১৩,৮০১.
ব্যাখ্যা
প্রশ্ন:
সমাধান:
সমাধান:
PrepBank · বিষয়ভিত্তিক প্রশ্ন
Bank Math
Bank Math
PrepBank · পাতা ১৩৯ / ১৬১ · ১৩,৮০১–১৩,৯০০ / ১৬,১২৪
১৩,৮০২.
If (x/y) + (y/x) = √7 then what is the value of (x4/y4) + (y4/x4) ?
ব্যাখ্যা
Question: If (x/y) + (y/x) = √7 then what is the value of (x4/y4) + (y4/x4) ?
Solution:
দেওয়া আছে,
(x/y) + (y/x) = √7
∴ x4/y4 + y4/x4
= (x/y)4 + (y/x)4
= {(x/y)2}2 + {(y/x)2}2
= {(x/y)2 + (y/x)2}2 - 2.(x2/y2).(y2/x2)
= {(x/y)2 + (y/x)2}2 - 2
= [{(x/y) + (y/x)}2 - 2.(x/y).(y/x)]2 - 2
= {(√7)2 - 2}2 - 2
= (7 - 2)2 - 2
= 52 - 2
= 25 - 2
= 23
Solution:
দেওয়া আছে,
(x/y) + (y/x) = √7
∴ x4/y4 + y4/x4
= (x/y)4 + (y/x)4
= {(x/y)2}2 + {(y/x)2}2
= {(x/y)2 + (y/x)2}2 - 2.(x2/y2).(y2/x2)
= {(x/y)2 + (y/x)2}2 - 2
= [{(x/y) + (y/x)}2 - 2.(x/y).(y/x)]2 - 2
= {(√7)2 - 2}2 - 2
= (7 - 2)2 - 2
= 52 - 2
= 25 - 2
= 23
১৩,৮০৩.
The profit percent gained if an object is sold at Tk. 464 is equals to the loss percent when the object is sold at Tk. 436. Find the cost price of this object?
ব্যাখ্যা
Question: The profit percent gained if an object is sold at Tk. 464 is equals to the loss percent when the object is sold at Tk. 436. Find the cost price of this object?
Solution:
Let cost price = x
Profit = selling price - cost price = 464 - x
Loss = cost price - selling price = x - 436
loss % = {(x - 436)/x} × 100
profit % = {(464 - x)/x} × 100
ATQ,
loss % = profit %
{(x - 436)/x} × 100 = {(464 - x)/x} × 100
⇒ (x - 436)/x = (464 - x)/x
⇒ x - 436 = 464 - x
⇒ 2x = 464 + 436
⇒ 2x = 900
∴ x = 450
Solution:
Let cost price = x
Profit = selling price - cost price = 464 - x
Loss = cost price - selling price = x - 436
loss % = {(x - 436)/x} × 100
profit % = {(464 - x)/x} × 100
ATQ,
loss % = profit %
{(x - 436)/x} × 100 = {(464 - x)/x} × 100
⇒ (x - 436)/x = (464 - x)/x
⇒ x - 436 = 464 - x
⇒ 2x = 464 + 436
⇒ 2x = 900
∴ x = 450
১৩,৮০৪.
, then find the value of 'm' if (a + b + c) ≠ 0
ব্যাখ্যা
Question: , then find the value of 'm' if (a + b + c) ≠ 0
Solution:
Given,
a/(2a + b + c) = m
⇒ a = m(2a + b + c) .................... (1)
b/(a + 2b + c) = m
⇒ b = m(a + 2b + c) .................... (2)
c/(a + b + 2c) = m
⇒ c = m(a + b + 2c) .................... (3)
(1) + (2) + (3)
a + b + c = m(2a + b + c + a + 2b + c + a + b + 2c)
⇒ a + b + c = m(4a + 4b + 4c)
⇒ a + b + c = 4m(a + b + c)
⇒ 4m = (a + b + c)/(a + b + c)
⇒ 4m = 1
∴ m = 1/4
Solution:
Given,
a/(2a + b + c) = m
⇒ a = m(2a + b + c) .................... (1)
b/(a + 2b + c) = m
⇒ b = m(a + 2b + c) .................... (2)
c/(a + b + 2c) = m
⇒ c = m(a + b + 2c) .................... (3)
(1) + (2) + (3)
a + b + c = m(2a + b + c + a + 2b + c + a + b + 2c)
⇒ a + b + c = m(4a + 4b + 4c)
⇒ a + b + c = 4m(a + b + c)
⇒ 4m = (a + b + c)/(a + b + c)
⇒ 4m = 1
∴ m = 1/4
১৩,৮০৫.
Complete the sentence:
Despite the objections from the team, he insisted on _______________.
Despite the objections from the team, he insisted on _______________.
ব্যাখ্যা
• Insist on/upon/ Insist on doing something:
- To keep doing something, even if it annoys other people, or people think it is not good for you.
- বাংলায়, কোনো কিছুতে জোর দেয়া বা পীড়াপীড়ি করা।
- Insist on/upon এর সাথে verb এর -ing form বসে।
• Example sentence:
- He insisted on my going there.
- She insists on doing everything her own way.
- He insisted on seeing her.
• Complete Sentence: Despite the objections from the team, he insisted on implementing the new strategy.
Source: Advanced Learner's Communicative English Grammar & Composition By Chowdhury & Hossain.
১৩,৮০৬.
Two trains are 150 meters and 180 meters long. They are running on parallel tracks in the same direction. The faster train runs at 90 km/h and the slower at 54 km/h. How much time will the faster train take to completely pass the slower train?
ব্যাখ্যা
Question: Two trains are 150 meters and 180 meters long. They are running on parallel tracks in the same direction. The faster train runs at 90 km/h and the slower at 54 km/h. How much time will the faster train take to completely pass the slower train?
Solution:
Given,
Length of first train = 150 meters
Length of second train = 180 meters
Speed of faster train = 90 km/h
= (90 × 1000)/3600 = 25 m/s
Speed of slower train = 54 km/h
= (54 × 1000)/3600 = 15 m/s
Since they are moving in the same direction,
∴ Relative speed = 25 - 15 = 10 m/s
To pass completely, the faster train must cover the length of both trains.
∴ Total lenght = 150 + 180 = 330 meters
Now,
Time = Distance / Speed
= 330/10 = 33 seconds
১৩,৮০৭.
A student loses 1 mark for every wrong answer and scores 2 marks for every correct answer. If he answers all the 60 questions in an exam and scores 39 marks, how many of them were correct?
ব্যাখ্যা
Question: A student loses 1 mark for every wrong answer and scores 2 marks for every correct answer. If he answers all the 60 questions in an exam and scores 39 marks, how many of them were correct?
Solution:
ধরি,
ক সংখ্যক উত্তর ভুল দিয়েছে।
যদি ৬০ টি উত্তর সঠিক হতো তাহলে পেত = ৬০ × ২ = ১২০ নম্বর
প্রতিটি ভুল উত্তরের জন্য প্রকৃতপক্ষে কাটা যায় = ২ + ১ = ৩ নম্বর
প্রশ্নমতে,
১২০ - ৩ক = ৩৯
৩ক = ১২০ - ৩৯
ক = ৮১/৩
ক = ২৭
সুতরাং, ২৭ টি উত্তর ভুল ছিল এবং (৬০ - ২৭) বা, ৩৩ টি উত্তর সঠিক ছিল।
Solution:
ধরি,
ক সংখ্যক উত্তর ভুল দিয়েছে।
যদি ৬০ টি উত্তর সঠিক হতো তাহলে পেত = ৬০ × ২ = ১২০ নম্বর
প্রতিটি ভুল উত্তরের জন্য প্রকৃতপক্ষে কাটা যায় = ২ + ১ = ৩ নম্বর
প্রশ্নমতে,
১২০ - ৩ক = ৩৯
৩ক = ১২০ - ৩৯
ক = ৮১/৩
ক = ২৭
সুতরাং, ২৭ টি উত্তর ভুল ছিল এবং (৬০ - ২৭) বা, ৩৩ টি উত্তর সঠিক ছিল।
১৩,৮০৮.
If the length of rectangle B and the width of rectangle A is one-half the width of rectangle B, what is the ratio of the area of rectangle A to the area of rectangle B?
ব্যাখ্যা
ধরি, B এর দৈর্ঘ্য x একক এবং প্রস্থ y একক
∴ A এর দৈর্ঘ (1/2)x একক এবং (1/2)y একক
B এর ক্ষেত্রফল xy বর্গএকক
A এর ক্ষেত্রফল (1/2 x × 1/2 y) বর্গএকক
= xy/4 বর্গএকক
∴ A:B = xy/4:xy
= 1:4
∴ A/B = 1/4
১৩,৮০৯.
A team of 3 men and 4 women can complete 1/3 of a job in 4 days. After that, 2 more women join the team and they complete the next 1/3 of the job in 3 days. How many women can complete the whole job in 6 days?
ব্যাখ্যা
Question: A team of 3 men and 4 women can complete 1/3 of a job in 4 days. After that, 2 more women join the team and they complete the next 1/3 of the job in 3 days. How many women can complete the whole job in 6 days?
Solution:
Let,
Men = M, Women = W
3M & 4W complete in 4 days = 1/3 part of the job
∴ 3M & 4W complete in 1 days = 1/12 part of the job
After joining 2 women,
3M & 6W complete in 3 days= 1/3 part of the job
3M & 6W complete in 1 days= 1/9 part of the job
Now,
2 women can complete in 1 day = (1/9) - (1/12) = (4 - 3)/36 = 1/36
∴ 1 women can complete in 1 day = 1/72
∴ 1 women dose 1(whole) of the job in = 1 × 72 = 72 day
∴ In 72 days the job completed by 1 women
∴ In 1 days the job completed by = 1 × 72 women
∴ In 6 days the job completed by = 72/6 = 12 women
Solution:
Let,
Men = M, Women = W
3M & 4W complete in 4 days = 1/3 part of the job
∴ 3M & 4W complete in 1 days = 1/12 part of the job
After joining 2 women,
3M & 6W complete in 3 days= 1/3 part of the job
3M & 6W complete in 1 days= 1/9 part of the job
Now,
2 women can complete in 1 day = (1/9) - (1/12) = (4 - 3)/36 = 1/36
∴ 1 women can complete in 1 day = 1/72
∴ 1 women dose 1(whole) of the job in = 1 × 72 = 72 day
∴ In 72 days the job completed by 1 women
∴ In 1 days the job completed by = 1 × 72 women
∴ In 6 days the job completed by = 72/6 = 12 women
১৩,৮১০.
The angle of elevation of a ladder leaning against a wall is 60º and the foot of the ladder is 3.5 m away from the wall. The length of the ladder is-
ব্যাখ্যা
Question: The angle of elevation of a ladder leaning against a wall is 60º and the foot of the ladder is 3.5 m away from the wall. The length of the ladder is-
Solution:
Let AB be the wall and BC be the ladder.
