2 , 3
খ
খ) - 3/2 , 3
গ
গ) - 3/2 , - 3
ঘ
ঘ) 3/2 , - 3

ব্যাখ্যা

Question: What are the solutions to the equation 2x²+ 9x + 9 = 0

Solution:
Given that
2x² + 9x + 9 = 0.........(1)
Comparing ax² + bx + c = 0 with (1) get, a = 2, b = 9 and c = 9

We know
x = {(- b) ± √(b² - 4ac)}/2a
= [{- (9)} ± √{(9)² - 4.2(9)]/2.2
= (- 9 ± √9)/4
=(- 9 ± 3)/4
= (- 9 + 3)/4, (- 9 - 3)/4
= - 3/2 , - 3

৬,৮৬৮.

A fort had provision of food for 200 men for 60 days. After 20 days, 40 men left the fort. The number of days for which the remaining food will last, is-

ক
40 days
খ
50 days
গ
45 days
ঘ
55 days

ব্যাখ্যা

Question: A fort had provision of food for 200 men for 60 days. After 20 days, 40 men left the fort. The number of days for which the remaining food will last, is-

Solution:
After 20 days : 200 men had food for 40 days.
Suppose 160 men had food for x days.
Now, Less men, More days (Indirect Proportion)
160 : 200 : : 40 : x
⇒ 160/200 = 40/x
⇒ 160 x = 200 × 40
⇒ x = (200 × 40)/160
∴ x = 50 days

৬,৮৬৯.

If sin θ = 3/5 and θ is an acute angle, find the value of tan θ.

ক
1/4
খ
3/4
গ
2/3
ঘ
0

ব্যাখ্যা

Question: If sin θ = 3/5 and θ is an acute angle, find the value of tan θ.

Solution:
Given sin θ = 3/5 and θ is acute.

We know,
sin²θ + cos²θ = 1
cos²θ = 1 - (3/5)² = 1 - 9/25 = 16/25
∴ cos θ = √(16/25) = 4/5

Now, tan θ = sin θ/cos θ
= (3/5) ÷ (4/5)
= 3/4

∴ tan θ = 3/4.

৬,৮৭০.

A wall 8 m long, 6 m high and 22.5 cm thick is made up of bricks, each measuring 25 cm × 11.25 cm × 6 cm. The number of bricks required is

ক
7,200
খ
6,400
গ
6,000
ঘ
5,600

ব্যাখ্যা

Question: A wall 8 m long, 6 m high and 22.5 cm thick is made up of bricks, each measuring 25 cm × 11.25 cm × 6 cm. The number of bricks required is

Solution:
দেয়ালের দৈর্ঘ্য l = 8m = 800cm
দেয়ালের উচ্চতা h = 6m = 600cm
দেয়ালের প্রস্থ b = 22.5cm

দেয়ালের আয়তন = lbh
=800 × 600 × 22.5
=10800000 cm3

ইটের দৈর্ঘ্য l = 25cm
ইটের প্রস্থ b = 11.25cm
ইটের উচ্চতা h = 6cm
ইটের আয়তন =lbh
=25 ×11.25 × 6
=1687.5cm3

ইটের সংখ্যা = 10800000/1687.5
= 6400 টি

৬,৮৭১.

At what angle the hands of a clock are inclined at 15 minutes past 5?

ক
ক) 67.5°
খ
খ) 72.5°
গ
গ) 58.5°
ঘ
ঘ) 69.5°

ব্যাখ্যা

Question: At what angle the hands of a clock are inclined at 15 minutes past 5?

Solution:
Hours hand moves in 15 past.
5 from 12 p.m = (5 + 15/60) hours = 21/4 hours
Angle of hours hand = (360/12) × (21/4)
= 157.5°

Minutes hands makes angle of = (360/60) × 15
= 90°

Angle between hours and minutes hands = (157.5° - 90°)
= 67.5°

৬,৮৭২.

Jobayer spent 2700 taka, which is 20 percent of his monthly salary. What is his monthly salary?

ক
13500 taka
খ
22500 taka
গ
16500 taka
ঘ
8000 taka

ব্যাখ্যা

Question: Jobayer spent 2700 taka, which is 20 percent of his monthly salary. What is his monthly salary?

