CAIE P1 (Pure Mathematics 1) 2014 November

1 In the expansion of \(( 2 + a x ) ^ { 6 }\), the coefficient of \(x ^ { 2 }\) is equal to the coefficient of \(x ^ { 3 }\). Find the value of the non-zero constant \(a\).

2
\includegraphics[max width=\textwidth, alt={}, center]{5b558b70-df50-437e-8425-47954f360765-2_618_700_431_721} In the diagram, \(O A D C\) is a sector of a circle with centre \(O\) and radius \(3 \mathrm {~cm} . A B\) and \(C B\) are tangents to the circle and angle \(A B C = \frac { 1 } { 3 } \pi\) radians. Find, giving your answer in terms of \(\sqrt { } 3\) and \(\pi\),

1. the perimeter of the shaded region,
2. the area of the shaded region.

3

1. Express \(9 x ^ { 2 } - 12 x + 5\) in the form \(( a x + b ) ^ { 2 } + c\).
2. Determine whether \(3 x ^ { 3 } - 6 x ^ { 2 } + 5 x - 12\) is an increasing function, a decreasing function or neither.

4 Three geometric progressions, \(P , Q\) and \(R\), are such that their sums to infinity are the first three terms respectively of an arithmetic progression. Progression \(P\) is \(2,1 , \frac { 1 } { 2 } , \frac { 1 } { 4 } , \ldots\).
Progression \(Q\) is \(3,1 , \frac { 1 } { 3 } , \frac { 1 } { 9 } , \ldots\).

1. Find the sum to infinity of progression \(R\).
2. Given that the first term of \(R\) is 4 , find the sum of the first three terms of \(R\).

5

1. Show that \(\sin ^ { 4 } \theta - \cos ^ { 4 } \theta \equiv 2 \sin ^ { 2 } \theta - 1\).
2. Hence solve the equation \(\sin ^ { 4 } \theta - \cos ^ { 4 } \theta = \frac { 1 } { 2 }\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
   \(6 \quad A\) is the point \(( a , 2 a - 1 )\) and \(B\) is the point \(( 2 a + 4,3 a + 9 )\), where \(a\) is a constant.
3. Find, in terms of \(a\), the gradient of a line perpendicular to \(A B\).
4. Given that the distance \(A B\) is \(\sqrt { } ( 260 )\), find the possible values of \(a\).

7 Three points, \(O , A\) and \(B\), are such that \(\overrightarrow { O A } = \mathbf { i } + 3 \mathbf { j } + p \mathbf { k }\) and \(\overrightarrow { O B } = - 7 \mathbf { i } + ( 1 - p ) \mathbf { j } + p \mathbf { k }\), where \(p\) is a constant.

1. Find the values of \(p\) for which \(\overrightarrow { O A }\) is perpendicular to \(\overrightarrow { O B }\).
2. The magnitudes of \(\overrightarrow { O A }\) and \(\overrightarrow { O B }\) are \(a\) and \(b\) respectively. Find the value of \(p\) for which \(b ^ { 2 } = 2 a ^ { 2 }\).
3. Find the unit vector in the direction of \(\overrightarrow { A B }\) when \(p = - 8\).

8 A curve \(y = \mathrm { f } ( x )\) has a stationary point at \(( 3,7 )\) and is such that \(\mathrm { f } ^ { \prime \prime } ( x ) = 36 x ^ { - 3 }\).

1. State, with a reason, whether this stationary point is a maximum or a minimum.
2. Find \(\mathrm { f } ^ { \prime } ( x )\) and \(\mathrm { f } ( x )\).

9
\includegraphics[max width=\textwidth, alt={}, center]{5b558b70-df50-437e-8425-47954f360765-3_842_695_1103_724} The diagram shows parts of the graphs of \(y = x + 2\) and \(y = 3 \sqrt { } x\) intersecting at points \(A\) and \(B\).

1. Write down an equation satisfied by the \(x\)-coordinates of \(A\) and \(B\). Solve this equation and hence find the coordinates of \(A\) and \(B\).
2. Find by integration the area of the shaded region.
   [0pt] [Question 10 is printed on the next page.]

10

1. The functions f and g are defined for \(x \geqslant 0\) by $$\begin{aligned} & \mathrm { f } : x \mapsto ( a x + b ) ^ { \frac { 1 } { 3 } } , \text { where } a \text { and } b \text { are positive constants, }
   & \mathrm { g } : x \mapsto x ^ { 2 } . \end{aligned}$$ Given that \(\mathrm { fg } ( 1 ) = 2\) and \(\operatorname { gf } ( 9 ) = 16\),
   1. calculate the values of \(a\) and \(b\),
   2. obtain an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
2. A point \(P\) travels along the curve \(y = \left( 7 x ^ { 2 } + 1 \right) ^ { \frac { 1 } { 3 } }\) in such a way that the \(x\)-coordinate of \(P\) at time \(t\) minutes is increasing at a constant rate of 8 units per minute. Find the rate of increase of the \(y\)-coordinate of \(P\) at the instant when \(P\) is at the point \(( 3,4 )\).