Algebra

Let, Fred bought x potatoes and (12 - x) tomatoes

ATQ, 
0.24x + 0.76(12 - x) = 6.52
Or, 0.24x + 9.12 - 0.76x = 6.52
Or, 0.52x = 2.6
Or, x = 5

Question: If {1/|2q - 7|} > 1/5, then what is the value of q?

Solution:
Given that, 
{1/|2q - 7|} > 1/5
⇒ |2q - 7| < 5
⇒ - 5 < 2q - 7 < 5
⇒ - 5 + 7 < 2q - 7 + 7 < 5 + 7
⇒ 2 < 2q < 12
∴ 1 < q < 6

Question: If a² - 8 = 2√15, Than what is the value of a. 

Solution: 
Given that, 
a² - 8 = 2√15
⇒ a² = 8 + 2√15
⇒ a² = 5 + 2 × √5 × √3 + 3
⇒ a² = (√5)² + + 2 × √5 × √3 + (√3)²
⇒ a² = (√5 + √3)² ; [(a + b)² = a² + 2ab + b²]
∴ a = √5 + √3

Question: If x = 1 + √2 and y = 1 - √2, find the value of x² + y².

Solution: 
Given that,
x = 1 + √2,
y = 1 - √2

∴ x + y = 1 + √2 + 1 - √2
= 2

And,
xy = (1 + √2)(1 - √2)
= 1² - (√2)²
= 1 - 2
= - 1 

Now,
x² + y² = (x + y)² - 2xy
= (2)² - 2(- 1)
= 4 + 2
= 6

Question: Find the equation of the line with x-intercept = 6 and y-intercept = 5.

Solution:
Given,
x-intercept = 6, So, the line passes through (6, 0).
y-intercept = 5, So, the line passes through (0, 5).

We know, The intercept form of a line is:
(x/a) + (y/b) = 1, where a = x-intercept, b = y-intercept.
⇒ (x/6) + (y/5) = 1
⇒ (5x + 6y)/30 = 1
⇒ 5x + 6y = 30
⇒ 5x + 6y - 30 = 0

∴ The equation of the line is 5x + 6y - 30 = 0

Question: When x = (y + 3)² , which of the following matches (- 2y - 6)²?

Solution:
Here,
x = (y + 3)²

∴ (- 2y - 6)² ={- 2 (y + 3)}²
= 4 × (y + 3)²
= 4x

Question: If a - b = 3 and ab = 2, then the value of a³ - b³ - 3ab will be?
 
Solution:
We know,
a³ - b³ = (a - b)(a² + ab + b²)

So,
a³ - b³ - 3ab = (a - b)(a² + ab + b²) - 3ab
= 3 × (a² + b² + ab) - 3ab
= 3(a² + b²) + 3ab - 3ab
= 3(a² + b²)

Also, a² + b² = (a - b)²+ 2ab = 3² + 2×2 = 9 + 4 = 13

Therefore,
∴ a³- b³ - 3ab = 3 × 13 = 39

Question: If px² + 24x + 16 is a perfect square number, then p = ?

Solution:
দেওয়া আছে, রাশিটি একটি পূর্ণবর্গ সংখ্যা।
px² + 24x + 16
= (3x)² + 2(3x)(4) + (4)²

একটি রাশি পূর্ণবর্গ হওয়ার জন্য, এটি (a² + 2ab + b²) অথবা (a² - 2ab + b²) আকারের হতে হবে। এখানে,
a = 3x এবং b = 4

যেহেতু প্রথম পদটি a² এর সমান, 
∴ px² = a²
⇒ px² = (3x)²
⇒ px² = 9x²
⇒ p = 9

অতএব, p এর মান হলো 9।

Question: If a + b + c = 13 and a² + b² + c² = 69, then what is the value of ab + bc + ca?

