1.02b Surds: manipulation and rationalising denominators

1. \(\quad \mathrm { f } ( x ) = ( \sqrt { x } + 3 ) ^ { 2 } + ( 1 - 3 \sqrt { x } ) ^ { 2 }\).

Show that \(\mathrm { f } ( x )\) can be written in the form \(a x + b\) where \(a\) and \(b\) are integers to be found.

4. Express each of the following in the form \(p + q \sqrt { 2 }\) where \(p\) and \(q\) are rational.

1. \(( 4 - 3 \sqrt { 2 } ) ^ { 2 }\)
2. \(\frac { 1 } { 2 + \sqrt { 2 } }\)

6. $$f ( x ) = x ^ { \frac { 3 } { 2 } } - 8 x ^ { - \frac { 1 } { 2 } }$$

1. Evaluate \(\mathrm { f } ( 3 )\), giving your answer in its simplest form with a rational denominator.
2. Solve the equation \(\mathrm { f } ( x ) = 0\), giving your answers in the form \(k \sqrt { 2 }\).

1. (i) Evaluate \(\left( 5 \frac { 4 } { 9 } \right) ^ { - \frac { 1 } { 2 } }\).
   (ii) Find the value of \(x\) such that

$$\frac { 1 + x } { x } = \sqrt { 3 } ,$$ giving your answer in the form \(a + b \sqrt { 3 }\) where \(a\) and \(b\) are rational.

1

1. Expand and simplify \(( 3 + 4 \sqrt { 5 } ) ( 3 - 2 \sqrt { 5 } )\).
2. Express \(\sqrt { 72 } + \frac { 32 } { \sqrt { 2 } }\) in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers and \(b\) is as small as possible.

2

1. Expand and simplify \(( 7 - 2 \sqrt { 3 } ) ^ { 2 }\).
2. Express \(\frac { 20 \sqrt { 6 } } { \sqrt { 50 } }\) in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers and \(b\) is as small as possible.

4

1. Express \(125 \sqrt { 5 }\) in the form \(5 ^ { k }\).
2. Simplify \(10 + 7 \sqrt { 5 } + \frac { 38 } { 1 - 2 \sqrt { 5 } }\), giving your answer in the form \(a + b \sqrt { 5 }\).

5

1. Express \(\sqrt { 48 } + \sqrt { 75 }\) in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers.
2. Simplify \(\frac { 7 + 2 \sqrt { 5 } } { 7 + \sqrt { 5 } }\), expressing your answer in the form \(\frac { a + b \sqrt { 5 } } { c }\), where \(a , b\) and \(c\) are integers.

7

1. Expand and simplify \(( 7 + 3 \sqrt { 2 } ) ( 5 - 2 \sqrt { 2 } )\).
2. Simplify \(\sqrt { 54 } + \frac { 12 } { \sqrt { 6 } }\).

8 The volume \(V\) of a cone with base radius \(r\) and slant height \(l\) is given by the formula $$V = \frac { 1 } { 3 } \pi r ^ { 2 } \sqrt { l ^ { 2 } - r ^ { 2 } }$$ Rearrange this formula to make \(l\) the subject.

9

1. Express \(\sqrt { 48 } + \sqrt { 27 }\) in the form \(a \sqrt { 3 }\).
2. Simplify \(\frac { 5 \sqrt { 2 } } { 3 - \sqrt { 2 } }\). Give your answer in the form \(\frac { b + c \sqrt { 2 } } { d }\).

10

1. Simplify \(\frac { \sqrt { 48 } } { 2 \sqrt { 27 } }\).
2. Expand and simplify \(( 5 - 3 \sqrt { 2 } ) ^ { 2 }\).

11

1. Express \(\sqrt { 75 } + \sqrt { 48 }\) in the form \(a \sqrt { 3 }\).
2. Express \(\frac { 14 } { 3 - \sqrt { 2 } }\) in the form \(b + c \sqrt { d }\).

12

1. Express \(\frac { 1 } { 5 + \sqrt { 3 } }\) in the form \(\frac { a + b \sqrt { 3 } } { c }\), where \(a , b\) and \(c\) are integers.
2. Expand and simplify \(( 3 - 2 \sqrt { 7 } ) ^ { 2 }\).

15

1. Simplify \(\sqrt { 98 } \quad \sqrt { 50 }\).
2. Express \(\frac { 6 \sqrt { 5 } } { 2 + \sqrt { 5 } }\) in the form \(a + b \sqrt { 5 }\), where \(a\) and \(b\) are integers.