Solution:
If 20% is equal to 2700 taka
Then 100% is equal to (2700 × 100)/20 taka
= 13500 taka

৬,৮৭৩.

The sum of the present ages of a father and his son is 50 years. Three years ago, father's age was three times the age of the son. After three years, father's age will be:

ক
36 years
খ
39 years
গ
42 years
ঘ
33 years

ব্যাখ্যা

Question: The sum of the present ages of a father and his son is 50 years. Three years ago, father's age was three times the age of the son. After three years, father's age will be:

Solution:
Let,
The present ages of son = a
then, the present age of father = 50 - a

Then, (50 - a) - 3 = 3(a - 3)
⇒ 47 - a = 3a - 9
⇒ 4a = 56
⇒ a = 14

∴ Father's age after 3 years = (50 - 14) + 3 = 39 years.

৬,৮৭৪.

There is provision of food in fort for 1,200 soldiers for 60 days. After 15 days, 200 soldiers leave the fort. Remaining food will last for how many days.

ক
ক) 56 days
খ
খ) 50 days
গ
গ) 48 days
ঘ
ঘ) 54 days

ব্যাখ্যা

১৫ দিন পর সৈন্য আছে= (১২০০ - ২০০) জন = ১০০০ জন

১২০০ জন সৈন্যের খাবার আছে ৪৫ দিনের
১ জন সৈন্যের খাবার আছে ৪৫ ×১২০০ দিনের
১০০০ জন সৈন্যের খাবার আছে (৪৫ ×১২০০)/১০০০ দিনের
=৫৪ দিনের

৬,৮৭৫.

The product of two co-prime numbers is 442. Then their LCM is =?

ক
ক) 442
খ
খ) 17
গ
গ) 26
ঘ
ঘ) 35

ব্যাখ্যা

Question: The product of two co-prime numbers is 442. Then their LCM is =?

Solution:
HCF of co-prime number is always 1
∴ Let number are = x & y respectively
Product of number = xy
xy = 442 (given)
∴ Product of number = LCM × HCF
⇒ LCM × 1 = 442
⇒ LCM = 442

৬,৮৭৬.

A, B and C completed a work costing Tk. 1800. A work for 6 days, B for 4 days and C for 9 days. If their daily wages are in the ratio of 5 : 6 : 4, how much amount will be received by A?

ক
ক) Tk. 420
খ
খ) Tk. 600
গ
গ) Tk. 750
ঘ
ঘ) Tk. 900

ব্যাখ্যা

Ratio of their daily wages = 5 : 6 : 4
Total wage = Tk. 1800
6 × 5x + 4 × 6x + 4 × 9x = 1800 [As A, B, C works for 6, 4, 9 days respectively]
Or, 90x = 1800
Or, x = 1800/90

Total amount of A
= 1800 × 30/90
= Tk. 600

৬,৮৭৭.

A train passes two bridges of length 800 m and 400 m in 100 seconds and 60 seconds respectively. The length of the train is -

ক
ক) 400 meters
খ
খ) 300 meters
গ
গ) 200 meters
ঘ
ঘ) 100 meters

ব্যাখ্যা

Question: A train passes two bridges of length 800 m and 400 m in 100 seconds and 60 seconds respectively. The length of the train is -

Solution:
Let length of the train be x m and speed of the train is s kmph.
Speed, s = (x + 800)/100 . . . . . (i)
Speed, s = (x + 400)/60. . . . . (ii)

Equating equation (i) and (ii),
we get,
(x + 800)/100 = (x + 400)/60
Or, (x + 800)/5 = (x + 400)/3
Or, 5x + 2000 = 3x + 2400
Or, 2x = 400
∴ x = 200m

∴ The length of the train is 200 meters.

৬,৮৭৮.

Find the number that should be placed in the gap of the series : 64, 80, 96, _______, 128

ক
110
খ
112
গ
114
ঘ
116

ব্যাখ্যা

Question: Find the number that should be placed in the gap of the series : 64, 80, 96, _______, 128

Solution:
64 + 16 = 80
80 + 16 = 96
96 + 16 = 112
112 + 16 = 128

৬,৮৭৯.

