ব্যাংক নিয়োগ বিষয়ভিত্তিক প্রস্তুতি

Question: If the n^th term of an arithmetic progression is 7n + 1, then what is the common difference?

Solution:
The nth term of an arithmetic progression is Tn = 7n + 1
n = 1 then, T₁ = 7 × 1 + 1 = 8
n = 2 then, T₂ = 7 × 2 + 1 = 15
n = 3 then, T₃ = 7 × 3 + 1 = 22
n = 4 then, T₄ = 7 × 4 + 1 = 29
............................

Common difference,
T₂ - T₁ = 15 - 8 = 7
T₄ - T₃ = 29 - 22 = 7

∴ The common difference is 7.

Question: A boat travels 18 km downstream in 45 minutes. If the speed of the stream is 5 km/h, what is the speed of the boat in still water?

Solution:
স্রোতের অনুকূলে 45 মিনিটে যায় 18 কিমি
স্রোতের অনুকূলে 1 মিনিটে যায় 18/45 কিমি
স্রোতের অনুকূলে 1 ঘণ্টা বা 60 মিনিটে যায় (18 × 60)/45 কিমি
= 24 কিমি

∴ স্রোতের অনুকূলে বেগ = 24 কিমি/ঘণ্টা

দেওয়া আছে,
স্রোতের বেগ = 5 কিমি/ঘণ্টা।

∴ স্থির পানিতে নৌকার বেগ = স্রোতের অনুকূলে বেগ - স্রোতের বেগ
= 24 - 5 = 19 কিমি/ঘণ্টা।

Question: P scored 30% marks and failed by 15 marks. Q scored 45% marks and obtained 30 marks more than the pass marks. What is the pass percentage?

Solution:
Let the total marks be x.

Given,
P scored 30% marks and failed by 15 marks:
0.30x + 15 = Pass marks
Q scored 45% marks and obtained 30 marks more than pass marks:
0.45x - 30 = Pass marks

Now,
0.30x + 15 = 0.45x - 30
⇒ 0.45x - 0.30x = 15 + 30
⇒ 0.15x = 45
⇒ x = 45/0.15
∴ x = 300

Pass marks = 0.30 × 300 + 15
= 90 + 15 = 105

∴ Pass percentage = (105/300) × 100% = 35%

Question: A train 150 meters long takes 40 seconds to cross a 350-meter-long bridge. How much time will the train take to cross a 250-meter-long platform?

Solution:
Length of train = 150 m
Length of bridge = 350 m
∴ Total distance to cross bridge = 150 + 350 = 500 m
Time taken = 40 seconds
∴ Speed of train = Total distance/Time
= 500/40 = 12.5 m/s

Length of platform = 250 m
∴ Total distance to cross platform = 150 + 250 = 400 m

∴ Time taken = Total distance/Speed
= 400/12.5 seconds
= 32 seconds

Question: A man buys an article for 25% more than its value and sells it for 20% less than its value. His gain or loss percentage is –

Solution:
Let the original value of the article = 100 
∴ Cost Price (CP) = 100 + 25% of 100 
= 100 + 25 = 125 

∴ Selling Price (SP) = 100 - 20% of 100 
= 100 - 20 = 80 

Since SP 80 is less than CP 125, there is a Loss.

∴ Loss = CP - SP = 125 - 80 = 45 টাকা
∴ Loss percentage = (Loss/CP) × 100%
= (45/125) × 100%
= 36% loss

Question: The supplement of an angle exceeds twice the angle by 30°. Then the angle is equal to-

Solution:
Let the angle be x
Then, its supplement = 180 - x

According to the question,
180 - x = 2x + 30
⇒ 180 - 30 = 3x
⇒ 150 = 3x
⇒ x = 50°

Question: In a 500 m race, the speeds of two runners, A and B are in the ratio 5 : 6. If A is given a start of 100m, by how many meters does A win the race?

Solution:
Total race length = 500 meters.
A is given a start of 100 meters, so A runs 500 - 100 = 400 meters.

Speed ratio A : B = 5 : 6.

Let, B runs = X meter

Therefore,
400/X = 5/6
⇒ X = (6 × 400)/5
∴ X = 480m

Remaining distance for B = 500 - 480 = 20 meters.
Therefore, A wins by 20 meters.

