1.02b Surds: manipulation and rationalising denominators

5

1. Solve the equation \(6 \sqrt { y } + \frac { 2 } { \sqrt { y } } - 7 = 0\).
2. Hence solve the equation \(6 \sqrt { \tan x } + \frac { 2 } { \sqrt { \tan x } } - 7 = 0\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

2 The circle with equation \(( x - 3 ) ^ { 2 } + ( y - 5 ) ^ { 2 } = 40\) intersects the \(y\)-axis at points \(A\) and \(B\).

1. Find the \(y\)-coordinates of \(A\) and \(B\), expressing your answers in terms of surds.
2. Find the equation of the circle which has \(A B\) as its diameter.

10 The function f is defined by \(\mathrm { f } : x \mapsto x ^ { 2 } - 3 x\) for \(x \in \mathbb { R }\).

1. Find the set of values of \(x\) for which \(\mathrm { f } ( x ) > 4\).
2. Express \(\mathrm { f } ( x )\) in the form \(( x - a ) ^ { 2 } - b\), stating the values of \(a\) and \(b\).
3. Write down the range of f .
4. State, with a reason, whether f has an inverse. The function g is defined by \(\mathrm { g } : x \mapsto x - 3 \sqrt { } x\) for \(x \geqslant 0\).
5. Solve the equation \(\mathrm { g } ( x ) = 10\).

7 \includegraphics[max width=\textwidth, alt={}, center]{1b9c6b41-69dd-4132-92c7-9507cbd741dd-10_551_657_274_735} The diagram shows the curves \(y = \sqrt { 2 \pi - 2 x }\) and \(y = \sin ^ { 2 } x\) for \(0 \leqslant x \leqslant \pi\). The shaded region is bounded by the two curves and the line \(x = 0\). Find the exact area of the shaded region.

7 \includegraphics[max width=\textwidth, alt={}, center]{712be8e6-e1e9-4662-b1f1-51c39c2c9df1-10_551_657_274_735} The diagram shows the curves \(y = \sqrt { 2 \pi - 2 x }\) and \(y = \sin ^ { 2 } x\) for \(0 \leqslant x \leqslant \pi\). The shaded region is bounded by the two curves and the line \(x = 0\). Find the exact area of the shaded region.

2 Using the substitution \(u = 4 ^ { x }\), solve the equation \(4 ^ { x } + 4 ^ { 2 } = 4 ^ { x + 2 }\), giving your answer correct to 3 significant figures.

3

1. Given that \(\sin \left( \theta + 45 ^ { \circ } \right) + 2 \cos \left( \theta + 60 ^ { \circ } \right) = 3 \cos \theta\), find the exact value of \(\tan \theta\) in a form involving surds. You need not simplify your answer.
2. Hence solve the equation \(\sin \left( \theta + 45 ^ { \circ } \right) + 2 \cos \left( \theta + 60 ^ { \circ } \right) = 3 \cos \theta\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).

5

1. Simplify \(( \sqrt { } ( 1 + x ) + \sqrt { } ( 1 - x ) ) ( \sqrt { } ( 1 + x ) - \sqrt { } ( 1 - x ) )\), showing your working, and deduce that $$\frac { 1 } { \sqrt { } ( 1 + x ) + \sqrt { } ( 1 - x ) } = \frac { \sqrt { } ( 1 + x ) - \sqrt { } ( 1 - x ) } { 2 x }$$
2. Using this result, or otherwise, obtain the expansion of $$\frac { 1 } { \sqrt { } ( 1 + x ) + \sqrt { } ( 1 - x ) }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).

3

1. By simplifying \(\left( x ^ { n } - \sqrt { x ^ { 2 n } + 1 } \right) \left( x ^ { n } + \sqrt { x ^ { 2 n } + 1 } \right)\), show that \(\frac { 1 } { x ^ { n } - \sqrt { x ^ { 2 n } + 1 } } = - x ^ { n } - \sqrt { x ^ { 2 n } + 1 }\). [1]
   Let \(u _ { n } = x ^ { n + 1 } + \sqrt { x ^ { 2 n + 2 } + 1 } + \frac { 1 } { x ^ { n } - \sqrt { x ^ { 2 n } + 1 } }\).
2. Use the method of differences to find \(\sum _ { \mathrm { n } = 1 } ^ { \mathrm { N } } \mathrm { u } _ { \mathrm { n } }\) in terms of \(N\) and \(x\).
3. Deduce the set of values of \(x\) for which the infinite series $$u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots$$ is convergent and give the sum to infinity when this exists.

7. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. $$f ( x ) = 2 x - 3 \sqrt { x } - 5 \quad x > 0$$

1. Solve the equation $$f ( x ) = 9$$
2. Solve the equation $$\mathrm { f } ^ { \prime \prime } ( x ) = 6$$

1. In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

1. $$f ( x ) = ( x + \sqrt { 2 } ) ^ { 2 } + ( 3 x - 5 \sqrt { 8 } ) ^ { 2 }$$ Express \(\mathrm { f } ( x )\) in the form \(a x ^ { 2 } + b x \sqrt { 2 } + c\) where \(a , b\) and \(c\) are integers to be found.
2. Solve the equation $$\sqrt { 3 } ( 4 y - 3 \sqrt { 3 } ) = 5 y + \sqrt { 3 }$$ giving your answer in the form \(p + q \sqrt { 3 }\) where \(p\) and \(q\) are simplified fractions to be found.

3. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

1. Show that \(\frac { \sqrt { 180 } - \sqrt { 80 } } { \sqrt { 5 } }\) is an integer and find its value.
2. Simplify $$\frac { 4 \sqrt { 5 } - 5 } { 7 - 3 \sqrt { 5 } }$$ giving your answer in the form \(a + b \sqrt { 5 }\) where \(a\) and \(b\) are rational numbers.

8. Solve, using algebra, the equation $$x - 6 x ^ { \frac { 1 } { 2 } } + 4 = 0$$ Fully simplify your answers, writing them in the form \(a + b \sqrt { c }\), where \(a , b\) and \(c\) are integers to be found.
(5)

1. Given that

$$y = 5 x ^ { 3 } + \frac { 3 } { x ^ { 2 } } - 7 x \quad x > 0$$ find, in simplest form,

1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)

1. Given that

$$a = \frac { 1 } { 64 } x ^ { 2 } \quad b = \frac { 16 } { \sqrt { x } }$$ express each of the following in the form \(k x ^ { n }\) where \(k\) and \(n\) are simplified constants.

1. \(a ^ { \frac { 1 } { 2 } }\)
2. \(\frac { 16 } { b ^ { 3 } }\)
3. \(\left( \frac { a b } { 2 } \right) ^ { - \frac { 4 } { 3 } }\)

4. Answer this question without the use of a calculator and show all your working.

1. Show that $$\frac { 4 } { 2 \sqrt { 2 } - \sqrt { 6 } } = 2 \sqrt { 2 } ( 2 + \sqrt { 3 } )$$
2. Show that $$\sqrt { 27 } + \sqrt { 21 } \times \sqrt { 7 } - \frac { 6 } { \sqrt { 3 } } = 8 \sqrt { 3 }$$

Simplify the following expressions fully.

1. \(\left( x ^ { 6 } \right) ^ { \frac { 1 } { 3 } }\)
2. \(\sqrt { 2 } \left( x ^ { 3 } \right) \div \sqrt { \frac { 32 } { x ^ { 2 } } }\)

2. (i) Given that \(\frac { 49 } { \sqrt { 7 } } = 7 ^ { a }\), find the value of \(a\).
(ii) Show that \(\frac { 10 } { \sqrt { 18 } - 4 } = 15 \sqrt { 2 } + 20\) You must show all stages of your working.

3. A curve has equation $$y = \sqrt { 2 } x ^ { 2 } - 6 \sqrt { x } + 4 \sqrt { 2 } , \quad x > 0$$ Find the gradient of the curve at the point \(P ( 2,2 \sqrt { 2 } )\).
Write your answer in the form \(a \sqrt { 2 }\), where \(a\) is a constant.
(Solutions based entirely on graphical or numerical methods are not acceptable.) \(L\)

2. Without using your calculator, solve $$x \sqrt { 27 } + 21 = \frac { 6 x } { \sqrt { 3 } }$$ Write your answer in the form \(a \sqrt { b }\) where \(a\) and \(b\) are integers. You must show all stages of your working.

3. Answer this question without a calculator, showing all your working and giving your answers in their simplest form.

1. Solve the equation $$4 ^ { 2 x + 1 } = 8 ^ { 4 x }$$
2. (a) Express $$3 \sqrt { 18 } - \sqrt { 32 }$$ in the form \(k \sqrt { 2 }\), where \(k\) is an integer.
   (b) Hence, or otherwise, solve $$3 \sqrt { 18 } - \sqrt { 32 } = \sqrt { n }$$