^{1. If 1/y + z + 1/z + x = 2/x +y, then find the value of x^2 + y^2.}

^{2. Find the value of :}

^{2a / a + b + 2b / b +c + 2c / c + a + (b -c) (c -a) (a - b) / (b + c) (c + a) (a + b).}

^{3. If the polynomial x^19 + x^17 + x^13 + x^11 + x^7 + x^5 + x^3 is divided by (x^2 + 1), then find the remainder.}

^{4. If pqr = 1, find the value of :}

^{1 / 1 + p + q^-1 + 1 / 1 + q + r^-1 + 1 / 1 + r + p^-1.}

^{5. A goat is tethered to one end of a rope of 20 m, while the other end is fixed at the centre of a large circular field. There is a square elevated plate with sides of 10 m on the field such that one corner of the elevated square platform coincides with centre of the circular field. If the goat is unable to mount the square elevated platform, what is the total area the goat will able to graze?}

Let's tackle these mathematical problems step by step. Each question presents a unique challenge, and I'll break them down for clarity and understanding.

1. Solving the Equation for x and y

We start with the equation:

1/y + z + 1/z + x = 2/x + y

To find the values of x and y, we can rearrange the equation. First, let's isolate the terms involving x and y:

1/y + z + 1/z - y = 2/x - x

Next, we can multiply through by xyz to eliminate the denominators:

z(xz + yz + x) = 2y - xy

Solving this can get complex, but let's assume specific values for z to simplify our calculations. If we set z = 1, we can find values for x and y that satisfy the equation. After some algebra, we can derive that:

x = 1, y = 1

Thus, we calculate:

x² + y² = 1² + 1² = 2

2. Evaluating the Expression

We need to find the value of:

2a / (a + b) + 2b / (b + c) + 2c / (c + a) + (b - c)(c - a)(a - b) / ((b + c)(c + a)(a + b))

Let's break this down into parts. The first three fractions can be simplified using common denominators. The last term can be evaluated separately. If we assume a = b = c, we can simplify the expression significantly:

Result = 2

3. Finding the Remainder of a Polynomial

We are given the polynomial:

x¹⁹ + x¹⁷ + x¹³ + x¹¹ + x⁷ + x⁵ + x³

We need to find the remainder when this polynomial is divided by (x² + 1). By the Remainder Theorem, we can evaluate the polynomial at the roots of the divisor, which are i and -i:

Calculating:

• For x = i: i¹⁹ + i¹⁷ + i¹³ + i¹¹ + i⁷ + i⁵ + i³ = 0
• For x = -i: (-i)¹⁹ + (-i)¹⁷ + (-i)¹³ + (-i)¹¹ + (-i)⁷ + (-i)⁵ + (-i)³ = 0

Thus, the remainder is:

0

4. Evaluating the Expression with p, q, and r

Given that pqr = 1, we need to find:

1 / (1 + p + q⁻¹) + 1 / (1 + q + r⁻¹) + 1 / (1 + r + p⁻¹)

Substituting q = 1/(pr) and simplifying each term leads us to a common denominator. After simplification, we find:

Result = 1

5. Grazing Area of the Goat

In this scenario, the goat is tethered to a 20 m rope at the center of a circular field, while a square platform of 10 m sides occupies part of the area. The goat can graze in a circular area with a radius of 20 m, but we need to subtract the area of the square platform:

The area the goat can graze is:

Area of circle = π * r² = π * (20)² = 400π

The area of the square platform is:

Area of square = side² = 10² = 100

Thus, the total area the goat can graze is:

400π - 100

Calculating this gives us the final grazing area.

Each of these problems showcases different mathematical principles, from algebra to geometry, and understanding these concepts will enhance your problem-solving skills. If you have any further questions or need clarification on any step, feel free to ask!