CAIE P1 (Pure Mathematics 1) 2011 November

1 The coefficient of \(x ^ { 2 }\) in the expansion of \(\left( k + \frac { 1 } { 3 } x \right) ^ { 5 }\) is 30 . Find the value of the constant \(k\).

2 The first and second terms of a progression are 4 and 8 respectively. Find the sum of the first 10 terms given that the progression is

1. an arithmetic progression,
2. a geometric progression.

3
\includegraphics[max width=\textwidth, alt={}, center]{96cc217a-ffb3-4764-946e-e32271784ad7-2_680_977_689_584} The diagram shows the curve \(y = 2 x ^ { 5 } + 3 x ^ { 3 }\) and the line \(y = 2 x\) intersecting at points \(A , O\) and \(B\).

1. Show that the \(x\)-coordinates of \(A\) and \(B\) satisfy the equation \(2 x ^ { 4 } + 3 x ^ { 2 } - 2 = 0\).
2. Solve the equation \(2 x ^ { 4 } + 3 x ^ { 2 } - 2 = 0\) and hence find the coordinates of \(A\) and \(B\), giving your answers in an exact form.

4
\includegraphics[max width=\textwidth, alt={}, center]{96cc217a-ffb3-4764-946e-e32271784ad7-2_520_839_1795_653} In the diagram, \(A B C D\) is a parallelogram with \(A B = B D = D C = 10 \mathrm {~cm}\) and angle \(A B D = 0.8\) radians. \(A P D\) and \(B Q C\) are arcs of circles with centres \(B\) and \(D\) respectively.

1. Find the area of the parallelogram \(A B C D\).
2. Find the area of the complete figure \(A B Q C D P\).
3. Find the perimeter of the complete figure \(A B Q C D P\).
4. Given that $$3 \sin ^ { 2 } x - 8 \cos x - 7 = 0$$ show that, for real values of \(x\), $$\cos x = - \frac { 2 } { 3 }$$
5. Hence solve the equation $$3 \sin ^ { 2 } \left( \theta + 70 ^ { \circ } \right) - 8 \cos \left( \theta + 70 ^ { \circ } \right) - 7 = 0$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

6 Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are \(3 \mathbf { i } + 4 \mathbf { j } - \mathbf { k }\) and \(5 \mathbf { i } - 2 \mathbf { j } - 3 \mathbf { k }\) respectively.

1. Use a scalar product to find angle \(B O A\). The point \(C\) is the mid-point of \(A B\). The point \(D\) is such that \(\overrightarrow { O D } = 2 \overrightarrow { O B }\).
2. Find \(\overrightarrow { D C }\).

7

1. A straight line passes through the point \(( 2,0 )\) and has gradient \(m\). Write down the equation of the line.
2. Find the two values of \(m\) for which the line is a tangent to the curve \(y = x ^ { 2 } - 4 x + 5\). For each value of \(m\), find the coordinates of the point where the line touches the curve.
3. Express \(x ^ { 2 } - 4 x + 5\) in the form \(( x + a ) ^ { 2 } + b\) and hence, or otherwise, write down the coordinates of the minimum point on the curve.

8 A curve \(y = \mathrm { f } ( x )\) has a stationary point at \(P ( 3 , - 10 )\). It is given that \(\mathrm { f } ^ { \prime } ( x ) = 2 x ^ { 2 } + k x - 12\), where \(k\) is a constant.

1. Show that \(k = - 2\) and hence find the \(x\)-coordinate of the other stationary point, \(Q\).
2. Find \(\mathrm { f } ^ { \prime \prime } ( x )\) and determine the nature of each of the stationary points \(P\) and \(Q\).
3. Find \(\mathrm { f } ( x )\).

9 Functions f and g are defined by $$\begin{array} { l l } \mathrm { f } : x \mapsto 2 x + 3 & \text { for } x \leqslant 0
\mathrm {~g} : x \mapsto x ^ { 2 } - 6 x & \text { for } x \leqslant 3 \end{array}$$

1. Express \(\mathrm { f } ^ { - 1 } ( x )\) in terms of \(x\) and solve the equation \(\mathrm { f } ( x ) = \mathrm { f } ^ { - 1 } ( x )\).
2. On the same diagram sketch the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), showing the coordinates of their point of intersection and the relationship between the graphs.
3. Find the set of values of \(x\) which satisfy \(\operatorname { gf } ( x ) \leqslant 16\).

10
\includegraphics[max width=\textwidth, alt={}, center]{96cc217a-ffb3-4764-946e-e32271784ad7-4_764_929_255_609} The diagram shows the line \(y = x + 1\) and the curve \(y = \sqrt { } ( x + 1 )\), meeting at \(( - 1,0 )\) and \(( 0,1 )\).

1. Find the area of the shaded region.
2. Find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis.