সিনিয়র অফিসার নিয়োগ প্রস্তুতি (আর্কাইভ)

Question: The difference between two positive numbers is 8 and the difference of their squares is 160. What is the smallest number?

Solution:
Let the numbers be x and (x + 8)

According to the question,
(x + 8)² - x² = 160
⇒ x² + 16x + 64 - x² = 160
⇒ 16x + 64 = 160
⇒ 16x = 96
⇒ x = 6

∴ The smallest number is = 6

Question: If the nth term of an arithmetic progression is 5n + 3, then what is the common difference?

Solution:
The nth term of an arithmetic progression is T_n = 5n + 3

n = 1 then, T₁ = 5 × 1 + 3 = 8
n = 2 then, T₂ = 5 × 2 + 3 = 13
n = 3 then, T₃ = 5 × 3 + 3 = 18
n = 4 then, T₄ = 5 × 4 + 3 = 23

Common difference, T₂ - T₁ = 13 - 8 = 5
T₄ - T₃ = 23 - 18 = 5

∴ The common difference is 5.

Question: A boat travels 24 km downstream in 40 minutes. If the speed of the stream is 6 km/h, what is the speed of the boat in still water?

Solution:
স্রোতের অনুকূলে 40 মিনিটে যায় 24 কিমি
স্রোতের অনুকূলে 1 মিনিটে যায় 24/40 কিমি
স্রোতের অনুকূলে 1 ঘণ্টা বা 60 মিনিটে যায় (24 × 60)/40 কিমি
= 36 কিমি

∴ স্রোতের অনুকূলে বেগ = 36 কিমি/ঘণ্টা

দেওয়া আছে,
স্রোতের বেগ = 6 কিমি/ঘণ্টা।

∴ স্থির পানিতে নৌকার বেগ = স্রোতের অনুকূলে বেগ - স্রোতের বেগ
= 36 - 6 = 30 কিমি/ঘণ্টা।

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.

Question: If 3^{x + y} = 81 and 3^{x - y} = 9, then what are the values of x and y respectively?

Solution:
Given,
3^{x + y} = 81
⇒ 3^{x + y} = 3⁴
⇒ x + y = 4 .......(1)

Again,
3^{x - y} = 9
⇒ 3^{x - y} = 3²
⇒ x - y = 2 ........(2)

Now, solving (1) and (2) we get,
x + y + x - y = 4 + 2
⇒ 2x = 6
⇒ x = 3

Now, x + y = 4
⇒ 3 + y = 4
⇒ y = 4 - 3
⇒ y = 1

∴ (x, y) = (3, 1)

Question: Karim started a business with Tk. 80,000. After some months, Rahim joined with Tk. 60,000. At the end of the year, the profit was divided in the ratio 8 : 3. For how many months was Rahim in the business?

Solution:
Let, Rahim joined for x months.

ATQ,
(80,000 × 12)/(60,000 × x) = 8/3
⇒ (80,000 × 12 × 3) = (60,000 × x × 8)
⇒ 2,880,000 = 480,000x
⇒ x = 2,880,000/480,000
⇒ x = 6

∴ Rahim joined for 6 months.

Question: The present ages of P and Q are in the ratio 2 : 7. After 8 years, the ratio of their ages will be 3 : 8. What is the difference in their present ages?

Solution:
Let the present ages be,
P = 2x and Q = 7x

Ages after 8 years,
P = 2x + 8, Q = 7x + 8

According to the problem, the ratio becomes 3:8
(2x + 8)/(7x + 8) = 3/8
⇒ 8(2x + 8) = 3(7x + 8)
⇒ 16x + 64 = 21x + 24
⇒ 64 - 24 = 21x - 16x
⇒ 40 = 5x
⇒ x = 8

P = 2 × 8 = 16 years
Q = 7 × 8 = 56 years

∴ Difference = 56 - 16 = 40 years

Question:

Solution:
কোনো বর্গ ম্যাট্রিক্সের ট্রেস (Trace) হলো তার প্রধান কর্ণ বরাবর উপাদানগুলির যোগফল।
এখানে ম্যাট্রিক্স A এর প্রধান কর্ণ বরাবর উপাদানগুলি হলো 2, 5, এবং 8.
∴ Trace(A) = 2 + 5 + 8 = 15

Question: A mobile phone is sold for Tk. 9,600, and the seller makes a profit of 25% on the cost price. What will be the new selling price if he reduces the profit to 10%?