A man invested Tk. 4455 in Tk. 10 shares quoted at Tk. 8.25. If the rate of dividend be 12%, his annual income is:

ক
ক) 207.40
খ
খ) 534.60
গ
গ) 648
ঘ
ঘ) 655.60

ব্যাখ্যা

Number of shares =4455/8.25= 540.
Face value = tk. (540 x 10) = tk. 5400.
Annual income = tk.12/100x 5400= tk. 648.

৬,৮৮০.

A rectangular water tank is 2 m high, 4m long and 2.5 m high wide. How many liters of water can it hold?

ক
ক) 15000 litre
খ
খ) 20000 litre
গ
গ) 22000 litre
ঘ
ঘ) 25000 litre

ব্যাখ্যা

Question: A rectangular water tank is 2 m high, 4m long and 2.5 m high wide. How many liters of water can it hold?

Solution:
Volume = length × width × height
= 2 × 2.5 × 4 m³
= 20 m³

1 m³ = 1000 litre
⇒ 20 m³ = 20 × 1000 litre
= 20000 litre

৬,৮৮১.

What is the volume of a cylindrical shape water container, that has a height of 7cm and diameter of 10cm?

ক
1100 cm³
খ
75.35 cm³
গ
550 cm³
ঘ
110 cm³

ব্যাখ্যা

Question: What is the volume of a cylindrical shape water container, that has a height of 7cm and diameter of 10cm?

Solution:
Given,
Diameter of the container = 10cm
Thus, radius of the container = 10/2 = 5cm
Height of container = 7cm

As we know, from the formula,
Volume of a cylinder = πr²h cubic units.

Therefore, volume of given container, V = π × 5² × 7
V = π × 25 × 7 = (22/7) × 25 × 7 = 22 × 25
V = 550 cm³

৬,৮৮২.

Sarah is 50 years old and Nazrul is 40 years old. How long ago was the ratio of their ages 3 : 2?

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

ব্যাখ্যা

Question: Sarah is 50 years old and Nazrul is 40 years old. How long ago was the ratio of their ages 3 : 2?

Solution:
Let us assume x years ago
At present: Sarah is 50 years and Nazrul is 40 years
x years ago: Sarah’s age = (50 - x) and Nazrul's age = (40 - x)

ATQ,
(50 - x)/(40 - x) = 3/2
⇒ 100 - 2x = 120 - 3x
⇒ x = 20

Therefore, the answer is 20 years.

৬,৮৮৩.

The average age of all the students in a class is 22 years. The average age of the boys in the class is 25 years, and that of the girls is 18 years. If the number of girls in the class is 24, find the number of boys in the class.

ক
30
খ
32
গ
36
ঘ
40

ব্যাখ্যা

Question: The average age of all the students in a class is 22 years. The average age of the boys in the class is 25 years, and that of the girls is 18 years. If the number of girls in the class is 24, find the number of boys in the class.

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

Then,
22 (x + 24) = 25x + (18 × 24)
⇒ 22x + 528 = 25x + 432
⇒ 3x = 96
⇒ x = 32

∴ The number of boys in the class is 32

৬,৮৮৪.

A jar contains milk and water in the ratio 5 : 1. If the quantity of milk is more than that of water by 8 liters, then what is the quantity of water?

ক
ক) 1.5 liter
খ
খ) 2 liter
গ
গ) 6 liter
ঘ
ঘ) 8 liter

ব্যাখ্যা

ধরি,
দুধ আছে = 5x লিটার এবং পানি আছে = x লিটার

শর্তমতে,
5x - x = 8
⇒ 4x = 8
⇒ x = 8/4
x = 2

পানি আছে = 2 লিটার

৬,৮৮৫.

Karim's salary is 5000 takas. If his salary increases by 5%, 10%, and 20% consecutively over three years, what will his salary be at the end of three years?

ক
Tk. 7560
খ
Tk. 6930
গ
Tk. 6875
ঘ
Tk. 7930
ঙ
Tk. 10250

ব্যাখ্যা

Question: Karim's salary is 5000 takas. If his salary increases by 5%, 10%, and 20% consecutively over three years, what will his salary be at the end of three years?

