Bank Math

Question: If C is the midpoint of the points A(2, - 1) and B(8, 5), find the length of AC.

Solution:
দেওয়া আছে,
 A(2, - 1) এবং B(8, 5), 
এবং C হলো AB-এর মধ্যবিন্দু।

দূরত্বের সূত্র ব্যবহার করে AB-এর দৈর্ঘ্য নির্ণয় করি,
AB = √(x₂ - x₁)² + (y₂ - y₁)²)
= √(8 - 2)² + {5 - (-1)}²
= √(6² + 6²)
= √(36 + 36)
= √72
= √(36 × 2)
= 6√2

C হলো AB এর মধ্যবিন্দু, তাই AC = AB/2
 = 6√2/2
 = 3√2

Question: What is the largest number that divides 43, 91, and 183, leaving the same remainder in all three cases?

Solution:
এখানে বৃহত্তম সংখ্যা হচ্ছে সংখ্যাগুলোর পার্থক্যগুলোর H.C.F. (গ.সা.গু)।

প্রথম দুটি সংখ্যার পার্থক্য: 91 - 43 = 48
পরের দুটি সংখ্যার পার্থক্য: 183 - 91 = 92

48 এবং 92 এর মৌলিক উৎপাদক:
48 = 2 × 2 × 2 × 2 × 3 = 2⁴ × 3
92 = 2 × 2 × 23 = 2² × 23

H.C.F. = 2² = 4

∴ নির্ণীত বৃহত্তম সংখ্যা = 4

25% of A = 35% of B
⇒ (25/100)A = (35/100)B
⇒ A/4 = 7B/20
⇒ A/B = (7/20)×4 = 7/5
⇒ A:B = 7:5

We know, Perimeter = 2(l+b) = 76
⇒ (l+b) = 38
⇒ l = 38 - b .....(1)
Given, the area, lb = 360 .....(2)
From (2)
(38 – b)b = 360
⇒ 38b – b² = 360
⇒ b² – 38b + 360 = 0
⇒ (b – 20)(b – 18) = 0
So, b = 20 or b = 18
When b = 20, l = 18 and when b = 18, l = 20.
∴ the shorter side is 18.

Question: Three unbiased coins are tossed. What is the probability of getting at least 2 tails?

Solution:
When three fair (unbiased) coins are tossed, the total number of possible outcomes = 2 × 2 × 2 = 8.
{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT} 

And all possible outcomes are, {HTT, THT, TTH, TTT} = 4

∴ Probability = (number of favorable outcomes)/(total possible outcomes)
= 4/8
= 1/2

So the probability of getting at least 2 tails is 1/2.

Let the number be = x
According to question,
x²=(75.15)²−(60.12)²
⇒x²=(75.15+60.12)(75.15−60.12)
⇒x²=135.27×15.03
⇒x²=2033.1081
⇒x=45.09

Question: In a factory, there are workers, executives and clerks. 70% of the employees are workers, 596 are executives and the remaining 334 employees are clerks. How many employees are there in the factory?

Solution: 
Let
Total employee 100%
Executive and clerks (100 - 70)% = 30%

executive and clerks employee = 596 + 334 = 930 

Now
30% employee = 930
∴ 1% employee = 930/30
∴ 100% employee = ( 930 × 100)/30 = 3100 

Question: Identify the smallest integer that, when doubled, can be evenly divided 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

Sum of 18 observations = 18 x 124
=2232

Correct sum of 18 observations = 2232 - 46 - 82 + 64 + 28
= 2196
Therefore,
Correct average =2196/18
= 122.

Area or outer space = 2a x 2a = 4a²
From ABCD rectangle, In ΔBAD,
BD² = (AD)² + (AB)²
⇒ (2a)² = x² + x²
⇒ 4a² = 2x² 
⇒ 2a² = x² 
∴ x = √2a

Area of inner square = √2a × √2a = 2a² 

∴ The difference between the areas of outer and inner squares is = 4a² - 2a² = 2a² 

Question: The length of a rectangle is thrice its breath, and its perimeter is 120 meters. What is its area?

Solution:
Let the breath = x
So, the Length = 3x

Perimeter of a rectangle = 2 (Length + Breadth)
So, 2(3x + x) = 120
⇒ 6x + 2x = 120
⇒ 8x = 120
∴ x = 120/8 = 15

Now, Breadth = 15
so, length = 15 × 3 = 45

So, its area = Length × Breadth = 45 × 15 = 675 sq. m.

Question: If TOPIC is written as POITC, then how would FINAL be written in the same code?

Solution:
The code contains the letters of the word in the order- third,second, fourth, first and fifth

Question: If tan 6A = √3, then find A.

