CAIE P1 (Pure Mathematics 1) 2014 June

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

2 The first term in a progression is 36 and the second term is 32 .

1. Given that the progression is geometric, find the sum to infinity.
2. Given instead that the progression is arithmetic, find the number of terms in the progression if the sum of all the terms is 0 .

3 \includegraphics[max width=\textwidth, alt={}, center]{0b047754-84f2-46ea-b441-7c68cef47641-2_485_623_790_760} The diagram shows part of a circle with centre \(O\) and radius 6 cm . The chord \(A B\) is such that angle \(A O B = 2.2\) radians. Calculate

1. the perimeter of the shaded region,
2. the ratio of the area of the shaded region to the area of the triangle \(A O B\), giving your answer in the form \(k : 1\).

4

1. Prove the identity \(\frac { \tan x + 1 } { \sin x \tan x + \cos x } \equiv \sin x + \cos x\).
2. Hence solve the equation \(\frac { \tan x + 1 } { \sin x \tan x + \cos x } = 3 \sin x - 2 \cos x\) for \(0 \leqslant x \leqslant 2 \pi\).

5 A function f is such that \(\mathrm { f } ( x ) = \frac { 15 } { 2 x + 3 }\) for \(0 \leqslant x \leqslant 6\).

1. Find an expression for \(\mathrm { f } ^ { \prime } ( x )\) and use your result to explain why f has an inverse.
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\), and state the domain and range of \(\mathrm { f } ^ { - 1 }\).

6 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 12 } { \sqrt { } ( 4 x + a ) }\), where \(a\) is a constant. The point \(P ( 2,14 )\) lies on the curve and the normal to the curve at \(P\) is \(3 y + x = 5\).

1. Show that \(a = 8\).
2. Find the equation of the curve.

7 The position vectors of points \(A , B\) and \(C\) relative to an origin \(O\) are given by $$\overrightarrow { O A } = \left( \begin{array} { l } 2 \\ 1 \\ 3 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 6 \\ - 1 \\ 7 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 2 \\ 4 \\ 7 \end{array} \right)$$

1. Show that angle \(B A C = \cos ^ { - 1 } \left( \frac { 1 } { 3 } \right)\).
2. Use the result in part (i) to find the exact value of the area of triangle \(A B C\).

8

1. Express \(2 x ^ { 2 } - 10 x + 8\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants, and use your answer to state the minimum value of \(2 x ^ { 2 } - 10 x + 8\).
2. Find the set of values of \(k\) for which the equation \(2 x ^ { 2 } - 10 x + 8 = k x\) has no real roots.

9 The base of a cuboid has sides of length \(x \mathrm {~cm}\) and \(3 x \mathrm {~cm}\). The volume of the cuboid is \(288 \mathrm {~cm} ^ { 3 }\).

1. Show that the total surface area of the cuboid, \(A \mathrm {~cm} ^ { 2 }\), is given by $$A = 6 x ^ { 2 } + \frac { 768 } { x }$$
2. Given that \(x\) can vary, find the stationary value of \(A\) and determine its nature.

10 \includegraphics[max width=\textwidth, alt={}, center]{0b047754-84f2-46ea-b441-7c68cef47641-3_812_720_1484_715} The diagram shows the curve \(y = - x ^ { 2 } + 12 x - 20\) and the line \(y = 2 x + 1\). Find, showing all necessary working, the area of the shaded region.

11 \includegraphics[max width=\textwidth, alt={}, center]{0b047754-84f2-46ea-b441-7c68cef47641-4_995_867_260_639} The diagram shows a parallelogram \(A B C D\), in which the equation of \(A B\) is \(y = 3 x\) and the equation of \(A D\) is \(4 y = x + 11\). The diagonals \(A C\) and \(B D\) meet at the point \(E \left( 6 \frac { 1 } { 2 } , 8 \frac { 1 } { 2 } \right)\). Find, by calculation, the coordinates of \(A , B , C\) and \(D\).