Then, ∠ACB = 60°
⇒ AC = 3.5 m
⇒ AC/BC= cos60∘
⇒ AC/BC= 1/2
⇒ BC = 2 × AC
⇒ BC = 2 × 3.5
⇒ BC = 7 m
Solution:
Let AB be the wall and BC be the ladder.
Then, ∠ACB = 60°
⇒ AC = 3.5 m
⇒ AC/BC= cos60∘
⇒ AC/BC= 1/2
⇒ BC = 2 × AC
⇒ BC = 2 × 3.5
⇒ BC = 7 m
১৩,৮১১.
A group of 10 people has their average weight increased by 3 kg after replacing a 50 kg person with someone new. What is the possible weight of the replacement?
ব্যাখ্যা
Question: A group of 10 people has their average weight increased by 3 kg after replacing a 50 kg person with someone new. What is the possible weight of the replacement?
Solution:
ধরি, 10 জন ব্যক্তির বর্তমান গড় ওজন = x কেজি।
∴ মোট ওজন = 10x কেজি।
50 কেজি ওজনের ব্যক্তি চলে যাওয়ার পর মোট ওজন = (10x - 50) কেজি।
মনে করি, নতুন ব্যক্তির ওজন = y কেজি।
নতুন ব্যক্তি প্রবেশ করার পর নতুন মোট ওজন হবে = (10x - 50 + y) কেজি।
এবং নতুন গড় = (x + 3) কেজি।
প্রশ্নমতে,
(10x - 50 + y)/10 = x + 3
⇒ 10x - 50 + y = 10(x + 3)
⇒ 10x - 50 + y = 10x + 30
⇒ 10x - 10x + y = 30 + 50
⇒ y = 80
সুতরাং, নতুন ব্যক্তির ওজন = 80 কেজি।
১৩,৮১২.
A dishonest dealer marks up the price of his goods by 20% and gives a discount of 10% to the customer. Besides, he also cheats both his supplier and his buyer by 100 grams while buying or selling 1 kilogram. Find the percentage profit earned by the shopkeeper?
ব্যাখ্যা
He buys 1100 grams instead of 1000 grams.
Suppose he bought 1100 grams for Tk. 1000.
He sells only 900 grams when he takes the money for 1 kg.
Now, according to the problem,
he sells at an 8% profit (20% markup, 10% discount).
Hence, his selling price is Tk. 1080 for 900 grams.
Now,
1100 grams for Tk. 1000
Hence, 1188 grams for Tk. 1080
Selling: 900 grams for Tk. 1080
Hence % profit = (288/900) × 100 = 32%
১৩,৮১৩.
A college hostel mess has provisions for 25 days for 350 boys. At the end of 10 days, when some boys were shifted to another hostel, it was found that now the provisions will last for 21 more days. How may boys were shifted to another hostel?
ব্যাখ্যা
Question: A college hostel mess has provisions for 25 days for 350 boys. At the end of 10 days, when some boys were shifted to another hostel, it was found that now the provisions will last for 21 more days. How may boys were shifted to another hostel?
Solution:
A college hostel mess has provisions for 25 days for 350 boys.
At the end of 10 days, when some boys were shifted to another hostel, it was found that now the provisions will last for 21 more days.
Available provisions = 25 × 350 = 8750 units
After 10 days the remaining provision = 8750 - 10 × 350 ⇒ 5250 units
Let N number of boys shifted to another hostel.
According to the question,
(350 - N) × 21 = 5250
⇒ 350 - N = 5250/21
⇒ 350 - N = 250
⇒ N = 350 - 250
⇒ N = 100
∴ 100 boys were shifted to another hostel.
Solution:
A college hostel mess has provisions for 25 days for 350 boys.
At the end of 10 days, when some boys were shifted to another hostel, it was found that now the provisions will last for 21 more days.
Available provisions = 25 × 350 = 8750 units
After 10 days the remaining provision = 8750 - 10 × 350 ⇒ 5250 units
Let N number of boys shifted to another hostel.
According to the question,
(350 - N) × 21 = 5250
⇒ 350 - N = 5250/21
⇒ 350 - N = 250
⇒ N = 350 - 250
⇒ N = 100
∴ 100 boys were shifted to another hostel.
১৩,৮১৪.
A started a business with Tk. 2100 and is joined afterwards by B with Tk. 3600. After how many months did B join if the profits at the end of the year are divided equally?
ব্যাখ্যা
Question: A started a business with Tk. 2100 and is joined afterwards by B with Tk. 3600. After how many months did B join if the profits at the end of the year are divided equally?
Solution:
Suppose,
B joined after x months.
Then,
2100 × 12 = 3600 × (12 - x)
⇒ 252 = 432 - 36x
⇒ 36x = 180
∴ x = 5
∴ B joined after 5 months
Solution:
Suppose,
B joined after x months.
Then,
2100 × 12 = 3600 × (12 - x)
⇒ 252 = 432 - 36x
⇒ 36x = 180
∴ x = 5
∴ B joined after 5 months
১৩,৮১৫.
A man swimming in a stream which flows (3/2) km/hr finds that in a given time he can swim twice as far with the stream as he can against it. At what rate does he swim?
ব্যাখ্যা
Question: A man swimming in a stream which flows (3/2) km/hr finds that in a given time he can swim twice as far with the stream as he can against it. At what rate does he swim?
Solution:
Let the rate of his swim x km/h
When he swim with the flow then speed =(x + 3/2) km/h
∴ S1 = (x + 3/2) × t
When he swim against the flow stream then speed = (x - 3/2) km/h
∴ S2 = (x - 3/2) × t
According to the question,
S1 = 2 × S2
⇒ (x + 3/2)t = 2(x - 3/2)t
⇒ (2x + 3)/2 = 2x - 3
⇒ 2x + 3 = 4x - 6
⇒ 9 = 2x
⇒ x = 9/2
∴ x = 4.5 km/hr
Therefore, the man swims at 9/2 km/h (or 4.5 km/h) in still water.
১৩,৮১৬.
In Live MCQ's class there are 12 boys and 18 girls. Write the ratio of girls : boys in its simplest form.
ব্যাখ্যা
Question: In Live MCQ's class there are 12 boys and 18 girls. Write the ratio of girls : boys in its simplest form.
Solution:
The question asks for the ratio girls:boys so girls must be first and boys second. It also asks for the answer in its simplest form.
Number of girl : Number of boys = 18 : 12 = 3 : 2
The ratio 18 : 12 represents the number of girls to boys but is not in its simplest form. Ratios are typically expressed in the simplest form by dividing both terms by their greatest common divisor.
Here’s the simplification process:
Original Ratio= 18 : 12
Greatest Common Divisor (GCD): The GCD of 18 and 12 is 6.
Simplify: Divide both terms by 6
18/6 : 12/6 = 3 : 2
Solution:
The question asks for the ratio girls:boys so girls must be first and boys second. It also asks for the answer in its simplest form.
Number of girl : Number of boys = 18 : 12 = 3 : 2
The ratio 18 : 12 represents the number of girls to boys but is not in its simplest form. Ratios are typically expressed in the simplest form by dividing both terms by their greatest common divisor.
Here’s the simplification process:
Original Ratio= 18 : 12
Greatest Common Divisor (GCD): The GCD of 18 and 12 is 6.
Simplify: Divide both terms by 6
18/6 : 12/6 = 3 : 2
১৩,৮১৭.
A pen was sold at 10% loss. If the selling price was Tk. 6 more, then the seller could have made a profit of 5%. What was the purchase price of the pen?
ব্যাখ্যা
মনে করি,
ক্রয়মূল্য = 100 টাকা
∴ 10% loss এ বিক্রয়মূল্য = 90 টাকা
এবং 5% profit এ বিক্রয়মূল্য = 105 টাকা
∴ বিক্রয়মূল্য (105 - 90) = 15 টাকা বেশি হলে,
5% profit হতো।
এখন ঐকিক নিয়ম ব্যবহার করে, বিক্রয়মূল্য 15 টাকা বেশি হলে ক্রয়মূল্য 100 টাকা
∴ বিক্রয়মূল্য 1 টাকা বেশি হলে ক্রয়মূল্য (100/15) টাকা
∴ বিক্রয়মূল্য 6 টাকা বেশি ক্রয়মূল্য (100/15) × 6
= 40 টাকা।
১৩,৮১৮.
Arif is 4 times as old as Bobby. In 6 years, Bobby will be twice as old as Sami. Four years from now Arif will be 36 years old, how old will Sami be in 6 years?
ব্যাখ্যা
ধরি,
ববির বর্তমান বয়স B বছর
আরিফের বর্তমান বয়স A বছর
সামির বর্তমান বয়স S বছর
প্রশ্নমতে,
A=4B ............(১)
A+4=36 ..................(২)
B+6=2(S+6) ...............(৩)
(২) নং সমীকরণ থেকে পাই,
A = 36−4 = 32
এখন, A = 4B প্রতিস্থাপন করলে,
32 = 4B
∴ B = 8
(৩) নং সমীকরণ থেকে পাই,
8+6 = 2(S+6)
14 = 2S+12
2S = 2
S = 1
∴ ৬ বছর পর সামির বয়স = S + 6
= 1 + 6
= 7 বছর।
১৩,৮১৯.
The angle of elevation of the top of a tower at a point on the ground 37 m away from the foot of the tower is 45°. What is the height of the tower?
ব্যাখ্যা
Question: The angle of elevation of the top of a tower at a point on the ground 37 m away from the foot of the tower is 45°. What is the height of the tower?
Solution:
Let AB be tower and C is a point on the ground 37 m away
From the foot of tower B
The angle of elevation is 45°
Let h be the height of the tower.
∴ tanθ = AB/BC
⇒ tan45∘ = AB/37
⇒ 1 = AB/37
⇒ AB = 37 m
Solution:
Let AB be tower and C is a point on the ground 37 m away
From the foot of tower B
The angle of elevation is 45°
Let h be the height of the tower.
∴ tanθ = AB/BC
⇒ tan45∘ = AB/37
⇒ 1 = AB/37
⇒ AB = 37 m
১৩,৮২০.
If 75% of a number is added to 75, then the result is the number itself. The number is -
ব্যাখ্যা
Let,
the number be x.
Then, 75% of x + 75 = x
⇒ x - 75x/100 = 75
⇒ x - 3x/4 = 75
⇒ x/4 = 75
⇒ x = 300
১৩,৮২১.
A is two years older than B who is twice as old as C. If the total of the ages of A, B and C be 32, then how old is B?
ব্যাখ্যা
Let
C's age be x years.
Then,
B's age = 2x years.
A's age = (2x + 2) years.
(2x + 2) + 2x + x = 32
5x + 2 = 32
5x = 32 - 2
5x = 30
x = 6.
Hence, B's age = 2x = 12 years.
C's age be x years.
Then,
B's age = 2x years.
A's age = (2x + 2) years.
