Algebra

Question: Which value of y will satisfy the given inequality, 2(y - 3) ≥ 3y - 4 ?

Solution:
Given,
2(y - 3) ≥ 3y - 4
⇒ 2y - 6 ≥ 3y - 4
⇒ 2y - 3y ≥ - 4 + 6
⇒ - y ≥ 2
⇒ y ≤ - 2

Question: Find the product of two consecutive numbers if three times the first number is 5 more than twice the second number.

Solution:
Let the numbers be a and a + 1.

According to the question:
3 × (first number) = 2 × (second number) + 5
⇒ 3a = 2(a + 1) + 5
⇒ 3a = 2a + 2 + 5
⇒ 3a = 2a + 7
⇒ 3a - 2a = 7
⇒ a = 7

∴ The numbers are 7 and 8.
Product = 7 × 8 = 56

Question: If A = {p, q, r, s, t}, then how many proper subsets does A have?

Solution:
Given that, 
A = {p, q, r, s, t}
The number of elements in set A is 5.

We know that,
Number of proper subsets = 2ⁿ - 1  ; [where n = number of elements in the set]
∴ Number of proper subsets of A = 2⁵ - 1
= 32 - 1
= 31

অপশন a থেকে, 
a = {300/(9 + 1)} × 1 = 30 [100 থেকে ছোট, তাই বাদ]

অপশন b থেকে,
a = {300/(5 + 2)} × 5 = 128.7  [ভগ্নাংশ, তাই বাদ]

অপশন c থেকে,
a = {300/(5 + 3)} × 5 = 187.5 [ভগ্নাংশ, তাই বাদ]
b = {300/(5 + 3)} × 3 = 112.5 [ভগ্নাংশ, তাই বাদ]

অপশন d থেকে,
a = {300/(3 + 2)} × 3 = 180
এবং, b = {300/(3 + 2)} × 2 = 120

দেয়া আছে, y = x - 2
যেহেতু, y = mx + c
সুতরাং, ঢাল = 1

a, c, d এই তিনটি অপশনের সমীকরণ দেখলে বুঝা যায় যে এগুলোর সমাধান করলে x এর সহগ 1 হবে না

সমান্তরাল হতে হলে দুটির ঢাল ই সমান হতে হবে। অপশনের মধ্য থেকে
2y = 2x - 6
⇒ y = x - 3

সুতরাং, ঢাল = 1

Question: The average of a natural number and Its cube Is 13 times the number. The cube of the number is:
(Janata RC 2022 অনুযায়ী) 

Solution:
let the natural number be = x

According to the Question,
(x + x³)/2 = 13x
⇒ x + x³ = 26x
⇒ x³ = 26x - x
⇒ x³ = 25x
⇒ x³/x = 25
⇒ x² = 25
⇒ x = ± 5

But since x is a natural number, the value of x must be positive.

Therefore, x = 5.
Hence, x³ = 5³ = 125

Let, y = f(x) = 2x - 1
or, y = 2x - 1
or, 2x = y + 1
or, x = (y + 1)/2
∴ y = f(x)
Or, f^-1(y) = x
or, f^-1(y) = (y + 1)/2
∴ f^-1(x) = (x + 1)/2

Question: If n(A) = 39, n(B) = 23 and n(A ∩ B) = 19, then n(A ∪ B) = ?

Solution:
We know that,
n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
= 39 + 23 - 19
= 62 - 19
= 43

Question: If x = 7 - 4√3, then 

Solution: 
Given,
x = 7 - 4√3
⇒ x = 4 + 3 - 4√3
⇒ x = 2² + (√3)² - 2 × 2√3
⇒ x = (2 - √3)²  [ a² - 2ab + b² = (a - b)² ]
∴ √x = 2 - √3

Again,
√x = 2 - √3
⇒ 1/√x = 1/(2 - √3)
⇒ 1/√x = (2 + √3)/{(2 - √3) (2 + √3)}
⇒ 1/√x = (2 + √3)/(4 - 3)
∴ 1/√x = 2 + √3

∴ √x + (1/√x) = 2 - √3 + 2 + √3
∴ √x + (1/√x) = 4

Question: The series is: (2/√5), - 2, 2√5, - 10, ......... What is the seventh term of this series?