Question: At what rate of compound interest per annum will a sum of Tk. 4000 becomes Tk. 4840 in 2 years?

​Solution:
Principal, P = Tk. 4000
Compound Amount, C = Tk. 4840
Time, n = 2 years
Rate, r = ?

We know,
C = P × (1 + r/100)ⁿ
⇒ 4840 = 4000 × (1 + r/100)²
⇒ (1 + r/100)² = 4840/4000 
⇒ (1 + r/100)² = 484/400
⇒ 1 + r/100 = 22/20 [উভয়পাশে বর্গমূল করে]
⇒ r/100 = (11/10) - 1
⇒ r/100 = (11 - 10)/10
⇒ r/100 = 1/10
⇒ r = (1 × 100)/10
∴ r = 10

∴ Interest Rate = 10%

Question: Which of the following is irrational?

Solution:
​√10 একটি অমূলদ সংখ্যা (irrational number)।
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 কোনো স্বাভাবিক সংখ্যার বর্গমূল কিংবা তার ভগ্নাংশ একটি অমূলদ সংখ্যা। যেমন, √2 = 1.414213..., √6 = 2.229489... ইত্যাদি অমূলদ সংখ্যা।
- কোনো অমূলদ সংখ্যাকে দুইটিপূর্ণ সংখ্যার অনুপাত হিসেবে প্রকাশ করা যায় না।
-অমূলদ সংখ্যাকে একটি মূলদ সংখ্যা দ্বারা গুণ করলে অমূলদ সংখ্যা পাওয়া যায়।
অর্থাৎ, non zero rational number × irrational number = irrational number.

Question: When 25% of the first number is added to the second number, the second number becomes 3/2 times the first number. What is the ratio of the first number to the second number?

Solution:
Let the first number = x
and the second number = y.

According to the question,
y + 25% of x = (3/2)x
⇒ y + (25/100)x = (3/2)x
⇒ y + (1/4)x = (3/2)x
⇒ y = (3/2)x - (1/4)x
⇒ y = (6 - 1)x/4
⇒ y = 5x/4
∴ y = 5x/4

Therefore, x : y = 4 : 5 

Question: A committee of 3 men and 2 women is to be formed from 5 men and 4 women. In how many ways can the committee be formed?

​Solution:
We have 5 men and 4 women.
We need to choose 3 men from 5 and 2 women from 4.

∴ Number of ways = ⁵C₃ × ⁴C₂
= {5!/3!(5 - 3)!} × {4!/2!(4 - 2)!}
= {(5 × 4)/2} × {(4 × 3)/2}
= 10 × 6
= 60 ways

Question: Tk. 8000 becomes Tk. 9600 in 2 years at a certain rate of simple interest. If the rate becomes double, what will be the total amount in 5 years on the same principal?
​
​Solution:
Principal = Tk. 8000
Amount after 2 years = Tk. 9600
∴ Interest in 2 years = 9600 - 8000 = Tk. 1600
∴Interest per year = 1600/2 = Tk. 800

Since the interest rate doubles, the yearly interest also doubles,
∴ Interest per year = 800 × 2 = Tk. 1600
⇒ Interest for 5 years = 1600 × 5 = Tk. 8000
⇒ New total amount = 8000 + 8000 = Tk. 16000

Question: Find the equation of the vertical line passing through the point (7, - 4).

Solution:
একটি উল্লম্ব রেখা (vertical line) হলো এমন একটি সরলরেখা যা Y-অক্ষের সমান্তরাল। এই ধরনের রেখার একটি বিশেষ বৈশিষ্ট্য হলো, এই রেখার উপর অবস্থিত প্রতিটি বিন্দুর x-স্থানাঙ্ক (x-coordinate) একই থাকে, কিন্তু y-স্থানাঙ্ক (y-coordinate) পরিবর্তিত হতে পারে।

উল্লম্ব রেখার (vertical line) সাধারণ সমীকরণ হলো: x = a
যেখানে a একটি ধ্রুবক সংখ্যা এবং রেখার প্রতিটি বিন্দুর x এর মান একই থাকে।