(2x + 2) + 2x + x = 32
5x + 2 = 32
5x = 32 - 2
5x = 30
x = 6.
Hence, B's age = 2x = 12 years.
১৩,৮২২.
A small company employs 3 men and 5 women. If a team of 4 employees is to be randomly selected to organize the company retreat, what is the probability that the team will have exactly 2 women?
ব্যাখ্যা
Question: A small company employs 3 men and 5 women. If a team of 4 employees is to be randomly selected to organize the company retreat, what is the probability that the team will have exactly 2 women?
Solution:
the probability that the team will have exactly 2 women is = (5C2 × 3C2)/8C4
= 30/70
= 3/7
১৩,৮২৩.
The length of room is 1.5 times of its breadth. If the area of the room is 216 sq. meters , what is the perimeter of the room?
ব্যাখ্যা
ধরি, আয়তাকার ঘরের প্রস্থ x মিটার এবং দৈর্ঘ্য ১.৫x মিটার বা ৩x/২ মিটার
প্রশ্নমতে,
x × ৩x/২ = ২১৬
⇒ ৩x² = ৪৩২
⇒ x² = ১৪৪
∴ x = ১২
অর্থাৎ প্রস্থ x = ১২ মিটার এবং দৈর্ঘ্য ৩x/২ = ৩×১২/২ = ১৮ মিটার
∴ আয়তাকার ঘরের পরিসীমা = ২(১২+১৮) = ৬০ মিটার
১৩,৮২৪.
Three pots contain 100ml of water, milk, and oil respectively. 10% of the first pot is mixed with the second pot and then 20% of the second pot is mixed with the third pot. How much water is there in the third pot?
ব্যাখ্যা
Question: Three pots contain 100ml of water, milk, and oil respectively. 10% of the first pot is mixed with the second pot and then 20% of the second pot is mixed with the third pot. How much water is there in the third pot?
Solution:
After mixing 10% water from the first pot to the second pot,
total mixture = 100 ml + (10% of 100ml) = 110ml
in 110 ml,
water = 10ml,
milk = 100 ml,
20% of the second pot is mixed with the third pot.
20% of the second pot contains = (20% of 110) ml = 22ml mixture.
in a 22ml mixture of the second pot,
water = (10/110)22 ml = 2ml
milk = (100/110)22 ml= 20ml
Solution:
After mixing 10% water from the first pot to the second pot,
total mixture = 100 ml + (10% of 100ml) = 110ml
in 110 ml,
water = 10ml,
milk = 100 ml,
20% of the second pot is mixed with the third pot.
20% of the second pot contains = (20% of 110) ml = 22ml mixture.
in a 22ml mixture of the second pot,
water = (10/110)22 ml = 2ml
milk = (100/110)22 ml= 20ml
১৩,৮২৫.
In a farm there are cows and hens. If heads are counted there are 180, if legs are counted there are 420. The number of cows in the farm is -
ব্যাখ্যা
Question: In a farm there are cows and hens. If heads are counted there are 180, if legs are counted there are 420. The number of cows in the farm is -
Solution:
Let, the number of cows be x.
Then, number of hens would be (180 - x)
ATQ,
(180 - x) × 2 + 4x = 420
⇒ 360 - 2x + 4x = 420
⇒ 2x = 420 - 360
⇒ 2x = 60
⇒ x = 60/2
∴ x = 30
∴ The number of cows 30.
Solution:
Let, the number of cows be x.
Then, number of hens would be (180 - x)
ATQ,
(180 - x) × 2 + 4x = 420
⇒ 360 - 2x + 4x = 420
⇒ 2x = 420 - 360
⇒ 2x = 60
⇒ x = 60/2
∴ x = 30
∴ The number of cows 30.
১৩,৮২৬.
If then find the value of
ব্যাখ্যা
Question: If then find the value of
Solution:
Given,
2a + b/a + 4b = 3
⇒ 2a + b = 3(a + 4b)
⇒ 2a + b = 3a + 12b
⇒ 3a - 2a = b - 12b
⇒ a = - 11b
∴ a + b/a + 2b = - 11b + b/- 11b + 2b
= - 10b/- 9b
= 10/9
Solution:
Given,
2a + b/a + 4b = 3
⇒ 2a + b = 3(a + 4b)
⇒ 2a + b = 3a + 12b
⇒ 3a - 2a = b - 12b
⇒ a = - 11b
∴ a + b/a + 2b = - 11b + b/- 11b + 2b
= - 10b/- 9b
= 10/9
১৩,৮২৭.
sin(17π/2 + θ) = ?
ব্যাখ্যা
Question: sin(17π/2 + θ) = ?
Solution:
sin(17π/2 + θ)
= sin{17 × (π/2) + θ}
= sin(π/2 + θ)
= cosθ
Solution:
sin(17π/2 + θ)
= sin{17 × (π/2) + θ}
= sin(π/2 + θ)
= cosθ
১৩,৮২৮.
The angle of elevation of the top of a tower at a point on the ground 37 m away from the foot of the tower is 45°. What is the height of the tower?
ব্যাখ্যা
Question: The angle of elevation of the top of a tower at a point on the ground 37 m away from the foot of the tower is 45°. What is the height of the tower?
Solution:
Let AB be tower and C is a point on the ground 37 m away
From the foot of tower B
The angle of elevation is 45°
Let h be the height of the tower.
∴ tanθ = AB/BC
⇒ tan45∘ = AB/37
⇒ 1 = AB/37
⇒ AB = 37 m
Solution:
Let AB be tower and C is a point on the ground 37 m away
From the foot of tower B
The angle of elevation is 45°
Let h be the height of the tower.
∴ tanθ = AB/BC
⇒ tan45∘ = AB/37
⇒ 1 = AB/37
⇒ AB = 37 m
১৩,৮২৯.
A 40-meter cable is attached from the top of a vertical pole to the ground. If the cable makes an angle of 30° with the ground, what is the height of the pole?
ব্যাখ্যা
Question: A 40-meter cable is attached from the top of a vertical pole to the ground. If the cable makes an angle of 30° with the ground, what is the height of the pole?
Solution:
ধরি, উচ্চতা(Height), AB = h
দেয়া আছে, AC = 40m
∠ACB = 30°
∴ sin30°= AB/AC
⇒ 1/2 = h/40
⇒ h = 40 × 1/2
∴ h = 20 m
১৩,৮৩০.
ব্যাখ্যা
Question:
Solution:
১৩,৮৩১.
The ratio of milk to water in a mixture is 5 : 3. If 6 liters of milk are added to the mixture, the new ratio of milk to water becomes 8 : 3. Find the final amount of milk in the new mixture.
ব্যাখ্যা
Question: The ratio of milk to water in a mixture is 5 : 3. If 6 liters of milk are added to the mixture, the new ratio of milk to water becomes 8 : 3. Find the final amount of milk in the new mixture.
Solution:
Let the initial amount of milk be 5x liters
and the amount of water 3x liters.
ATQ,
Ratio of milk and water after adding 6 liters of milk
(5x + 6)/3x = 8/3
⇒ 3(5x + 6) = 8 × 3x
⇒ 15x + 18 = 24x
⇒ 18 = 9x
⇒ x = 2
∴ Final amount of milk in mixture = 5x + 6
= (5 × 2) + 6
= 10 + 6 = 16 liters.
১৩,৮৩২.
Which of the following is the average of first five prime numbers?
ব্যাখ্যা
First five prime numbers = 2, 3, 5, 7, 11
∴ Average = (2 + 3 + 5 + 7 + 11)/5 = 28/5 = 5.6
১৩,৮৩৩.
A pipe can fill up an empty tank in 15 minutes. Another pipe flows out 10 liters of water per minute. If the two pipes are opened together and the empty tank is filled up in 30 minutes, how much water does the tank contain?
ব্যাখ্যা
Question: A pipe can fill up an empty tank in 15 minutes. Another pipe flows out 10 liters of water per minute. If the two pipes are opened together and the empty tank is filled up in 30 minutes, how much water does the tank contain?
Solution:
Let the tank empty in x minute
ATQ,
(1/15) - (1/x) = 1/30
(1/15) - (1/30) = 1/x
1/30 = 1/x
x = 30
So the tank emptied by the other pipe in 30 minute
∴ The tank contain = 30 × 10 liter
= 300 liters
১৩,৮৩৪.
If A = 3, B = 5, C = 7, and so on, what is the meaning of the following numbers 39, 29, 3, 19, 25?
ব্যাখ্যা
Question: If A = 3, B = 5, C = 7, and so on, what is the meaning of the following numbers 39, 29, 3, 19, 25?
Solution:
Given,
A = 3, B = 5, C = 7, .....
∴ প্রতিটি কোড = (অক্ষরের অবস্থান × 2) + 1
So,
(39 - 1)/2 = 19 → S
(29 - 1)/2 = 14 → N
(3 - 1)/2 = 1 → A
(19 - 1)/2 = 9 → I
(25 - 1)/2 = 12 → L
∴ The meaning of the following number = SNAIL
১৩,৮৩৫.
The radius of a wheel is 7 cm. How many revolutions will it make in travelling 88 kilometers?
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
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
১৩,৮৩৬.
A sum of Tk. 25000 amounts to Tk. 31000 in 4 years at the rate of simple interest what is the rate of interest?
ব্যাখ্যা
Question: A sum of Tk. 25000 amounts to Tk. 31000 in 4 years at the rate of simple interest what is the rate of interest?
Solution:
Interest earned in 4years = 31000 - 25000 = 6000
Simple Interest per year = 6000/4
PRT = 1500
⇒ 25000 × R × 1 = 1500
⇒ R = 1500/25000 = 0.06
Thus Simple Interest rate is 6 p.c.p a
Solution:
Interest earned in 4years = 31000 - 25000 = 6000
Simple Interest per year = 6000/4
PRT = 1500
⇒ 25000 × R × 1 = 1500
⇒ R = 1500/25000 = 0.06
Thus Simple Interest rate is 6 p.c.p a
১৩,৮৩৭.
Probability of 3 students solving a question are 1/2, 1/3, and 1/4. Probability to solve the question is:
ব্যাখ্যা
Question: Probability of 3 students solving a question are 1/2, 1/3, and 1/4. Probability to solve the question is:
Solution:
Probability of 3 students,
P(A) = 1/2, ∴ P(A′) = 1/2
P(B) = 1/3, ∴ P(B′) = 2/3
P(C) = 1/4, ∴ P(C′) = 3/4
So, Probability of no one solve the question is = (1/2) × (2/3) × (3/4)
= 1/4
∴ P(None) = 1/4
Then, The probability to solve the question is = 1 - 1/4
= 3/4
Hence, the correct answer is 3/4.
Solution:
Probability of 3 students,
P(A) = 1/2, ∴ P(A′) = 1/2
P(B) = 1/3, ∴ P(B′) = 2/3
P(C) = 1/4, ∴ P(C′) = 3/4
So, Probability of no one solve the question is = (1/2) × (2/3) × (3/4)
= 1/4
∴ P(None) = 1/4
Then, The probability to solve the question is = 1 - 1/4
= 3/4
Hence, the correct answer is 3/4.
