ব্যাংক নিয়োগ বিষয়ভিত্তিক প্রস্তুতি

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

Solution:
The nth term of an arithmetic progression is T_n = 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: If 5^{x + y} = 25 and 5^{x - y} = 5, then what are the values of x and y respectively?

Solution:
Given,
5^{x + y} = 25
⇒ 5^{x + y} = 5²
⇒ x + y = 2 .......(1)

Again,
5^{x - y} = 5
⇒ 5^{x - y} = 5¹
⇒ x - y = 1 ........(2)

Now, solving (1) and (2) we get,
x + y + x - y = 2 + 1
⇒ 2x = 3
∴ x = 3/2

Now,
x + y = 2
⇒ 3/2 + y = 2
⇒ y = 2 - (3/2)
​⇒ y = 1/2
​
​(x, y) = (3/2, 1/2)

Question: The present ages of A and B are in the ratio 3 : 5. After 10 years, the ratio of their ages will be 4 : 6. What is the difference in their present ages?
​
​Solution:
​Let the present ages be,
​A = 3x and B = 5x
​
​Ages after 10 years,
​A = 3x + 10
​B = 5x + 10
​
​According to the problem, the ratio becomes 4 : 6
​(3x + 10) : (5x + 10) = 4 : 6
⇒ ​​(3x + 10)/(5x + 10) = 4/6
​⇒ ​​3(3x + 10) =2(5x + 10)
​⇒ ​​9x + 30 = 10x + 20
⇒ ​​​10x - 9x = 30 - 20
∴ ​x = 10
​
​A = 3 × 10 = 30 years
​B = 5  × 10 = 50 years
​
∴ Difference = 50 - 30 = 20 years

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 the 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: 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: 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:
​√15 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: 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 units
∴ Cost Price (CP) = 100 + 25% of 100 
= 100 + 25 = 125 units

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

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

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

Question: The ratio of length and breadth of a rectangular park is 7 : 5. A man runs along its boundary at 8 km/hr and takes 9 minutes for one round. Find its area in sq. meters.

Solution:
One round of the park is equal to the perimeter of the park.
So, by completing one round, the man covers a distance equal to the perimeter of the park.
Now,
Distance or perimeter = speed × time
= 8 × (9/60)
= 1.2 km
= 1200 meters

Let,
Length = 7x and breadth = 5x
So, Perimeter,
2(7x + 5x) = 1200
⇒ 24x = 1200
∴ x =1200/24 = 50 meters

So, Length = 7 × 50 = 350 meters
And, Breadth = 5 × 50 = 250 meters

Area = Length × Breadth
= 350 × 250
= 87500 sq. m.

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: 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: 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: 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: 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: What should be the value of "Q" so that the expression (49 - 28x + Qx²) becomes a perfect square?

Solution:
(49 - 28x + Qx²)
= (7)² - 2 × 7 × 2x + (2x)² + Qx² - (2x)²
= (7 - 2x)² + Qx² - 4x²

The expression becomes a perfect square if,
Qx² - 4x² = 0
⇒ Qx² = 4x² 
∴ Q = 4

Thus, when Q = 4, the expression is (7 - 2x)², which is a perfect square.

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

ব্যবধি আকারে লিখলে হয়: (- 1, 7]

(- 1, 7] বলতে বোঝায় যে, - 1 এর চেয়ে বড় এবং 7 বা তার চেয়ে ছোট সব বাস্তব সংখ্যা এই সমাধানের অন্তর্ভুক্ত।

Question: A ladder is leaning against a wall. It makes a 60° angle with the wall. If the distance between the foot of the ladder and the 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: If the points A(1, 5), B(k, 1), and C(4, 11) are collinear, then find the value of k.