প্রশ্নে বলা হয়েছে রেখাটি (7, - 4) বিন্দুর মধ্য দিয়ে যায়।
যেহেতু, আমরা জানি, একটি উল্লম্ব রেখার প্রতিটি বিন্দুর x-স্থানাঙ্ক একই থাকে, এবং এই বিন্দুর x-স্থানাঙ্ক হলো 7, সুতরাং, রেখাটির সমীকরণ হবে:
x = 7

Question: If the average of m numbers is n² and the average of n numbers is m², find the average of all (m + n) numbers.
​
​Solution:
​Given that,
​The average of m numbers is n²
​Sum of m numbers = m⋅n²
​
​And
The average of n numbers is m²
​Sum of n numbers = n⋅m²

​∴ ​Total sum = mn² + nm² = mn(m + n)
​
​∴ ​​Average = mn(m + n)/(m + n) = mn​

Question: A metal sphere weighing 120 kilograms is melted and recast into 6000 identical nails. Calculate the weight of each nail in grams.

Solution:
দেওয়া আছে,
ধাতুর বলের ওজন = 120 = 120 × 1000 = 120000 গ্রাম
পেরেকের সংখ্যা = 6000 টি

এখন,
6000 পেরেকের ওজন = 120000 গ্রাম
∴ 1 টি পেরেকের ওজন = (120000/6000) = 20 গ্রাম

Question: If (a - 18)² + (b - 12)² + (c - 6)² = 0 then, What is the value of (a + b + c)^1/2 = ?
​
​Solution:
​দেওয়া আছে,
​(a - 18)² + (b - 12)² + (c - 6)² = 0
​
​আমরা জানি, যেকোনো বর্গের যোগফল শূন্য হলে প্রতিটি বর্গই শূন্য হবে।
​সুতরাং,
​​(a - 18)² = 0
⇒ ​a - 18 = 0
∴ ​a = 18
​
​আবার,
​(b - 12)² = 0
⇒ ​​b - 12 = 0
∴ ​​b = 12
​
​এবং
​(c - 6)² = 0
⇒ ​​c - 6 = 0
∴ ​c = 6
​
​প্রদত্ত রাশি,
​ (a + b + c)^1/2
​= (18 + 12 + 6)^1/2
​= (36)^1/2
​= 6

Question: A square garden is surrounded by a path of uniform width 2 meters. If the area of the path is 48 square meters, find the side length of the garden.

Solution:
Let the side of the garden = x meters.
Then, the side of the garden including the path = x + (2 × 2)
= x + 4 meters.

Area of path = Area of garden with path - Area of garden
⇒ 48 = (x + 4)² - x²
​⇒ 48 = x² + 8x + 16 - x²
​⇒ 8x + 16 = 48
⇒ 8x = 48 - 16
⇒ x = 32/8
∴ x = 4 meters

∴ Therefore, side length of the garden is 4 meters.

Question: If log⁡_m243 + log⁡_m81 = 9, find the value of m.
​
​Solution:
​Given that,
​log⁡_m243 + log⁡_m81 = 9
​⇒ ​​log⁡_m(243 × 81) = 9
⇒ ​​log⁡_m19683 = 9
⇒ ​m⁹ = 19683
​⇒ ​m⁹ = 3⁹
∴ m = 3​

Question: A train 150 meters long passes a signal post in 15 seconds. How long will it take to pass a bridge that is 450 meters long?

Solution:
Train's speed = Distance/Time
= 150/15 = 10 m/s

Total distance to pass the bridge,
= Length of train + Length of bridge
= 150 m + 450 m
= 600 m

∴ Required time = Distance/Speed
= 600/10
= 60 seconds
​= 1 minute

∴ The train will take 60 seconds or 1 minute to pass platform.

Question: A bag contains 7 green balls, 8 blue balls, and 5 yellow balls. One ball is drawn at random. What is the probability that the ball drawn is neither green nor yellow?
​
​Solution:
​Total balls = 7 + 8 + 5 = 20
​
​Favorable outcomes = balls that are neither green nor yellow, that mean blue balls = 8
​
​∴ P(blue) = Favorable outcomes​/total outcomes​ = 8/20 = 2/5

Question: How many terms are there in the geometric progression,
3, 6, 12, 24, …, 1536