১৩,৮৩৮.
Find the greatest number-
ব্যাখ্যা
Question: Find the greatest number-
Solution:
(1/9)2 = 1/81 = 0.012
(√0.3)2 = 0.3
(2/5)2 = (0.4)2 = 0.16
(0.3)2 = 0.09
∴ √0.3 সংখ্যাটি বৃহত্তম।
Solution:
(1/9)2 = 1/81 = 0.012
(√0.3)2 = 0.3
(2/5)2 = (0.4)2 = 0.16
(0.3)2 = 0.09
∴ √0.3 সংখ্যাটি বৃহত্তম।
১৩,৮৩৯.
A man and a boy received Tk. 800 as wages for 5 days for the work they did together. The man's efficiency in the work was thrice times that of the boy. What are the daily wages of the boy?
ব্যাখ্যা
Man = 3 boy
Daily wages for them = 800/5
= 160
4 boy (3 Boy + 1 Boy) = 160
4 boy = 160
Or, boy = Tk. 40
Daily wages for them = 800/5
= 160
4 boy (3 Boy + 1 Boy) = 160
4 boy = 160
Or, boy = Tk. 40
১৩,৮৪০.
A merchant has three different types of oil: 240 litres, 300 litres, and 420 litres. Find the minimum number of containers of equal size that can store the oil without mixing.
ব্যাখ্যা
Question: A merchant has three different types of oil: 240 litres, 300 litres, and 420 litres. Find the minimum number of containers of equal size that can store the oil without mixing.
Solution:
The size of each container should be the greatest possible that divides all three quantities, i.e., H.C.F of 240, 300, and 420.
H.C.F(240, 300, 420) = 60 litres
Total oil = 240 + 300 + 420 = 960 litres
Number of containers required = 960/60 = 16
১৩,৮৪১.
The monthly income of a person was Tk. 13500 and his monthly expenditure was Tk. 9000. Next year, his income increased by 14% and his expenditure by 7%. Find the percentage increase in his savings.
ব্যাখ্যা
Question: The monthly income of a person was Tk. 13500 and his monthly expenditure was Tk. 9000. Next year, his income increased by 14% and his expenditure by 7%. Find the percentage increase in his savings.
Solution:
Savings before increased income = Tk. (13500 - 9000)
= Tk. 4500
Increased income = 114% of 13500 = Tk. (114/100) × 13500
= Tk. 15390
Increased expenditure = 107% of 9000 = Tk. (107/100) × 9000
= Tk. 9630
Saving after Increased income = Tk. (15390 - 9630) = Tk. 5760
Increased Savings = Tk. (5760 - 4500)
= Tk. 1260
∴ Increase % in savings = (1260/4500) × 100 %
= 28%
Solution:
Savings before increased income = Tk. (13500 - 9000)
= Tk. 4500
Increased income = 114% of 13500 = Tk. (114/100) × 13500
= Tk. 15390
Increased expenditure = 107% of 9000 = Tk. (107/100) × 9000
= Tk. 9630
Saving after Increased income = Tk. (15390 - 9630) = Tk. 5760
Increased Savings = Tk. (5760 - 4500)
= Tk. 1260
∴ Increase % in savings = (1260/4500) × 100 %
= 28%
১৩,৮৪২.
The sum of four consecutive even integers is 68. What is the value of the largest number?
ব্যাখ্যা
Let, the numbers be = x, x+2, x+4, x+6
ATQ, x + x + 2 + x + 4 + x + 6 = 68
Or, 4x = 68 - 12 = 56
Or, x = 14
So, the largest number is = 14 + 6 = 20
১৩,৮৪৩.
A bicycle company makes five styles of bikes in seven different colors. How many different bicycles can the company make when considering both style and color?
ব্যাখ্যা
Color of bicycle = 7
Style of bicycle = 5
The company make = 5×7 = 35 different bicycles
Style of bicycle = 5
The company make = 5×7 = 35 different bicycles
১৩,৮৪৪.
A man buys Tk. 50 shares paying 9% dividend. The man wants to have an interest of 12% on his money. The market value of each share is-
ব্যাখ্যা
Question: A man buys Tk. 50 shares paying 9% dividend. The man wants to have an interest of 12% on his money. The market value of each share is-
Solution:
Dividend on Tk. 50 = Tk. (9/100) × 50
= Tk 9/2
Tk.12 is an income of Tk.100.
∴ Tk. 9/2 is an income on = Tk. (100/12) × (9/2)
= Tk. 37.5
Solution:
Dividend on Tk. 50 = Tk. (9/100) × 50
= Tk 9/2
Tk.12 is an income of Tk.100.
∴ Tk. 9/2 is an income on = Tk. (100/12) × (9/2)
= Tk. 37.5
১৩,৮৪৫.
X can complete a task in 12 days and Y in 18 days. If they work together for 3 days, what fraction of work is left?
ব্যাখ্যা
Question: X can complete a task in 12 days and Y in 18 days. If they work together for 3 days, what fraction of work is left?
Solution:
The work rate of X is 1/12 (since X can complete the work in 12 days).
The work rate of Y is 1/18 (since Y can complete the work in 18 days).
In 3 days they complete = 3{(1/12) + (1/18)}
= 3 × (5/36)
= 5/12
Then the fraction of work that is left is = 1 - (5/12)
= (12 - 5)/12
= 7/12
Solution:
The work rate of X is 1/12 (since X can complete the work in 12 days).
The work rate of Y is 1/18 (since Y can complete the work in 18 days).
In 3 days they complete = 3{(1/12) + (1/18)}
= 3 × (5/36)
= 5/12
Then the fraction of work that is left is = 1 - (5/12)
= (12 - 5)/12
= 7/12
১৩,৮৪৬.
If a student of the mathematics department sells all his books for Tk. 7,500, he loses 25%. To gain 10%, at what price should he sell the books?
ব্যাখ্যা
Question: If a student of the mathematics department sells all his books for Tk. 7,500, he loses 25%. To gain 10%, at what price should he sell the books?
Solution:
Let,
The cost price of the books x.
Selling price = x - 25% of x
⇒ 7500 = x - (25x/100)
⇒ 7500 = 75x/100
⇒ 75x = 750000
∴ x = 10000
To gain 10% = 10000 + 10% of 10000
= 10000 + 1000
= Tk. 11000
To gain 10%, he should sell the books for Tk. 11000.
১৩,৮৪৭.
A container holds 4 quarts of chemical and 4 quarts of water. How many quarts of water must be added to the container to create a mixture that is 3 parts chemical to 5 parts water by volume?
ব্যাখ্যা
ধরি,
x quants পানি যোগ করতে হবে।
প্রশ্নমতে
4/(4 + x) = 3/5
12 + 3x = 20
3x = 8
x = 8/3
x quants পানি যোগ করতে হবে।
প্রশ্নমতে
4/(4 + x) = 3/5
12 + 3x = 20
3x = 8
x = 8/3
১৩,৮৪৮.
The average of runs of a cricket player of 10 innings was 32. How many runs must he make in his next innings so as to increase his average of runs by 4?
ব্যাখ্যা
Question: The average of runs of a cricket player of 10 innings was 32. How many runs must he make in his next innings so as to increase his average of runs by 4?
Solution:
Average = 32
So, total = 32 × 10 = 320.
Now increase in average = 4
So, new average = 32 + 4 = 36
Total runs after next innings = 36 × 11 = 396
∴ Runs made in the 11th inning = 396 - 320 = 76
১৩,৮৪৯.
A tank with a length of 3m and width of 2m can store 9000 liters. The height of the tank is?
ব্যাখ্যা
Question: A tank with a length of 3m and width of 2m can store 9000 liters. The height of the tank is?
Solution:
Let,
the height of the tank is h
here,
length, l = 3m
width, b = 2m
volume, V = l × b × h
= 3 × 2 × h
= 6h m3
as, 1m3 is equal 1000 liters
∴ (6h × 1000) = 9000
6h = 9
h = 1.5m
Solution:
Let,
the height of the tank is h
here,
length, l = 3m
width, b = 2m
volume, V = l × b × h
= 3 × 2 × h
= 6h m3
as, 1m3 is equal 1000 liters
∴ (6h × 1000) = 9000
6h = 9
h = 1.5m
১৩,৮৫০.
If readymade shirts need alterations 2 out of 3 in sleeves, 3 out of 4 in collar and 4 out of 5 in the body, how many alterations will be required for 60 shirts?
ব্যাখ্যা
প্রশ্ন: If readymade shirts need alterations 2 out of 3 in sleeves, 3 out of 4 in collar and 4 out of 5 in the body, how many alterations will be required for 60 shirts?
সমাধান:
2/3th of 60 + 3/4th of 60 + 4/5th of 60
Thus, the number of alterations required = 40 + 45 + 48 = 133
সমাধান:
2/3th of 60 + 3/4th of 60 + 4/5th of 60
Thus, the number of alterations required = 40 + 45 + 48 = 133
১৩,৮৫১.
A contractor undertakes to do a piece of work in 40 days. He engages 100 men at the beginning and 100 more after 35 days and completes the work in stipulated time. If he had not engaged the additional men, how many days behind schedule would it be finished?
ব্যাখ্যা
[(100 × 35) + (200 × 5)]men can finish the work in 1 day.
Therefore, 4500 men can finish the work in 1 day. 100 men can finish it in = 4500/100 = 45 days.
This is 5 days behind Schedule
Therefore, 4500 men can finish the work in 1 day. 100 men can finish it in = 4500/100 = 45 days.
This is 5 days behind Schedule
১৩,৮৫২.
An alloy of copper and nickel contains 65% copper. A second alloy contains copper and nickel in the ratio 17 : 3. In what ratio should the two alloys be mixed so that the new mixture contains 4 times as much copper as nickel?
ব্যাখ্যা
Question: An alloy of copper and nickel contains 65% copper. A second alloy contains copper and nickel in the ratio 17 : 3. In what ratio should the two alloys be mixed so that the new mixture contains 4 times as much copper as nickel?
Solution:
First alloy - 65% copper. ∴ 35% Nickel.
Ratio is 65 : 35 = 13 : 7.
2nd alloy,
Ratio is 17 : 3
∴ Proportion of copper in the 1st alloy = 13/20 and
proportion of copper in the 2nd alloy = 17/20.
Also proportion of copper in resulting alloy = 4/5.
Hence required ratio is (17/20 - 4/5) : (4/5 – 13/20) = 1 : 3.
Solution:
First alloy - 65% copper. ∴ 35% Nickel.
Ratio is 65 : 35 = 13 : 7.
2nd alloy,
Ratio is 17 : 3
∴ Proportion of copper in the 1st alloy = 13/20 and
proportion of copper in the 2nd alloy = 17/20.
