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### Question 10
Write the following in the form \( a + \frac{b}{c} \), where \( c \) is an integer and \( a \) and \( b \) are rational numbers.

- (i) \(\frac{3 + \sqrt{2}}{4}\)
- (ii) \(\frac{2 - \sqrt{3}}{1 + \sqrt{3}}\)
- (iii) \(\frac{\sqrt{5}}{\sqrt{5} + 5}\)
- (iv) \(\frac{4 + \sqrt{2}}{3}\)

### Question 11
- (i) A square has sides of length \( x \) cm and a diagonal of length 8 cm. Use Pythagoras' theorem to find the exact value of \( x \) and the area of the square.
- (ii) The diagonal of a square is 6 cm. Find the exact value of the perimeter of the square.

### Question 12
A ladder of length 6 m is placed on horizontal ground with the foot of the ladder 2 m from the vertical side of a house. How far up the wall does the ladder reach? Give your answer in the simplest possible form.

### Question 13
The cost of building a lighthouse is proportional to the cube of its height, \( h \).
The distance, \( d \), that the top of the lighthouse can be seen from a point at sea level is modeled by \( d = \sqrt{2Rh} \), where \( R \) is the radius of the Earth and \( d \), \( R \), and \( h \) are in the same units.
Three possible designs \( X \), \( Y \), and \( Z \) are considered, in which the top of the lighthouse can be seen at 20 km, 40 km, and 60 km respectively. Find the ratios of the costs of designs \( X \), \( Y \), and \( Z \).

### Question 14
Simplify the following by rationalizing the denominator.
- (i) \(\frac{\sqrt{3}}{\sqrt{3} - \sqrt{2}}\)
- (ii) \(\frac{\sqrt{5} + 2\sqrt{2}}{\sqrt{5} - \sqrt{2}}\)

### Question 15
- (i) On a single sheet of graph paper, or using graphing software, draw the curves with equations \( y = \sqrt{x} \) and \( y = \sqrt{x + 2} \) for \( 0 \leq x \leq 20 \).
- (ii) Choose two values of \( x \) with \( 0 \leq x \leq 20 \) and show that for these values, the point \( \left( x, \frac{1}{\sqrt{x} - 2 - \sqrt{x}} \right) \) lies between the two curves.

### Question 

The image contains a set of math problems related to simplifying expressions, solving geometric problems, rationalizing denominators, and graphing equations. It also includes questions about the cost ratios of lighthouses and their relationship to height and visibility.

This page provides solutions to complex mathematical problems, including simplifying algebraic expressions, solving geometric problems using the Pythagorean theorem, rationalizing denominators, and calculating the cost ratios of lighthouses based on height and visibility. Ideal for Grades 10-12 students.

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