Algebra

Let the number be x and y
Then,
x²−y² = 63 & x−y = 3
On dividing, we get: x + y = 21
Solving x + y = 21 and x - y = 3,
We get: x = 12 and y = 9
∴ Larger number = 12

Question: What is the distance of (5, 12) from the origin?

Solution:
Since the distance of the coordinate (5, 12) is taken from the origin, then the coordinates of the origin are (0, 0).
Therefore,
x₁ = 0, y₁ = 0
x₂ = 5, y₂ = 12

We know,
The distance between two points = √[(x₂ - x₁)² + (y₂ - y₁)²]
= √[(5 - 0)² + (12 - 0)²]
= √[25 + 144] 
= √169
= 13 units

Let X be the number which is added to 80
80% of X = 0.8X
Now,
80 + 0.8X = X
0.2X = 80
X = 80/0.2 = 400

Question: [(289)^0.17 × (17)^0.16]² = ? 
(Janata RC 22 এর অনুরূপ)

Solution:
(289)^0.17 × (17)^0.16
= {(17)²}^0.17 × (17)^0.16
= 17^{(2 × 0.17)} × (17)^0.16
= (17)^0.34 × (17)^0.16
= (17)^{0.34 + 0.16}
= (17)^0.50
= (17)^50/100
= (17)^1/2
= √17

Hence, (√17)² = 17

Question: Solution set of the inequality: p - 5 > 4p + 7 Is
(Janata RC 2022 অনুযায়ী)

Solution:
p - 5 > 4p + 7
⇒ - 5 > 4p - p + 7
⇒ - 5 > 3p + 7
⇒ - 5 - 7 > 3p
⇒ - 12 > 3p
⇒ - 12/3 > 3p/3
⇒ - 4 > p
⇒ p < - 4

∴ নির্ণেয় সমাধান সেট: (- ∞, - 4)

Given, 3√x = 2√3
⇒ √x/√3 = 2/3
⇒ x/3 = 4/9
∴ x = (4×3)/9 = 1.33

Question: If |2x - 2| ≤ 8, what is the maximum value of x?

Solution:
Given that,
|2x - 2| ≤ 8
⇒ - 8 ≤ 2x - 2 ≤ 8 
⇒ - 8 + 2 ≤ 2x - 2 + 2 ≤ 8 + 2 
⇒ - 6 ≤ 2x ≤ 10
⇒ (- 6/2) ≤ (2x/2) ≤ (10/2)
⇒ - 3 ≤ x ≤ 5

∴ The maximum value of x is 5

3x² + 2x² - 6x + k = 0
⇒ 3(1)² + 2(1)² - 6(1) + k = 0 [As, x - 1 is a factor]
⇒ 3 + 2 - 6 + k  = 0
⇒ k = 1

Question: The square root of (4 + 3√5)(4 - 3√5) is:

Solution:
Using the identity (a + b)(a - b) = a² - b²

We get,
(4 + 3√5)(4 - 3√5) = 4² - (3√5)²
= 16 - 45
= -29

Since the result is negative,
√(-29) = i√29

Therefore, the square root is i√29.

Question: In a class, 54 students are good in Bangla only, 63 students are good in Mathematics only and 41 students are good in English only. There are 18 students who are good in both Bangla and Mathematics. 10 students are good in all three subjects. What is the number of students who are good in either Bangla or Mathematics but not in English?

Solution:

No. of students who are good in either Bangla or Mathematics but not in English = 54 + 18 + 63 = 135
Let B₁M₁E₁ denote the set of students studying Bangla, Mathematics and English.
No. of students of English only = 41
No. of students of Bangla only = 63
No. of students of Maths only = 54
n(B ∩ M ∩ E) = 10
n(B ∩ M) = 18
No. of students who study 'B' or 'M' but not 'E' = 63 + 54 + 18 - 10 = 125

Question: If x + y = 8 and xy = 20, then what is the value of x³ + y³ = ?

Solution:
Given that, 
x + y = 8 and xy = 20

We know, 
x³ + y³ = (x + y)³ - 3xy(x + y)
= (8)³ - 3 × 20 × 8
= 512 - 480
= 32

Question: If (√11 - 2)/(√11 + 2) = a√11 + b, then the value of a is-

Solution:
L.H.S = (√11 - 2)/(√11 + 2)
= {(√11 - 2)/(√11 + 2)} × (√11 - 2)/(√11 - 2)
= (√11 - 2)²/{(√11)²- 2²}
= (11 + 4 - 2 × 2 × √11)/(11 - 4)
= (15 - 4√11)/7
= (15/7) - (4/7) × √11
= - (4/7) × √11 + (15/7)
= a√11 + b (R.H.S)
(Compare the coefficients of √11 and constant term)
a = - (4/7)
b = (15/7)

∴ the value of a =  - (4/7)

Question: If x ≥ 7 and y ≤ 4 which of the following must be true?

