CAIE P1 (Pure Mathematics 1) 2012 June

1

1. Prove the identity \(\tan ^ { 2 } \theta - \sin ^ { 2 } \theta \equiv \tan ^ { 2 } \theta \sin ^ { 2 } \theta\).
2. Use this result to explain why \(\tan \theta > \sin \theta\) for \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).

2 Relative to an origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 2
- 1

4 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 4
2
- 2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 1
3
p \end{array} \right)$$ Find

1. the unit vector in the direction of \(\overrightarrow { A B }\),
2. the value of the constant \(p\) for which angle \(B O C = 90 ^ { \circ }\). 3 The first three terms in the expansion of \(( 1 - 2 x ) ^ { 2 } ( 1 + a x ) ^ { 6 }\), in ascending powers of \(x\), are \(1 - x + b x ^ { 2 }\). Find the values of the constants \(a\) and \(b\). 4
3. Solve the equation \(\sin 2 x + 3 \cos 2 x = 0\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
4. How many solutions has the equation \(\sin 2 x + 3 \cos 2 x = 0\) for \(0 ^ { \circ } \leqslant x \leqslant 1080 ^ { \circ }\) ?

5
\includegraphics[max width=\textwidth, alt={}, center]{1b5d8cb1-fd1b-4fcf-8975-b5d020991c9a-2_570_1050_1393_550} The diagram shows part of the curve \(x = \frac { 8 } { y ^ { 2 } } - 2\), crossing the \(y\)-axis at the point \(A\). The point \(B ( 6,1 )\) lies on the curve. The shaded region is bounded by the curve, the \(y\)-axis and the line \(y = 1\). Find the exact volume obtained when this shaded region is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis.

6 The first term of an arithmetic progression is 12 and the sum of the first 9 terms is 135 .

1. Find the common difference of the progression. The first term, the ninth term and the \(n\)th term of this arithmetic progression are the first term, the second term and the third term respectively of a geometric progression.
2. Find the common ratio of the geometric progression and the value of \(n\).

7 The curve \(y = \frac { 10 } { 2 x + 1 } - 2\) intersects the \(x\)-axis at \(A\). The tangent to the curve at \(A\) intersects the \(y\)-axis at \(C\).

1. Show that the equation of \(A C\) is \(5 y + 4 x = 8\).
2. Find the distance \(A C\).

8
\includegraphics[max width=\textwidth, alt={}, center]{1b5d8cb1-fd1b-4fcf-8975-b5d020991c9a-3_554_385_641_879} In the diagram, \(A B\) is an arc of a circle with centre \(O\) and radius \(r\). The line \(X B\) is a tangent to the circle at \(B\) and \(A\) is the mid-point of \(O X\).

1. Show that angle \(A O B = \frac { 1 } { 3 } \pi\) radians. Express each of the following in terms of \(r , \pi\) and \(\sqrt { } 3\) :
2. the perimeter of the shaded region,
3. the area of the shaded region.

9 A curve is such that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - 4 x\). The curve has a maximum point at (2,12).

1. Find the equation of the curve. A point \(P\) moves along the curve in such a way that the \(x\)-coordinate is increasing at 0.05 units per second.
2. Find the rate at which the \(y\)-coordinate is changing when \(x = 3\), stating whether the \(y\)-coordinate is increasing or decreasing.

10 The equation of a line is \(2 y + x = k\), where \(k\) is a constant, and the equation of a curve is \(x y = 6\).

1. In the case where \(k = 8\), the line intersects the curve at the points \(A\) and \(B\). Find the equation of the perpendicular bisector of the line \(A B\).
2. Find the set of values of \(k\) for which the line \(2 y + x = k\) intersects the curve \(x y = 6\) at two distinct points.

11 The function f is such that \(\mathrm { f } ( x ) = 8 - ( x - 2 ) ^ { 2 }\), for \(x \in \mathbb { R }\).

1. Find the coordinates and the nature of the stationary point on the curve \(y = \mathrm { f } ( x )\). The function g is such that \(\mathrm { g } ( x ) = 8 - ( x - 2 ) ^ { 2 }\), for \(k \leqslant x \leqslant 4\), where \(k\) is a constant.
2. State the smallest value of \(k\) for which g has an inverse. For this value of \(k\),
3. find an expression for \(\mathrm { g } ^ { - 1 } ( x )\),
4. sketch, on the same diagram, the graphs of \(y = \mathrm { g } ( x )\) and \(y = \mathrm { g } ^ { - 1 } ( x )\).