10
\includegraphics[max width=\textwidth, alt={}, center]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-13_696_931_269_607}
The diagram shows the graph of \(y = 1 - 3 x ^ { - \frac { 1 } { 2 } }\).
- Verify that the curve intersects the \(x\)-axis at \(( 9,0 )\).
- The shaded region is enclosed by the curve, the \(x\)-axis and the line \(x = a\) (where \(a > 9\) ). Given that the area of the shaded region is 4 square units, find the value of \(a\).
June 2007
1 A geometric progression \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by
$$u _ { 1 } = 15 \quad \text { and } \quad u _ { n + 1 } = 0.8 u _ { n } \text { for } n \geqslant 1 \text {. }$$
- Write down the values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\).
- Find \(\sum _ { n = 1 } ^ { 20 } u _ { n }\).
2 Expand \(\left( x + \frac { 2 } { x } \right) ^ { 4 }\) completely, simplifying the terms.
3 Use logarithms to solve the equation \(3 ^ { 2 x + 1 } = 5 ^ { 200 }\), giving the value of \(x\) correct to 3 significant figures.
4
\includegraphics[max width=\textwidth, alt={}, center]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-14_543_855_1155_646}
The diagram shows the curve \(\mathrm { y } = \sqrt { 4 \mathrm { x } + 1 }\). - Use the trapezium rule, with strips of width 0.5 , to find an approximate value for the area of the region bounded by the curve \(\mathrm { y } = \sqrt { 4 \mathrm { x } + 1 }\), the x -axis, and the lines \(\mathrm { x } = 1\) and \(\mathrm { x } = 3\). Give your answer correct to 3 significant figures.
- State with a reason whether this approximation is an under-estimate or an over-estimate.
5
- Show that the equation
$$3 \cos ^ { 2 } \theta = \sin \theta + 1$$
can be expressed in the form
$$3 \sin ^ { 2 } \theta + \sin \theta - 2 = 0 .$$
- Hence solve the equation
$$3 \cos ^ { 2 } \theta = \sin \theta + 1 ,$$
giving all values of \(\theta\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).
6 (a) (i) Find \(\int x \left( x ^ { 2 } - 4 \right) d x\)
- Hence evaluate \(\int _ { 1 } ^ { 6 } x \left( x ^ { 2 } - 4 \right) d x\).
(b) Find \(\int \frac { 6 } { x ^ { 3 } } d x\)
7 (a) In an arithmetic progression, the first term is 12 and the sum of the first 70 terms is 12915 . Find the common difference.
(b) In a geometric progression, the second term is - 4 and the sum to infinity is 9 . Find the common ratio.
8
\includegraphics[max width=\textwidth, alt={}, center]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-15_305_744_1043_703}
The diagram shows a triangle ABC , where angle BAC is 0.9 radians. BAD is a sector of the circle with centre \(A\) and radius \(A B\). - The area of the sector \(B A D\) is \(16.2 \mathrm {~cm} ^ { 2 }\). Show that the length of \(A B\) is 6 cm .
- The area of triangle \(A B C\) is twice the area of sector \(B A D\). Find the length of \(A C\).
- Find the perimeter of the region \(B C D\).
9 The polynomial \(f ( x )\) is given by
$$f ( x ) = x ^ { 3 } + 6 x ^ { 2 } + x - 4$$
- (a) Show that \(( x + 1 )\) is a factor of \(f ( x )\).
(b) Hence find the exact roots of the equation \(f ( x ) = 0\). - (a) Show that the equation
$$2 \log _ { 2 } ( x + 3 ) + \log _ { 2 } x - \log _ { 2 } ( 4 x + 2 ) = 1$$
can be written in the form \(\mathrm { f } ( \mathrm { x } ) = 0\).
(b) Explain why the equation
$$2 \log _ { 2 } ( x + 3 ) + \log _ { 2 } x - \log _ { 2 } ( 4 x + 2 ) = 1$$
has only one real root and state the exact value of this root.