Solution:
১ম বছর ৫% বৃদ্ধিতে,
নতুন বেতন = ৫০০০ +{৫০০০ × (৫/১০০)}
=৫০০০ + ২৫০ = ৫২৫০ টাকা

২য় বছর ১০% বৃদ্ধিতে,
নতুন বেতন = ৫২৫০ +{৫২৫০ × (১০/১০০)}
=৫২৫০ + ৫২৫ = ৫৭৭৫ টাকা

৩য় বছর ৫% বৃদ্ধিতে,
নতুন বেতন = ৫৭৭৫ +{৫৭৭৫ × (২০/১০০)}
=৫৭৭৫ + ১১৫৫ = ৬৯৩০ টাকা

∴ তিন বছর শেষে করিম সাহেবের বেতন হবে ৬৯৩০ টাকা

৬,৮৮৬.

If the LCM of two numbers is 70 and their HCF is 2, find the numbers.

ক
2, 35
খ
6, 70
গ
4, 70
ঘ
14, 10

ব্যাখ্যা

Question: If the LCM of two numbers is 70 and their HCF is 2, find the numbers.

Solution:
HCF × LCM = First number × Second number
⇒ 2 × 70 = 140 (product of two numbers)

as 14 × 10 = 140. So, the required numbers are 14 and 10.

৬,৮৮৭.

A and B can do a piece of work in 40 days, B and C can do it in 120 days. If B alone can do it in 180 days, in how many days will A and C do it together?

ক
18 days
খ
25 days
গ
22.5 days
ঘ
45 days

ব্যাখ্যা

Question: A and B can do a piece of work in 40 days, B and C can do it in 120 days. If B alone can do it in 180 days, in how many days will A and C do it together?

Solution:
A + B take 40 days. B alone takes 180 days.
∴ A will take 1/40 - 1/180 = 7/360 i.e. 360/7 days.

B + C take 120 days.
∴ C alone will take 1/120 - 1/180 = 1/360 360 days.

∴ A & C together will take 7/360 + 1/360 = 8/360 i.e. 360/8 = 45 days to complete the work.

৬,৮৮৮.

The cost of Type 1 rice is Tk. 15 per kg and Type 2 rice is Tk. 20 per kg. If both Type 1 and Type 2 are mixed in the ratio of 2 : 3, then the price per kg of the mixed variety of rice is-

ক
Tk. 19.50
খ
Tk. 19
গ
Tk. 18
ঘ
Tk. 18.50

ব্যাখ্যা

Question: The cost of Type 1 rice is Tk. 15 per kg and Type 2 rice is Tk. 20 per kg. If both Type 1 and Type 2 are mixed in the ratio of 2 : 3, then the price per kg of the mixed variety of rice is-

Solution:
Let the price of the mixed variety be Rs. x per kg.
By the rule of alligation, we have :
Cost of 1 kg of type 1 rice Cost of 1 kg of type 2 rice

∴(20 - x)/(x - 15) = 2/3
⇒ 60 - 3x = 2x - 30
⇒ x = 18.

৬,৮৮৯.

A box contains 200 marbles, 25% of them are blue and the rest are black. From the box you gave your brother a certain number of marbles of which 60% are black. You then found that among the remaining marbles, only 20% are blue. How many marbles did you give to your brother?

ক
40
খ
50
গ
60
ঘ
70
ঙ
None

ব্যাখ্যা

Question: A box contains 200 marbles, 25% of them are blue and the rest are black. From the box you gave your brother a certain number of marbles of which 60% are black. You then found that among the remaining marbles, only 20% are blue. How many marbles did you give to your brother?

Solution:
বাক্সে মার্বেল আছে = 200 টি
Blue মার্বেল আছে = 200 এর 25%
= 200 এর 25/100 = 50 টি
Black মার্বেল আছে = (200 - 50)টি = 150টি

ধরি, ভাইকে দেওয়া হয়েছিলো = a টি

প্রশ্নমতে,
(200 - a)20% + a . 40% = 50
⇒ 0.2 (200 - a) + 0.4a = 50
⇒ 40 - 0.2a + 0.4a = 50
⇒ 0.2a = 10
∴ a = 50

ভাইকে দেওয়া হয়েছিলো 50টি

৬,৮৯০.

If log₁₀(2^m + m - 4) = m(1 - log₁₀5), then m =?

ক
2
খ
3
গ
4
ঘ
5

ব্যাখ্যা

Question: If log₁₀(2^m + m - 4) = m(1 - log₁₀5), then m =?

