CAIE P1 (Pure Mathematics 1) 2008 November

1 Find the value of the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( \frac { x } { 2 } + \frac { 2 } { x } \right) ^ { 6 }\).

2 Prove the identity $$\frac { 1 + \sin x } { \cos x } + \frac { \cos x } { 1 + \sin x } \equiv \frac { 2 } { \cos x }$$

3 The first term of an arithmetic progression is 6 and the fifth term is 12 . The progression has \(n\) terms and the sum of all the terms is 90 . Find the value of \(n\).

4 \includegraphics[max width=\textwidth, alt={}, center]{08729aab-586b-4210-94c9-77b1f6b1d873-2_558_1488_863_331} The diagram shows a semicircular prism with a horizontal rectangular base \(A B C D\). The vertical ends \(A E D\) and \(B F C\) are semicircles of radius 6 cm . The length of the prism is 20 cm . The mid-point of \(A D\) is the origin \(O\), the mid-point of \(B C\) is \(M\) and the mid-point of \(D C\) is \(N\). The points \(E\) and \(F\) are the highest points of the semicircular ends of the prism. The point \(P\) lies on \(E F\) such that \(E P = 8 \mathrm {~cm}\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O D , O M\) and \(O E\) respectively.

1. Express each of the vectors \(\overrightarrow { P A }\) and \(\overrightarrow { P N }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
2. Use a scalar product to calculate angle \(A P N\).

5 The function f is such that \(\mathrm { f } ( x ) = a - b \cos x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\), where \(a\) and \(b\) are positive constants. The maximum value of \(\mathrm { f } ( x )\) is 10 and the minimum value is - 2 .

1. Find the values of \(a\) and \(b\).
2. Solve the equation \(\mathrm { f } ( x ) = 0\).
3. Sketch the graph of \(y = \mathrm { f } ( x )\).

6 \includegraphics[max width=\textwidth, alt={}, center]{08729aab-586b-4210-94c9-77b1f6b1d873-3_597_417_274_865} In the diagram, the circle has centre \(O\) and radius 5 cm . The points \(P\) and \(Q\) lie on the circle, and the arc length \(P Q\) is 9 cm . The tangents to the circle at \(P\) and \(Q\) meet at the point \(T\). Calculate

1. angle \(P O Q\) in radians,
2. the length of \(P T\),
3. the area of the shaded region.

7 \includegraphics[max width=\textwidth, alt={}, center]{08729aab-586b-4210-94c9-77b1f6b1d873-3_385_360_1379_561} \includegraphics[max width=\textwidth, alt={}, center]{08729aab-586b-4210-94c9-77b1f6b1d873-3_364_369_1379_1219} A wire, 80 cm long, is cut into two pieces. One piece is bent to form a square of side \(x \mathrm {~cm}\) and the other piece is bent to form a circle of radius \(r \mathrm {~cm}\) (see diagram). The total area of the square and the circle is \(A \mathrm {~cm} ^ { 2 }\).

1. Show that \(A = \frac { ( \pi + 4 ) x ^ { 2 } - 160 x + 1600 } { \pi }\).
2. Given that \(x\) and \(r\) can vary, find the value of \(x\) for which \(A\) has a stationary value.

8 The equation of a curve is \(y = 5 - \frac { 8 } { x }\).

1. Show that the equation of the normal to the curve at the point \(P ( 2,1 )\) is \(2 y + x = 4\). This normal meets the curve again at the point \(Q\).
2. Find the coordinates of \(Q\).
3. Find the length of \(P Q\).

9 \includegraphics[max width=\textwidth, alt={}, center]{08729aab-586b-4210-94c9-77b1f6b1d873-4_719_670_264_735} The diagram shows the curve \(y = \sqrt { } ( 3 x + 1 )\) and the points \(P ( 0,1 )\) and \(Q ( 1,2 )\) on the curve. The shaded region is bounded by the curve, the \(y\)-axis and the line \(y = 2\).

1. Find the area of the shaded region.
2. Find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Tangents are drawn to the curve at the points \(P\) and \(Q\).
3. Find the acute angle, in degrees correct to 1 decimal place, between the two tangents.

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

1. Sketch, in a single diagram, the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), making clear the relationship between the two graphs. The function g is defined by $$\mathrm { g } : x \mapsto 6 x - x ^ { 2 } \text { for } x \in \mathbb { R }$$
2. Express \(\operatorname { gf } ( x )\) in terms of \(x\), and hence show that the maximum value of \(\operatorname { gf } ( x )\) is 9 . The function h is defined by $$\mathrm { h } : x \mapsto 6 x - x ^ { 2 } \text { for } x \geqslant 3$$
3. Express \(6 x - x ^ { 2 }\) in the form \(a - ( x - b ) ^ { 2 }\), where \(a\) and \(b\) are positive constants.
4. Express \(\mathrm { h } ^ { - 1 } ( x )\) in terms of \(x\).