CAIE P1 (Pure Mathematics 1) 2016 June

1 Find the coefficient of \(x\) in the expansion of \(\left( \frac { 1 } { x } + 3 x ^ { 2 } \right) ^ { 5 }\).

2
\includegraphics[max width=\textwidth, alt={}, center]{8c358a10-a3e1-47b5-ae62-30ba6b76c167-2_627_551_429_790} The diagram shows part of the curve \(y = \left( x ^ { 3 } + 1 \right) ^ { \frac { 1 } { 2 } }\) and the point \(P ( 2,3 )\) lying on the curve. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

3 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { 2 } + \frac { k } { x ^ { 3 } }\) and passes through the point \(P ( 1,9 )\). The gradient of the curve at \(P\) is 2 .

1. Find the value of the constant \(k\).
2. Find the equation of the curve.

4 The 1st, 3rd and 13th terms of an arithmetic progression are also the 1st, 2nd and 3rd terms respectively of a geometric progression. The first term of each progression is 3 . Find the common difference of the arithmetic progression and the common ratio of the geometric progression.

5 A curve has equation \(y = 8 x + ( 2 x - 1 ) ^ { - 1 }\). Find the values of \(x\) at which the curve has a stationary point and determine the nature of each stationary point, justifying your answers.

6
\includegraphics[max width=\textwidth, alt={}, center]{8c358a10-a3e1-47b5-ae62-30ba6b76c167-3_655_1011_255_566} The diagram shows triangle \(A B C\) where \(A B = 5 \mathrm {~cm} , A C = 4 \mathrm {~cm}\) and \(B C = 3 \mathrm {~cm}\). Three circles with centres at \(A , B\) and \(C\) have radii \(3 \mathrm {~cm} , 2 \mathrm {~cm}\) and 1 cm respectively. The circles touch each other at points \(E , F\) and \(G\), lying on \(A B , A C\) and \(B C\) respectively. Find the area of the shaded region \(E F G\).

7 The point \(P ( x , y )\) is moving along the curve \(y = x ^ { 2 } - \frac { 10 } { 3 } x ^ { \frac { 3 } { 2 } } + 5 x\) in such a way that the rate of change of \(y\) is constant. Find the values of \(x\) at the points at which the rate of change of \(x\) is equal to half the rate of change of \(y\).

8

1. Show that \(3 \sin x \tan x - \cos x + 1 = 0\) can be written as a quadratic equation in \(\cos x\) and hence solve the equation \(3 \sin x \tan x - \cos x + 1 = 0\) for \(0 \leqslant x \leqslant \pi\).
2. Find the solutions to the equation \(3 \sin 2 x \tan 2 x - \cos 2 x + 1 = 0\) for \(0 \leqslant x \leqslant \pi\).

9 The position vectors of \(A , B\) and \(C\) relative to an origin \(O\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 2
3
- 4 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { c } 1
5
p \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 5
0
2 \end{array} \right) ,$$ where \(p\) is a constant.

1. Find the value of \(p\) for which the lengths of \(A B\) and \(C B\) are equal.
2. For the case where \(p = 1\), use a scalar product to find angle \(A B C\).

10 The function f is such that \(\mathrm { f } ( x ) = 2 x + 3\) for \(x \geqslant 0\). The function g is such that \(\mathrm { g } ( x ) = a x ^ { 2 } + b\) for \(x \leqslant q\), where \(a , b\) and \(q\) are constants. The function fg is such that \(\operatorname { fg } ( x ) = 6 x ^ { 2 } - 21\) for \(x \leqslant q\).

1. Find the values of \(a\) and \(b\).
2. Find the greatest possible value of \(q\). It is now given that \(q = - 3\).
3. Find the range of fg.
4. Find an expression for \(( \mathrm { fg } ) ^ { - 1 } ( x )\) and state the domain of \(( \mathrm { fg } ) ^ { - 1 }\).

11 Triangle \(A B C\) has vertices at \(A ( - 2 , - 1 ) , B ( 4,6 )\) and \(C ( 6 , - 3 )\).

1. Show that triangle \(A B C\) is isosceles and find the exact area of this triangle.
2. The point \(D\) is the point on \(A B\) such that \(C D\) is perpendicular to \(A B\). Calculate the \(x\)-coordinate of \(D\).