Algebra

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: What will come at the place of question mark ?
7, 26, 63, 124, 215, 342, ?

Solution:
The terms are given in a series
(2³ - 1) = 7
(3³ - 1) = 26
(4³ - 1) = 63
(5³ - 1) = 124
(6³ - 1) = 215
(7³ - 1) = 342
So, the missing term is,
(8³ - 1) = 511.

Question: Find the equation of the line with x-intercept = 6 and y-intercept = 2.

Solution: 
x- intercept = 6, so, the line passes through (6, 0)
y- intercept = 2, so, the line passes through (0, 2)

We know, the intercept form of a line is:
(x/a) + (y/b) = 1, where a = x- intercept and b = y- intercept  
⇒ (x/6) + (y/2) = 1
⇒ (x + 3y)/6 = 1
⇒ x + 3y = 6
∴ x + 3y - 6 = 0

so, the equation of the line is  x + 3y - 6 = 0.

4x² + 20xy + 16y²
= 4x² + 4xy + 16xy + 16y²
= 4x(x + y) + 16y(x + y)
= (x + y)(4x + 16y)
= 4(x + y)(x + 4y)
= 4.5.4
= 80

a² + b² + c² - ab - bc - ca = (a + b + c)² - 3(ab + bc + ca)
= (7 + 5 + 3)² - 3{(7 × 5) + (5 × 3) + (3 × 7)}
= 15² + 3(35 + 15 + 21)
= 225 - 3 × 71
=225 - 213
= 12.

HCF of the given numbers will be the greatest number which can divide 48, 84 and 144
18 = 2 × 3 × 3
84 = 2 × 2 × 3 × 7
144 = 2 × 2 × 2 × 2 × 3 × 3
∴ HCF = 2 × 3 = 6
Hence 6 is the greatest number which divides 18, 84 and 144

When m is divided by n, the remainder is 8.
⇒m = nk + 8, where k is the quotient and n>8 (since Divisor > Remainder)
⇒m/n = (nk + 8)/n
⇒m/n = k + 8/n … (i)
In the above relation, 8/n must be a fractional value such that:
0 < 8/n < 1 ..… (ii)
We know that:
m/n = 89.32 = 89 + 0.32 ..… (iii)
Thus, from (i), (ii) and (iii), we have:
k = 89
8/n = 0.32
⇒ n = 8/0.32 = 100/4
⇒ n = 25

Question: If a + (1/a) = 4, what is the value of a³ + (1/a³)?

Solution:
দেওয়া আছে, a + (1/a) = 4

আমরা জানি,
a³ + (1/a³)
= {a + (1/a)}³ - 3 × a × (1/a) × (a + 1/a)
= (4)³ - 3 × 4
= 64 - 12
= 52

Question: What is the absolute value (modulus) of x if - 8 < x < 2?

Solution:
Given that, 
- 8 < x < 2
⇒ - 8 + 3 < x + 3 < 2 + 3
⇒ - 5 < x + 3 < 5
∴ |x + 3| < 5

Question: What should be the value of "P" so that the expression (9 − 12x + Px²) becomes a perfect square?

Solution:
(9 − 12x + Px²)
= (3)² − 2 × 3 × 2x + (2x)² + Px² - (2x)²
= (3 - 2x)² + Px² - 4x²

∴ the expression becomes a perfect square if,
Px² - 4x² = 0
⇒ Px² = 4x²
∴ P = 4

Question: n(A\B) + n(A ∩ B) = ?

Solution:
We know that,
A\B = {x : x ∈ A and x ∉ B} 

Example: A = {a, b, c} and B = {c, d}, 
then A ∩ B = {a, b, c} ∩ {c, d} = {c}
and A\B = {a, b, c} \ {c, d} = {a, b}

We can say that,
n(A\B) = n(A) - n(A ∩ B)

Now,
n(A\B) + n(A ∩ B) = n(A) - n(A ∩ B) + n(A ∩ B) = n(A)

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: The slope of the line 4x - 8y = 16 is not the same as the slope of which one of the following lines?

Solution:
প্রথমে, প্রদত্ত রেখাটির ঢাল নির্ণয় করতে হবে। রেখাটির সমীকরণকে y = mx + c আকারে রূপান্তর করতে হবে। এখানে 'm' হলো ঢাল (Slope)।

প্রদত্ত রেখার সমীকরণ:
4x - 8y = 16
⇒ - 8y = - 4x + 16
 ⇒ y = (- 4/- 8)x + (16/- 8)
⇒ y = (1/2)x - 2
∴ এই রেখাটির ঢাল (m) হলো 1/2.