Also proportion of copper in resulting alloy = 4/5.
Hence required ratio is (17/20 - 4/5) : (4/5 – 13/20) = 1 : 3.
১৩,৮৫৩.
Rafi and Shafi can paint a room in 4 days. Shafi can do it alone in 6 days. How many days will Rafi take to paint the room alone?
ব্যাখ্যা
Question: Rafi and Shafi can paint a room in 4 days. Shafi can do it alone in 6 days. How many days will Rafi take to paint the room alone?
Solution:
ধরা যাক,
Rafi একা কাজটি করতে x দিন সময় নেন।
∴ Rafi-র 1 দিনের কাজ = 1/x
∴ Shafi-র 1 দিনের কাজ = 1/6 (কারণ সে একা ৬ দিনে কাজটি করে)
তারা দুজনে মিলে ৪ দিনে কাজ শেষ করে।
∴ তাদের 1 দিনের একত্রে কাজ = 1/4
এখন,
(1/x) + (1/6) = 1/4
⇒ (6 + x) / (6x) = 1/4
⇒ 4(6 + x) = 6x
⇒ 24 + 4x = 6x
⇒ 24 = 6x - 4x = 2x
⇒ x = 24 / 2 = 12
∴ Rafi একা কাজটি করতে 12 দিন সময় নেবে।
১৩,৮৫৪.
If logx (1/√32) = - 5/2 what is the value of x?
ব্যাখ্যা
Question: If logx (1/√32) = - 5/2 what is the value of x?
সমাধান:
দেওয়া আছে,
logx (1/√32) = - 5/2
⇒ x- 5/2 = 1/√32 [logaM = x হলে, ax = M হয়]
⇒ x- 5/2 = 1/(321/2)
⇒ x- 5/2 = 32- 1/2
⇒ x- 5/2 = (25)- 1/2
⇒ x- 5/2 = 2- 5/2
∴ x = 2
১৩,৮৫৫.
A train travels 180 km in 3 hours. What is its average speed?
ব্যাখ্যা
Question: A train travels 180 km in 3 hours. What is its average speed?
Solution:
ট্রেনটির অতিক্রান্ত দূরত্ব = 180 কি.মি.
মোট সময় = 3 ঘণ্টা
গড় গতিবেগ = 180/3 km/h
= 60 km/h
Solution:
ট্রেনটির অতিক্রান্ত দূরত্ব = 180 কি.মি.
মোট সময় = 3 ঘণ্টা
গড় গতিবেগ = 180/3 km/h
= 60 km/h
১৩,৮৫৬.
The cost of a book was Tk.75. The cost was first increased by 20% and later on it was reduced by 20%. The present cost of the book is:
ব্যাখ্যা
Question: The cost of a book was Tk.75. The cost was first increased by 20% and later on it was reduced by 20%. The present cost of the book is:
Solution:
Solution:
Here, the initial Cost = Tk. 75
After 20% increase in the cost, it becomes,
(75 + 20% of 75) = Tk. 90
Now, the Cost is decreased by 20%, So the cost will become,
(90 - 20% of 90) = Tk. 72
Hence, the present cost is Rs. 72
১৩,৮৫৭.
A train 220 m long is moving at 45km/h. The time taken by the train to cross a tunnel 260m long is-
ব্যাখ্যা
নির্ণেয় সময় = (220+260) / 45×(5/18)
= (480×18) / (45×5)
= 38.4 সেকেন্ড
সঠিক উত্তর হবে 38.4 সেকেন্ড কিন্তু অপশনে এই মান না থাকলেও কাছাকাছি মান আছে 38। তাই উত্তর হবে 38 সেকেন্ড।
১৩,৮৫৮.
A motorboat, whose speed is 15 km/hr in still water goes 30 km downstream and comes back in a total of 4 hours 30 minutes. The speed of the stream (in km/hr) is:
ব্যাখ্যা
Let the speed of the stream be x km/hr. Then,
Speed downstream = (15 + x) km/hr,
Speed upstream = (15 - x) km/hr
30/(15 + x) +30/(15 - x) = 4(1/2)
900/(225 - X2) = 9/2
9X2 = 225
X2 = 25
X = 5 km/hr
১৩,৮৫৯.
A starts a business with Tk. 3800 and after 4 months, B joins with some capital. At the end of the year, the profit is divided in the ratio 3 : 4. What is B's contribution to the business?
ব্যাখ্যা
Question: A starts a business with Tk. 3800 and after 4 months, B joins with some capital. At the end of the year, the profit is divided in the ratio 3 : 4. What is B's contribution to the business?
Solution:
Let the Contribution of B is Tk. x
A invested for 12 months and B invested for (12 - 4) = 8 months
Now
⇒ (3800 × 12)/(x × 8) = 3/4
⇒ (3800 × 12 × 4)/(3 × 8)= x
⇒ x = 7600
∴ Contribution of B to the business is 7600
Solution:
Let the Contribution of B is Tk. x
A invested for 12 months and B invested for (12 - 4) = 8 months
Now
⇒ (3800 × 12)/(x × 8) = 3/4
⇒ (3800 × 12 × 4)/(3 × 8)= x
⇒ x = 7600
∴ Contribution of B to the business is 7600
১৩,৮৬০.
If selling price is doubled, the profit fourples. Find the profit percent?
ব্যাখ্যা
Question: If selling price is doubled, the profit fourples. Find the profit percent?
Solution:
Let,
The cost price be Tk.100 and sell price be Tk .x,
Then
The profit is (x - 100)
Now The sell price is doubled, then the new sell price is 2x
New profit is (2x - 100)
ATQ,
4(x - 100) = 2x - 100
or, 4x - 400 = 2x - 100
or, 4x - 2x = 400 - 100
∴ x = 150
Then the Profit = (150 - 100) = 50
Hence the profit percentage is = (50 × 100)/100 %
= 50%
Solution:
Let,
The cost price be Tk.100 and sell price be Tk .x,
Then
The profit is (x - 100)
Now The sell price is doubled, then the new sell price is 2x
New profit is (2x - 100)
ATQ,
4(x - 100) = 2x - 100
or, 4x - 400 = 2x - 100
or, 4x - 2x = 400 - 100
∴ x = 150
Then the Profit = (150 - 100) = 50
Hence the profit percentage is = (50 × 100)/100 %
= 50%
১৩,৮৬১.
The maximum number of students among whom 1001 pens and 910 pencils can be distributed in such a way that each student gets the same number of pens and same number of pencils is-
ব্যাখ্যা
Question: The maximum number of students among whom 1001 pens and 910 pencils can be distributed in such a way that each student gets the same number of pens and same number of pencils is-
Solution:
If each student gets the same number of pens and same number of pencils, the maximum number of students will be the HCF of the total numbers.
∴ Maximum number of students = HCF(1001, 910)
Now,
Factorize the numbers,
1001 = 7 × 11 × 13 and 910 = 2 × 5 × 7 × 13
Common factors are 7 and 13
∴ Multiply them = 7 × 13 = 91
So the maximum number of students = 91
১৩,৮৬২.
In how many ways can 6 examination papers be arranged so that the best and the worst papers never come together?
ব্যাখ্যা
Question: In how many ways can 6 examination papers be arranged so that the best and the worst papers never come together?
Solution:
Total ways = 6!
= 720 ways
if two papers come together, we can consider them one.
ways that they will come together = 5! × 2!
= 120 × 2
= 240 ways
∴ ways the best and the worst papers never come together = 720 - 240 ways
= 480 ways
Solution:
Total ways = 6!
= 720 ways
if two papers come together, we can consider them one.
ways that they will come together = 5! × 2!
= 120 × 2
= 240 ways
∴ ways the best and the worst papers never come together = 720 - 240 ways
= 480 ways
১৩,৮৬৩.
Three runners A, B and C run a race, with runner A finishing 12 metres ahead of runner B and 18 metres ahead of runner C, in another race of same type runner B finished 8 metres ahead of runner C. Each runner travels the entire distance at a constant speed. The length of the race is:
ব্যাখ্যা
Question: Three runners A, B and C run a race, with runner A finishing 12 metres ahead of runner B and 18 metres ahead of runner C, in another race of same type runner B finished 8 metres ahead of runner C. Each runner travels the entire distance at a constant speed. The length of the race is:
Solution:
Let A finish the race of x metres
B finish the race of = x - 12 meters.
C finish the race of = x - 18 - - - - - - - - (1)
In another race of B & C
B finish race of = x meters.
C finish race of = x - 8 - - - - - - - - (2)
Ratio of speeds of B & C
(x - 12)/(x - 18) = x/(x - 8)
⇒ (x - 12)(x - 8) = x(x -18)
⇒ x2 - 12x - 8x + 96 = x2 - 18x
⇒ x2 - 20x + 96 = x2 - 18x
⇒ 96 = 2x
∴ x = 48 metres.
∴ The length of the race is 48 meters.
Solution:
Let A finish the race of x metres
B finish the race of = x - 12 meters.
C finish the race of = x - 18 - - - - - - - - (1)
In another race of B & C
B finish race of = x meters.
C finish race of = x - 8 - - - - - - - - (2)
Ratio of speeds of B & C
(x - 12)/(x - 18) = x/(x - 8)
⇒ (x - 12)(x - 8) = x(x -18)
⇒ x2 - 12x - 8x + 96 = x2 - 18x
⇒ x2 - 20x + 96 = x2 - 18x
⇒ 96 = 2x
∴ x = 48 metres.
∴ The length of the race is 48 meters.
১৩,৮৬৪.
Pens that used to cost Tk. 150 for a package of 3 now cost Tk. 300 for a package of 5. What is the percent increase in the price of these pens?
ব্যাখ্যা
Question: Pens that used to cost Tk. 150 for a package of 3 now cost Tk. 300 for a package of 5. What is the percent increase in the price of these pens?
Solution:
Original cost: Tk. 150 for a package of 3 pens.
Cost per pen originally: Original cost per pen = 150/3 = Tk. 50 per pen
New cost: Tk. 300 for a package of 5 pens.
Cost per pen now: New cost per pen=300/5 = Tk. 60 per pen
Increase in cost = 60 - 50 = Tk. 10 per pen
Percent increase = (10/50) × 100 = 20%
Solution:
Original cost: Tk. 150 for a package of 3 pens.
Cost per pen originally: Original cost per pen = 150/3 = Tk. 50 per pen
New cost: Tk. 300 for a package of 5 pens.
Cost per pen now: New cost per pen=300/5 = Tk. 60 per pen
Increase in cost = 60 - 50 = Tk. 10 per pen
Percent increase = (10/50) × 100 = 20%
১৩,৮৬৫.
If the product of two positive number is 1575 and the ratio is 7:9 then the greatest number is:
ব্যাখ্যা
Let the numbers are 7x and 9x
According to the question,
⇒ 7x × 9x = 1575
⇒ 63x2 = 1575
⇒ x2 = 25
⇒ x = 5
Then greater number
= 9x
= 9 × 5
= 45
১৩,৮৬৬.
