CAIE P1 (Pure Mathematics 1) 2016 November

1 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 8 } { \sqrt { } ( 4 x + 1 ) }\). The point \(( 2,5 )\) lies on the curve. Find the equation of the curve.

2

1. Express the equation \(\sin 2 x + 3 \cos 2 x = 3 ( \sin 2 x - \cos 2 x )\) in the form \(\tan 2 x = k\), where \(k\) is a constant.
2. Hence solve the equation for \(- 90 ^ { \circ } \leqslant x \leqslant 90 ^ { \circ }\).

3 A curve has equation \(y = 2 x ^ { 2 } - 6 x + 5\).

1. Find the set of values of \(x\) for which \(y > 13\).
2. Find the value of the constant \(k\) for which the line \(y = 2 x + k\) is a tangent to the curve.

4 In the expansion of \(( 3 - 2 x ) \left( 1 + \frac { x } { 2 } \right) ^ { n }\), the coefficient of \(x\) is 7 . Find the value of the constant \(n\) and hence find the coefficient of \(x ^ { 2 }\).

5 The line \(\frac { x } { a } + \frac { y } { b } = 1\), where \(a\) and \(b\) are positive constants, intersects the \(x\) - and \(y\)-axes at the points \(A\) and \(B\) respectively. The mid-point of \(A B\) lies on the line \(2 x + y = 10\) and the distance \(A B = 10\). Find the values of \(a\) and \(b\).

6
\includegraphics[max width=\textwidth, alt={}, center]{3a631b88-5ba5-49e7-a312-dfd8a6d8a24e-2_615_809_1535_667} The diagram shows a metal plate \(A B C D\) made from two parts. The part \(B C D\) is a semicircle. The part \(D A B\) is a segment of a circle with centre \(O\) and radius 10 cm . Angle \(B O D\) is 1.2 radians.

1. Show that the radius of the semicircle is 5.646 cm , correct to 3 decimal places.
2. Find the perimeter of the metal plate.
3. Find the area of the metal plate.

7 The equation of a curve is \(y = 2 + \frac { 3 } { 2 x - 1 }\).

1. Obtain an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Explain why the curve has no stationary points. At the point \(P\) on the curve, \(x = 2\).
3. Show that the normal to the curve at \(P\) passes through the origin.
4. A point moves along the curve in such a way that its \(x\)-coordinate is decreasing at a constant rate of 0.06 units per second. Find the rate of change of the \(y\)-coordinate as the point passes through \(P\).

8

1. A cyclist completes a long-distance charity event across Africa. The total distance is 3050 km . He starts the event on May 1st and cycles 200 km on that day. On each subsequent day he reduces the distance cycled by 5 km .
   1. How far will he travel on May 15th?
   2. On what date will he finish the event?
2. A geometric progression is such that the third term is 8 times the sixth term, and the sum of the first six terms is \(31 \frac { 1 } { 2 }\). Find
   1. the first term of the progression,
   2. the sum to infinity of the progression.

9 Relative to an origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 2
- 2
- 1 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } - 2
3
6 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 2
6
5 \end{array} \right)$$

1. Use a scalar product to find angle \(A O B\).
2. Find the vector which is in the same direction as \(\overrightarrow { A C }\) and of magnitude 15 units.
3. Find the value of the constant \(p\) for which \(p \overrightarrow { O A } + \overrightarrow { O C }\) is perpendicular to \(\overrightarrow { O B }\).

10 A function f is defined by \(\mathrm { f } : x \mapsto 5 - 2 \sin 2 x\) for \(0 \leqslant x \leqslant \pi\).

1. Find the range of f .
2. Sketch the graph of \(y = \mathrm { f } ( x )\).
3. Solve the equation \(\mathrm { f } ( x ) = 6\), giving answers in terms of \(\pi\). The function g is defined by \(\mathrm { g } : x \mapsto 5 - 2 \sin 2 x\) for \(0 \leqslant x \leqslant k\), where \(k\) is a constant.
4. State the largest value of \(k\) for which g has an inverse.
5. For this value of \(k\), find an expression for \(\mathrm { g } ^ { - 1 } ( x )\).