Solution:
At 25% profit,
Selling Price = 125% of Cost Price
∴ Cost Price = (100/125) × Selling Price
= (100/125) × 9,600
= 7,680 Tk.

Now, if the seller wants 10% profit,
∴ New Selling Price = 7,680 + (10% of 7,680)
= 7,680 + {(10/100) × 7,680}
= 7,680 + 768
= 8,448 Tk.

Question: A train 200 meters long takes 50 seconds to cross a 300-meter-long bridge. How much time will the train take to cross a 150-meter-long platform?

Solution:
Length of train = 200 m
Length of bridge = 300 m
∴ Total distance to cross bridge = 200 + 300 = 500 m

Time taken = 50 seconds

∴ Speed of train = Total distance/Time
= 500/50
= 10 m/s

Length of platform = 150 m
∴ Total distance to cross platform = 200 + 150 = 350 m

∴ Time taken = Total distance/Speed
= 350/10
= 35 seconds

Question: What should be the value of "Q" so that the expression (25 - 30x + Qx²) becomes a perfect square?

Solution:
(25 - 30x + Qx²)
= (5)² - 2 × 5 × 3x + (3x)² + Qx² - (3x)²
= (5 - 3x)² + Qx² - 9x²

∴ The expression becomes a perfect square if,
Qx² - 9x² = 0
⇒ Qx² = 9x²
∴ Q = 9

Question: If α, β are the roots of the equation x² - 9x + 20 = 0, then αβ equals:

Solution:
x² - 9x + 20 = 0
⇒ x² - 5x - 4x + 20 = 0
⇒ x(x - 5) - 4(x - 5) = 0
⇒ (x - 5)(x - 4) = 0
⇒ x = 5, 4

Hence, α = 5, β = 4

Hence, The value of α × β = 5 × 4 = 20
∴ αβ = 20

Shortcut:
দ্বিঘাত সমীকরণ ax² + bx + c = 0 এর মূলদ্বয় α এবং β হলে,
αβ = c/a [যেখানে, a হলো x² এর সহগ এবং c ধ্রুবক পদ]
∴ αβ = 20/1 = 20

Question: An amount of Tk. 8,000 yields a simple interest of Tk. 1,440 in 3 years. What is the annual rate of interest?

Solution:
Given,
Principal, P = 8000
Simple Interest, SI = 1440
Time, n = 3 years
Rate of interest, r = ?

We know,
I = Pnr/100
⇒ r = (I × 100)/(P × n)
⇒ r = (1440 × 100)/(8000 × 3) 
⇒ r = 144000/24000 
∴ r = 6%

So, the annual rate of interest is 6%.

Question: What is the distance between the points A (- 1, 3) and B (5, - 5)?

Solution:

Question: If log_x(8/125) = - 3, then x = ?

Solution:
log_x(8/125) = - 3
⇒ x^{- 3} = 8/125
⇒ x^{- 3} = (2/5)³
⇒ x^{- 3} = (5/2)^{- 3}
⇒ x = 5/2
∴ x = 5/2

Question: Today is Monday. After 45 days, it will be:

Solution:
Each day of the week is repeated after 7 days.

So, after 42 days, it will be Monday.

After 43 days, it will be Tuesday.
After 44 days, it will be Wednesday.
∴ After 45 days, it will be Thursday.

Question:

Solution:

Question: If tan(θ + 30°) = √3, what is the value of sinθ?

Solution:
Given that,
tan(θ + 30°) = √3
⇒ tan(θ + 30°) = tan 60°
⇒ θ + 30° = 60°
⇒ θ = 60° - 30°
⇒ θ = 30°

Now,
sinθ 
= sin 30°
= 1/2

Question: Two pipes X and Y can fill a cistern in 10 and 15 hours respectively. Both pipes are opened together. After how many hours should pipe X be turned off so that the cistern is filled in 9 hours?