If x = √3 + √2, then the value of x + (1/x) is?
ব্যাখ্যা
Question: If x = √3 + √2, then the value of {x + (1/x)} is?
Solution:
Solution:
১৩,৮৬৭.
The students in Parimal’s class walk, cycle or drive to school in the ratio 2 : 1 : 4. If 8 students walk, how many students are there in Parimal’s class altogether?
ব্যাখ্যা
Question: The students in Parimal’s class walk, cycle or drive to school in the ratio 2 : 1 : 4. If 8 students walk, how many students are there in Parimal’s class altogether?
Solution:
Let,
Number of students walk = 2x
Number of student having cycle = x
Number of student driving = 4x
∴ Total number of students = 2x + x + 4x = 7x
ATQ,
2x = 8
∴ x = 4
∴ Number of students = 7 × 4 = 28
Solution:
Let,
Number of students walk = 2x
Number of student having cycle = x
Number of student driving = 4x
∴ Total number of students = 2x + x + 4x = 7x
ATQ,
2x = 8
∴ x = 4
∴ Number of students = 7 × 4 = 28
১৩,৮৬৮.
If tan53° = 4/3, then, what is the value of tan8°?
ব্যাখ্যা
Question: If tan53° = 4/3, then, what is the value of tan8°?
Solution:
We know,
tan(A - B) = (tanA - tanB)/(1 + tanAtanB)
Now,
tan8° = tan(53° - 45°)
⇒ tan8° = (tan53° - tan45°)/(1 + tan53° tan45°)
⇒ tan8° = (4/3 - 1)/{1 + (4/3) × 1}
⇒ tan8° = (1/3)/(7/3)
∴ tan8° = 1/7
Solution:
We know,
tan(A - B) = (tanA - tanB)/(1 + tanAtanB)
Now,
tan8° = tan(53° - 45°)
⇒ tan8° = (tan53° - tan45°)/(1 + tan53° tan45°)
⇒ tan8° = (4/3 - 1)/{1 + (4/3) × 1}
⇒ tan8° = (1/3)/(7/3)
∴ tan8° = 1/7
১৩,৮৬৯.
What is the square root of 0.16?
ব্যাখ্যা
Question: What is the square root of 0.16?
Solution:
√0.16
= √(16/100)
= 4/10
= 0.4
Solution:
√0.16
= √(16/100)
= 4/10
= 0.4
১৩,৮৭০.
A shopkeeper marks an item at Tk 1500 and gives a 20% discount. If he still makes a 25% profit, what was the cost price?
ব্যাখ্যা
Question: A shopkeeper marks an item at Tk 1500 and gives a 20% discount. If he still makes a 25% profit, what was the cost price?
Solution:
Selling price after 20% discount
= 1500 - 20% of 1500
= 1500 - {(20/100) × 1500}
= 1500 - 300
= 1200
Let the cost price = x
Profit = 25%
Now,
⇒ x + 25% of x = 1200
⇒ x + (25/100)x = 1200
⇒ x + (x/4) = 1200
⇒ 5x = 4800
⇒ x = 4800/5
∴ x = 960
So, the cost price of that item is Tk. 960
Solution:
Selling price after 20% discount
= 1500 - 20% of 1500
= 1500 - {(20/100) × 1500}
= 1500 - 300
= 1200
Let the cost price = x
Profit = 25%
Now,
⇒ x + 25% of x = 1200
⇒ x + (25/100)x = 1200
⇒ x + (x/4) = 1200
⇒ 5x = 4800
⇒ x = 4800/5
∴ x = 960
So, the cost price of that item is Tk. 960
১৩,৮৭১.
A number is 5 more than its one-sixth part. The number is-
ব্যাখ্যা
Question: A number is 5 more than its one-sixth part. The number is-
Solution:
Given that,
A number is 5 more than its one-sixth part.
Let the number be x.
According to the question,
x = (1/6)x + 5
⇒ x - (1/6)x = 5
⇒ (6x - x)/6 = 5
⇒ 5x/6 = 5
⇒ x = 5 × (6/5)
∴ x = 6
So the number is 6.
১৩,৮৭২.
ব্যাখ্যা
Question:
Solution:
দেয়া আছে,
- ট্রেস(Trace): Matrix এর trace হলো একটি বর্গাকার ম্যাট্রিক্সের প্রধান কর্ণের উপাদানগুলোর যোগফল।
- প্রধান কর্ণ হলো ম্যাট্রিক্সের উপরের বাম কোণ থেকে নিচের ডান কোণ পর্যন্ত উপাদানগুলো।
প্রদত্ত ম্যাট্রিক্স A এর প্রধান কর্ণের উপাদানগুলো হলো 5, 2, 3
তাহলে, প্রদত্ত ম্যাট্রিক্সের ট্রেস (Trace) B = 5 + 2 + 3 = 10
১৩,৮৭৩.
A 10% stock yields 8%. The market value of the stock is-
ব্যাখ্যা
Question: A 10% stock yields 8%. The market value of the stock is-
Solution:
For an income of Tk. 8, investment = Tk. 100.
For an income of Tk. 10, investment = Tk. (100/8) × 10
= Tk. 125
Solution:
For an income of Tk. 8, investment = Tk. 100.
For an income of Tk. 10, investment = Tk. (100/8) × 10
= Tk. 125
১৩,৮৭৪.
y2 - 15y + 56 = 0, then y = ?
ব্যাখ্যা
y2 - 15y + 56 = 0
y2 - 7y - 8y + 56 = 0
y(y - 7) - 8(y - 7) = 0
(y - 7)(y - 8) = 0
হয়
y - 7 = 0
y = 7
অথবা
y - 8 = 0
y = 8
y2 - 7y - 8y + 56 = 0
y(y - 7) - 8(y - 7) = 0
(y - 7)(y - 8) = 0
হয়
y - 7 = 0
y = 7
অথবা
y - 8 = 0
y = 8
১৩,৮৭৫.
A train 300 metres long is running at a speed of 25 metre per second. It will cross a bridge of 200 metres long in:
ব্যাখ্যা
Question: A train 300 metres long is running at a speed of 25 metre per second. It will cross a bridge of 200 metres long in:
Solution:
Speed of train = 25 m/s.
Total distance covered while passing bridge = 300 + 200 = 500 m
∴ Time = 500/25
= 20 seconds.
Solution:
Speed of train = 25 m/s.
Total distance covered while passing bridge = 300 + 200 = 500 m
∴ Time = 500/25
= 20 seconds.
১৩,৮৭৬.
The size of the wooden block is 5 x 10 x 20 cm. How many such blocks will be required to construct a solid wooden cube of minimum size?
ব্যাখ্যা
Side of smallest cube = L.C.M of 5, 10, 20 = 20cm
Volume of the cube = (20 x 20 x 20) cm3 = 8000 cm3
Volume of the block= (5 x 10 x 20) cm3 = 1000 cm3
∴ Required number of blocks = (8000/1000) = 8
১৩,৮৭৭.
A can contains a mixture of two liquids A and B in the ratio 5 : 3. When 12 litres of mixture are drawn off and the can is filled with B, the ratio of A and B becomes 5 : 8. How many litres of liquid A was contained by the can initially?
ব্যাখ্যা
Question: A can contains a mixture of two liquids A and B in the ratio 5 : 3. When 12 litres of mixture are drawn off and the can is filled with B, the ratio of A and B becomes 5 : 8. How many litres of liquid A was contained by the can initially?
Solution:
Let liquids A is initially 5x, liquid B is 3x litres
12 litres of mixture are drawn, remaining mixture A = 5x - (5/8) × 12 = 5x - 7.5
Remaining mixture B = 3x - (3/8) × 12 = 3x - 4.5
After filling, mixture becomes = 3x - 4.5 + 12 = 3x + 7.5
5x - 7.5/3x + 7.5 = 5/8
⇒ (40x - 60)/(24x + 60) = 5/8
⇒ 320x - 480 = 120x + 300
⇒ 200x = 780
⇒ x = 3.9
Liquid A is (5 × 3.9) or 19.5 litres
Solution:
Let liquids A is initially 5x, liquid B is 3x litres
12 litres of mixture are drawn, remaining mixture A = 5x - (5/8) × 12 = 5x - 7.5
Remaining mixture B = 3x - (3/8) × 12 = 3x - 4.5
After filling, mixture becomes = 3x - 4.5 + 12 = 3x + 7.5
5x - 7.5/3x + 7.5 = 5/8
⇒ (40x - 60)/(24x + 60) = 5/8
⇒ 320x - 480 = 120x + 300
⇒ 200x = 780
⇒ x = 3.9
Liquid A is (5 × 3.9) or 19.5 litres
১৩,৮৭৮.
A sum of Tk. 10,000 amounts to Tk. 12,000 in 4 years at the rate of simple interest. What is the rate of interest?
ব্যাখ্যা
Question: A sum of Tk. 10,000 amounts to Tk. 12,000 in 4 years at the rate of simple interest. What is the rate of interest?
Solution:
Interest = 12000 - 10000 = Tk. 2000
Let, interest rate = r%
We know,
I = Pnr
⇒ 2000 = 10000 × 4 × r/100
⇒ r = 5
Interest rate 5%
Solution:
Interest = 12000 - 10000 = Tk. 2000
Let, interest rate = r%
We know,
I = Pnr
⇒ 2000 = 10000 × 4 × r/100
⇒ r = 5
Interest rate 5%
১৩,৮৭৯.
If |x + 2| ≤ 6, then which of the following intervals represents all possible values of the expression 3x - 4 ?
ব্যাখ্যা
Question: If |x + 2| ≤ 6, then which of the following intervals represents all possible values of the expression 3x - 4 ?
Solution:
|x + 2| ≤ 6
⇒ - 6 ≤ x + 2 ≤ 6
⇒ - 6 - 2 ≤ x + 2 - 2 ≤ 6 - 2
⇒ - 8 ≤ x ≤ 4
⇒ - 24 ≤ 3x ≤ 12
⇒ - 24 - 4 ≤ 3x - 4 ≤ 12 - 4
⇒ - 28 ≤ 3x - 4 ≤ 8
∴ All possible values of 3x - 4 lie in the interval [- 28, 8].
১৩,৮৮০.
The average age of any and Any's twin two sisters is 8 years. The age of each of twin two sisters is 6 years. What is the any's age?
ব্যাখ্যা
The total age of twin two sisters 6 × 2 = 12 years
The total age of Any and Any's twin two sisters is 8 × 3 = 24 years.
Any's age = (24 - 12) years = 12 years
--------------------------------------------------------
এনি ও তার যমজ দুই বোনের গড় বয়স ৮ বছর। যমজ দুই বোনের প্রত্যেকের বয়স ৬ বছর হলে, এনির বয়স কত?
