Explanation

Every subgroup of a cyclic group is also cyclic.

This is a key result in group theory under abstract algebra.

Explanation

Let's denote the cost of a geometry box as x. Since the cost of a bag is three times the cost of a geometry box, the cost of a bag is 3x.

Given that the cost of 2 bags and 3 geometry boxes is Rs.3,960, we can set up the equation:

2(3x) + 3(x) = 3960

6x + 3x = 3960

9x = 3960

Divide both sides by 9:

x = 3960 / 9

x = 440

Explanation

Step 1: Perform polynomial long division or synthetic division

To find the quotient when (2x³ - 5x²y + 5xy² - 3y³) is divided by (2x - 3y), let's try factoring or direct division.

Step 2: Factor the numerator if possible

Notice that (2x³ - 5x²y + 5xy² - 3y³) can be factored as:

(2x³ - 3x²y) + (-2x²y + 3xy²) + (2xy² - 3y³)

= x²(2x - 3y) - xy(2x - 3y) + y²(2x - 3y)

= (2x - 3y)(x² - xy + y²)

Step 3: Identify the quotient

Given the factorization (2x - 3y)(x² - xy + y²), the quotient when (2x³ - 5x²y + 5xy² - 3y³) is divided by (2x - 3y) is x² - xy + y².

Explanation

Given:

μ + v = 8

μv = 15

The formula for (μ - v)² is:

(μ - v)² = (μ + v)² - 4μv

= 8² - 4 × 15

= 64 - 60

= 4

μ - v = ±√4

μ - v = ±2

Explanation

Given:

α + β = 5 (sum of roots)

αβ = k (product of roots)

3α + 2β = 16

From α + β = 5, we get:

α = 5 - β

Substitute α in 3α + 2β = 16:

3(5 - β) + 2β = 16

15 - 3β + 2β = 16

-β = 1

β = -1

Now, α = 5 - β

α = 5 - (-1)

α = 6

k = αβ

k = 6 × (-1)

k = -6

Explanation

Given:

a - b = 7

ab = 6

The formula for a³ - b³ is:

a³ - b³ = (a - b)(a² + ab + b²)

= (a - b)((a - b)² + 3ab)

Substitute the values:

a³ - b³ = 7((7)² + 3(6))

= 7(49 + 18)

= 7 × 67

= 469

Explanation

x - 8/3 + x - 3/2 = 0

Multiply both sides by 6:

2(x - 8) + 3(x - 3) = 0

2x - 16 + 3x - 9 = 0

5x = 25

x = 5

Explanation

Given x = 1 is a solution, let's find 'a':

a(1)² + 5(1) - 3 = 0

a + 2 = 0

a = -2

The equation is -2x² + 5x - 3 = 0.

Let's factor or solve:

-2x² + 5x - 3 = 0

-2x² + 2x + 3x - 3 = 0

-2x(x - 1) + 3(x - 1) = 0

(-2x + 3)(x - 1) = 0

x = 1 (given) or x = 3/2

The other solution is x = 3/2.

Explanation

Given:

1. Sum of roots: α + β = 2

2. Sum of cubes of roots: α³ + β³ = 98

Step-by-Step Solution

Step 1: Use the identity for α³ + β³

α³ + β³ = (α + β)³ - 3αβ(α + β)

Step 2: Substitute known values

98 = (2)³ - 3αβ(2)

98 = 8 - 6αβ

Step 3: Solve for αβ

98 - 8 = -6αβ

90 = -6αβ

αβ = -15

Step 4: Form the quadratic equation

x² - (α + β)x + αβ = 0

x² - 2x - 15 = 0

The answer is x² - 2x - 15 = 0.

Explanation

tan θ = sin θ/cos θ and cot θ = cos θ/sin θ

tan θ + cot θ = sin θ/cos θ + cot θ = cos θ/sin θ

= sin^2 θ + cos^2 θ/sin θ cos θ

= 1/sin θ cos θ (since sin² θ + cos² θ = 1)

Now, 1/sin θ cos θ = 1/sin θ × 1/cos θ = cosec θ × sec θ = sec θ cosec θ

The answer is sec θ cosec θ.