Solution:
First term, a = 3
Common ratio, r = 6/3 = 2

Last term or nth term of GP = ar^{n - 1}
⇒ 1536 = 3 × (2^{n - 1})
⇒ 2^{n - 1} = 1536/3
⇒ 2^{n - 1} = 512
⇒ 2^{n - 1} = 2⁹
So, comparing the power,
Thus, n - 1 = 9
∴ n = 10

∴ Number of terms = 10

Question: If (7x - 3y) : (x - 3y) = 5 : 11, find the value of x/y.
​
​Solution:
​Given that,
​(7x - 3y) : (x - 3y) = 5 : 11
​⇒ (7x - 3y)/(x - 3y) = 5/11
 ​⇒ ​77x - 33y = 5x - 15y
 ​⇒ ​77x - 5x = 33y - 15y
 ​⇒ ​72x = 18y
 ​⇒ ​x/y = 18/72
​∴ x/y = 1/4​

Question: Two pipes A and B can fill the tank in 24 and 36 minutes, respectively. Both the pipes are opened together. After how many minutes should the pipe B be turned off, so that the tank be fill in 18 minutes?

Solution:
Given that,
Pipe A fills the tank in 24 minutes.
Pipe B fills the tank in 36 minutes.
Total time to fill the tank = 18 minutes.
Now,
LCM of 24 and 36 = 72 (Total capacity of the tank).
Efficiency of pipe A = 72/24 = 3 units/minute.
Efficiency of pipe B = 72/36 = 2 units/minute.

Let,
pipe B be turned off after x minutes.
Pipe A works for 18 minutes.
Pipe B works for x minutes.
Work done by A in 18 minutes = 3 × 18 = 54 units.
​Work done by B in x minutes = 2x = 2x units.

Total work done = 54 + 2x = 72
⇒ 2x = 72 - 54
⇒ 2x = 18
⇒ x = 18/2
∴ x = 9

∴ Pipe B should be turned off after 9 minutes.

Question: If cosA sinA = 0,then (cosA + sinA)² =?

Solution:
(cosA + sinA)²
= cos²A + 2 cosA sinA + sin²A
= 1 + 2.0 [sin²A + cos²A = 1]
= 1 + 0
= 1

Question: In how many ways can 5 people from a group of 8 people be seated around a circular table?

Solution:
5 people out of 8 = ⁸C₅
= 8!/5!(8 - 5)!
= 8!/(3! × 5!)
​= (8 × 7 × 6 × 5!)/(6  × 5!)
= 56

And 5 people around a circular table = (5 - 1)! = 4! = 24

∴ Total ways = 24 × 56 = 1344

Question: If a + b + c = 6 and a² + b² + c² = 40 then, a³ + b³ + c³ - 3abc = ?
​
​Solution:
Given that,
​ a + b + c = 6
​a² + b² + c² = 40

​Now,
​a + b + c = 6
​⇒ (a + b + c)² = 6²
​⇒ a² + b² + c² + 2ab + 2bc + 2ac = 36
​⇒ 40 + 2(ab + bc + ca) = 36
​⇒ 2(ab + bc + ca) = - 4
​⇒ ab + bc + ca = - 2

​Then,
​a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ac)
= 6[40 - (-2)]
​= 6[40 + 2]
​= 6 × 42
​= 252

∴ ​a³ + b³ + c³ - 3abc = 252

Question: ∠M and ∠N are complementary to each other. If ∠M = 20° + 4x and ∠N = 6x, find the value of ∠N.

Solution:
Here,
∠M = 20° + 4x and ∠N = 6x

For complementary angles,
∠M + ∠N = 90°
⇒ (20° + 4x) + 6x = 90°
⇒ 20° + 4x + 6x = 90°
⇒ 20° + 10x = 90°
⇒ 10x = 90° - 20°
⇒ 10x = 70°
∴ x = 7°

So, ∠N = 6 × 7° = 42°

Question: What is the solution of the inequality, 
- 12 < 4x - 8 ≤ 20 ?
​
​Solution:
- 12 < 4x - 8 ≤ 20
⇒ - 12 + 8 < 4x - 8 + 8 ≤ 20 + 8
⇒ - 4 < 4x ≤ 28
⇒ - 4/4 < 4x/4 ≤ 28/4
⇒ - 1 < x ≤ 7

∴ solution of the inequality: (- 1, 7]

Question: Find the value of .