Solution:
tan 6A = √3
⇒ tan 6A = tan60°
⇒ 6A = 60°
⇒ A = 60°/6
∴ A = 10°

According to question,
√(0.25/0.0009) × √(0.09/0.36)
⇒ √((25/9)×100) × √(9/36)
⇒ ((5×10)/3) × (3/6)
⇒ 25/3
⇒ 8(1/3)

6(-3)(1/3)(-0.25)
= 6×0.25
= 1.5

Income ratio = Ratul : Sajib = 5 : 6 = 5/6
Common factor helps in finding actual values easily
So, take 'A' as a common factor.
Income of Ratul = 5A; Income of Sajib = 6A
(Expenses of Ratul)/(Expenses of Sajib) = (Ratul Income - Ratul Saving)/(Sajib Income -Sajib Saving) = 7/9
Since Ratul save, Tk. 1000/- more than Sajib, Ratul's savings = Tk. 4000/-
5A - 4000/6A - 3000 = 7/9
∴ 9(5A-4000) = 7(6A-3000)
∴ A = 5000
Income of Ratul = 5A = 25000 ; Income of Sajib = 6A = 30000
Spending of Ratul =25000 - 4000 = 21000
Spending of Sajib = 30000 - 3000 = 27000.

Question: If y = sin(sinx) then what is the value of dy/dx?

Solution:

Question: A swimmer can swim at a speed of 4 km/h in still water. If the river current flows at 2 km/h, how long will it take to swim 12 km downstream and then return to the starting point?

Solution: 
Speed of the swimmer in still water = 4 km/h
Speed of the current = 2 km/h
Distance one way = 12 km

Downstream Speed = 4 + 2 = 6 km/h
Upstream Speed = 4 - 2 = 2 km/h 

Time taken downstream = 12/6 = 2 hours
Time taken upstream = 12/2 = 6 hours

∴ Total time = 6 + 2 
= 8 hours

Question: A batsman makes a score of 80 runs in the 16th innings and increases average by 3. What is his average after 16th innings?

Solution: 
Assume his initial average after 15 innings = x
His total runs after 15 innings = 15x

After scoring 80 runs his average got increased by 3 to x + 3
So his total runs after 16 innings= 16 × (x +3)

But it was given that the difference in the total scores after 15 innings and 16 innings =80
16×(x + 3) - 15X=80
16x + 48 - 15x = 80 
x + 48 = 80
x = 32 

His average after 16th innings is = 32 + 3 = 3 = 35 

Question: The difference between two integers is 4. If their product is 221, then the sum of the two numbers is?

Solution:
Let the two integers be x and y
The difference between the two integers is 4,
∴ x - y = 4 and their product is, xy = 221

We know,
⇒ (x + y)² = (x - y)² + 4xy
⇒ (x + y)² = 4² + (4 × 221)
⇒ (x + y)² = 16 + 884
⇒ (x + y)² = 900
⇒ x + y = √900
∴ x + y = 30

∴ The sum of the two numbers is 30

Let,
The distance be x km.
Then,
Time taken to cover x km by cycle + Time taken to cover x km by scooter = 6 hr 30 min
⇒ (Time taken to cover 2x km by cycle) + (Time taken to cover 2x km by scooter) = 13 hrs
But,
Time taken to cover 2x km by cycle = 8 hr 40 min.
∴ Time Taken to cover 2x km by scooter = 13 hrs - 8 hr 40 min
= 4 hr 20 min.
Hence, required time = 4 hours 20 min.

ত্রিভুজের যেকোন মধ্যমা ত্রিভুজটিকে সমান ক্ষেত্রফলবিশিষ্ট দুটি ত্রিভুজক্ষেত্রে বিভক্ত করে।

Question: If tomorrow is Friday, what day of the week will it be after 46 days?

Solution: 
Given that,
Tomorrow is Friday, then today is Thursday

Days of the week repeat every 7 days. So, find the remainder when 46 is divided by 7.
So 46 ÷ 7 = 6 weeks + 4 days remainder.
So the day will move 4 days forward. 

Count 4 days from Thursday,
Thursday → Friday → Saturday → Sunday → Monday

Therefore, 46 days from today (Thursday) is Monday.

Question: A man on tour travels first 60 km at 20 km/hr and the next 60 km at 30 km/hr. The average speed for the first 120 km of the tour is :

সমাধান:
প্রথম অংশের জন্য সময় = দূরত্ব/গতিবেগ
= 60 কিমি/20 কিমি/ঘন্টা
= 3 ঘন্টা

দ্বিতীয় অংশের জন্য সময় = দূরত্ব/গতিবেগ
= 60 কিমি/30 কিমি/ঘন্টা
= 2 ঘন্টা

মোট অতিক্রান্ত দূরত্ব = 60 কিমি + 60 কিমি = 120 কিমি
মোট সময় = 3 ঘন্টা + 2 ঘন্টা = 5 ঘন্টা

∴ গড় গতিবেগ = মোট দূরত্ব/মোট সময় 
= 120 কিমি/5 ঘন্টা
= 24 কিমি/ঘন্টা

Mobin's age:Xam's age = 4:5 = 1:5/4
Mobin's age:Faisal's age = 5:6 = 1:6/5
Let, Mobin's age be x years. Then, Xami's age = 5x/4
And, Faisal's age = 6x/5 years
∴ x + 5x/4 + 6x/5 = 69
⇒ 20x + 25x + 24x = 69 × 20
⇒ 20x + 25x + 24x = 1380
⇒ 69x = 1380
⇒ x = 1380/69
= 20
Xami's age = 5x/4 = (5 × 20)/4
= 25 years

Total cost of the computer = Tk. 39000
Down payment = Tk. 17000
Balance = Tk. (39000 - 17000) = Tk. 22000.
Let the rate of interest be R% p.a.
Amount of Tk. 22000 for 5 months
= {22000 + 22000 × (5/12) × R/100}
= {22000 + (275R/3)}

The customer pays the shopkeeper Tk. 4800 after 1 month,
Tk. 4800 after 2 months, ...... and Tk. 4800 after 5 months.