এবার, প্রদত্ত অপশনগুলোর প্রত্যেকটির ঢাল নির্ণয় করি:

ক) x - 2y = 8
⇒ - 2y = - x + 8
⇒ y = (- x/- 2) + (8/- 2)
⇒ y = (1/2)x - 4
∴ ঢাল, m = 1/2

খ) 3x - 6y = 12
⇒ - 6y = - 3x + 12
⇒ y = (- 3/- 6)x + (12/- 6)
⇒ y = (1/2)x - 2
∴ ঢাল, m = 1/2

গ) y = 3x + 5
∴ ঢাল, m = 3

ঘ) y = x/2 + 4
⇒ y = (1/2)x + 4
∴ ঢাল, m = 1/2

সুতরাং, দেখা যাচ্ছে যে শুধুমাত্র অপশন (গ) এর রেখার ঢাল মূল রেখার ঢাল থেকে ভিন্ন।
∴ সঠিক উত্তর: গ) y = 3x + 5

Question: If a + (1/a) = √7, what is the value of a³ + (1/a³)?

Solution:
দেওয়া আছে, a + (1/a) = √7

এখন,
a³ + (1/a³)
= (a + 1/a)³ - 3 . a . 1/a(a + 1/a)
= (√7)³ - 3 × (√7)
= 7√7 − 3√7
= 4√7

Given, 4x - 3y = 24.5 ..... (i) and 5x + 2y = -1 ...... (ii)
Lets, (ii)×3 + (i)×2
⇒ 15x + 6y + 8x - 6y = - 3 + 49
⇒ 23x = 46
⇒ x = 2

Question: If x = 3t, y = 1/2 × (t + 1), then the value of t for which x = 2y is?

Solution:
Here given,
x = 3t -------(1)
and, y = 1/2 × (t + 1) ----(2)

Now, when:
x = 2y
⇒ x = 2 × 1/2 × (t + 1) [from equation 2]
⇒ x = t + 1 -------(3)
∴ 3t = t + 1 [From equation (1)]
⇒ 2t = 1
⇒ t = 1/2

Question: For the inequality |x - 1| < 5, where x ∈ N. What is the solution set?

Solution:
দেওয়া আছে, 
|x - 1| < 5 এবং x ∈ N
⇒ - 5 < x - 1 < 5
⇒ - 5 + 1 < x - 1 + 1 < 5 + 1
∴ - 4 < x < 6

এখন, 
 x ∈ N এর অর্থ হলো x স্বাভাবিক সংখ্যা। 
সুতরাং, - 4 < x < 6 সীমার মধ্যে স্বাভাবিক সংখ্যা গুলো হলো 1, 2, 3, 4, 5

সুতরাং, সমাধান সেট = {1, 2, 3, 4, 5}

Question: (p² - 7p + 10)/(p² - 8p + 15) = ?

Solution: 
(p² - 7p + 10)/(p² - 8p + 15) 
= (p² - 2p - 5p + 10)/(p² - 3p - 5p + 15) 
= {p(p - 2) - 5(p - 2)}/{p(p - 3) - 5(p - 3)} 
= (p - 2)(p - 5)/(p - 3)(p - 5) 
= (p - 2)/(p - 3)

Given,
5 < x < 10 and y = x + 5
Possible value of x = 6, 7, 8, 9
Take greatest value of x which is 9
So, y = 9+5 = 14
∴ x + y = 9 + 14 = 23

Question: If 7m² - 16mn + 4n² is divided by 7m - 2n, the result is-

Solution:
দেওয়া আছে,
7m² - 16mn + 4n²
= 7m² - 14mn - 2mn + 4n²
= 7m(m - 2n) - 2n(m - 2n)
= (m - 2n) (7m - 2n)

∴ (m - 2n) (7m - 2n)/(7m - 2n) = (m - 2n)

Question: In a group of 150 people, 90 people read Newspaper A, 65 people read Newspaper B, and 30 people read both Newspaper A and Newspaper B. How many people read neither Newspaper A nor Newspaper B?

Solution:
মোট লোক = 150
A পত্রিকা পড়ে, n(A) = 90
B পত্রিকা পড়ে, n(B) = 65
উভয়টি পড়ে, n(A ∩ B) = 30

∴ কমপক্ষে একটি পত্রিকা পড়ে, n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
= 90 + 65 - 30
= 125

∴ যারা কোনটিই পড়ে না = মোট লোক - যারা কমপক্ষে একটি পড়ে
= 150 - 125
= 25

∴ 25 জন কোন পত্রিকাই পড়ে না।

Question: P = {x ∈ N : x³ < 216}. Then, how many elements are there in set P?

Solution:
Here, N = the set of natural numbers
= {1, 2, 3, 4, 5, 6, 7, 8, ……}

Given that, 
P = {x ∈ N : x³ < 216}

Now check each value.
When x = 1, 1³ = 1 < 216
When x = 2, 2³ = 8 < 216
When x = 3, 3³ = 27 < 216
When x = 4, 4³ = 64 < 216
When x = 5, 5³ = 125 < 216
When x = 6, 6³ = 216 < 216 ; false (not true)

Therefore, the set P contains only the values that satisfy the condition.
P = {1, 2, 3, 4, 5}
∴ The number of elements in set P = 5

For option a, let's assume that A = 25, so option a can’t be entirely true
For option b, let's assume that A = 12, then option b is true
For option c, let's assume that A = 20, so option c is true
For option d, let's assume that A = 20, so option d is also true.
From the given condition of the options and question, option a can’t be inferred.