Solution: 
We know,
(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
⇒ 2(ab + bc + ca) = (a + b + c)² - ( a² + b² + c²)
⇒ 2(ab + bc + ca) = 13² - 69 [given, a + b + c = 13 and a² + b² + c² = 69]
⇒ 2(ab + bc + ca) = 169 - 69 = 100
⇒ (ab + bc + ca) = 100/2
∴ (ab + bc + ca) = 50

5808 = (4 × 4) × (11 × 11) × 3 = 4² × 11² × 3 
∴ 5808 should be multiplied by another 3 to make it a full square. 

Question: If b is one-fourth of a, then what is the value of  

Solution:
Given, b = 1/4 of a = a/4

Now,

98.98 ÷ 11.03 + 7.014 × 15.99
= 121.128 ≈ 121 

Another approach:
98.98 ÷ 11.03 + 7.014 × 15.99
Let consider the equation as = 99 ÷ 11 + 7 × 16 = 9 + 112 = 121

(2√27 – √75 + √12)
= 2√(3².3) - √(3 × 5²) + √(3 × 2²)
= 6√3 – 5√3 + 2√3
= 3√3

Question: If x² + y² = 50 and xy = 21, what is the value of (x - y)²?
 
Solution:
We are given:
x² + y² = 50
xy = 21

Use the identity:
(x - y)² = x² + y² - 2xy

Substitute the values:
⇒ (x - y)² = x² + y² - 2xy
⇒ (x - y)² = 50 - 2 × 21
⇒ (x - y)² = 50 - 42
∴ (x - y)² = 8

Question: If x² - √7x + 1 = 0, then the value of x² + x^{- 2} = ?

Solution:
দেয়া আছে,
x² - √7x + 1 = 0
⇒ x² + 1 = √7x
⇒ x + (1/x) = √7 [উভয়পক্ষকে x দ্বারা ভাগ করে]

প্রদত্ত রাশি = x² + x^{- 2}
= x² + (1/x²)
= (x + 1/x)² - 2 . x . (1/x)
= (√7)² - 2
= 7 - 2
= 5

∴ নির্ণেয় মান হলো 5

Question: If a + b = √11 and a - b = √5, what is the value of 8ab(a² + b²)?

Solution:
দেওয়া আছে, a + b = √11 এবং a - b = √5

আমরা জানি,
4ab = (a + b)² - (a - b)²
2(a² + b²) = (a + b)² + (a - b)²

এখন,
8ab(a² + b²) = (4ab) × 2(a² + b²)
= [(a + b)² - (a - b)²] × [(a + b)² + (a - b)²]
= [(√11)² - (√5)²] × [(√11)² + (√5)²]
= (11 - 5) × (11 + 5)
= 6 × 16
= 96

According to the figure x and y are in negative relation but as they are in (+ +) coordination, so they have built a positive correlation. Which is satisfied only by y = 1/x equation. And equation a, b and d are equation of straightline. 

Question: If P = {1, 4, 9, 16, 25, 36, 49, 64}, the number of proper subsets of P is
(Janata RC 2022 অনুযায়ী)

Solution:
দেওয়া আছে,
P = {1, 4, 9, 16, 25, 36, 49, 64}

সেটের উপাদান সংখ্যা = 8

∴ প্রকৃত উপসেট সংখ্যা = 2ⁿ - 1
= 2⁸ - 1
= 256 - 1
= 255

Question: If x/y + y/x = √7, what is the value of (x⁴/y⁴) + (y⁴/x⁴)?

Solution: 
Given that, 
x/y + y/x = √7

Now, 
(x⁴/y⁴) + (y⁴/x⁴)
= (x²/y²)² + (y²/x²)²
= (x²/y² + y²/x²)² - 2 ; [a² + b² = (a + b)² - 2ab]
= {(x/y)² + (y/x)²}² - 2
= {(x/y + y/x)² - 2}² - 2 ; [a² + b² = (a + b)² - 2ab]
= {(√7)² - 2}² - 2
= (7 - 2)² - 2
= 25 - 2
= 23

Question: If x + y = a, x² + y² = b² and x³ + y³ = c³ then find the value of a³ + 2c³ = ? 