Solution:
log₁₀(2^m + m - 4) = m(1 - log₁₀5)
⇒ log₁₀(2^m + m - 4) = m(log₁₀10 - log₁₀5)
⇒ log₁₀(2^m + m - 4) = m log₁₀(10/5)
⇒ log₁₀(2^m + m - 4) = mlog₁₀2
⇒ log₁₀(2^m + m - 4) = log₁₀2^m
⇒ 2^m + m - 4 = 2^m
⇒ m = 4

৬,৮৯১.

Identify the irrational number from the following options.

ক
3/5
খ
1.2
গ
√2
ঘ
0.75

ব্যাখ্যা

Question: Identify the irrational number from the following options.
(Officer General 22 এর অনুরূপ)

Soluiton:
অমূলদ সংখ্যা (irrational number):
- যে সংখ্যাকে p/q আকারে প্রকাশ করা যায় না, যেখানে p ও q পূর্ণসংখ্যা এবং q ≠ 0, সে সংখ্যাকে অমূলদ সংখ্যা বলা হয়।
- পূর্ণবর্গ নয় এরূপ যে কোনাে স্বাভাবিক সংখ্যার বর্গমূল কিংবা তার ভগ্নাংশ একটি অমূলদ সংখ্যা।
যেমন√2 = 1.414213..., √3 = 1.732 ..., ইত্যাদি অমূলদ সংখ্যা।

- কোনাে অমূলদ সংখ্যাকে দুইটিপূর্ণ সংখ্যার অনুপাত হিসেবে প্রকাশ করা যায় না।
- অমূলদ সংখ্যাকে একটি মূলদ সংখ্যা দ্বারা গুণ করলে অমূলদ সংখ্যা পাওয়া যায়।

৬,৮৯২.

X can do 1/5 of a work in 8 days, Y can do 25% of the work in 50 days and Z can do 1/4 of the work in 15 days. Who will complete the work first?

ক
X
খ
Y
গ
Z
ঘ
X and Y both
ঙ
None of these

ব্যাখ্যা

Question: X can do 1/5 of a work in 8 days, Y can do 25% of the work in 50 days and Z can do 1/4 of the work in 15 days. Who will complete the work first?

Solution:
X এর সম্পূর্ণ কাজ করতে সময় লাগবে = (5 × 8) = 40 দিন

Y এর সম্পূর্ণ কাজ করতে সময় লাগবে = {50 × (100/25)} = 200 দিন

Z এর সম্পূর্ণ কাজ করতে সময় লাগবে = (4 × 15) = 60 দিন

∴ X পুরো কাজ সবচেয়ে কম সময়ে, অর্থাৎ 40 দিনে শেষ করতে পারবে।

৬,৮৯৩.

Find the LCM of 2.5, 0.5 and 0.175.

ক
7.5
খ
17.5
গ
2.5
ঘ
0.75

ব্যাখ্যা

Question: Find the LCM of 2.5, 0.5 and 0.175.

Solution:
Given that,
2.5 = 25/10
0.5 = 5/10
0.175 = 175/1000

Now,
LCM of Numerators is 25, 5, 175 = 175
and HCF of Denominators is 10, 10, 1000 = 10

We know,
LCM of two or more fractions is given by,
LCM = LCM of Numerators/HCF of Denominators
= 175/10
= 17.5

৬,৮৯৪.

A tank has water for 72 trees and can last for 54 days for them. If each tree is given 10% less water than 90 trees can get the water for how many days?

ক
ক) 24 days
খ
খ) 36 days
গ
গ) 42 days
ঘ
ঘ) 48 days

ব্যাখ্যা

Let each tree take T amount of water every day.
So 72 trees take 72T water in one day.

With 10% reduction each tree will consume 90T/100 amount each day.
So 90 trees take 90 (90T/100) amount in one day
Total water quantity is constant
∴ 72T × 54 = 90 × (90T/100) × D
∴ D = 48 days = Number of days 90 trees can use the water.

৬,৮৯৫.

The radius of a wheel is 7 cm. How many revolutions will it make in travelling 88 kilometers?

ক
100000
খ
200000
গ
250000
ঘ
100200

ব্যাখ্যা

Question: The radius of a wheel is 7 cm. How many revolutions will it make in travelling 88 kilometers?

Solution:
আমরা জানি,
চাকার পরিধি = 2πr = 2 × (22/7) × 7 = 44 সে. মি.