সমাধান:
ধরি, মোট সময় 9 ঘন্টা পর চৌবাচ্চাটি পূর্ণ হয়। এই সম্পূর্ণ সময়ে কেবল নল Y খোলা ছিল।

নল Y, 15 ঘন্টায় চৌবাচ্চাটি পূর্ণ করতে পারে।
∴ 1 ঘন্টায় Y পূর্ণ করে 1/15 অংশ।
∴ 9 ঘন্টায় Y পূর্ণ করে = 9/15 অংশ
= 3/5 অংশ।

অবশিষ্ট অংশ যা X পূর্ণ করেছিল = 1 - 3/5 অংশ
= 2/5 অংশ।

নল X, 10 ঘন্টায় পূর্ণ করে 1 অংশ।
∴ 1 অংশ পূর্ণ করে 10 ঘন্টায়।
∴ 2/5 অংশ পূর্ণ করে = (10 × 2/5) ঘন্টা
= 4 ঘন্টা।

∴ নল X কে 4 ঘন্টা পর বন্ধ করতে হবে।

Question: Find the midpoint of the line segment joining the points A₁(2, 5) and A₂(8, - 3).

Solution:

Question: If 8 men or 12 boys can make 60 tables in 6 days, then how many tables will be made by 4 men and 6 boys in 8 days?

Solution:
Here, 8 men = 12 boys
∴ 4 men = (12/2) boys = 6 boys

∴ 4 men and 6 boys = (6 + 6) boys = 12 boys

Now,
12 boys can make 60 tables in 6 days
In 1 day, 12 boys can make (60/6) = 10 tables
∴ In 8 days, 12 boys can make = (10 × 8) tables
= 80 tables

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

Solution:
15 minutes past 3 অর্থাৎ, ৩ টা 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 ঘন্টায় 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: What number should come next in the series:
3, 7, 15, 31, 63, .......?

Solution: দেওয়া আছে,
সিরিজটি হলো: 3, 7, 15, 31, 63, .......

প্রতিটি পার্থক্য আগের পার্থক্যের 2 গুণ।

3 থেকে 7 পর্যন্ত পার্থক্য: 4
7 থেকে 15 পর্যন্ত পার্থক্য: 8 (4 × 2)
15 থেকে 31 পর্যন্ত পার্থক্য: 16 (8 × 2)
31 থেকে 63 পর্যন্ত পার্থক্য: 32 (16 × 2)
সুতরাং, পরবর্তী পার্থক্যটি হবে: 32 × 2 = 64

পরবর্তী সংখ্যাটি হবে শেষ সংখ্যা এবং এই পার্থক্যের যোগফল: 63 + 64 = 127

অতএব, পরবর্তী সংখ্যাটি হলো 127

Question: Which of the following is irrational?

Solution:
একটি সংখ্যা অমূলদ (irrational) হয় যদি এটি p/q আকারে প্রকাশ করা না যায়, যেখানে p এবং q পূর্ণসংখ্যা এবং q ≠ 0।

ক) 0.75 = 75/100 = 3/4 = এটি p/q আকারে প্রকাশ করা যায়, তাই এটি মূলদ সংখ্যা।

খ) √256 = 16 একটি পূর্ণবর্গ সংখ্যা (16² = 256), তাই √256 = 16 একটি মূলদ সংখ্যা।

গ) 2/5 = এটি ইতিমধ্যে p/q আকারে আছে, তাই এটি মূলদ সংখ্যা।

ঘ) √27 = √(9 × 3) = 3√3 একটি পূর্ণবর্গ সংখ্যা নয়, তাই √27 একটি অমূলদ সংখ্যা। এটি p/q আকারে প্রকাশ করা যায় না।

উত্তর: ঘ) √27 একটি অমূলদ (irrational) সংখ্যা।

Question: A team of 4 men and 3 women is to be formed from 6 men and 5 women. In how many ways can the team be formed?

Solution:
We have 6 men and 5 women.
We need to choose 4 men from 6 and 3 women from 5.

