Bank Math

Question: The next term of the series: 36, 81, 144, 225, ____ is

Solution: 
Given, 36, 81, 144, 225,
The series is = 6² , 9², 12², 15², 18²
So, next term is 18² = 324

Question: In a town, the ratio of the number of men to the number of women is 3 : 5. If 120 men and 80 women shift to the town, the new ratio of men to women becomes 2 : 3. What was the initial number of men in the town?

Solution:
Let the initial number of men and women be 3x and 5x, respectively.

According to the question,
(3x + 120)/(5x + 80) = 2/3
⇒ 3(3x + 120) = 2(5x + 80)
⇒ 9x + 360 = 10x + 160
⇒ 360 - 160 = 10x - 9x
⇒ 200 = x
∴ x = 200

∴ The initial number of men = 3x = 3(200) 
= 600 men

Question: Solve for x: log₃(x + 5) = 3

Solution:
Given that,
log₃(x + 5) = 3
⇒ x + 5 = 3³  [log_ab = c ⇒ b = a^c] 
⇒ x + 5 = 27
⇒ x = 27 - 5
∴ x = 22

Let A, B, C represent their respective weights. Then, we have:
A + B + C = (45 x 3) = 135 .... (i)
A + B = (40 x 2) = 80 .... (ii)
B + C = (43 x 2) = 86 ....(iii)
Adding (ii) and (iii), we get: A + 2B + C = 166 .... (iv)
Subtracting (i) from (iv), we get : B = 31
∴ B's weight = 31 kg.

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

Solution:
The events when three unbiased coins are tossed = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
Total number of events 8

The events of getting at least two heads {HHH, HHT, HTH, THH}

Number of expected events = 4

∴ The probability of getting at least two heads is = 4/8 = 1/2

‘LOGARITHM’ contains 10 different letters.

Required number of word
= Number of arrangements of 10 letters, taking 4 at a time
= ¹⁰P₄ = (10 × 9 × 8 × 7) = 5040.

Question: The ratio of two numbers is 3 : 4 and their H.C.F is 4. Their L.C.M is:
 
Solution:
ধরি,
সংখ্যা দুইটি যথাক্রমে 3x, 4x
 3x, 4x এর লসাগু = 12x
3x, 4x এর গসাগু = x

প্রশ্নমতে,
x = 4

∴  3x, 4x এর লসাগু = 12x
= 12 × 4
= 48

Let Rajeev's present age be  x year.
Rajeev's age after 15 year = (x + 15) year.
Rajeev's age 5 year back = (x - 5) year
Then ATQ, 
x + 15 = 5 (x - 5)
x + 15 = 5x - 25
=> x = 10
Hence, Rajeev's present age = x = 10 year.

Question: If C is the midpoint of the points A(1, 2) and B(7, 10), find the length of AC.

Solution:
দেওয়া আছে,
A(1, 2) এবং B(7, 10), এবং C হলো AB-এর মধ্যবিন্দু।

প্রথমে, দূরত্বের সূত্র ব্যবহার করে AB-এর দৈর্ঘ্য নির্ণয় করি।
AB = √{(x₂ - x₁)² + (y₂ - y₁)²}
AB = √{(7 - 1)² + (10 - 2)²}
AB = √(6² + 8²)
AB = √(36 + 64)
AB = √100
AB = 10

যেহেতু C হলো AB-এর মধ্যবিন্দু, তাই AC হবে AB-এর অর্ধেক।
∴ AC = AB/2
= 10/2
= 5

ধরি,
AB হচ্ছে দেয়াল এবং BC হচ্ছে মই।
এখানে ∠ACB = 60° এবং AC = 4.6 মিটার
তাহলে, AC/BC = cos 60° = 1/2 [ যেহেতু, cosΘ = ভূমি/অতিভুজ]
⇒ BC = 2 × AC
∴ BC = 2 × 4.6
= 9.2m

Question: Find the solution of the inequality |2x - 3| ≤ 1.

Solution:
Given that,  
|2x - 3| ≤ 1
⇒ - 1 ≤ 2x - 3 ≤ 1  
⇒ - 1 + 3 ≤ 2x ≤ 1 + 3  ; [adding 3 to all parts]
⇒ 2 ≤ 2x ≤ 4  
⇒ 1 ≤ x ≤ 2 ; [dividing all parts by 2]

Therefore, the solution is 1 ≤ x ≤ 2 or in interval notation [1, 2]

Let the rate of 3^rd type of wheat be Tk. W
Quantity ratio = 2 : 1 : 3
This means if we take 2kg of Type 1 and 1 kg of Type 2, then we must take 3kg of Type 3.
Also, the mix will have 2kg + 1kg + 3kg = 6 kg quantity.
∴ (145 x 2) + (165 x 1) + (3 x W) = 6 x 180
⇒ 290 + 165 + 3W = 1080
⇒ 3W = 1080 - 455
⇒ 3W = 625
⇒ W = 625/3
⇒ W = 208.33

∴ W = Tk. 208.33 per kg = 3^rd type price per kg

Question: Twelve distinct points are randomly placed on the circumference of a circle. At most how many triangles can be formed using these points?

