ব্যাংক নিয়োগ প্রস্তুতি ⎯ লং কোর্স

Question: Find out the value of the term, (2x + 3)².

Solution:
Using algebraic formula,
(a + b)² = a² + 2ab + b²

∴ (2x + 3)² = (2x)² + 2 × 2x × 3 + 3² 
⇒ (2x + 3)² = 4x² + 12x + 9

Question: Find the value of 17² - 4².

Solution:
17² - 4²
= (17 + 4)(17 - 4)
= 21 × 13
= 273

Question: Solve the following equation.


Solution:

Question: If a³ - 8b³ = - 2 and a = - 1, then b =?

Solution:
a = - 1

a³ - 8b³ = - 2
⇒ (- 1)³ - 8b³ = - 2
⇒ - 1 - 8b³ = - 2
⇒ - 8b³ = - 2 + 1
⇒ - 8b³ = - 1
⇒ 8b³ = 1
⇒ b³ = 1/8
⇒ b³ = (1/2)³
∴ b = 1/2

Question: What will be the equation form for the following statement?
The sum of 3 times x and 11 is 32.

Solution:
3 times x = 3x
The sum of 3 times x and 11 is 32.
3x + 11 = 32

Question: Factor completely: x³ - 8.

Solution:
We know that,
a³ - b³ = (a - b)(a² + ab + b²)

∴ x³ - 8
= x³ - 2³
= (x - 2)(x² + x.2 + 2²)
= (x - 2)(x² + 2x + 4)

Question: Evaluate y² - y - 6 where y = - 4.

Solution:
y = - 4

y² - y - 6
= (- 4)² - (- 4) - 6
= 16 + 4 - 6
= 20 - 6
= 14

Question: What is the sum of the roots of the equation x² - 6x + 9 = 0.

Solution:
x² - 6x + 9 = 0
⇒ x² - 2.x.3 + 3² = 0
⇒ (x - 3)² = 0
⇒ (x - 3)(x - 3) = 0
∴ x = 3  or  x = 3

∴ The sum of the roots = 3 + 3 = 6

Question: Simplify: z² + 11z² - 5z - 11z² + 5z.

Solution:
z² + 11z² - 5z - 11z² + 5z
= z²

Question: Find the solution to the equation (x² + 4x + 4)/(x + 2) = 0.

Solution:
(x² + 4x + 4)/(x + 2) = 0
⇒ (x² + 2.x.2 + 2²)/(x + 2) = 0
⇒ (x + 2)²/(x + 2) = 0
⇒ x + 2 = 0
∴ x = - 2

Question: Find the value of the expression a² - 2ab + b² for a = 1, b = 1.

Solution:
a² - 2ab + b²
= (a - b)²
= (1 - 1)²
= (0)²
= 0

Question: Which of the following is not a quadratic equation?

Solution:
Option B is not a quadratic equation Since it has degree 3.

Question: The factors of the polynomial x² - 9x + 18 are:

Solution:
x² - 9x + 18
= x² - 3x - 6x + 18
= x(x - 3) - 6(x - 3)
= (x - 3)(x - 6)

Question: When x³ - 3x² + x + 2 is divided by polynomial g(x), then quotient and remainder are x - 2 and - 2x + 4, respectively, g(x) is:

Solution:
According to the division algorithm,
Dividend = Divisor × Quotient + Remainder

We have,
Dividend = x³ - 3x² + x + 2,
Divisor = g(x),
Quotient = x - 2
and Remainder = -2x + 4

Put the given values in the below equation and simplify it, to get the value of g (x).

Dividend = Divisor × Quotient + Remainder
(x³ - 3x² + x + 2) = g(x) × (x - 2) + (- 2x + 4)
⇒ (x³ - 3x² + x + 2) - (- 2x + 4) = g(x) × (x - 2)
⇒ (x³ - 3x² + x + 2x + 2 - 4) = g(x) × (x - 2)
⇒ (x³ - 3x² + 3x - 2) = g(x) × (x - 2)
⇒ g(x) = (x³ - 3x² + 3x - 2)/(x - 2)

Therefore, g(x) = x² - x + 1

Question: The quadratic equation whose roots are 1 and (- 1/2):

Solution:
The quadratic equation whose roots are 1 and (- 1/2)

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: The equation 2x² + kx + 3 = 0 has two equal roots, then the value of k is-

Solution:
Here a = 2, b = k, c = 3
Since the equation has two equal roots
∴ b² -  4ac = 0
⇒ (k)² - 4 × 2 × 3 = 0
⇒ k² = 24
⇒ k = ± √24
∴ k = ± √(4 × 6) = ± 2√6

Question: If x + y = 5 and x - y = 1 then what is the value of xy?

Solution:
Given that,
x + y = 5
x - y = 1

∴ xy = {(x + y)/2}² - {(x - y)/2}²
= (5/2)² - (1/2)²
= 25/4 - 1/4
= (25 - 1)/4
= 24/4
= 6

Question: α and β are the roots of 4x² + 3x + 7 = 0, then the value of (1/α) + (1/β) is-

Solution:

Question: If x - 1/x = 4; what is the value of x + 1/x?

Solution: 
x - 1/x = 4
⇒ (x - 1/x)² = (4)²
⇒ x² + 1/x² - 2 = 16
⇒ x² + 1/x² = 18
⇒ x² + 1/x² + 2 = 18 + 2
⇒ (x + 1/x)² = 20
⇒ x + 1/x = √20
∴ x + 1/x = 2√5

Question: If the roots of px² + qx + 2 = 0 are reciprocal of each other, then

Solution:
Let,
α and β are the roots
here
α = 1/β
∴ αβ = 1
⇒ 2/p = 1
∴ p = 2

Question: If x - 1/x = 4, then x³ - 1/x³ =?

Solution:
x³ - 1/x³
= (x - 1/x)³ + 3x.(1/x)(x - 1/x)
= 4³ + 3 × 4
= 64 + 12
= 76