Solution:
Here,
First term, a = 2/√5
Common ratio, r = - 2/(2/√5)
= - 2 × √5/2
= - √5

We know that,
The nth term of a geometric progression is given by = ar^{n - 1}
∴ Seventh term = ar^{7 - 1}
= ar⁶
= (2/√5) × (-√5)⁶
= (2/√5) × {(-√5)2}3
= (2/√5) × (5)3
= (2/√5) × 125
= 250/√5
= (250 × √5)/5
= 50√5

∴ 7^th term = 50√5

Question: Which of the following describes all values of x for which 1 - x² ≥ 0?

Solution:
1 - x² ≥ 0
⇒ - x² ≥ - 1
⇒ x² ≤ 1
⇒ x² ≤ 1²
∴ - 1 ≤ x ≤ 1

Question: What is the sum of the following sequence: 5, 12, 19, 26, ... , 54?

Solution:
এটি একটি সমান্তর ধারা (arithmetic series)।
প্রথম পদ, a = 5
সাধারণ অন্তর, d = 12 - 5 = 7
শেষ পদ= 54

আমরা জানি,
n তম পদ = a + (n - 1)d
⇒ 54 = 5 + (n - 1)7
⇒ 49 = 7(n - 1)
⇒ n - 1 = 7
⇒ n = 8

সমষ্টি, S_n = n/2{2a + (n - 1)d}
∴ S₈ = (8/2){2(5) + (8 - 1)7}
= 4{10 + (7 × 7)}
= 4{10 + 49}
= 4 × 59
= 236

অতএব, প্রদত্ত ধারাটির সমষ্টি হলো 236

Let, A³ = 27 = 3³
So, A = 3
and, A² = 9

So the maximum value of a² b² c² = (1/3 × 1/3 × 1/3) = 1/27
( When the sum of the three positive quantities is fixed, the product will be maximum when the quantities are equal)
Hence, the maximum value of ABC = 1/√27
= 1/(3√3)

প্রদত অসমতাটি হলো,
1 - 3x ≤ 4
⇒ 1 - 3x - 1 ≤ 4 - 1
⇒ -3x ≤ 3
⇒ 3x ≥ -3 [উভয়পক্ষে (-1) দ্বারা গুণ করে]
⇒ x ≥ -3/3
⇒ x ≥ -1
Answer: x ≥ -1.

Question: If x + 1/x = 2, then what is the value of x¹⁰ + x¹⁰⁰?

Solution:
দেওয়া আছে,
x + 1/x = 2
⇒ x² + 1 = 2x [উভয় পক্ষকে x দ্বারা গুণ]
⇒ x² - 2x + 1 = 0
⇒ (x - 1)² = 0
⇒ x - 1 = 0
⇒ x = 1

এখন,
x¹⁰ + x¹⁰⁰
= (1)¹⁰ + (1)¹⁰⁰
= 1 + 1
= 2

সুতরাং, নির্ণেয় মান হলো 2।

Given, 3x - 7y = 0 .... (i)
⇒ 3x = 7y
and x + 2y = 13 .... (ii)
(ii)×3 ⇔ 3x + 6y = 39
⇒ 7y + 6y = 39
⇒ 13y = 39
∴ y = 3

প্রশ্ন:

সমাধান:

Given,
(2a + b)/(a + 4b) = 3
2a + b = 3a + 12b
-a = 11b
a = -11b

∴ (a + b)/(a + 2b)
= (-11b + b)/(-11b + 2b)
= -10b/-9b
= 10/9.

Question: What will be the result if (4x + 20)/4 is subtracted from (x + 10)?

Solution:
Expression = (x + 10) - {(4x + 20)/4}
= (x + 10) - {4(x + 5)/4}
= (x + 10) - (x + 5)
= x + 10 - x - 5
= 5

Question: If 4x² - 6x + 1 = 0, then the value of 8x³ + 1/8x³ is- 

Solution:

a² + b²
= 1/2{(a + b)² + (a - b)²}
= 1/2(13² + 3²)
= 1/2(169 + 9)
= 89

Question: The number of subsets of a set with 6 elements is:

Solution:
- কোনো সেট থেকে যতগুলো সেট গঠন করা যায়, এদের প্রত্যেকটি সেটকে ঐ সেটের উপসেট (subset) বলা হয়।
কোনো সেটের উপাদানের সংখ্যা, n = 6
ঐ সেটের উপসেট (subset) সংখ্যা = 2ⁿ
=2⁶
= 64

Question: Nine times a whole number is equal to five less than twice the square of the number. Find the number?