Question: What should be the value of "P" so that the expression (16 − 24x + Px²) becomes a perfect square?

Solution:
(16 − 24x + Px²)
= (4)² − 2 × 4 × 3x + (3x)²+ Px² − (3x)²
= (4 − 3x)² + Px² − 9x²

∴ The expression becomes a perfect square if,
Px² − 9x² = 0
⇒ Px² = 9x²
∴ P = 9

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

Solution: Let the required whole number be x.

According to the question,
5x = 2x² – 3
⇒ 2x² – 5x – 3 = 0
⇒ 2x² - 6x + x - 3 = 0
⇒ 2x(x - 3) + 1(x - 3)
⇒ (2x + 1)(x – 3) = 0

So,
2x + 1 = 0 or x – 3 = 0
⇒ x = –1/2 or x = 3

Since x must be a whole number,
∴ the required number is 3.

Question:  (32)^2/5 + (243)^1/5 = 3k , the value of k is:

Solution:
32^(2/5) = (2⁵)^2/5 = 2² = 4,
243^(1/5) = (3⁵)^1/5 = 3

∴ (32)^2/5 + (243)^1/5 = 3k
 ⇒ 4 + 3 = 3k
⇒ 7 = 3k
⇒ 3k = 7
⇒ k = 7/3

Question: A technician charges a Tk. 50 service fee plus Tk. 30 per hour for labur. If a customer's total cost is Tk. 200, what is the maximum number of full hours the technician can work?

Solution:
Given that, 
Fixed service fee = Tk. 50
Charge per hour = Tk. 30
Total bill = Tk. 200

Let h = number of full hours the technician works.

Now, total cost equation,
50 + 30h ≤ 200
⇒ 30h ≤ 200 - 50
⇒ 30h ≤ 150
⇒ h ≤ 150 / 30
∴ h ≤ 5

So the technician can work a maximum of 5 full hours.

Question: If 7 - 3x ≤ 16, then what is the value of x?

Solution:
7 - 3x ≤ 16
⇒ - 3x ≤ 16 - 7
⇒ - 3x ≤ 9
⇒  3x ≥ - 9
⇒ x ≥ - 9/3
⇒ x ≥ - 3

সমাধানটিকে ব্যবধি (Interval) আকারে প্রকাশ করলে হয়: [- 3, ∞)
এখানে তৃতীয় বন্ধনী [ দ্বারা বোঝায় - 3 সমাধান সেটের অন্তর্ভুক্ত, এবং ∞ এর পাশে প্রথম বন্ধনী) বোঝায় যে এটি অসীম পর্যন্ত।

Question: The price of 10 chairs is equal to that of 4 tables. The price of 15 chairs and 2 tables together is Tk. 4,000. The total price of 12 chairs and 3 tables is:

Solution: 
Let the cost of a chair and that of table be Rs. x and Rs. y respectively.
Then,10x = 4y or y = 5x/2
∴15x + 2y = 4000
⇒ 15x + 2 × 5x/2 = 4000
⇒ 20x = 4000
∴ x = 200
So, y = 5 × 200/2 = 500
Hence, the cost of 12 chairs and 3 tables = 12x + 3y = Tk. (2400 + 1500) = Tk.3900

Question: Solution set of the inequality: 3y + 4 ≥ 2y - 5 is-

Solution:
Given that,
3y + 4 ≥ 2y - 5
⇒ 3y + 4 - 2y ≥ 2y - 5 - 2y
⇒ y + 4 ≥ - 5
⇒ y ≥ - 5 - 4
⇒ y ≥ - 9

∴ Solution set of the inequality is  [- 9,  ∞)

Question: A geometric series has its first term as 1 divided by square root of 2, and its common ratio is √2. Which term in the sequence is 32√2?

Solution:
First term, a = 1/√2
Common ratio, r = √2

Let, the n-th term be = ar^{n - 1} = 32√2
⇒ (1/√2) (√2)^{n - 1} = 32√2
⇒ (√2)^{n - 1} = 32√2 × √2
⇒ (√2)^{n - 1} = 32 × 2
⇒ (√2)^{n - 1} = 64
⇒ (√2)^{n - 1} = (√2)¹²
⇒ n - 1 = 12
∴ n = 13

∴ The 13th term is 32√2.

let x = 5 and y = 3

Then,
(8x - 5y) : (8x + 5y)
= {(8 × 5) - (5 ×3)} : {(8 × 5} + (5 × 3)}
= (40 - 15) : (40 + 15)
= 25 : 55
= 5 : 11.

Question: Solve the inequality: 5(x - 3) + 7 < 3(2x - 1).

Solution:
Given inequality,
5(x - 3) + 7 < 3(2x - 1)
⇒ 5x - 15 + 7 < 6x - 3
⇒ 5x - 8 < 6x - 3
⇒ 5x - 6x < - 3 + 8
⇒ - x < 5
∴ x > - 5