Solution: 
Given that, 
x + y = a .........(1)
x² + y² = b² .........(2)
And x³ + y³ = c³

Now, 
a³ + 2c³
= (x + y)³ + 2(x³ + y³)
= x³ + 3x²y + 3xy² + y³ + 2x³ + 2y³   ; [(a + b)³ = a³ + 3a²b + 3ab² + b³]
= 3x³ + 3y³ + 3xy(x + y)
= 3(x³ + y³) + 3xy(x + y)
= 3{(x + y)(x² - xy + y²)} + 3xy(x + y)  ; [a³ + b³ = (a + b)(a² - ab + b²)]
= 3(x + y)(x² - xy + y² + xy)
= 3(x + y)(x² + y²)
= 3ab² ; [From 1 and 2]

Let the missing number be x.
Given,
20 + 8 × 0.5/(20 - x) = 12
⇒ (20 + 4)/(20 - x) = 12
⇒ 24/(20 - x) = 12
⇒ 20 - x = 24/12
⇒ 20 - x = 2
⇒ x = 20 - 2
⇒ x = 18.

Question: If A = {a, b, c, d, e, f, g} and B = {d, e, f, g}, then A - B = ?

Solution:
A - B = {a, b, c, d, e, f, g} - {d, e, f, g}
= {a, b, c}

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 y = √8 + √7, then what is the value of y³ + (1/y³)?

Solution:
দেওয়া আছে,
y = √8 + √7
⇒ 1/y = 1/(√8 + √7)
⇒ 1/y = (√8 - √7)/(√8 + √7)(√8 - √7)
⇒ 1/y = (√8 - √7)/{(√8)² - (√7)²}
⇒ 1/y = (√8 - √7)/(8 - 7)
∴ 1/y = √8 - √7

এখন, y + 1/y = (√8 + √7) + (√8 - √7)
= 2√8 = 2 × 2√2 = 4√2

এখন,
y³ + (1/y³)
= (y + 1/y)³ - 3(y)(1/y)(y + 1/y)
= (y + 1/y)³ - 3(y + 1/y)
= (4√2)³ - 3(4√2)
= (4³)(√2)³ - 12√2
= 64(2√2) - 12√2
= 128√2 - 12√2
= 116√2

সুতরাং, নির্ণেয় মান হলো 116√2

Cloth is required for 1 shirt
= 2 m, 60 cm or 260 cm
Cloth is required for 7 shirt
= 260 × 7
= 1820 cm or 18 m 20 cm

Question: If (4P - 4)/(3P - 3) = (4/P) where P ≠ 1, what is the value of P² - 4P + 3 = ?

Solution:
Given that,
(4P - 4)/(3P - 3) = 4/P
or, P(4P - 4) = 4(3P - 3)
or, 4P²- 4P = 12P - 12
or, 4P²- 4P - 12P + 12 = 0
or, 4P²- 16P + 12 = 0
or, 4(P² - 4P + 3) = 0
or, P² - 4P + 3 = 0/4
∴ P² - 4P + 3 = 0

a - b ≥ a + b
⇒ a - b - b ≥ a
⇒ -2b ≥ 0
⇒ 2b ≤ 0 [-1 দ্বারা গুণ করে] 
∴ b ≤ 0

Question: Solve the inequality |x - 2| < 5

Solution:
|x - 2| < 5
⇒ - 5 < x - 2 < 5
⇒ - 5 + 2 < x - 2 + 2 < 5 + 2
⇒ - 3 < x < 7

Question: P = {x ∈ N : 2 < x ≤ 6} and Q = {x ∈ N : x is an even number and x ≤ 8}. Find the value of P ∩ Q.