যমজ দুই বোনের প্রত্যেকের বয়স ৬ বছর হলে, তাদের মোট বয়স ৬ × ২ বছর = ১২ বছর।
এনি ও তার যমজ দুই বোনের গড় বয়স ৮ বছর হলে, তাদের মোট বয়স ৮ × ৩ বছর = ২৪ বছর
এনির বয়স (২৪ - ১২) বছর = ১২ বছর
The total age of Any and Any's twin two sisters is 8 × 3 = 24 years.
Any's age = (24 - 12) years = 12 years
--------------------------------------------------------
এনি ও তার যমজ দুই বোনের গড় বয়স ৮ বছর। যমজ দুই বোনের প্রত্যেকের বয়স ৬ বছর হলে, এনির বয়স কত?
যমজ দুই বোনের প্রত্যেকের বয়স ৬ বছর হলে, তাদের মোট বয়স ৬ × ২ বছর = ১২ বছর।
এনি ও তার যমজ দুই বোনের গড় বয়স ৮ বছর হলে, তাদের মোট বয়স ৮ × ৩ বছর = ২৪ বছর
এনির বয়স (২৪ - ১২) বছর = ১২ বছর
১৩,৮৮১.
২২৩৬ টাকা A, B ও C এর মধ্যে এমনভাবে ভাগ করে দেওয়া হলো যেন A, C থেকে ২৫% বেশি এবং C, B থেকে ২৫% কম পায়। তাহলে A কত টাকা পাবে?
ব্যাখ্যা
প্রশ্ন: ২২৩৬ টাকা A, B ও C এর মধ্যে এমনভাবে ভাগ করে দেওয়া হলো যেন A, C থেকে ২৫% বেশি এবং C, B থেকে ২৫% কম পায়। তাহলে A কত টাকা পাবে?
সমাধান:
মনে করি, A পায় = x টাকা
যেহেতু, A এর টাকা = C এর ১২৫% টাকা
∴ C পায় = {(x/১২৫) × ১০০} = ৪x/৫ টাকা
আবার, C পায় = B এর (১০০ - ২৫)%
⇒ ৪x/৫ = B এর ৭৫%
⇒ ৪x/৫ = B × (৭৫/১০০)
⇒ B = {(৪x/৫) × (১০০/৭৫)}
⇒ B = (৮০x/৭৫) = (১৬x/১৫) টাকা
প্রশ্নমতে,
x + (৪x/৫) + (১৬x/১৫) = ২২৩৬
⇒ (৯x/৫) + (১৬x/১৫) = ২২৩৬
⇒ (২৭x + ১৬x)/১৫ = ২২৩৬
⇒ ৪৩x/১৫ = ২২৩৬
⇒ x = (২২৩৬ × ১৫)/৪৩
∴ x = ৭৮০
সুতরাং, A পায় ৭৮০ টাকা।
সমাধান:
মনে করি, A পায় = x টাকা
যেহেতু, A এর টাকা = C এর ১২৫% টাকা
∴ C পায় = {(x/১২৫) × ১০০} = ৪x/৫ টাকা
আবার, C পায় = B এর (১০০ - ২৫)%
⇒ ৪x/৫ = B এর ৭৫%
⇒ ৪x/৫ = B × (৭৫/১০০)
⇒ B = {(৪x/৫) × (১০০/৭৫)}
⇒ B = (৮০x/৭৫) = (১৬x/১৫) টাকা
প্রশ্নমতে,
x + (৪x/৫) + (১৬x/১৫) = ২২৩৬
⇒ (৯x/৫) + (১৬x/১৫) = ২২৩৬
⇒ (২৭x + ১৬x)/১৫ = ২২৩৬
⇒ ৪৩x/১৫ = ২২৩৬
⇒ x = (২২৩৬ × ১৫)/৪৩
∴ x = ৭৮০
সুতরাং, A পায় ৭৮০ টাকা।
১৩,৮৮২.
A man and a boy can finish a job working together for 6 days. The man can work as much as two boys can do. Then in how many days can the man do it alone?
ব্যাখ্যা
Question: A man and a boy can finish a job working together for 6 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 1/6 part
2 man and 2 boy can do in 1 day 2/6 part = 1/3 part
Now 1 man = 2 boys
(2 + 1) man can do in 1 day is 1/3 part
3 man can do in 1 day is 1/3 part
1 man can do in 1 day is 1/(3 × 3) part = 1/9 part
∴ Man alone can do the work in 9 days
Solution:
1 man and 1 boy can do in 1 day 1/6 part
2 man and 2 boy can do in 1 day 2/6 part = 1/3 part
Now 1 man = 2 boys
(2 + 1) man can do in 1 day is 1/3 part
3 man can do in 1 day is 1/3 part
1 man can do in 1 day is 1/(3 × 3) part = 1/9 part
∴ Man alone can do the work in 9 days
১৩,৮৮৩.
A man walk a certain distance and rides back in 4 hours 30 minutes. He could ride both ways in 3 hours. The time required by the man to walk both ways is :
ব্যাখ্যা
Time taken to ride one way = 3/2 = 1.5 hrs
Time taken to walk one way:
= 4.5 - 1.5
= 3 hrs
Time taken to walk both way :
= 3×2
= 6 hours
১৩,৮৮৪.
Fahim rows a boat at 8 km/h in still water. He finds that it takes him 2 hours longer to row upstream than downstream for a certain distance. If the speed of the stream is s km/h, and the distance is 30 km, find s.
ব্যাখ্যা
Question: Fahim rows a boat at 8 km/h in still water. He finds that it takes him 2 hours longer to row upstream than downstream for a certain distance. If the speed of the stream is s km/h, and the distance is 30 km, find s.
সমাধান:
স্থির পানিতে নৌকার গতিবেগ = 8 কিমি/ঘন্টা
স্রোতের গতিবেগ = s কিমি/ঘন্টা
মোট দূরত্ব = 30 কিমি
স্রোতের অনুকূলে গতিবেগ (Downstream) = (8 + s) কিমি/ঘন্টা
স্রোতের প্রতিকূলে গতিবেগ (Upstream) = (8 - s) কিমি/ঘন্টা
স্রোতের অনুকূলে সময় = দূরত্ব/গতিবেগ = 30/(8 + s) ঘন্টা
স্রোতের প্রতিকূলে সময় = দূরত্ব/গতিবেগ = 30 / (8 - s) ঘন্টা
স্রোতের প্রতিকূলে যেতে 2 ঘন্টা বেশি সময় লাগে।
প্রশ্নমতে,
{30/(8 - s)} - {30/(8 + s)} = 2
⇒ 30[{(1/(8 - s)} - {1/(8 + s)}] = 2
⇒ {1/(8 - s)} - {1/(8 + s)} = 2/30
⇒{(8 + s) - (8 - s)}/{(8 - s)(8 + s)} = 1/15
⇒ 2s /(64 - s2) = 1/15
⇒ 30s = 64 - s2
⇒ s2 + 30s - 64 = 0
⇒ s2 + 32s - 2s - 64 = 0
⇒ s(s + 32) - 2(s + 32) = 0
⇒ (s - 2)(s + 32) = 0
সুতরাং, s = 2 অথবা s = - 32
যেহেতু স্রোতের গতিবেগ ঋণাত্মক হতে পারে না, তাই গ্রহণযোগ্য মান হলো s = 2 কিমি/ঘন্টা।
সুতরাং, স্রোতের গতিবেগ হলো 2 কিমি/ঘন্টা।
১৩,৮৮৫.
Ellmi bought 5 pints of frozen yogurt and a tray of jumbo shrimp from The Food Place for a total of Tk. 4500. If the price of a tray of jumbo shrimp is Tk. 2500, what is the price of a pint of frozen yogurt?
ব্যাখ্যা
Question: Ellmi bought 5 pints of frozen yogurt and a tray of jumbo shrimp from The Food Place for a total of Tk. 4500. If the price of a tray of jumbo shrimp is Tk. 2500, what is the price of a pint of frozen yogurt?
Solution:
Ellmi bought 5 pints of frozen yogurt and a shrimp tray.
We can use x to represent price of a pint of frozen yogurt.
The price of 5 pints of frozen yogurt, then, is 5x
The price of a shrimp tray is 2500
Ellmi spent a total of Tk. 4500, which is the sum of 5x and 2500. We can now write an equation and solve it:
5x + 2500 = 4500
⇒ 5x = 2000
∴ x = 400
Solution:
Ellmi bought 5 pints of frozen yogurt and a shrimp tray.
We can use x to represent price of a pint of frozen yogurt.
The price of 5 pints of frozen yogurt, then, is 5x
The price of a shrimp tray is 2500
Ellmi spent a total of Tk. 4500, which is the sum of 5x and 2500. We can now write an equation and solve it:
5x + 2500 = 4500
⇒ 5x = 2000
∴ x = 400
১৩,৮৮৬.
If logx(1/8) = - 3, then x = ?
ব্যাখ্যা
logx(1/8) = - 3
⇒ x- 3 = 1/8
⇒ x- 3 = (1/2)3
⇒ x- 3 = 2- 3
⇒ x = 2
⇒ x- 3 = 1/8
⇒ x- 3 = (1/2)3
⇒ x- 3 = 2- 3
⇒ x = 2
১৩,৮৮৭.
The next term of the series 9 + 36 + 81 + 144 + ____ is
ব্যাখ্যা
Given, 9 + 36 + 81 + 144
The series is = 32 + 62 + 92 + 122 + 152
So, next term is 152 = 225
১৩,৮৮৮.
The 2nd term of a geometric sequence is 4/3, and the 5th term is 32/81. What is the common ratio?
ব্যাখ্যা
Question: The 2nd term of a geometric sequence is 4/3, and the 5th term is 32/81. What is the common ratio?
Solution:
আমরা জানি, একটি গুণোত্তর ধারার n-তম পদ, an = a.rn - 1
দেওয়া আছে,
2য় পদ, a2 = 4/3
⇒ ar = 4/3 …...(1)
5ম পদ, a5 = 32/81
⇒ ar4 = 32/81 …....(2)
এখন, সমীকরণ (2) ÷ সমীকরণ (1) ⇒
(ar4)/(ar) = (32/81)/(4/3)
⇒ r3 = (32/81) × (3/4)
⇒ r3 = 8/27
⇒ r3 = (2/3)3
⇒ r = 2/3
∴ সাধারণ অনুপাত (common ratio) হলো 2/3
১৩,৮৮৯.
An article is sold for Tk. 3600 at a profit of 25%. What would have been the actual profit or loss if it had been sold at Tk. 2700?
ব্যাখ্যা
Question: An article is sold for Tk. 3600 at a profit of 25%. What would have been the actual profit or loss if it had been sold at Tk. 2700?
Solution:
The cost price = (3600 × 100)/125 = 2880
New selling price = Tk. 2700
∴ Loss = 2880 - 2700 = 180
∴ Loss percentage = (100 × 180)/2880 = 6.25%.