∴ মোট দূরত্ব = 88 কি. মি. = 88 × 1000 × 100 = 8800000 সে. মি.

∴ ঘূর্ণন সংখ্যা = 8800000/44 = 200000 টি

৬,৮৯৬.

Nila started a boutique shop by investing Tk. 25,000. After 4 months, Tania joined her with an investment of Tk. 35,000. After 2 year, the business earned a profit of Tk. 16,770. What was Nila's share in the profit?

ক
Tk. 7740
খ
Tk. 6850
গ
Tk. 6180
ঘ
Tk. 5900

ব্যাখ্যা

Question: Nila started a boutique shop by investing Tk. 25,000. After 4 months, Tania joined her with an investment of Tk. 35,000. After 2 year, the business earned a profit of Tk. 16,770. What was Nila's share in the profit?

Solution:
Ratio of capitals of Nila and Tania = (25000 × 24) : (35000 × 20)
= 600000 : 700000
= 6 : 7

Nila's share is = Tk. (16770 × 6)/13 = Tk. 7740.

৬,৮৯৭.

What is the area of an isosceles triangle if two of its sides measure 6 cm and 12 cm?

ক
7√5 cm²
খ
9√15 cm²
গ
9√11 cm²
ঘ
12√5 cm²
ঙ
None

ব্যাখ্যা

Question: What is the area of an isosceles triangle if two of its sides measure 6 cm and 12 cm?

Solution:
In an isosceles triangle, two sides are equal. The possible third side can be either 6 cm or 12 cm.

If the equal sides are 6, the sides become 6, 6, and 12 — which violates the triangle inequality rule.
If the equal sides are 12, the sides become 6, 12, and 12 — which satisfies the triangle inequality.
∴ The valid third side is 12 cm.

Now,
a = 6 cm, b = 12 cm, c = 12 cm

∴ Semi-perimeter s =(6 + 12 + 12)/2 =15 cm

We know
from "Heron’s formula"
Area of the triangle = √{s(s - a)(s - b)(s - c)}
= √{15(15 - 6)(15 - 12)(15 - 12)}
= √(15 × 9 × 3 × 3)
= 9√15 cm²

৬,৮৯৮.

A purse contains 342 coins consisting of one Taka, 50 cents and 25 cents coins. If their values are in the ratio of 11 : 9 : 5 then find the number of 50 cents coins?

ক
180
খ
150
গ
162
ঘ
99

ব্যাখ্যা

Let the value of one taka, 50 cents, and 25 cents be 11x, 9x, 5x respectively.

No. of 1 taka coins = (11x / 1) =11x
No. of 50 cents coins = (9x / 0.5) = 18x
No. of 25 cents coins = (5x / 0.25) = 20x

11x + 18x + 9x = 342
⇒ 38x = 342
⇒ x = 9

Therefore, no. of 1 taka coins = 11 x 9 = 99 coins
No. of 50 cents coins = 18 x 9 = 162 coins
No. of 25 cents coins = 20 x 9 = 180 coins.

৬,৮৯৯.

The average marks obtained by 22 candidates in an examination are 45. The average marks of the first ten are 55 and that last eleven are 40. The number of marks obtained by the 11^th candidate is

ক
ক) 0
খ
খ) 45
গ
গ) 48
ঘ
ঘ) 52

ব্যাখ্যা

Total marks scored by 22 candidates = 22 × 45
= 990
Total marks scored by first 10 candidates =10 × 55
= 550
Total marks scored by last 11 candidates =11 × 40
=440
∴ Marks scored by 11th candidate = 990 - (550 + 440)
= 0

৬,৯০০.

If a right-angled isosceles triangle has height 5 cm, then base is:

ক
5√2
খ
5 cm
গ
12 cm
ঘ
13 cm

ব্যাখ্যা

Question: If a right-angled isosceles triangle has height 5 cm, then base is:

Solution:

আমরা জানি,
সমকোণী সমদ্বিবাহু ত্রিভুজে সমকোণ সংলগ্ন দুইটি বাহু সমান হয়।

দেওয়া আছে, উচ্চতা = 5 cm

যেহেতু ত্রিভুজটি সমকোণী সমদ্বিবাহু,
∴ উচ্চতা = ভূমি
∴ ভূমি = 5 cm