CAIE P1 (Pure Mathematics 1) 2007 June

1 Find the value of the constant \(c\) for which the line \(y = 2 x + c\) is a tangent to the curve \(y ^ { 2 } = 4 x\).

2 \includegraphics[max width=\textwidth, alt={}, center]{b24ed4c7-ab07-45f4-adf2-027734c36b62-2_633_787_402_680} The diagram shows the curve \(y = 3 x ^ { \frac { 1 } { 4 } }\). The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 4\). Find the volume of the solid obtained when this shaded region is rotated completely about the \(x\)-axis, giving your answer in terms of \(\pi\).

3 Prove the identity \(\frac { 1 - \tan ^ { 2 } x } { 1 + \tan ^ { 2 } x } \equiv 1 - 2 \sin ^ { 2 } x\).

4 Find the real roots of the equation \(\frac { 18 } { x ^ { 4 } } + \frac { 1 } { x ^ { 2 } } = 4\).

5 \includegraphics[max width=\textwidth, alt={}, center]{b24ed4c7-ab07-45f4-adf2-027734c36b62-2_586_682_1726_733} In the diagram, \(O A B\) is a sector of a circle with centre \(O\) and radius 12 cm . The lines \(A X\) and \(B X\) are tangents to the circle at \(A\) and \(B\) respectively. Angle \(A O B = \frac { 1 } { 3 } \pi\) radians.

1. Find the exact length of \(A X\), giving your answer in terms of \(\sqrt { } 3\).
2. Find the area of the shaded region, giving your answer in terms of \(\pi\) and \(\sqrt { } 3\).

6 \includegraphics[max width=\textwidth, alt={}, center]{b24ed4c7-ab07-45f4-adf2-027734c36b62-3_593_878_269_635} The diagram shows a rectangle \(A B C D\). The point \(A\) is \(( 2,14 ) , B\) is \(( - 2,8 )\) and \(C\) lies on the \(x\)-axis. Find

1. the equation of \(B C\),
2. the coordinates of \(C\) and \(D\).

7 The second term of a geometric progression is 3 and the sum to infinity is 12 .

1. Find the first term of the progression. An arithmetic progression has the same first and second terms as the geometric progression.
2. Find the sum of the first 20 terms of the arithmetic progression.

8 The function f is defined by \(\mathrm { f } ( x ) = a + b \cos 2 x\), for \(0 \leqslant x \leqslant \pi\). It is given that \(\mathrm { f } ( 0 ) = - 1\) and \(\mathrm { f } \left( \frac { 1 } { 2 } \pi \right) = 7\).

1. Find the values of \(a\) and \(b\).
2. Find the \(x\)-coordinates of the points where the curve \(y = \mathrm { f } ( x )\) intersects the \(x\)-axis.
3. Sketch the graph of \(y = \mathrm { f } ( x )\).

9 Relative to an origin \(O\), the position vectors of the points \(A\) and \(B\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 4 \\ 1 \\ - 2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { r } 3 \\ 2 \\ - 4 \end{array} \right) .$$

1. Given that \(C\) is the point such that \(\overrightarrow { A C } = 2 \overrightarrow { A B }\), find the unit vector in the direction of \(\overrightarrow { O C }\). The position vector of the point \(D\) is given by \(\overrightarrow { O D } = \left( \begin{array} { l } 1 \\ 4 \\ k \end{array} \right)\), where \(k\) is a constant, and it is given that \(\overrightarrow { O D } = m \overrightarrow { O A } + n \overrightarrow { O B }\), where \(m\) and \(n\) are constants.
2. Find the values of \(m , n\) and \(k\).

10 The equation of a curve is \(y = 2 x + \frac { 8 } { x ^ { 2 } }\).

1. Obtain expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Find the coordinates of the stationary point on the curve and determine the nature of the stationary point.
3. Show that the normal to the curve at the point \(( - 2 , - 2 )\) intersects the \(x\)-axis at the point \(( - 10,0 )\).
4. Find the area of the region enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\).

11 \includegraphics[max width=\textwidth, alt={}, center]{b24ed4c7-ab07-45f4-adf2-027734c36b62-4_862_892_932_628} The diagram shows the graph of \(y = \mathrm { f } ( x )\), where \(\mathrm { f } : x \mapsto \frac { 6 } { 2 x + 3 }\) for \(x \geqslant 0\).

1. Find an expression, in terms of \(x\), for \(\mathrm { f } ^ { \prime } ( x )\) and explain how your answer shows that f is a decreasing function.
2. Find an expression, in terms of \(x\), for \(\mathrm { f } ^ { - 1 } ( x )\) and find the domain of \(\mathrm { f } ^ { - 1 }\).
3. Copy the diagram and, on your copy, sketch the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\), making clear the relationship between the graphs. The function g is defined by \(\mathrm { g } : x \mapsto \frac { 1 } { 2 } x\) for \(x \geqslant 0\).
4. Solve the equation \(\operatorname { fg } ( x ) = \frac { 3 } { 2 }\).