Algebra

A/Q, x+1 + 2x-1 + 3x+1 = 25
So, x = 4
Hence, 1st side = 4 + 1 = 5
2nd side = 2 × 4 - 1 = 7
3rd side = 3 × 4 + 1 = 13

Question: If y : x = 1 : 5 and 2x + y = 22, then what is the value of y?

Solution:
Given,
y : x = 1 : 5
⇒ y/x = 1/5
∴ y = x/5 ..........(1)

and,
2x + y = 22
⇒ 2x + (x/5) = 22
⇒ (10x + x)/5 = 22
⇒ 11x = 22 × 5
⇒ 11x = 110
⇒ x = 110/11
∴ x = 10

From equation (1),
y = 10/5 = 2

Question: What values of x satisfy the inequality 2 - 3x > 1?

Solution:
দেওয়া আছে, 
2 - 3x > 1
⇒ 2 - 3x - 2 > 1 - 2 ; [অসমতার উভয় পক্ষ থেকে ২ বিয়োগ করে পাই]
⇒ - 3x > - 1
⇒ - 3x/- 3 < - 1/- 3  ; [একটি ঋণাত্মক সংখ্যা দ্বারা অসমতাকে ভাগ করলে, তখন অসমতার চিহ্নটি উল্টে যায়।]
∴ x < 1/3

Given,
a/b + b/a = 2
⇒ (a² + b²)/ab = 2
⇒ (a² + b²) = 2ab
⇒ a² + b² - 2ab = 0
⇒ (a - b)² = 0
⇒ a - b = 0.

1 - 2x ≤ 3
⇒ 1 - 2x -1 ≤ 3 - 1
⇒- 2x ≤ -2
⇒ -2x/2 ≥ 2/-2 [-2 দ্বারা ভাগ করে]
∴ x ≥ -1

(x + y > 5 
(x – y  > 3 
^___________
      2x > 8
     ∴ x > 4

If t is odd, then 3t will always be odd
Thus, odd + odd = even (3t + 1 = even number)

Question: The members of a club participate in at least one game. Twenty of them play football, 10 play cricket, 12 play hokey. Three of them play cricket only, 4 of them play both the cricket and football but not hockey, 2 of them participate all games. How many people play both cricket and hockey but not football?

Solution:
ফুটবল খেলে = 20 জন
ক্রিকেট খেলে = 10 জন
হকি খেলে = 12 জন
শুধু ক্রিকেট খেলে = 3 জন
ক্রিকেট ও ফুটবল খেলে = 4 জন
ক্রিকেট, ফুটবল ও হকি খেলে = 2 জন
 
হকি ও ক্রিকেট খেলে কিন্তু ফুটবল খেলে না এদের সংখ্যা = { 10 - ( 3 + 4 + 2)} 
= 1 জন।

Let the three numbers be x, y, and z.

Given:Sum of squares of three numbers is 138 and sum of their products taken two at a time is 131
Therefore,
x²+y²+z²=138
xy + yz + zx=131

Formula:
(a + b + c)²= a² + b² + c²+ 2 (ab + bc + ca)
This formula can be used to easily find the sum of three numbers.

Substituting the values, we get
(x + y + z)²= x²+ y²+ z²+ 2 (xy + yz + zx)
(x + y + z )²= 138 + 2(131)
(x + y + z )²= 400
Hence, (x + y + z) = 20.

Question: If 3x + 2y = 12 and xy = 6 , then find the value of 27x³ + 8y³ = ?

Solution:
দেওয়া আছে, 3x + 2y = 12 এবং xy = 6
এখন,
27x³ + 8y³
= (3x)³ + (2y)³
= (3x + 2y)³ - 3 × 3x × 2y(3x + 2y) [a³ + b³ = (a + b)³ - 3ab(a + b)]
= (3x + 2y)³ - 18xy(3x + 2y)
= (12)³ - 18 × 6 × 12
= 1728 - 1296
= 432

সুতরাং, নির্ণেয় মান হলো 432।

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: If (x + 5)² = 64, which of the following can be the value of (x + 2)?