Solution:

Question: In a class, 30 students play basketball, 20 students play volleyball, and 8 students play both. 12 students play neither basketball nor volleyball. What is the total number of students in the class?

​​Solution:
Number of students who play basketball, n(B) = 30
Number of students who play volleyball, n(V) = 20
Number of students who play both basketball and volleyball, n(B ∩ V) = 8
Number of students who play neither = 12

n(B ∪ V) = n(B) + n(V) - n(B ∩ V)
= 30 + 20 - 8 = 42

Total students in the class = students who play basketball or volleyball + students who play neither
n(U) = n(B ∪ V) + neither = 42 + 12 = 54

∴ Total 54 students in the class.

Question: In a mixture of milk and water, the ratio is 7 : 5. If 6 liters of water is added, the new ratio becomes 7 : 6. What was the original amount of milk in the mixture?

Solution:
ধরি, শুরুতে দুধ ছিল = 7x লিটার,
পানি ছিল = 5x লিটার।

এখন 6 লিটার পানি যোগ করলে,
নতুন পানি = 5x + 6 লিটার

ATQ,
7x/(5x + 6) = 7/6
⇒ 6 × 7x = 7 × (5x + 6)
⇒ 42x = 35x + 42
⇒ 7x = 42
⇒ x = 6

∴ দুধের পরিমাণ = 7x = 7 × 6 = 42 লিটার

Question: A worker union contract specifies a 6% salary increase plus a Tk. 450 bonus for each worker. For a worker, this is equivalent to an 8% salary increase. What was this worker's salary before the new contract?

Solution:
ধরি, কর্মীর পূর্বের বেতন = x টাকা।

6% বৃদ্ধিতে বেতন = x + x এর 6%
= x + (6x/100) = 106x/100

বোনাস হিসেবে 450 টাকা যোগ করলে মোট বেতন = (106x/100) + 450

8% বৃদ্ধিতে বেতন = x + x এর 8%
= x + (8x/100) = (108x/100)

প্রশ্নমতে,
(106x/100) + 450 = (108x/100)
⇒ 450 = (108x/100) - (106x/100)
⇒ 450 = (2x/100)
⇒ x = (450 × 100)/2
∴ x = 22500

অর্থাৎ, কর্মীর পূর্ববর্তী বেতন ছিল 22500 টাকা।

Question: Find the greatest number that exactly divides each of the numbers 48, 72, and 108.

Solution:
We know,
The HCF (Highest Common Factor) of two or more numbers is the greatest number that divides each of them exactly.

Now,
Prime factorization of 48 = 2 × 2 × 2 × 2 × 3

Prime factorization of 72 = 2 × 2 × 2 × 3 × 3

Prime factorization of 108 = 2 × 2 × 3 × 3 × 3

∴ HCF of 48, 72, and 108 = 2 × 2 × 3 = 12

Therefore, the greatest number is 12.

Question: A ladder is leaning against a wall. It makes a 60° angle with the wall. If the distance between foot of ladder and wall is 7.5 meters, find the length of the ladder.

Solution:

Let BC be the wall and AC be the ladder.
∠BAC = 60° and AB = 7.5 meter
In ΔABC,
cos60° = AB/AC
⇒ 1/2 = 7.5/AC
⇒ AC = 7.5 × 2
∴ AC = 15

Question: Find the value of 5(m + 4) - 2(3m - 1) + m.

​Solution:
​Given that,
​5(m + 4) - 2(3m - 1) + m
​= 5m + 20 - 6m + 2 + m
​= 6m - 6m + 22
​= 22

Question: X buys a product for Tk. 400 and sells it to Y at a profit of 30%. Y then sells it to Z at a profit of 15%. How much does Z pay to Y?

সমাধান:
X এর 30% লাভে বিক্রয়মূল্য = 400 + 400 এর 30%
= 400 + (400 × 30/100)
= 400 + 120
= 520

X এর বিক্রয়মূল্য = Y এর ক্রয়মূল্য

Y এর 15% লাভে বিক্রয়মূল্য = 520 + 520 এর 15%
= 520 + (520 × 15/100)
= 520 + 78
= 598

সুতরাং, Y এর বিক্রয়মূল্য = Z এর ক্রয়মূল্য = Tk. 598