Solution:
The cost price = (3600 × 100)/125 = 2880
New selling price = Tk. 2700
∴ Loss = 2880 - 2700 = 180
∴ Loss percentage = (100 × 180)/2880 = 6.25%.
১৩,৮৯০.
If a + 1/a = 7, what is a3 + 1/a3?
ব্যাখ্যা
a3 + 1/a3
= (a + 1/a)3 - 3.a.1/a(a + 1/a)
= 73 - 3 × 7
= 343 - 21
= 322
= (a + 1/a)3 - 3.a.1/a(a + 1/a)
= 73 - 3 × 7
= 343 - 21
= 322
১৩,৮৯১.
The total of three consecutive multiples of 3 is 396. Determine the largest number.
ব্যাখ্যা
Question: The total of three consecutive multiples of 3 is 396. Determine the largest number.
Solution:
Let,
First multiple: 3x
Second multiple: 3(x + 1) = 3x + 3
Third multiple: 3(x + 2) = 3x + 6
ATQ,
3x + (3x + 3) + (3x + 6) = 396
⇒ 9x + 9 = 396
⇒ 9x = 387
⇒ x = 387/9
∴ x = 43
∴ The largest number = 3x + 6 = 3 × 43 + 6 = 135
Solution:
Let,
First multiple: 3x
Second multiple: 3(x + 1) = 3x + 3
Third multiple: 3(x + 2) = 3x + 6
ATQ,
3x + (3x + 3) + (3x + 6) = 396
⇒ 9x + 9 = 396
⇒ 9x = 387
⇒ x = 387/9
∴ x = 43
∴ The largest number = 3x + 6 = 3 × 43 + 6 = 135
১৩,৮৯২.
Find the lowest value of x if lx - 3l ≤ 7?
ব্যাখ্যা
Question: Find the lowest value of x if lx - 3l ≤ 7?
Solution:
Given that,
lx - 3l ≤ 7
⇒ - 7 ≤ x - 3 ≤ 7
⇒ - 7 + 3 ≤ x - 3 + 3 ≤ 7 + 3
⇒ - 4 ≤ x ≤ 10
This means that the value of x is between - 4 and 10,
So the lowest value of x that satisfies the inequality is - 4.
১৩,৮৯৩.
If x = 2 + √3, y = 2 - √3 and z = 1 then what is the value of {(x/yz) + (y/xz) + (z/xy)} + 2[(1/x) + (1/y) + (1/z)] =?
ব্যাখ্যা
Question: If x = 2 + √3, y = 2 - √3 and z = 1 then what is the value of {(x/yz) + (y/xz) + (z/xy)} + 2[(1/x) + (1/y) + (1/z)] =?
Solution:
Given,
x = 2 + √3
y = 2 - √3
z = 1
Now,
{(x/yz) + (y/xz) + (z/xy)} + 2[(1/x) + (1/y) + (1/z)]
= {(x2 + y2 + z2)/xyz} + 2[(1/x) + (1/y) + (1/z)]
= {(2 + √3)2 + (2 - √3)2 + (1)2/(2 + √3)(2 - √3)} + 2[{1/(2 + √3)} + {1/(2- √3)} +1]
= {(4 + 3 + 4√3 + 4 + 3 - 4√3 + 1)/1} + 2[2 - √3 + 2 + √3 +1]
= 15 + 10
= 25
Solution:
Given,
x = 2 + √3
y = 2 - √3
z = 1
Now,
{(x/yz) + (y/xz) + (z/xy)} + 2[(1/x) + (1/y) + (1/z)]
= {(x2 + y2 + z2)/xyz} + 2[(1/x) + (1/y) + (1/z)]
= {(2 + √3)2 + (2 - √3)2 + (1)2/(2 + √3)(2 - √3)} + 2[{1/(2 + √3)} + {1/(2- √3)} +1]
= {(4 + 3 + 4√3 + 4 + 3 - 4√3 + 1)/1} + 2[2 - √3 + 2 + √3 +1]
= 15 + 10
= 25
১৩,৮৯৪.
The ratio of the number of sides of a square to the number of edges of a cube is
ব্যাখ্যা
Question: The ratio of the number of sides of a square to the number of edges of a cube is
Solution:
the number of sides of a square = 4
the number of edges of a cube = 12
∴ The ratio of the number of sides of a square to the number of edges of a cube is = 4 : 12
= 1 : 3
Solution:
the number of sides of a square = 4
the number of edges of a cube = 12
∴ The ratio of the number of sides of a square to the number of edges of a cube is = 4 : 12
= 1 : 3
১৩,৮৯৫.
If n is a positive integer, then n(n + 1)(n + 2) is-
ব্যাখ্যা
Question: If n is a positive integer, then n(n + 1)(n + 2) is-
Solution:
There's a nice rule says:
The product of k consecutive integers is divisible by k, k - 1, k - 2, ..., 2, and 1
So, for example, the product of any 5 consecutive integers will be divisible by 5, 4, 3, 2 and 1
Likewise, the product of any 11 consecutive integers will be divisible by 11, 10, 9, . . . 3, 2 and 1
NOTE: the product may be divisible by other numbers as well, but these divisors are guaranteed.
Notice that n, n + 1, and n + 2 are three consecutive integers.
This means the product of n, n + 1, and n + 2 will be divisible by 3, 2 and 1
Since n(n + 1)(n + 2) is divisible by 2, then the product is ALWAYS EVEN.
This means we can eliminate answer choices A and B, because they put restrictions on when the product is even.
We can also eliminate C because it suggests that the product can be odd.
Likewise, since n(n + 1)(n + 2) is ALWAYS divisible by 3, we can eliminate answer choice D, because it puts a restriction on when the product is divisible by 3
If n is even ⇒ even × odd × even (Prod of two even numbers always divisible by 2 × 2 = 4)
Solution:
There's a nice rule says:
The product of k consecutive integers is divisible by k, k - 1, k - 2, ..., 2, and 1
So, for example, the product of any 5 consecutive integers will be divisible by 5, 4, 3, 2 and 1
Likewise, the product of any 11 consecutive integers will be divisible by 11, 10, 9, . . . 3, 2 and 1
NOTE: the product may be divisible by other numbers as well, but these divisors are guaranteed.
Notice that n, n + 1, and n + 2 are three consecutive integers.
This means the product of n, n + 1, and n + 2 will be divisible by 3, 2 and 1
Since n(n + 1)(n + 2) is divisible by 2, then the product is ALWAYS EVEN.
This means we can eliminate answer choices A and B, because they put restrictions on when the product is even.
We can also eliminate C because it suggests that the product can be odd.
Likewise, since n(n + 1)(n + 2) is ALWAYS divisible by 3, we can eliminate answer choice D, because it puts a restriction on when the product is divisible by 3
If n is even ⇒ even × odd × even (Prod of two even numbers always divisible by 2 × 2 = 4)
১৩,৮৯৬.
Six consecutive whole numbers are given. The sum of the first three numbers is 27. What is the sum of the last three numbers?
ব্যাখ্যা
Question: Six consecutive whole numbers are given. The sum of the first three numbers is 27. What is the sum of the last three numbers?
Solution:
let the numbers be x - 2, x - 1, x, x + 1, x + 2, x + 3
x - 2 + x - 1 + x = 27
⇒ 3x - 3 = 27
⇒ 3x = 27 + 3 = 30
⇒ x = 30/3
∴ x = 10
the sum of the last three numbers is = x + 1 + x + 2 + x + 3
= 10 + 1 + 10 + 2 + 10 + 3
= 36
Solution:
let the numbers be x - 2, x - 1, x, x + 1, x + 2, x + 3
x - 2 + x - 1 + x = 27
⇒ 3x - 3 = 27
⇒ 3x = 27 + 3 = 30
⇒ x = 30/3
∴ x = 10
the sum of the last three numbers is = x + 1 + x + 2 + x + 3
= 10 + 1 + 10 + 2 + 10 + 3
= 36
১৩,৮৯৭.
If the total length of diagonals of a cube is 12 cm, then what is the total surface area of the cube ?
ব্যাখ্যা
Question: If the total length of diagonals of a cube is 12 cm, then what is the total surface area of the cube ?
Solution:
a cube has 4 diagonals
∴ Length of a diagonal
= (12/4)cm
= 3cm
Let the length of each edge of the cube is = a cm
Then,
a√3 = 3
⇒ a = 3/√3
⇒ a = √3
∴ Total surface area of the cube
= 6a2
= 6(√3)2
= 18cm2
Solution:
a cube has 4 diagonals
∴ Length of a diagonal
= (12/4)cm
= 3cm
Let the length of each edge of the cube is = a cm
Then,
a√3 = 3
⇒ a = 3/√3
⇒ a = √3
∴ Total surface area of the cube
= 6a2
= 6(√3)2
= 18cm2
১৩,৮৯৮.
If the annual income from a 5% stock at 90 is Tk. 40 more than the income from a 6% stock at 120, then what is the total investment?
ব্যাখ্যা
Question: If the annual income from a 5% stock at 90 is Tk. 40 more than the income from a 6% stock at 120, then what is the total investment?
Solution:
Let,
The investment = Tk. x
5% stock at 90,
income from Tk. 90 = Tk. 5
∴ income from Tk. x = Tk. 5x/90
= Tk. x/18
6% stock at 120,
income from Tk. 120 = Tk. 6
∴ income from Tk. x = Tk. 6x/120
= Tk. x/20
ATQ,
x/18 - x/20 = 40
⇒ (10x - 9x)/180 = 40
⇒ x = 40 × 180
∴ x = 7200
Solution:
Let,
The investment = Tk. x
5% stock at 90,
income from Tk. 90 = Tk. 5
∴ income from Tk. x = Tk. 5x/90
= Tk. x/18
6% stock at 120,
income from Tk. 120 = Tk. 6
∴ income from Tk. x = Tk. 6x/120
= Tk. x/20
ATQ,
x/18 - x/20 = 40
⇒ (10x - 9x)/180 = 40
⇒ x = 40 × 180
∴ x = 7200
১৩,৮৯৯.
If A = {1, 4, 5, 6, 7, 9}, then the number of proper subsets of A is-
ব্যাখ্যা
Question: If A = {1, 4, 5, 6, 7, 9}, then the number of proper subsets of A is-
Solution:
কোন সেটের উপাদান n সংখ্যক হলে, ঐ সেটের প্রকৃত উপসেটের সংখ্যা = 2n - 1
এখানে,
n = 6
∴ প্রকৃত উপসেটের সংখ্যা = 26 - 1
= 64 - 1
= 63
১৩,৯০০.
The ratio of 1/5 to 2/7 is-
ব্যাখ্যা
Question: The ratio of 1/5 to 2/7 is-
Solution:
Here,
The ratio of 1/5 to 2/7 = 1/5 : 2/7
= (1/5) × 35 : (2/7) × 35
= 7 : 10
Solution:
Here,
The ratio of 1/5 to 2/7 = 1/5 : 2/7
= (1/5) × 35 : (2/7) × 35
= 7 : 10