Solution:
Given that,
Number of distinct points, n = 12
To form a triangle, we need to select 3 points out of n points.

Maximum number of triangles = ¹²C₃
= 12!/{3!(12 - 3)!}
= (12 × 11 × 10 × 9!)/(3 × 2 × 1 × 9!)
= (12 × 11 × 10)/6
= 220

A solid cube of side 6 is first painted pink and then cut into smaller cubes of side 2.

This means, we get a 3 × 3 × 3 cube
To get the the sides with just 2 faces painted, picture the edges of the each of the sides of the cube.
Apart from the corners (which have 3 faces painted), all other small cubes have 2 faces painted.

In total, we have 12 cubes of side 2 with exactly 2 sides painted

Question: A train can travel 25% faster than a car. Both start from point A at the same time and reach point B 100 kms away from A at the same time. On the way, however, the train lost about 15 minutes while stopping at the stations. The speed of the car is:

Solution:
ধরি, গাড়ির গতিবেগ = x কিমি/ঘন্টা।

যেহেতু ট্রেনের গতিবেগ গাড়ির গতিবেগের চেয়ে 25% বেশি, 
∴ ট্রেনের গতিবেগ = x + (x × 25/100)
= x + 0.25x = 1.25x কিমি/ঘন্টা

গাড়ির মোট সময় = দূরত্ব/গতিবেগ
= 100/x ঘন্টা

ট্রেনের মোট সময় (স্টপেজ ছাড়া) = দূরত্ব/গতিবেগ
= 100/(1.25x) ঘন্টা

ট্রেনটি স্টেশনে 15 মিনিট থেমেছিল।
15 মিনিট = 15/60 ঘন্টা = 1/4 ঘন্টা

যেহেতু ট্রেন এবং গাড়ি একই সময়ে গন্তব্যে পৌঁছায়, তাই গাড়ির মোট সময় এবং ট্রেনের স্টপেজ সহ মোট সময় সমান।

∴গাড়ির সময় = (ট্রেনের সময় + স্টপেজ সময়)
⇒ 100/x = 100/(1.25x) + 1/4
⇒ 100/x - 80/x = 1/4
⇒ 20/x = 1/4
⇒ x = 20 × 4
⇒ x = 80 কিমি/ঘন্টা

সুতরাং, গাড়িটির গতিবেগ হলো 80 কিমি/ঘন্টা।

Question: A person invests Tk. 3100 at a rate of 4% per annum under simple interest. After how many years will the total interest earned be Tk. 372?

Solution:
Given that,
Principal, P = Tk. 3100
Rate, r = 4%
Simple Interest, SI = Tk. 372

We know,
SI = (P × R × T) / 100
⇒ 372 = (3100 × 4 × n)/100
⇒ 372 = 124 × n
⇒ n = 372 ÷ 124
∴ n = 3 years

∴ Number of years = 3

 Question: 'A' can do a work in 10 days, and 'B' in 15 days. They work together for 4 days. How much of the work is left?

Solution:
মনে করি,
সম্পূর্ণ কাজ = 1 অংশ

∴ A একা একদিনে করে = 1/10​ অংশ।
B একা একদিনে করে = 1/15​ অংশ।

∴ A ও B একসাথে একদিনে করে = (1/10) + (1/15) অংশ
= (3 + 2)/30 
= 5/30
= 1/6 অংশ 

∴ A ও B একসাথে 4 দিনে করে = 4 × (1/6) অংশ
= 2/3 অংশ

∴ কাজ বাকি থাকে = 1 - (2/3) অংশ
= (3 - 2)/3 
= 1/3 অংশ 

log_x9/16 = – 1/2 
⇒ x^-1/2 = 9/16
⇒ √x = 16/9
⇒ x = (16/9)²   
∴ x = 256/81

Question: In a class of 150 students, the average score in mathematics is 82. If the 90 girls scored an average of 85, what is the average score of the remaining boys?

Solution:
ধরি, ছেলেদের গড় নম্বর = x

150 জন শিক্ষার্থীর মোট নম্বর = 150 × 82 = 12300
90 জন ছাত্রীর মোট নম্বর = 90 × 85 = 7650

প্রশ্নমতে,
7650 + (150 - 90) × x = 12300
⇒ 7650 + 60x = 12300
⇒ 60x = 12300 - 7650
⇒ 60x = 4650
⇒ x = 4650/60
⇒ x = 77.5

∴ 60 জন ছেলের গড় নম্বর = 77.5

Question: The product of two numbers is 300, and the sum of their squares is 625. What is the sum of two numbers?

Solution:
Let the numbers be a and b.
As per the question:
ab = 300
a² + b² = 625

So,
(a + b)² = a² + b² + 2ab
= 625 + 2 × 300
= 625 + 600
= 1225

∴ a + b = √1225 = 35

Question: A man looks into a mirror placed on the ground and sees the top of a tower. The mirror is 120 m away from the tower. If the man stands 0.6 m away from the mirror and his height is 1.8 m, find the height of the tower.

Solution:

Given that,
Distance from the mirror to the tower = 120 m
Distance from the man to the mirror = 0.6 m
Height of the man = 1.8 m
Height of the tower = H ?