Solution:
Given that,
x ≥ 7
and y ≤ 4
⇒ - y ≥ - 4

Now,
x - y ≥ 7 - 4
∴ x - y ≥ 3

Question: How many terms of the arithmetic should be progression 3, 7, 11, ... taken to make its sum equals to 820?

Solution:
এটি একটি সমান্তর ধারা,
যার ১ম পদ, a = 3
সাধারণ অন্তর, d = 4

আমরা জানি,
n- তম পদের সমষ্টি,
Sn = (n/2)[2a + (n - 1)d]

প্রশ্নমতে,
(n/2)[2a + (n - 1)d] = 820
⇒ (n/2)[6 + (n - 1)4] = 820
⇒ (n/2)[6 + 4n - 4] = 820
⇒ (n/2)[2(1 + 2n)] = 820
⇒ n(2n + 1) = 820
⇒ 2n² + n - 820 = 0
⇒ 2n² - 40n + 41n - 820 = 0
⇒ n(2n - 40) + 41(n - 40) = 0
⇒ (2n - 40)(n + 41) = 0
হয়, 
⇒ 2n - 40 = 0
⇒ 2n = 40 
n = 20

অথবা,
n + 41 = 0
n = - 41   ;[যা গ্রহণযোগ্য নয়]

সুতরাং, প্রদত্ত ধারাটিতে পদ আছে 20 টি। 

Question: If n(U) = 50, n(A) = 28, n(B) = 26 and n(A ∩ B) = 12 then n(A ∪ B)′ = ?

Solution:
আমরা জানি,
n(A ∪ B)= n(A) + n(B) - (A ∩ B)
= 28 + 26 - 12
= 42

এখন,
n(A ∪ B)′= n(U) -  n(A ∪ B)
= 50 - 42
= 8

সুতরাং, n(A ∪ B)′ = 8

Question: Express the following inequality using absolute value notation:
- 18 < x < - 6

Solution:
Given: - 18 < x < - 6
The midpoint (average) of - 18 and - 6 is,
Midpoint = {- 18 + (- 6)}/2
= - 24/2
= - 12

Now add 12 to all parts of the inequality to center it at zero.
- 18 + 12 < x + 12 < - 6 + 12
⇒ - 6 < x + 12 < 6
This is equivalent to |x + 12| < 6

Question: If p and q are the roots of the equation 3x²− 7x + 2 = 0, then what is the value of (1/p) + (1/q)?
 
Solution:

3x²− 7x + 2 = 0
⇒ 3x²- 6x - x + 2 = 0
⇒ 3x (x - 2) - 1 (x - 2) = 0
⇒ (3x - 1)(x - 2) = 0
⇒ x = 1/3 = p
∴ x = 2 = q

Now,
1/p + 1/q
= 1/(1/3) + 1/2
= 3 + 1/2
= 7/2

We know,
n(B U E U F) = n(B) + n(E) + n(F) - n(B ∩ E) - n(B ∩ F) - n(E ∩ F) + n(B ∩ E ∩ F)
Or, 24 = 6 + 12 + 15 + 1 - 2 - 2 - n(E ∩ F)
Or, n(E ∩ F) = 6

Question: If the sum of 3 consecutive integers is 210, then the sum of the two smaller integer is-

Solution:
Let,
Three consecutive integers is, x - 1 , x, x + 1

ATQ,
x - 1 + x + x + 1 = 210
⇒ 3x = 210
∴ x = 70

The sum of the two smaller integer is = x - 1 + x
= 70 - 1 + 70
= 140 - 1
= 139

এখানে,
(2x² + 3x - 5)(x + 2) = 2x³ + 4x² + 3x² + 6x - 5x - 10 = ax³+bx²+cx+d
⇒ 2x³ + 7x² + x - 10 = ax³ + bx² + cx + d
উভয় দিকে তুলনা করে পাই
a = 2, b = 7, c = 1, d = -10  
 অতএব, ac - bd = 2×1 -7(-10)
                    = 2 + 70
                    = 72

Let total boys is 4 and participated in sports is 1
Total girls is 8 and participated in sports 3
So, total student 4 + 8 = 12,
and participant = 1 + 3 = 4
Therefore, the proportion = 4/12

Question: If a - (1/a) = √5, what is the value of a³ - (1/a³)?