Solution:
আমরা জানি,
তিনটি বিন্দু A(1, 5), B(k, 1) এবং C(4, 11) সরলরেখায় অবস্থিত হলে, তাদের ঢাল (slope) সমান হবে।

আমরা জানি, 
ঢাল = (y₂​ - y₁​​)/(x₂ - x₁)
 
এখন, 
A এবং B-এর মধ্যে ঢাল, m_AB = (1 - 5)/(k - 1)
= - 4/(k - 1)

B এবং C-এর মধ্যে ঢাল, m_BC = (11 - 1)/(4 - k)
= 10/(4 - k)

শর্তমতে,
m_AB = m_BC
⇒ - 4/(k - 1) = 10/(4 - k)
⇒ 10(k - 1) = - 4(4 - k)
⇒ 10k - 10 = - 16 + 4k
⇒ 10k - 4k = - 16 + 10
⇒ 6k = - 6
∴ k = - 1

Question: A box contains 20 electric bulbs, out of which 4 are defective. Two balls are chosen at random from this box. The probability that at least one of them is defective, is:

Solution:
Given that,
Total bulbs = 20
Defective bulbs = 4
Non-defective bulbs = 20 - 4 = 16
Two bulbs are chosen at random (without replacement)

Now,
P(both non-defective) = (16/20) × (15/19) = 240/380 = 12/19

And,
∴ P(at least one defective) = 1 - P(both non-defective)
= 1 - (12/19)
= (19 - 12)/19
= 7/19
∴ The probability that at least one of them is defective is 7/19

Question: In a class of 92 students, 40 are taking English, 24 are taking Arabic and 10 are taking both courses. How many students are not enrolled in either course?

Solution:
Total students = 92
Students taking English n(E) = 40
Students taking Arabic n(A) = 24
Students taking both English and Arabic = 10

We know,
n(E ∪ A) = n(E) + n(A) - n(E ∩ A)
n(E ∪ A) = 40 + 24 - 10 = 54

∴ Not enrolled = Total students - n(E ∪ A) = 92 - 54 = 38

Question: A can complete a work in 24 days and B in 16 days. They work together for 6 days. How many more days will A take alone to finish the remaining work?

Solution:
A একা কাজটি করতে পারে = 24 দিনে
∴ A এর একদিনের কাজ = 1/24 অংশ
এবং, 
   B একা কাজটি করতে পারে = 16 দিনে
∴ B এর একদিনের কাজ = 1/16 অংশ

∴ A ও B একসাথে একদিনের কাজ = (1/24) + (1/16) = (2 + 3)/48 = 5/48 অংশ
তারা 6 দিনে একসাথে কাজ করে = 6 × (5/48) = 5/8 অংশ

বাকি কাজ = 1 - (5/8) = 3/8 অংশ

অতএব,
A, 1/24 অংশ কাজ করে 1 দিনে 
∴ 3/8  অংশ কাজ করে = (24 × 3)/8 = 9 দিনে 

অতএব, A একা বাকি কাজ শেষ করতে ৯ দিন লাগবে।

Question: What is the angle between the hour and minute hands of a clock when it is 3 : 15 pm?

Solution:
3টা 15 মিনিট = 3 + (15/60) ঘন্টা = 3 + 1/4 = 13/4 ঘন্টা

আমরা জানি, ঘণ্টার কাঁটা 12 ঘণ্টায় 360° ঘোরে।
∴ 1 ঘণ্টায় ঘোরে = 360°/12 = 30°
∴ 13/4 ঘণ্টায় ঘোরে = (30° × 13)/4
= 390°/4
= 97.5°

আবার, মিনিটের কাঁটা 60 মিনিটে 360° ঘোরে।
∴ 1 মিনিটে ঘোরে = 360°/60 = 6°
∴ 15 মিনিটে ঘোরে = 15 × 6° = 90°

∴ ঘড়ির কাঁটা দুটির মধ্যবর্তী কোণ = |97.5° - 90°|
= 7.5°

Question: In a geometric progression, the 4^th term is 16 and the 7^th term is 128. Find the 10^th term.