17

1. Simplify \(5 \sqrt { 8 } + 4 \sqrt { 50 }\). Express your answer in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers and \(b\) is as small as possible.
2. Express \(\frac { \sqrt { 3 } } { 6 \sqrt { 3 } }\) in the form \(p + q \sqrt { 3 }\), where \(p\) and \(q\) are rational.

1 Use an isosceles right-angled triangle to show that \(\cos 45 ^ { \circ } = \frac { 1 } { \sqrt { 2 } }\).

9 The curve in Fig. 9.1 has equation \(\sqrt { x } + \sqrt { y } = 1\). \begin{figure}[h] \end{figure}

1. Show that this is part, but not all of the curve \(y = 1 - 2 \sqrt { x } + x\). Sketch the full curve \(y = 1 - 2 \sqrt { x } + x\).
2. Fig.9.2 shows a star shape made up of four parts, one of which is given in part (i) above. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2f403099-2813-40d8-a9ae-1f7e64d41f80-4_380_681_1197_651} \captionsetup{labelformat=empty} \caption{Fig. 9.2}
   \end{figure} For each of the sections of the shape labelled \(\mathrm { A } , \mathrm { B }\) and C , state the equation of the curve and the domain.
3. The shape shown in Fig.9.2 is made into that in Fig. 10.3 by stretching the part of the figure for which \(y > 0\) by a scale factor of 2 . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2f403099-2813-40d8-a9ae-1f7e64d41f80-4_405_686_1996_605} \captionsetup{labelformat=empty} \caption{Fig. 9.3}
   \end{figure} Find the area of this shape.

3 Given that \(y = \ln \left( \sqrt { \frac { 2 x - 1 } { 2 x + 1 } } \right)\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 2 x - 1 } - \frac { 1 } { 2 x + 1 }\).

3.Given that \(\phi = \frac { 1 } { 2 } ( \sqrt { 5 } + 1 )\) ,

1. show that
   1. \(\phi ^ { 2 } = \phi + 1\)
   2. \(\frac { 1 } { \phi } = \phi - 1\)
2. The equations of two curves are $$\begin{array} { r l r l } y & = \frac { 1 } { x } & x > 0 \\ \text { and } & y & = \ln x - x + k & x > 0 \end{array}$$ where \(k\) is a positive constant.
   The curves touch at the point \(P\) .
   Find in terms of \(\phi\)
   1. the coordinates of \(P\) ,
   2. the value of \(k\) .

1.(a)Write down the exact value of \(\cos 405 ^ { \circ }\) (b)Hence,using a double angle identity for cosine,or otherwise,determine the exact value of \(\cos 101.25 ^ { \circ }\) ,giving your answer in the form $$a \sqrt { b + c \sqrt { 2 + \sqrt { 2 } } }$$ where \(a\) ,\(b\) and \(c\) are rational numbers.

6.(a)Show that $$\sqrt { 2 + \sqrt { 3 } } - \sqrt { 2 - \sqrt { 3 } } = \sqrt { 2 }$$ (b)Hence prove that $$\log _ { \frac { 1 } { 8 } } ( \sqrt { 2 + \sqrt { 3 } } - \sqrt { 2 - \sqrt { 3 } } ) = - \frac { 1 } { 6 } .$$ (c)Find all possible pairs of integers \(a\) and \(n\) such that $$\log _ { \frac { 1 } { n } } ( \sqrt { a + \sqrt { 15 } } - \sqrt { a - \sqrt { 15 } } ) = - \frac { 1 } { 2 } .$$

1.(a)Solve the equation $$\sqrt { } ( 3 x + 16 ) = 3 + \sqrt { } ( x + 1 )$$ (b)Solve the equation $$\log _ { 3 } ( x - 7 ) - \frac { 1 } { 2 } \log _ { 3 } x = 1 - \log _ { 3 } 2$$

6

1. Write \(\sqrt { } 80\) in the form \(c \sqrt { } 5\), where \(c\) is a positive constant. A rectangle \(R\) has a length of ( \(1 + \sqrt { } 5\) ) cm and an area of \(\sqrt { 80 } \mathrm {~cm} ^ { 2 }\).
2. Calculate the width of \(R\) in cm . Express your answer in the form \(p + q \sqrt { 5 }\), where \(p\) and \(q\) are integers to be found.