\section*{Jan 2008}
1
The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius 11 cm . The angle \(A O B\) is 0.7 radians. Find the area of the segment shaded in the diagram.
2 Use the trapezium rule, with 3 strips each of width 2, to estimate the value of
$$\int _ { 1 } ^ { 7 } \sqrt { x ^ { 2 } + 3 } \mathrm {~d} x$$
3 Express each of the following as a single logarithm: - \(\log _ { a } 2 + \log _ { a } 3\),
- \(2 \log _ { 10 } x - 3 \log _ { 10 } y\).
4
\includegraphics[max width=\textwidth, alt={}, center]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-16_515_713_1567_715}
In the diagram, angle \(B D C = 50 ^ { \circ }\) and angle \(B C D = 62 ^ { \circ }\). It is given that \(A B = 10 \mathrm {~cm} , A D = 20 \mathrm {~cm}\) and \(B C = 16 \mathrm {~cm}\). - Find the length of \(B D\).
- Find angle \(B A D\).
5 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 12 \sqrt { x }\). The curve passes through the point (4,50). Find the equation of the curve.
6 A sequence of terms \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by
$$u _ { n } = 2 n + 5 , \quad \text { for } n \geqslant 1$$
- Write down the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
- State what type of sequence it is.
- Given that \(\sum _ { n = 1 } ^ { N } u _ { n } = 2200\), find the value of \(N\).
7
\includegraphics[max width=\textwidth, alt={}, center]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-17_588_569_854_788}
The diagram shows part of the curve \(y = x ^ { 2 } - 3 x\) and the line \(x = 5\). - Explain why \(\int _ { 0 } ^ { 5 } \left( x ^ { 2 } - 3 x \right) \mathrm { d } x\) does not give the total area of the regions shaded in the diagram.
- Use integration to find the exact total area of the shaded regions.
8 The first term of a geometric progression is 10 and the common ratio is 0.8.
- Find the fourth term.
- Find the sum of the first 20 terms, giving your answer correct to 3 significant figures.
- The sum of the first \(N\) terms is denoted by \(S _ { N }\), and the sum to infinity is denoted by \(S _ { \infty }\). Show that the inequality \(S _ { \infty } - S _ { N } < 0.01\) can be written as
$$0.8 ^ { N } < 0.0002$$
and use logarithms to find the smallest possible value of \(N\).
9
- \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-18_378_770_274_731}
\captionsetup{labelformat=empty}
\caption{Fig. 1}
\end{figure}
Fig. 1 shows the curve \(y = 2 \sin x\) for values of \(x\) such that \(- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\). State the coordinates of the maximum and minimum points on this part of the curve. - \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-18_378_771_954_730}
\captionsetup{labelformat=empty}
\caption{Fig. 2}
\end{figure}
Fig. 2 shows the curve \(y = 2 \sin x\) and the line \(y = k\). The smallest positive solution of the equation \(2 \sin x = k\) is denoted by \(\alpha\). State, in terms of \(\alpha\), and in the range \(- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\),
(a) another solution of the equation \(2 \sin x = k\),
(b) one solution of the equation \(2 \sin x = - k\). - Find the \(x\)-coordinates of the points where the curve \(y = 2 \sin x\) intersects the curve \(y = 2 - 3 \cos ^ { 2 } x\), for values of \(x\) such that \(- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
10
- Find the binomial expansion of \(( 2 x + 5 ) ^ { 4 }\), simplifying the terms.
- Hence show that \(( 2 x + 5 ) ^ { 4 } - ( 2 x - 5 ) ^ { 4 }\) can be written as
$$320 x ^ { 3 } + k x$$
where the value of the constant \(k\) is to be stated.
- Verify that \(x = 2\) is a root of the equation
$$( 2 x + 5 ) ^ { 4 } - ( 2 x - 5 ) ^ { 4 } = 3680 x - 800$$
and find the other possible values of \(x\).