Solution:
Given, (x + 5)² = 64
⇒ (x + 5)² = 8²
∴ x + 5 = ± 8

Case 1: x + 5 = 8
x = 8 - 5 = 3
x + 2 = 3 + 2 = 5

Case 2: x + 5 = - 8
x = -8 - 5 = -13
x + 2 = -13 + 2 = - 11

Possible values of (x + 2) are 5 or - 11.

Correct Answer: ক) 5

Question: If (m + 4)² = 36, which of the following can be the value of (m - 3)?

Solution:
Given that,
(m + 4)² = 36
or, (m + 4)² = 6²
or, m + 4 = ± 6

Case 1 : m + 4 = 6
m = 6 - 4 = 2
m - 3 = 2 - 3 = - 1

Case 2 : m + 4 = - 6
m = - 6 - 4 = - 10
m - 3 = - 10 - 3 = - 13

Possible values of (m - 3) are - 1 or - 13

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

Solution:

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

প্রশ্ন: যদি a = 0.202 হয়, তাহলে 

এর মান কত?

সমাধান:

সঠিক উত্তর 1.202 হবে, যেহেতু (+) যোগ চিহ্ন দিয়ে বের করা রাশির উত্তর নেই।

According to math,
If,
x = y
Then, 1 - q = 2q + 1
⇒ 2q + q = 1 - 1
⇒ 3q = 0
⇒ q = 0.

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

Given that 1² + 2² + 3²..... + 10² = 385

Now, 3² + 6² + 9² + .....30²
= 3²×1² + 2²×3² + 3²×3² + ........ + 3²×10²
=  3²×(1² + 2² + 3²..... + 10²)
= 9 × 385
= 3465

Question: If A = {x ∈ N : 3 ≤ x < 8} and B = {x ∈ N: x is an odd number and x < 10}, what is the value of A ∩ B?

Solution:
দেওয়া আছে,
A = {x ∈ N : 3 ≤ x < 8}
এখানে, x এর মান 3 এর সমান বা বড় এবং 8 এর ছোট স্বাভাবিক সংখ্যা।
∴ A = {3, 4, 5, 6, 7}

আবার,
B = {x ∈ N : x বিজোড় সংখ্যা এবং x < 10}
x স্বাভাবিক বিজোড় সংখ্যা যা 10 এর ছোট।
∴ B = {1, 3, 5, 7, 9}

প্রদত্ত রাশি, A ∩ B
= {3, 4, 5, 6, 7} ∩ {1, 3, 5, 7, 9}
= {3, 5, 7}

অতএব, A ∩ B এর মান হলো {3, 5, 7}।

Question: If (x/y) + (y/x) = √8 then what is the value of (x⁴/y⁴) + (y⁴/x⁴) ?

Solution:
Given that, 
(x/y) + (y/x) = √8

∴ x⁴/y⁴ + y⁴/x⁴
= (x/y)⁴ + (y/x)⁴
= {(x/y)²}² + {(y/x)²}²
= {(x/y)² + (y/x)²}² - 2.(x²/y²).(y²/x²)
= {(x/y)² + (y/x)²}² - 2
= [{(x/y) + (y/x)}² - 2.(x/y).(y/x)]² - 2
= {(√8)² - 2}² - 2
= (8 - 2)² - 2
= 6² - 2
= 36 - 2
= 34

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: A geometric series has its first term as 1 divided by square root of 3, and its common ratio is √3. Which term in the sequence is 81√3?

Solution:
First term, a = 1/√3
Common ratio, r = √3
Let, the n-th term = ar^{n - 1}
⇒ (1/√3) . (√3)^{n - 1} = 81√3
⇒ (√3)^{n - 1} = 81√3 × √3
⇒ (√3)^{n - 1} = 243
⇒ (√3)^{n - 1} = (√3)¹⁰
⇒ n - 1 = 10
∴ n = 11

So, the 11th term is 81√3.

Question: If 1 < p < 4 and 2 < q < 6, which of the following best describes p - q?

