Comparing above two equation

Now,

First of all, we should not think that because of the 4 roots, the degree of the polynomial will be 4.

We need to remember that complex roots always occur in pairs.

Now,

Now, to find the value of the given pair roots,

1) 2ω² = -1 - √3 i

∴ -1 + √3i will also be a root.

2) 3 + 4ω = 3 - 2 + 2√3i

= 1 + 2√3i

∴ 1 - 2√3i will also be a root.

3) 3 + 4ω² = 3 - 2 - 2√3i

= 1 - 2√3i

i.e, 3 + 4ω and 3 + 4ω² are conjugate pairs.

i.e. 6 is a root.

Total we have 5 roots.

The minimum degree of the polynomial is 5.

now,

i) if x ≥ 0

x² + x = 0

x(x + 1) = 0

x = 0, x = -1

∴ x = 0 (∵ x ≥ 0)

y = 0, x = 0 is solution.

ii) if x < 0

x² - x = 0

x(x - 1) = 0

∴ x = 0, x = 1

Both are not possible

Case: 2

x = 0

-y² + |y| = 0

now,

i) if y ≥ 0

-y² + y = 0

y(1 - y) = 0

y = 0, y = 1

Two solutions:

(a) x = 0, y = 0 is a solution (z = 0)

(b) x = 0, y = 1 is a solution (z = i)

ii) if y < 0

-y² - y = 0

-y(1 + y) = 0

y = 0, y = -1

Two solutions:

(a) x = 0, y = 0 is a solution (z = 0)

(b) x = 0, y = -1 is a solution (z = -i)

Concept:

If 1, ω, ω² are the cube roots of unity,

⇒ 1 - ω² + 1 - ω

⇒ 2 - (ω² + ω)

⇒ 2 - (-1)       (∵ 1 + ω + ω2 = 0)

⇒ 2 + 1

⇒ 3

Concept:

Convert decimal to binary

Conversion steps:

• Divide the number by 2.
• Get the integer quotient for the next iteration.
• Get the remainder for the binary digit.
• Repeat the steps until the quotient is equal to 0.

Calculation:

Here, we have to find the binary equivalent of 45

So, the binary equivalent of 45 is 101101

∴ (45)₁₀ = (101101)₂

Concept used:

The middle position of an AP series is the average of the series,

Example: 1, 2, 3, 4, 5 are in AP, the average of the AP = (1 + 5)/2 = 3 is the middle position

Calculation:

Here 7^th term is the middle position of 5^th term and 9^th term then the 7th term = (16 + 22)/2 = 19

∴ The correct answer is 19

Given:

a₁, a₂, ..., a8 are eight numbers whose  average = 40

Average of numbers at even places = 25

Formula used:

Average = (Sum of terms)/(Number of terms)

Calculation:

The sum of these eight numbers = 8 × 40 = 320

The average of the numbers at even places (a2, a4, a6, a8) = 25.

The sum of these four numbers = 4 × 25 = 100

Now, we subtract the sum of the numbers at even places from the total sum, and then divide by the number of odd places.

The sum of the numbers at odd places = 320 - 100 = 220

There are 4 numbers at odd places, their average is 220/4 = 55

∴ Average of numbers at odd places is 55.

comparing 1) and (2), we get

Hence, x = 1

Concept used:

Euation Ax² + Bx + C = 0

If roots are equal B² - 4AC = 0

Formula Used:

(x - y)² = x² - 2xy + y²

(x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx

Calculation:

a(b - c) x² + b(c - a) x + c(a - b) = 0

Where A = a(b - c) , B = b(c - a) & C =  c(a - b)

Now, 

⇒ [b(c - a)]² - 4 × a(b - c) × c(a - b) = 0 

⇒ b²(c² + a² - 2ac) - 4ac(ab - b² -ac + bc) = 0

⇒ b²c² + b²a² - 2acb² - 4a²bc + 4acb² + 4a²c² - 4abc² = 0

⇒ a²b² + b²c² + 4a2c2 + 2acb2 - 4a2bc - 4abc²  = 0

By using the above identiy

⇒ (ab + bc - 2ac)² = 0

⇒ ab + bc - 2ac = 0 

⇒ b(a + c) = 2ac 

⇒ b = 2ac/(a + c)

So, a, b & c are in harmonic progression.

∴ The correct answer is Harmonic progression.

Formula used:

The formula used for summation for G.P: (r < 1 and series is of infinite terms)

Where, 

S, is the summation of G.P

a, is the first term

r, is the common ratio

Calculation:

Let S be the sum of the given series.

The required series can be written as:

Now, the above series becomes Geometric series with a common ratio of 10/20

Additional Information

(1) The formula used for summation for G.P: (r < 1 and series is of finite terms)

Where,

S = the summation of G.P

a = the first term

r = the common ratio

n = the number of terms