Number of ways to choose 4 men from 6:
⁶C₄ = 6!/(4!(6 - 4)!) 
= (6 × 5)/(2 × 1)
= 15

Number of ways to choose 3 women from 5:
⁵C₃ = 5!/(3!(5 - 3)!)
= (5 × 4)/(2 × 1)
= 10

Total number of ways to form the team = 15 × 10 = 150

Therefore, the team can be formed in 150 different ways.

Question: Find the value of 3(p + 5) - 2(2p - 3) + p

Solution: Given that,
3(p + 5) - 2(2p - 3) + p
= 3p + 15 - 4p + 6 + p
= (3p - 4p + p) + (15 + 6)
= 0 + 21
= 21

Question: What is the H.C.F. of the following fractions?
3/6, 6/9, 9/12

Solution:
আমরা জানি,
ভগ্নাংশের গসাগু = (লবের গসাগু)/(হরের লসাগু)

এখানে লব = 3, 6 এবং 9
3 = 3 × 1
6 = 3 × 2
9 = 3 × 3
∴ লবের গসাগু (H.C.F.) = 3

হর = 6, 9 এবং 12
6 = 2 × 3
9 = 3²
12 = 2² × 3
∴ হরের লসাগু (L.C.M.) = 2² × 3²
= 4 × 9 = 36

ভগ্নাংশের গসাগু = লবের গসাগু/হরের লসাগু
= 3/36
= 1/12

Question: In a class, 25 students play football, 15 students play cricket, and 5 students play both. 10 students play neither football nor cricket. What is the total number of students in the class?

Solution:
Number of students who play football, n(F) = 25
Number of students who play cricket, n(C) = 15
Number of students who play both football and cricket, n(F ∩ C) = 5
Number of students who play neither = 10

n(F ∪ C) = n(F) + n(C) - n(F ∩ C)
= 25 + 15 - 5
= 35

Total students in the class = students who play football or cricket + students who play neither
= 35 + 10
= 45

∴ There are 45 students in the class.

Question: What is the solution of the inequality, -10 < 3x - 4 ≤ 8 ?

Solution:
-10 < 3x - 4 ≤ 8
⇒ -10 + 4 < 3x - 4 + 4 ≤ 8 + 4
⇒ - 6 < 3x ≤ 12
⇒ - 6/3 < 3x/3 ≤ 12/3
⇒ - 2 < x ≤ 4

∴ Solution of the inequality: (- 2, 4]

The parenthesis "(" means - 2 is not included (open interval).
The bracket "]" means 4 is included (closed interval).

Question: ∠A and ∠B are complementary to each other. If ∠A = 30° + 3x and ∠B = 5x, find the value of ∠B.

Solution:
Here,
∠A = 30° + 3x and ∠B = 5x

For complementary angles,
∠A + ∠B = 90°
⇒ (30° + 3x) + 5x = 90°
⇒ 30° + 8x = 90°
⇒ 8x = 90° - 30°
⇒ 8x = 60°
⇒ x = 60°/8 = 7.5°

∴ ∠B = 5x = 5 × 7.5° = 37.5°

Question: If a + b + c = 5 and a² + b² + c² = 35, find the value of a³ + b³ + c³ - 3abc.

Solution:
Given, a + b + c = 5 and a² + b² + c² = 35

We know,
(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
⇒ (5)² = 35 + 2(ab + bc + ca)
⇒ 25 = 35 + 2(ab + bc + ca)
⇒ 2(ab + bc + ca) = 25 - 35 
⇒ 2(ab + bc + ca) = - 10
∴ ab + bc + ca = - 5

Now,
a³ + b³ + c³ - 3abc
= (a + b + c){a² + b² + c² - (ab + bc + ca)}
= 5 × {35 - (- 5)}
= 5 × 40
= 200

Therefore, the value is 200.

Question: A bag contains 6 red balls, 9 black balls, and 5 white balls. One ball is drawn at random. What is the probability that the ball drawn is neither red nor white?

Solution:

Total balls = 6 + 9 + 5 = 20

Favorable outcomes = balls that are neither red nor white, that is, black balls = 9

∴ P(black) = Favorable outcomes/Total outcomes
= 9/20

Therefore, the probability is 9/20.

Question: A man completes a journey in 10 hours. He travels the first half of the journey at the rate of 30 km/hr and the second half at the rate of 50 km/hr. Find the total journey in km.

