CAIE P1 (Pure Mathematics 1) 2010 November

1 Find \(\int \left( x + \frac { 1 } { x } \right) ^ { 2 } \mathrm {~d} x\).

2 In the expansion of \(( 1 + a x ) ^ { 6 }\), where \(a\) is a constant, the coefficient of \(x\) is - 30 . Find the coefficient of \(x ^ { 3 }\).

3 Functions f and g are defined for \(x \in \mathbb { R }\) by $$\begin{aligned} & \mathrm { f } : x \mapsto 2 x + 3
& \mathrm {~g} : x \mapsto x ^ { 2 } - 2 x \end{aligned}$$ Express \(\operatorname { gf } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.

4

1. Prove the identity \(\frac { \sin x \tan x } { 1 - \cos x } \equiv 1 + \frac { 1 } { \cos x }\).
2. Hence solve the equation \(\frac { \sin x \tan x } { 1 - \cos x } + 2 = 0\), for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

5
\includegraphics[max width=\textwidth, alt={}, center]{73c0c113-8f35-4e7f-ad5d-604602498b71-2_741_533_1279_808} The diagram shows a pyramid \(O A B C\) with a horizontal base \(O A B\) where \(O A = 6 \mathrm {~cm} , O B = 8 \mathrm {~cm}\) and angle \(A O B = 90 ^ { \circ }\). The point \(C\) is vertically above \(O\) and \(O C = 10 \mathrm {~cm}\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O B\) and \(O C\) as shown. Use a scalar product to find angle \(A C B\).

6

1. The fifth term of an arithmetic progression is 18 and the sum of the first 5 terms is 75 . Find the first term and the common difference.
2. The first term of a geometric progression is 16 and the fourth term is \(\frac { 27 } { 4 }\). Find the sum to infinity of the progression.

7 A function f is defined by f : \(x \mapsto 3 - 2 \tan \left( \frac { 1 } { 2 } x \right)\) for \(0 \leqslant x < \pi\).

1. State the range of f .
2. State the exact value of \(\mathrm { f } \left( \frac { 2 } { 3 } \pi \right)\).
3. Sketch the graph of \(y = \mathrm { f } ( x )\).
4. Obtain an expression, in terms of \(x\), for \(\mathrm { f } ^ { - 1 } ( x )\).

8
\includegraphics[max width=\textwidth, alt={}, center]{73c0c113-8f35-4e7f-ad5d-604602498b71-3_314_803_751_671} The diagram shows a metal plate consisting of a rectangle with sides \(x \mathrm {~cm}\) and \(y \mathrm {~cm}\) and a quarter-circle of radius \(x \mathrm {~cm}\). The perimeter of the plate is 60 cm .

1. Express \(y\) in terms of \(x\).
2. Show that the area of the plate, \(A \mathrm {~cm} ^ { 2 }\), is given by \(A = 30 x - x ^ { 2 }\). Given that \(x\) can vary,
3. find the value of \(x\) at which \(A\) is stationary,
4. find this stationary value of \(A\), and determine whether it is a maximum or a minimum value.

9
\includegraphics[max width=\textwidth, alt={}, center]{73c0c113-8f35-4e7f-ad5d-604602498b71-4_837_1020_255_559} The diagram shows two circles, \(C _ { 1 }\) and \(C _ { 2 }\), touching at the point \(T\). Circle \(C _ { 1 }\) has centre \(P\) and radius 8 cm ; circle \(C _ { 2 }\) has centre \(Q\) and radius 2 cm . Points \(R\) and \(S\) lie on \(C _ { 1 }\) and \(C _ { 2 }\) respectively, and \(R S\) is a tangent to both circles.

1. Show that \(R S = 8 \mathrm {~cm}\).
2. Find angle \(R P Q\) in radians correct to 4 significant figures.
3. Find the area of the shaded region.

10 The equation of a curve is \(y = 3 + 4 x - x ^ { 2 }\).

1. Show that the equation of the normal to the curve at the point \(( 3,6 )\) is \(2 y = x + 9\).
2. Given that the normal meets the coordinate axes at points \(A\) and \(B\), find the coordinates of the mid-point of \(A B\).
3. Find the coordinates of the point at which the normal meets the curve again.

11 The equation of a curve is \(y = \frac { 9 } { 2 - x }\).

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and determine, with a reason, whether the curve has any stationary points.
2. Find the volume obtained when the region bounded by the curve, the coordinate axes and the line \(x = 1\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
3. Find the set of values of \(k\) for which the line \(y = x + k\) intersects the curve at two distinct points. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }