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### Solved Examples

**Question 1:**The sum of the square of the two consecutive positive even integers is 164. Find the integers.

**Solution:**

Let two consecutive even integers, x and x + 2

sum of the square of the two consecutive positive even integers = 164

=> $x^2$ + $(x + 2)^2$ = 164

=> $x^2$ + $x^2$ + $2^2$ + 2 * $x$ * 2 = 164

=> 2$x^2$ + 4 + 4x = 164

=> 2$x^2$ + 4x = 164 - 4

=> 2$x^2$ + 4x = 160

=> 2$x^2$ + 4x - 160 = 0

which is quadratic equation

Solve for x, 2$x^2$ + 4x - 160 = 0

=> 2$x^2$ + 20x - 16x - 160 = 0

=> 2x(x + 10) - 16(x + 10) = 0

=> (2x - 16)(x + 10) = 0

either 2x - 16 = 0 or x + 10 = 0

=> 2x = 16 or x = - 10

Since integers are consecutive positive, so neglect x = -10

=> 2x = 16

=> x = 8, first integer

and second integer = 8 + 2 = 10

Hence consecutive positive integer are, 8 and 10.

Step 1:Step 1:

sum of the square of the two consecutive positive even integers = 164

=> $x^2$ + $(x + 2)^2$ = 164

=> $x^2$ + $x^2$ + $2^2$ + 2 * $x$ * 2 = 164

=> 2$x^2$ + 4 + 4x = 164

=> 2$x^2$ + 4x = 164 - 4

=> 2$x^2$ + 4x = 160

=> 2$x^2$ + 4x - 160 = 0

which is quadratic equation

Step 2:Step 2:

Solve for x, 2$x^2$ + 4x - 160 = 0

=> 2$x^2$ + 20x - 16x - 160 = 0

=> 2x(x + 10) - 16(x + 10) = 0

=> (2x - 16)(x + 10) = 0

Step 3:Step 3:

either 2x - 16 = 0 or x + 10 = 0

=> 2x = 16 or x = - 10

Since integers are consecutive positive, so neglect x = -10

=> 2x = 16

=> x = 8, first integer

and second integer = 8 + 2 = 10

Hence consecutive positive integer are, 8 and 10.

**answer****Question 2:**Simplify the expression

$\frac{4i}{5 - i}$ * $\frac{3 - i}{5 + i}$

**Solution:**

Given $\frac{4i}{5 - i}$ * $\frac{3 - i}{5 + i}$

Multiply numerators and denominators of both the fractions

=> $\frac{4i}{5 - i}$ * $\frac{3 - i}{5 + i}$ = $\frac{(4i)(3 - i)}{(5 - i)(5 + i)}$

= $\frac{4i * 3 - 4i * i}{5^2 - i ^2}$

= $\frac{12 i - 4 i^2}{25 + 1}$

[(a - ib)(a + ib) = $a^2$ + $b^2$]

= $\frac{12 i + 4}{26}$

Reduce the fraction

=> $\frac{12 i + 4}{26}$

= $\frac{6 i + 2}{13}$

or $\frac{ 2 + 6i}{13}$

=> $\frac{4i}{5 - i}$ * $\frac{3 - i}{5 + i}$ = $\frac{2 + 6i}{13}$

**Step 1:**Multiply numerators and denominators of both the fractions

=> $\frac{4i}{5 - i}$ * $\frac{3 - i}{5 + i}$ = $\frac{(4i)(3 - i)}{(5 - i)(5 + i)}$

= $\frac{4i * 3 - 4i * i}{5^2 - i ^2}$

= $\frac{12 i - 4 i^2}{25 + 1}$

[(a - ib)(a + ib) = $a^2$ + $b^2$]

= $\frac{12 i + 4}{26}$

[$i^2$ = - 1]

Step 2:[$i^2$ = - 1]

Step 2:

Reduce the fraction

=> $\frac{12 i + 4}{26}$

**= $\frac{2(6 i + 2)}{2 * 13}$**= $\frac{6 i + 2}{13}$

or $\frac{ 2 + 6i}{13}$

=> $\frac{4i}{5 - i}$ * $\frac{3 - i}{5 + i}$ = $\frac{2 + 6i}{13}$

**Question 3:**The ratio of land and water area in the earth is 1 : 2. In the northern hemisphere this ratio is 2 : 3. Find the land and water area in the southern hemisphere.

**Solution:**

Let the area of each hemisphere be x.

Therefore total area of the earth is 2x.

Area of land in the northern hemisphere = $\frac{2}{2 + 3}$ * x = $\frac{2}{5}$x

and area of water in northern hemisphere = $\frac{3}{2 + 3}$ * x = $\frac{3}{5}$x

Area of land in the whole earth = $\frac{1}{1+2}$ 2x = $\frac{2}{3}$x

and area of water in the whole earth = $\frac{2}{1+2}$ * 2x = $\frac{4}{3}$x

Therefore area of land in the southern hemisphere = $\frac{2x}{3} - \frac{2x}{5} = \frac{4x}{15}$

and area of water in the southern hemisphere = $\frac{4x}{3} - \frac{3x}{5} = \frac{11x}{15}$

Thus the ratio of land and area of water in the southern hemisphere = $\frac{4x}{15}$ : $\frac{11x}{15}$ = 4 : 11.

Therefore total area of the earth is 2x.

Area of land in the northern hemisphere = $\frac{2}{2 + 3}$ * x = $\frac{2}{5}$x

and area of water in northern hemisphere = $\frac{3}{2 + 3}$ * x = $\frac{3}{5}$x

Area of land in the whole earth = $\frac{1}{1+2}$ 2x = $\frac{2}{3}$x

and area of water in the whole earth = $\frac{2}{1+2}$ * 2x = $\frac{4}{3}$x

Therefore area of land in the southern hemisphere = $\frac{2x}{3} - \frac{2x}{5} = \frac{4x}{15}$

and area of water in the southern hemisphere = $\frac{4x}{3} - \frac{3x}{5} = \frac{11x}{15}$

Thus the ratio of land and area of water in the southern hemisphere = $\frac{4x}{15}$ : $\frac{11x}{15}$ = 4 : 11.