Solution:
দেওয়া আছে,
a - 1/a = √5

এখন, 
a³ - (1/a³)
= {a - (1/a)}³ + 3 . a . 1/a . {(a - 1/a)}
= (√5)³ + 3
= 5√5 + 3√5
= 8√5

Question: If Set A = {1, 2, 3} and Set B = {1}, which of the following is true?

Solution:
A = {1, 2, 3}, B = {1}
∴ A - B = {1, 2, 3} - {1}
= {2, 3} ; true

খ)
A ∩ B ; common elements
A ∩ B = {1}
not equal to {1, 2, 3} ;  false

গ) 
A × B ; set of all ordered pairs (a, b) where a ∈ A and b ∈ B
A × B = {(1, 1), (2, 1), (3, 1)}
This is not equal to {1, 2, 3} ; false

ঘ) 
A ∪ B ; all elements from A or B
A ∪ B = {1, 2, 3}
not equal to {1} ; false

Final answer ক) A - B = {2, 3}

Question: How many terms are there in the geometric progression (GP) is 5 + 20 + 80 + 320 +........... + 20480?

Solution:
First term, a = 5
Common ratio, r = 20/5 = 4
And
Last term ,l = 20480

We know,
a_n = a⋅r^{n - 1}
⇒ 20480 = 5 × (4^{n - 1})
⇒ 4^{n - 1} = 20480/5 = 4096
⇒ 4^{n - 1}  = 4⁶
⇒ n - 1 = 6
∴ n = 7
So the number of terms is 7

Question: If X ∈ N and 17 < x < 23, and x is a prime number, then which of the following represents the list form of the set of such numbers?

Solution:
দেয়া আছে:
X ∈ N and 17 < x < 23

List all natural numbers between 17 and 23
⇒ 18, 19, 20, 21, 22

∴ Identify the prime numbers among them
⇒ 18 → divisible by 2; not prime.
⇒ 19 → prime.
⇒ 20 → divisible by 2; not prime.
⇒ 21 → divisible by 3 and 7; not prime.
⇒ 22 → divisible by 2; not prime.

∴ List of prime numbers in this range
{19}.

এখানে, 2x² - 4x + p = 0 সমীকরণকে ax^2 + bx + c = 0 সমীকরণের সাথে তুলনা করলে বাস্তব মূলের জন্য b² - 4ac ≥ 0 হবে
∴ (-4)² - 4(2)(p)  ≥  0
⇒ 16 - 8p ≥ 0
⇒ 16 ≥ 8p
⇒ 8p ≤ 16
∴ p ≤ 2

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)

প্রশ্ন: If a + b + c = 6 and a² + b² + c² = 14 find the value of (ab + bc + ca).

সমাধান:
দেওয়া আছে,
a + b + c = 6 এবং a² + b² + c² = 14

আমরা জানি,
(a + b + c)² = ( a² + b² + c²) + 2(ab + bc + ca)
বা, (6)² =14 + 2(ab + bc + ca)
বা, 36 = 14 + 2(ab + bc + ca)
বা, 36 - 14 = 2(ab + bc + ca)
বা, 22 = 2(ab + bc + ca)
বা, ab + bc + ca = 22/2
বা, ab + bc + ca = 11

Question: Determine x for which x² − 8x +15 is less than zero.

Solution:
Given,
x² − 8x +15 < 0
⇒ x² - 3x - 5x + 15 < 0
⇒ x(x - 3) - 5(x - 3) < 0
⇒ (x - 3)(x - 5) < 0

The inequality will be true if x - 3 > 0 and x - 5 < 0 .
x - 3 > 0
or, x > 3

x - 5 < 0
or, x < 5
The inequality will be true if 3 < x < 5

∴ The solution of the inequality is 3 < x < 5

Question: If a + (1/a) = 3, what is a³ + (1/a³)?

Solution:
দেওয়া আছে
a + (1/a) = 3

a³ + 1/a³ = (a + 1/a)³ - 3.a.1/a(a + 1/a)
= 3³ - 3 × 3
= 27 - 9
= 18