Thus, the shopkeeper keeps Tk. 4800 for 4 months, Tk. 4800 for 3 months, Tk. 4800 for 2 months, Tk. 4800 for 1 months and Tk. 4800 at the end.

∴ sum of the amounts of these installments
= (Tk. 4800 + S.I. on Tk. 4800 for 4 months) + (Tk. 4800 + S.I. on Tk. 4800 for 3 months) + ...... + (Tk. 4800 + S.I. on Tk. 4800 for 1 month) + Tk. 4800
= Tk. (4800 × 5) + S.I. on Tk. 4800 for (4 + 3 + 2 + 1) months
= Tk. 24000 + S.I. on Tk. 4800 for 10 months
= 24000 × 4800 × R × (10/12) × (1/100)
= (24000 + 40 R)

∴ {22000 + (275R/3)} = (24000 + 40 R)
⇒ 155R/3 = 2000
⇒ R = (2000 × 3)/155
= 38.71

Question: If log_x(1/125) = - 3, then x = ?

Solution:
Given that,
log_x(1/125) = - 3
⇒ x^{- 3} = 1/125  [log_a(b) = c  ⇒ a^c = b]
⇒ 1/x³ = 1/125
⇒ x³ = 125
∴ x = 5

Question: If your car runs from A to B at 60 km/hr and on returning from B to A, it is 40 km/hr, then average speed in km/hr of the car will be-

Solution:
Given that,
The speed of the car is travelling and the returning are 60 km/hr and 40 km/hr

We know,
Average speed = Total distance/Total time

Time = Distance/Speed

Now,
Let the distance between A and B be x

∴ Total time = (x/60) + (x/40)
= (2x + 3x)/120
= 5x/120 = x/24
∴ Total time = x/24

And total distance = x + x = 2x 

∴ Average speed = 2x/(x/24) = 48 km/hr

Question: Traveling at 108 kilometers per hour, a 120-meter train will cross a railway platform that is 210 meters long in…

Solution:
Here,
Speed of the running train = 108 km/hr
= {108 × (5/18)} m/sec
= 30 m/sec

And length of the train is = 120 metres
Length of platform = 210 m

So, the time will taken by the train = (Length of train + Length of platform)/Speed
= (120 + 210)/30
= 330/30 
= 11 sec

The difference in speed due to stopping = Speed without stoppage - Speed with stoppage
∴ Difference = 40-35 = 5km/hr
Thus, in 1-hour train covers 5 km less.

Time taken to cover 4km = 5km/(40 km/hr)
= 1/8 hours.
= (1/8 × 60) minutes.
= 7.5 minutes.

Hence, The train halts 7.5 minutes per hour.

Question: One year ago, the ratio between Fahim and Rohan's ages was 3 : 2. One year hence, the ratio of their age will be 4 : 3. What is the sum of their present ages in years?

Solution:
ধরি, এক বছর আগে ফাহিমের বয়স ছিল 3x বছর এবং রোহানের বয়স ছিল 2x বছর।
বর্তমানে ফাহিমের বয়স = (3x + 1) বছর
বর্তমানে রোহানের বয়স = (2x + 1) বছর

এক বছর পর তাদের বয়সের অনুপাত হবে 4 : 3
∴ এক বছর পর ফাহিমের বয়স হবে = (3x + 1) + 1 = (3x + 2) বছর
∴ এক বছর পর রোহানের বয়স হবে = (2x + 1) + 1 = (2x + 2) বছর

প্রশ্নমতে,
(3x + 2)/(2x + 2) = 4/3
⇒ 3(3x + 2) = 4(2x + 2)
⇒ 9x + 6 = 8x + 8
⇒ 9x - 8x = 8 - 6
⇒ x = 2

সুতরাং, বর্তমানে ফাহিমের বয়স = (3 × 2) + 1 = 6 + 1 = 7 বছর
এবং বর্তমানে রোহানের বয়স = (2 × 2) + 1 = 4 + 1 = 5 বছর
∴ তাদের বর্তমান বয়সের যোগফল = 7 + 5 = 12 বছর।

HCF of the given numbers will be the greatest number which can divide 48, 84 and 144
18 = 2 × 3 × 3
84 = 2 × 2 × 3 × 7
144 = 2 × 2 × 2 × 2 × 3 × 3
∴ HCF = 2 × 3 = 6
Hence 6 is the greatest number which divides 18, 84 and 144 without leaving any remainder