Now,
Height of the man/Distance from man to mirror = Height of the tower/Distance from tower to mirror
⇒ 1.8/0.6 = H/120
⇒ 3 = H/120
⇒ H = 120 × 3 = 360 m

Question: Find the sum of the reciprocals of two numbers whose total is 36, HCF is 3, and LCM is 105.

Solution:
Let, the numbers be a and b.

Then, a + b = 36 and ab =  3 × 105 = 315 [∵ Product of the numbers = HCF×LCM]

∴ sum of their reciprocals
= (1/a) + (1/b)
= (a + b)/ab
= 36/315
= 4/35

4x⁴ + 1
= (2x²)² + 1
= (2x²)² + 2.2x².1 + 1² - 4x²
= (2x² + 1)² - (2x)²
= (2x² + 2x + 1) (2x² - 2x + 1)

Question: А bоx contains 3 blue, 2 white, and 4 red marbles. If one marble is drawn at random, what is the probability that it will not be a white marble?

Solution:
Given that,
Blue marbles = 3
White marbles = 2
Red marbles = 4

∴ Total marbles = 3 + 2 + 4 = 9
And, number of non-white marbles = Blue + Red = 3 + 4 = 7

We know,
P(not white) = favorable outcomes​/total outcomes
= 7/9​ 

The answer is divisible by 987.
So we can use the hit and trial method to find out the number divisible by 987 from the given choices.
553681/987 gives a remainder not equal to 0
555181/987 gives a remainder not equal to 0
556581/987 gives a remainder not equal to 0
But 555681/987 gives 0 as a remainder. Hence this is the answer

Question: ⁶P_m = 120, ⁶C_m = 20, what is the value of m?

Solution:
Given,
⁶P_m = 120
⇒ 6!/(6 - m)! = 120 ..........(1)

⁶C_m = 20
⇒ 6!/{m!(6 - m)!} = 20 ..........(2)

(1) ÷ (2),
{6!/(6 - m)!} / [{6!/{m!(6 - m)!}] = 120/20
⇒ m! = 6

We know,
3! = 3 × 2 × 1 = 6
∴ m = 3

Question: The product of two consecutive even positive number is 528. Find the numbers.

Solution:
Let the integers be x and x + 2
So, The equation:
x(x + 2) = 528
⇒ x² + 2x - 528 = 0
⇒ x² + 24x - 22x - 528 = 0 [528 = (2 × 2 × 2 × 3) × (2 × 11) = 24 × 22]
⇒ x(x + 24) - 22(x + 24) = 0
⇒ (x - 22)(x + 24) = 0
⇒ x = 22 or x = -24
x = 22 [Ignoring the negative value]

so, x + 2 = 24

∴ Numbers = 22, 24

Question: If the simple interest on 6,000 Taka for 3 years is 1,800 Taka, what is the rate of interest per annum?

Solution:
Given, I = 1,800 Taka
P = 6,000 Taka
n = 3 years

We know, 
I = Pnr
⇒ r = I/Pn
⇒ r = 1800/(6000×3)
⇒ r = 0.1
∴ r = 10%

Question: Which number replaces the question mark?


Solution:
এখানে,
বাম হাতের উপাদানগুলোর পার্থক্য × ডান হাতের উপাদানগুলোর পার্থক্য = উপরের সংখ্যা।

অতএব, ১ম চিত্রে,
(14 - 9) × (13 - 7) = 5 × 6 = 30

২য় চিত্রে,
(12 - 9) × (17 - 11) = 3 × 6 = 18

∴ প্রশ্নবোধক স্থানে সংখ্যাটি হবে 18

Question: The volume of a cylinder is 100π and its radius is 5. What is the lateral surface area?

Solution:
Given that,
Volume = 100π cubic units
Radius (r) = 5 units

We know,
Volume of a cylinder, V = πr²h
⇒ 100π = π × (5)² × h 
⇒ 100π = π × 25 × h
⇒ h = 100π/(25π)
⇒ h = 100/25
∴ h = 4 units

∴ Lateral surface area of a cylinder = 2πrh
= 2 × π × 5 × 4
= 40π

So the lateral surface area is 40π square units.

Question: The price of a stock increased by 20% in January and then decreased by 10% in February. If the price of the stock was Tk. 108 at the end of February, what was the price at the beginning of January?

Solution:
Let the beginning price of January = 100x
The price of a stock increased by 20% in January.
Price of stock become = 100x + 100x of 20%
= 100x + (100x × 20)/100
= 120x

And then the stock price decreased by 10% in February
Price of stock become = 120x - 120x of 10%
= 120x - (120x × 10)/100
= 108x

ATQ,
108x = 108
⇒ x = 108/108
∴ x = 1

Therefore the price at the beginning of January
= 100 × 1
= Tk. 100

The price at the beginning of January was Tk. 100.

Let, each train’s length x
Relative speed = 108 + 90 = 198 km/h = 198 × (5/18) m/s
Distance covered in 3 minutes = 198 × (5/18) × 3 × 60 = 9900 m
∴ length of a train = 9900/2 = 4950 m