Solution: Let the required whole number be x.

According to the question,
9x = 2x² - 5
⇒ 2x² - 9x - 5 = 0
⇒(x - 5)(2x + 1) = 0
⇒ x - 5 = 0 or 2x + 1 = 0
⇒ x = 5 or x = - 1/2

Since x is supposed to be a whole number, the answer, i.e., the required whole number is 5.

Question: Identify the irrational number from the following options.
(Officer General 22 এর অনুরূপ)

Soluiton:
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 কোনাে স্বাভাবিক সংখ্যার বর্গমূল কিংবা তার ভগ্নাংশ একটি অমূলদ সংখ্যা।
   যেমন√2 = 1.414213..., √3 = 1.732 ...,  ইত্যাদি অমূলদ সংখ্যা।

- কোনাে অমূলদ সংখ্যাকে দুইটিপূর্ণ সংখ্যার অনুপাত হিসেবে প্রকাশ করা যায় না।
-  অমূলদ সংখ্যাকে একটি মূলদ সংখ্যা দ্বারা গুণ করলে অমূলদ সংখ্যা পাওয়া যায়।

Question:

Solution:

Question: a + b = √7, a - b = √5. Find the value of 17ab(a² + b²) = ?

Solution:
Given,
a + b = √7
a - b = √5

ATQ,
17ab(a² + b²)
= (17/8) × 8ab(a² + b²)
= (17/8) × 4ab × 2(a² + b²)
= (17/8) × {(a + b)²- (a - b)²)} {(a + b)²+(a - b)²)} 
= (17/8) × {(√7)²- (√5)²)} {(√7)²+(√5)²)}
= (17/8) × (7 - 5) × (7 + 5)
= (17/8) × 2 × 12
= (17/8) × 24
= 17 × 3
= 51

Question: The next term of the series: 25, 49, 81, ____ is

Solution:
Given: 25, 49, 81, ____
The series is: 5², 7², 9², 11²
So, the next term is 11² = 121 

Question: If (x + 7)² = 81, which of the following can be the value of (x - 5)? 

Solution: 
Given that,
(x + 7)² = 81
⇒ x + 7 = ± √81
x + 7 = ± 9
So there are two possible solutions.

Case 1: (Positive value) 
x + 7 = 9
⇒ x = 9 - 7
⇒ x = 2

Case 2: (Negative value)
⇒ x + 7 = - 9
⇒ x = - 9 - 7
∴ x = - 16

So x = 2, - 16

Now, x - 5 = 2 - 5 = - 3 ; [x = 2]

প্রশ্ন: x + y = x - y হলে, y এর মান নিচের কোনটি?

সমাধান:
দেওয়া আছে,
x + y = x - y
⇒ y = - y
⇒ y + y = 0
⇒ 2y = 0
∴ y = 0

Total Number = Club A + Club B - both club (van Diagram)
or, 42 = 20 + 28 - both
or, both = 6.

Question: If the sum of two numbers is 18 and the sum of their squares is 234, then what is the product of the two numbers?

Solution:
সংখ্যা দুটি যথাক্রমে x এবং y
∴ x + y = 18 এবং x² + y² = 234

আমরা জানি,
(x + y)² = x² + y² + 2xy
⇒ (18)² = 234 + 2xy
⇒ 324 = 234 + 2xy
⇒ 2xy = 324 - 234
⇒ 2xy = 90
⇒ xy = 90/2
∴ xy = 45

Given, x² + 9/x² = 31
Or, x²  + (3/x)²  = 31
Or, (x - 3/x)² + 2.x.3/x = 31
Or, (x - 3/x)²  = 31 - 6
Or, (x - 3/x)²  = 25
So, x - 3/x = 5