Surds (Edexcel IGCSE Further Maths): Exam Questions

Expand $\frac{3}{\sqrt{1-2x}}$ in ascending powers of $x$ up to and including the term in $x^{3}$

and simplifying each term as far as possible.

Write down the range of values of $x$ for which this expansion is valid.

Show that $\frac{3}{\sqrt{0.9}}=\sqrt{10}$

Express $\frac{1}{\sqrt{10}-3}$ in the form $a\sqrt{10}+b$ , where $a$ and $b$ are integers.

Hence, using your expansion with a suitable value for $x$, obtain an approximation to $5$ decimal places of $\frac{1}{\sqrt{10}-3}$

Given that $\frac{a+2\sqrt{5}}{3-\sqrt{5}}=\frac{11+b\sqrt{5}}{2}$ where $a$ is an integer and $b$ is prime,

find the value of $a$ and the value of $b$ Show your working clearly.

Figure 1 shows the triangle $ABC$

$\text{angle} A B C = 90^{\circ} A B = ( 2 + 4 \sqrt{5} ) \text{ cm} B C = ( a + b \sqrt{5} ) \text{ \text{cm} }$ where $a$ and $b$ are integers.

The area of triangle $A B C = ( 34 + 11 \sqrt{5} ) cm^{2}$

Without using a calculator, find the value of $a$ and the value of $b$