Solution: দেয়া আছে:
1 < p < 4 -------------(1)
2 < q < 6 -------------(2)

এখন, আমরা p - q এর সীমা বের করতে চাই।

(2)⇒
2 < q < 6
⇒ - 2 > -q > - 6 (যদি -1 দ্বারা গুণ করি)
⇒ - 6 < -q < - 2 ----------(3)

(1) এবং (3) যোগ করি,
⇒ (1 + (- 6 )) < p - q < (4 + (- 2))
⇒ - 5 < p - q < 2

Question: Find 

Solution:

Question: What is the solution of the inequality: - 10 < 2x - 4 ≤ 6 = ?

Solution:
Given that,
- 10 < 2x - 4 ≤ 6
⇒ - 10 + 4 < 2x - 4 + 4 ≤ 6 + 4
⇒ - 6 < 2x ≤ 10
⇒ - 6/2 < 2x/2 ≤ 10/2
⇒ - 3 < x ≤ 5

∴ Solution of the inequality: (-3, 5]

Question: If |x - 3| < 4, then for which values of m and n will m < 3x + 5 < n hold?

Solution:
|x - 3| < 4
⇒ - 4 < x - 3 < 4
⇒ - 4 + 3 < x < 4 + 3
⇒ - 1 < x < 7
⇒ - 1 × 3 < 3x < 7 × 3  ; [Now multiply all parts by 3]
⇒ - 3 < 3x < 21
⇒ - 3 + 5 < 3x + 5 < 21 + 5 ; [Now add 5 to all parts]
⇒ 2 < 3x + 5 < 26

Comparing this with m < 3x + 5 < n, then we get,
m = 2 and n = 26.

Question: Solve the inequality: 4(x - 3) > 2(x + 5) + 6 

Solution:
Given that,
4(x - 3) > 2(x + 5) + 6
⇒ 4x - 12 > 2x + 10 + 6
⇒ 4x - 12 > 2x + 16
⇒ 4x - 12 - 2x > 2x + 16 - 2x  ; [Subtract 2x from both sides]
⇒ 2x - 12 > 16
⇒ 2x - 12 + 12 > 16 + 12  ; [Add 12 to both sides]
⇒ 2x > 28
⇒ 2x/2 > 28/2 ; [Divide both sides by 2]
∴ x > 14 

Question:

Solution:

Question: If B = {1, 2, 3, 4} and C = {x, y, z}, then B ∪ C = ?

Solution:
B ∪ C = {1, 2, 3, 4} ∪ {x, y, z}
= {1, 2, 3, 4, x, y, z}

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: If D is the midpoint of the points P(4, 1) and Q(10, 9), find the length of PD.

Solution:
দেওয়া আছে, P(4, 1) এবং Q(10, 9), এবং D হলো PQ-এর মধ্যবিন্দু।

প্রথমে PQ-এর দৈর্ঘ্য নির্ণয় করি:
PQ = √{(x₂ - x₁)² + (y₂ - y₁)²}
⇒ PQ = √{(10 - 4)² + (9 - 1)²}
⇒ PQ = √(6² + 8²)
⇒ PQ = √(36 + 64)
⇒ PQ = √100
∴ PQ = 10

যেহেতু D হলো PQ-এর মধ্যবিন্দু, তাই PD হবে PQ-এর অর্ধেক।
∴ PD = PQ/2
= 10/2
= 5

Question: If a² - √5a + 1 = 0, then the value of a² + a^{- 2} = ?

Solution:
দেয়া আছে, 
a² - √5a + 1 = 0
⇒ a² + 1 = √5a
⇒ a + (1/a) = √5 [উভয়পক্ষকে a দ্বারা ভাগ করে]

প্রদত্ত রাশি= (a² + a^{- 2})
= a² + (1/a²)
= {a + (1/a)²} - 2. a. (1/a)
= (√5)² - 2
= 5 - 2
= 3

∴ নির্ণেয় মান = 3

Question: If a + b = √3 and a = √2 + b, what is the value of 4ab?

Solution: 
given,
a + b = √3
a = √2 + b
∴ a - b = √2

4ab = (a + b)² - (a - b)²
= 3 - 2
= 1