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.