ব্যাংক নিয়োগ প্রস্তুতি ⎯ লং কোর্স

Question: Which of the following is the polynomial equation 2x⁴ - 5x³ + 9x² - 4 = 0?

Solution:
The given polynomial equation is in terms of x.
The highest power of x is 4 and hence the degree of the equation is 4.
Hence, it is a biquadratic equation.

Question: (3a + 5b)/(3a - 5b) = 5 then a : b is equal to?

Solution:
(3a + 5b)/(3a - 5b) = 5
⇒ 3a + 5b = 15a - 25b
⇒ 12a = 30b
⇒ 2a = 5b
∴ a : b  = 5 : 2

Question: If x = 3 + 2√2, then the value of (√x + 1/√x) is?

Solution:
x = 3 + 2√2
⇒ x = 2 +1 + 2√2
⇒ x = (√2)² + 2 . √2 . 1 + (1)²
⇒ x = (√2 + 1)²
⇒ √x = √2 + 1
⇒ 1/√x = 1/ (√2 + 1)
⇒ 1/√x =1(√2 - 1)/(√2 + 1)(√2 - 1)
⇒ 1/√x = √2 - 1

∴ √x + 1/√x =√2 + 1 + √2 - 1
∴ √x + 1/√x = 2√2

Question: If x + 1/x = 1 then, find the value of x² + (1/x²) + 2 is -

Solution:
Given that
x + 1/x = 1

Now,
x² + 1/x² + 2
= x² + (1/x)² + 2 
= (x + 1/x)² - 2 . x .1/x + 2
= 1² - 2 + 2
= 1

Question: What is the value of expression (1 + x) (1 + x²) (1 + x⁴) (1 + x⁸) (1 - x)?

Solution:
Given expression
(1 + x) (1 + x²) (1 + x⁴) (1 + x⁸) (1 - x)
= (1 + x) (1 - x) (1 + x²) (1 + x⁴) (1 + x⁸)
= (1 - x²) (1 + x²) (1 + x⁴) (1 + x⁸)
= (1 - x⁴) (1 + x⁴) (1 + x⁸)
= (1 - x⁸) (1 + x⁸)
= (1 - x¹⁶)

Question: If x⁴ + y⁴ = 17 and x + y = 1, then the value of x²y² - 2xy?

Solution:
Given, 
x + y = 1
Or, (x + y)² = 1²
Or, x² + 2xy + y² = 1
Or, x² + y² = 1 - 2xy
Or, (x² + y²)² = (1 - 2xy)²
Or, x⁴ + 2 . x²y² + y⁴ = 1 - 4xy + (2xy)²
Or, x⁴ + y⁴ + 2x²y² = 1 - 4xy + 4x²y²
Or, 17 = 1 - 4xy + 2x²y²
Or, 2x²y² - 4xy = 17 - 1
Or, 2x²y² - 4xy = 16
∴ x²y² - 2xy = 8

Question: If (2x - 2y)/(x - 4y) = 4, then find the value of (x + y)/(x + 3y) = ? 

Solution: 
(2x - 2y)/(x - 4y) = 4
⇒ 2x - 2y = 4(x - 4y) 
⇒ 2x - 2y = 4x - 16y
⇒ 2x - 4x = - 16y + 2y
⇒ - 2x = - 14y
∴ x = 7y

∴ (x + y)/(x + 3y) = (7y + y)/(7y + 3y)
= 8y/10y
= 4/5

Solution:
Given Equation 2x² - 7x + 12 = 0
roots are α , β

We know,
∴ αβ = c/a
α + β = - b/a

∴ α + β = - (- 7/2) = 7/2
∴ αβ = 12/2 = 6

Now, α/β + β/α = (α² + β²)/αβ
= {(α + β)² - 2αβ}/αβ
= {(7/2)² - 2 . 6}/6
= {(49/4) - 12}/6
= {(49 - 48)/4}/6
= 1/24

Question: If x = 2 then the value of x³ + 27x² + 243x + 631 is -

Solution:
Given,
x = 2
Now, 
 x³ + 27x² + 243x + 631
= x³ + 3 . x² . 9 + 3 . x . 9² + 9³ - 98
= (x + 9)³ - 98
= (2 + 9)³ - 98
= 1331 - 98
= 1233

Question: If the sum of two digits of a two-digit number is 9, the number obtained by interchanging two digits is 45 less than the given number. What will the number be?

Solution:
ধরি, একক স্থানীয় অংক = x
দশক স্থানীয় অংক = y
সংখ্যাটি = 10y + x
স্থান বিনিময়কৃত সংখ্যাটি = 10x + y

প্রশ্নমতে,
x + y = 9
y = 9 - x -----------(1)

প্রশ্নমতে,
10x + y + 45 = 10y + x
বা, 10x + 9 - x + 45 = 10(9 - x) + x
বা, 9x + 9x = 90 - 54
বা, 18x = 36
∴ x = 2
এবং y = 9 - 2 = 7

∴ নির্ণেয় সংখ্যাটি = (10 × 7) + 2 = 72

Question: If x² - 2x + 1 = 0 then the value of (x⁴ + 2x² + 1)/x² is -

Solution:
Given, 
x² - 2x + 1 = 0
⇒ (x - 1)² = 0
⇒ x - 1 = 0
∴ x = 1

(x⁴ + 2x² + 1)/x² = (1⁴ + 2 . 1² + 1)/1²
= 4/1
= 4

Question: ‍If a + b = 7, a² + b² = 25 then what is the value of ab?

Solution:
Given, 
a + b = 7,
a² + b² = 25

We know,
a² + 2ab + b² = (a + b)²
Or, 2ab = (a + b)² - (a² + b²)
Or, 2ab = (7)² - 25
Or, 2ab = 49 - 25
Or, ab = 24/2
∴ ab = 12

Question: What is the value of a if 9 - 12x + ax² is an integer -

Solution: 
9 - 12x + ax²
= 3² - 2 . 3 . √ax + (√ax)²
∴ 6√ax = 12x
⇒ √a = 2
⇒ (√a)² = (2)²
∴ a = 4

Question: What is the solution of the equation 3/(y + 1) = 4/(y - 2)?

Solution:
3/(y + 1) = 4/(y - 2)
⇒ 3/(y + 1) - 4/(y - 2) = 0
⇒ (3y - 6 - 4y - 4)/(y + 1) (y - 2) = 0
⇒ - y - 10 = 0
∴ y = - 10

Question: A number is 2/5 times of another number. If the sum of two numbers is 98, what is the two numbers?

Solution:
Let, a number is x
another number is 2x/5

ATQ,
x + 2x/5 = 98
⇒ (5x + 2x)/5 = 98
⇒ 7x/5 = 98
⇒ 7x = 98 × 5
⇒ 7x = 490
∴ x = 70

So, a number is 70
another number is 2x/5 = (2 × 70)/5 = 28

Question: If (y - 1) (y + 2) = (y + 4) (y - 2) then what is the value of y?

Solution:
(y - 1) (y + 2) = (y + 4) (y - 2)
⇒ y² - y + 2y - 2 = y² + 4y - 2y - 8
⇒ y - 2 = 2y - 8
⇒ y - 2y = - 8 + 2
∴ y = 6

Question: If {(5x/6) + 3} and {(x/3) + 10} are equal, then what is the value of x?

Solution:
ATQ,
{(5x/6) + 3} = {(x/3) + 10}
⇒ (5x/6) - (x/3) = 10 - 3
⇒ (5x - 2x)/6 = 7
⇒ 3x/6 = 7
⇒ x = 42/3
∴ x = 14

Question: (1/2) {(a + b)² + (a - b)²} = ?

Solution:
(1/2) {(a + b)² + (a - b)²} 
= (1/2) (a² + 2ab + b² + a² - 2ab + b²)
= (1/2) {2 (a² + b²)}
= a² + b²

Question: Which of the following is the solution to x² - 2x - 2 = 0?

Solution:
ax² + bx + c = 0 সমীকরণের সাথে তুলনা করে পাই,
a = 1, b = - 2, c = - 2

∴ x = {- ( - 2) ± √( - 2)² - 4 . 1 . (- 2)}/ 2 . 1
= {2 ± √(4 + 8)}/2
= (2 ± √12)/2
= {2(1 ± √3)}/2
= 1 ± √3

অর্থাৎ, x₁ = 1 + √3, x₂ = 1 - √3