Solution:
Given that, 
P = {x ∈ N : 2 < x ≤ 6}
∴ P = {3, 4, 5, 6}

Q = {x ∈ N : x is even and x ≤ 8}
∴ Q = {2, 4, 6, 8}

Now,
P ∩ Q = {3, 4, 5, 6} ∩ {2, 4, 6, 8}
= {4, 6}

Question: Find an equation for the line with x-intercept = 2, Y - intercept = - 1

Solution:
দেওয়া আছে, 
x-অক্ষের ছেদবিন্দু 2   ∴ বিন্দু (2, 0)
y-অক্ষের ছেদবিন্দু - 1  ∴ বিন্দু (0, - 1)

এখন, (2, 0) এবং (0, - 1) বিন্দুর ঢাল, 
m = (y₂ - y₁)/(x₂ - x₁)
⇒ m = (- 1 - 0)/(0 - 2) = 1/2
∴ m = 1/2

এখন, রেখার সমীকরণ
y = mx + b, [যেখানে, ঢাল m এবং y-অক্ষের ছেদবিন্দু, b ].
⇒ y = (1/2)x - 1  ; [m = 1/2, y-অক্ষের ছেদবিন্দু, b = - 1]
⇒ 2y = x - 2
∴ x - 2y = 2

Question: If y + (3/y) = 5, what is the value of y³ + (27/y³)?

Solution:
দেওয়া আছে y + 3/y = 5

∴ y³ + 27/y³ = (y + 3/y)³ - 3 × (y + 3/y) × 3
= 5³ - 9 × 5
= 125 - 45
= 80

Let geese be denoted by 'G' and Dogs by 'D'
Geese have 2 legs; Dogs have 4 legs.

Total Heads = G + D = 84 ------------------------- (1)
Total Legs = 2G + 4D = 282 --------------------- (2)

Divide equation 2 by 2, we get,
G + 2D = 141 -------------------------------------- (3)
Equation 3 - Equation 2
G + 2D - G - D = 141 - 84
∴ D = 57

So, Geese = 84 - 57 = 27

Question: If x² + y² = 40 and xy = 12, what is the value of (x - y)²?
 
Solution:
We are given:
x² + y² = 40
xy = 12

Use the identity:
(x - y)² = x² + y² - 2xy

Substitute the values:
⇒ (x - y)² = x² + y² - 2xy
⇒ (x - y)² = 40 - 2 × 12
⇒ (x - y)² = 40 - 24
∴ (x - y)² = 16

Question: Solve the inequality |1 - 2x| < 7

Solution:
Given that, 
|1 - 2x| < 7
⇒ - 7 < 1 - 2x < 7
⇒ - 7 - 1 < 1 - 1 - 2x < 7 - 1
⇒ - 8 < - 2x < 6
⇒ - 4 < - x < 3    (dividing by - 2 and reversing the inequality signs)
⇒ 4 > x > - 3
∴ - 3 < x < 4

Question: Solve the inequality, (x/4) < (4x - 1)/15

Solution: 
Given that, 
x/4 < (4x - 1)/15
⇒ 15x < 16x - 4
⇒ 15x - 16x < - 4
⇒ - x < - 4
⇒ x > 4 

Question: Find the equation of the line with x-intercept = 4 and y-intercept = 3.

Solution:
Given, x-intercept = 4,
So, the line passes through (4, 0).
y-intercept = 3,
So, the line passes through (0, 3).

We know,
The intercept form of a line is:
(x/a) + (y/b) = 1, where a = x-intercept, b = y-intercept.
⇒ (x/4) + (y/3) = 1
⇒ (3x + 4y)/12 = 1
⇒ 3x + 4y = 12
⇒ 3x + 4y - 12 = 0

∴ The equation of the line is 3x + 4y - 12 = 0

3/4 = 0.75
5/6 = 0.83
2/3 = 0.667
4/5 = 0.80
9/10 = 0.9
1/2 = 0.50

So, the fraction 4/5 is greater than 3/4 and less than 5/6

Product of 1^st and 4^th terms (extremes) = product of 2^nd and 3^rd terms (means)
⇒ 2.5x = 40
⇒ x = 40/2.5 = 16

We know,
Total = n(A) + n(B) - both + none
⇒ 36 = 20 + 28 - both + 0
⇒ 36 = 48 - both
⇒ both = 48 - 36
⇒ both = 12