Solution:
Let the first term = a
Common ratio = r
We know,
n-term = ar^{n - 1}

Then,
4^th term, ar³ = 16  ........(1)  
7th term, ar⁶ = 128  ........(2)

Now, divide equation (2) by equation (1) then we get,
(ar⁶)/(ar³) = 128/16  
⇒ r³ = 8  
⇒ r³ = 2³  
∴ r = 2  

Then substitute r = 2 into equation (1) 
a.(2)³ = 16   
⇒ a × 8 = 16  
∴ a = 2

Now, 10^th term
= ar⁹  
= 2 × 2⁹ 
= 2 × 2⁹  
= 2¹⁰ 
= 1024

∴ The 10th term is 1024

Question: The least number, which when divided by 12, 15, 20 and 54 leaves in each case a remainder of 8 is-

Solution:
The number leaves a remainder 8 when divided by 12, 15, 20 and 54.
So the required number = LCM(12, 15, 20, 54) + 8

Now, 
12 = 2 × 2 × 3
15 = 3 × 5
20 = 2 × 2 × 5
54 = 2 × 3 × 3 × 3

∴ LCM(12, 15, 20, 54) = 540

∴ Required Number = 540 + 8 = 548 

Question: There is 60% increase in an amount in 6 years at simple interest. What will be the compound interest of Tk. 12000 after 3 years at the same rate?

Solution:
Given that,
Increase in amount after 6 years = 60%

We know,
A = P(1 + r/100)n
simple Interest = (principal × rate × time)/100

Now,
Let the amount at the 1^st year be 100x
∴ Increased in amount = 60% of 100x = 60x
⇒ 60x = (100x × 6 × r)/100
⇒ 6r = 60
⇒ r = 60/6 = 10
∴ r = 10% 

∴ Compound Interest after 3 years = P(1 + r/100)ⁿ - P
= 12000(1 + 10/100)³ - 12000
= 12000 × (11/10)³ - 12000
= 12000 × (11/10) × (11/10) × (11/10) - 12000
= 15972 - 12000
= Tk. 3972

∴ The required answer is Tk. 3972

Question: Which of the following is equivalent to the pair of inequalities 2x - 5 ≤ 7 and 3x + 4 > 10?

Solution:
Solve the first inequality,
2x - 5 ≤ 7 
⇒ 2x ≤ 7 + 5
⇒ 2x ≤ 12
∴ x ≤ 6
And,
Solve the second inequality,
3x + 4 > 10 
⇒ 3x > 10 - 4
⇒ 3x > 6
∴ x > 2

∴ We get 2 < x ≤ 6

Question: In a tourist group of 100 people, 55 speak French, 40 speak Spanish, and 20 speak none of the languages. How many of them speak just one language?

Solution:
 

Let,
Number of people who can speak both languages = x persons
∴ Number of people who speak only French = (55 - x) persons
∴ Number of people who speak only Spanish = (40 - x) persons

Given that,
Number of people who speak none of the languages = 20 persons

According to the question,
Only French + Both + Only Spanish = Total students - Those who speak none
⇒ (55 - x) + x + (40 - x) = 100 - 20 
⇒ 95 - x = 80
⇒ x = 95 - 80
∴ x = 15

∴ Only French = (55 - 15) = 40 persons
∴ Only Spanish = (40 - 15) = 25 persons

∴ Number of people who speak only one language (French or Spanish) = (40 + 25) = 65 persons

Question: What is the slope of a line perpendicular to the line whose equation is 20x - 2y = 6?

Solution:
The general equation of a straight line is
y = mx + c ......(1) (Where, m = slope)

If the slope of a line is m, then the slope of the line perpendicular to it is,
m' = - (1/m)

Now,
20x - 2y = 6
⇒ 2y = 20x - 6
∴ y = 10x - 3
Comparing with equation (1), we get,
∴ m = 10

∴ The slope of the perpendicular line is, m' = - (1/10)

Question: A and B are two positive integers such that AB = 60. Which of the following cannot be the value of A + B?

Solution:
Factor pairs of 60:
(1, 60) → A + B = 61
(2, 30) → A + B = 32
(3, 20) → A + B = 23
(4, 15) → A + B = 19
(5, 12) → A + B = 17
(6, 10) → A + B = 16

So, possible values of A + B are: 61, 32, 23, 19, 17, 16.