Solution:
ধরা যাক, মোট যাত্রার দূরত্ব হলো d কিমি।
তাহলে, যাত্রার প্রথম অর্ধেকের দূরত্ব হবে d/2 কিমি
এবং দ্বিতীয় অর্ধেকের দূরত্বও হবে d/2 কিমি।

প্রথম অর্ধেক যাত্রায়,
সময় = দূরত্ব/গতিবেগ
= (d/2)/30 ঘন্টা
= d/60 ঘন্টা

দ্বিতীয় অর্ধেক যাত্রায়,
সময় = দূরত্ব/গতিবেগ
= (d/2)/50 ঘন্টা
= d/100 ঘন্টা

প্রশ্নমতে,
(d/60) + (d/100) = 10
⇒ (5d + 3d)/300 = 10
⇒ 8d/300 = 10
⇒ 8d = 10 × 300
⇒ 8d = 3000
⇒ d = 3000/8
⇒ d = 375 কিমি

∴ মোট যাত্রার দূরত্ব 375 কিলোমিটার।

Question: If 4^{3x + 5} = 1/16^{x + 1}, Find the value of x.

Solution
4^{3x + 5} = 1/16^{x + 1}
⇒ 2^{2(3x + 5)} = 1/2^{4(x + 1)}
⇒ 2^{6x + 10} = 2^{- 4(x + 1)}
⇒ 6x + 10 = - 4(x + 1)
⇒ 6x + 10 = - 4x - 4
⇒ 6x + 4x = - 4 - 10
⇒ 10x = - 14
⇒ x = - 14/10
∴ x = - 7/5

Question: A square park is surrounded by a path of uniform width 3 meters. If the area of the path is 75 square meters, find the side length of the park.

Solution:
Let the side of the park = x meters.
Then, the side of the park including the path = x + (2 × 3)
= x + 6 meters.

Area of the path = Area of park with path - Area of park
⇒ 75 = (x + 6)² - x²
⇒ 75 = x² + 12x + 36 - x²
⇒ 75 = 12x + 36
⇒ 12x = 75 - 36 = 39
⇒ x = 39/12 = 3.25 meters

Therefore, the side length of the park is 3.25 meters.

Question: An observer who is 1.6 meters tall is standing 25 meters away from a tower. If the angle of elevation from his eye to the top of the tower is 45°, what is the height of the tower?

Solution:


পর্যবেক্ষকের উচ্চতা, CD = 1.6 মিটার
এখানে, CD = EB
টাওয়ারের উচ্চতা = AB
এখন,
tan∠C = AE/CE
⇒ tan45° = AE/25
⇒ 1 = AE/25
∴ AE = 25

∴ AB = AE + BE
= 25 + 1.6 
= 26.6
∴ টাওয়ারটির উচ্চতা 26.6 মিটার।

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

Solution:
ধরি, ঘনকের বাহুর দৈর্ঘ্য = a মিটার।

আমরা জানি,
ঘনকের সম্পূর্ণ পৃষ্ঠের ক্ষেত্রফল = 6a²

প্রশ্নমতে,
6a² = 384
⇒ a² = 384/6
⇒ a² = 64
∴ a = 8 মিটার

এখন,
ঘনকের আয়তন = a³
= 8³
= 512 ঘন মিটার

অতএব, ঘনকটির আয়তন = 512 ঘন মিটার।

Question: If the average of p numbers is q² and the average of q numbers is p², find the average of all (p + q) numbers.

Solution:
Sum of p numbers = p × q²
Sum of q numbers = q × p²
Total sum = pq² + qp² = pq(p + q)

Total numbers = p + q

∴ Average of all (p + q) numbers = Total sum/Total numbers 
= [pq(p + q)]/(p + q) 
= pq

Question: Find the equation of the vertical line passing through the point (- 3, 5).

Solution:

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

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

প্রশ্নে বলা হয়েছে রেখাটি (- 3, 5) বিন্দুর মধ্য দিয়ে যায়।
 যেহেতু এই বিন্দুর x-স্থানাঙ্ক হলো - 3,
সুতরাং রেখাটির সমীকরণ হবে: x = - 3