Among the options, 18 is not possible.

Question: If C is the midpoint of the points A(2, 3) and B(8, 11), find the length of AC.

Solution:
দেওয়া আছে, A(2, 3) এবং B(8, 11), এবং C হলো AB-এর মধ্যবিন্দু।

দূরত্বের সূত্র ব্যবহার করে AB-এর দৈর্ঘ্য নির্ণয় করি।
AB = √{(x₂ - x₁)² + (y₂ - y₁)²}
AB = √{(8 - 2)² + (11 - 3)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

Question:

Solution:
আমরা জানি, কোনো বর্গ ম্যাট্রিক্সের প্রধান কর্ণের ভুক্তিগুলোর সমষ্টিকে ওই ম্যাট্রিক্সের ট্রেস (trace) বলা হয়।

প্রদত্ত ম্যাট্রিক্সটির প্রধান কর্ণের ভুক্তিগুলো হলো: 2, 5, 8.
অতএব, A ম্যাট্রিক্সের ট্রেস, Tr(A) = 2 + 5 + 8 = 15

∴ ম্যাট্রিক্সটির ট্রেস, Tr(A) = 15

Question:

Solution:

Question: Find the equation of the line with x-intercept = 4 and y-intercept = 3.

Solution:
Given, x-intercept = 4,
So, the line passes through (4, 0).
y-intercept = 3,
So, the line passes through (0, 3).

We know,
The intercept form of a line is:
(x/a) + (y/b) = 1, where a = x-intercept, b = y-intercept.
⇒ (x/4) + (y/3) = 1
⇒ (3x + 4y)/12 = 1
⇒ 3x + 4y = 12
⇒ 3x + 4y - 12 = 0

∴ The equation of the line is 3x + 4y - 12 = 0

Question:

Solution:

Question: An observer 1.6 m tall stands 20 meters away from a tree. The angle of elevation from his eye to the top of the tree is 45°. What is the height of the tree?

Solution:
 

মনে করি, 
গাছটির উচ্চতা AB। পর্যবেক্ষকের চোখ C বিন্দুতে আছে এবং তার উচ্চতা CD = 1.6 m
পর্যবেক্ষক থেকে গাছটির দূরত্ব BD = 20 m
এখানে, A, C এবং E বিন্দু দ্বারা গঠিত ACE হলো একটি সমকোণী ত্রিভুজ, যার ∠C = 45°।

আমরা জানি,
tan θ = লম্ব/ভূমি
এখানে, লম্ব = AE এবং ভূমি = CE
∴ tan 45° = AE/20
∴ 1 = AE/20
∴ AE = 20 মিটার

গাছটির মোট উচ্চতা, AB = AE + EB
= 20 + 1.6
= 21.6 মিটার

সুতরাং, গাছটির উচ্চতা হলো 21.6 মিটার।

Question: A cube has a total surface area of 384 square units. What is the volume of the cube?

Solution:
Given, total surface area of the cube, S = 384 square units.
We know, surface area of a cube, S = 6a²

According to the question,
6a² = 384
⇒ a² = 384 / 6
⇒ a² = 64
⇒ a² = 8²
⇒ a = 8

Again, we know, volume of the cube, V = a³
= 8³
= 512

Therefore, the volume of the cube is 512 cubic units.

Question: A square and a circle have the same perimeter. The side of the length of square is 44 cm, what is the area of the circle?

Solution:
Perimeter of the square = 4 × side length
= 4 × 44 cm
= 176 cm

As per the question, the square and circle have the same perimeter.
∴ Circumference of the circle = 176 cm
We know that, Circumference of the circle = 2πr
∴ 2πr = 176
⇒ r = 176/(2π)
⇒ r = 88/π
⇒ r = 88/(22/7)
⇒ r = 88 × 7/22
⇒ r = 4 × 7
⇒ r = 28 cm

Area of the circle = πr²
= (22/7) × 28²
= (22/7) × (28 × 28)
= 22 × 4 × 28
= 2464 sq. cm

∴ The area of the circle is 2464 sq. cm.