CAIE P1 (Pure Mathematics 1) 2017 June

1 The coefficients of \(x ^ { 2 }\) and \(x ^ { 3 }\) in the expansion of \(( 3 - 2 x ) ^ { 6 }\) are \(a\) and \(b\) respectively. Find the value of \(\frac { a } { b }\).

2 Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r }

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p \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { r } 2
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- 7 \end{array} \right)$$ and angle \(A O B = 90 ^ { \circ }\).

1. Find the value of \(p\).
   The point \(C\) is such that \(\overrightarrow { O C } = \frac { 2 } { 3 } \overrightarrow { O A }\).
2. Find the unit vector in the direction of \(\overrightarrow { B C }\).
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3. Prove the identity \(\frac { 1 + \cos \theta } { \sin \theta } + \frac { \sin \theta } { 1 + \cos \theta } \equiv \frac { 2 } { \sin \theta }\).
4. Hence solve the equation \(\frac { 1 + \cos \theta } { \sin \theta } + \frac { \sin \theta } { 1 + \cos \theta } = \frac { 3 } { \cos \theta }\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

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1. An arithmetic progression has a first term of 32, a 5th term of 22 and a last term of - 28 . Find the sum of all the terms in the progression.
2. Each year a school allocates a sum of money for the library. The amount allocated each year increases by \(2.5 \%\) of the amount allocated the previous year. In 2005 the school allocated \(
   ) 2000$. Find the total amount allocated in the years 2005 to 2014 inclusive.

5 The equation of a curve is \(y = 2 \cos x\).

1. Sketch the graph of \(y = 2 \cos x\) for \(- \pi \leqslant x \leqslant \pi\), stating the coordinates of the point of intersection with the \(y\)-axis. Points \(P\) and \(Q\) lie on the curve and have \(x\)-coordinates of \(\frac { 1 } { 3 } \pi\) and \(\pi\) respectively.
2. Find the length of \(P Q\) correct to 1 decimal place.
   The line through \(P\) and \(Q\) meets the \(x\)-axis at \(H ( h , 0 )\) and the \(y\)-axis at \(K ( 0 , k )\).
3. Show that \(h = \frac { 5 } { 9 } \pi\) and find the value of \(k\).

6 The horizontal base of a solid prism is an equilateral triangle of side \(x \mathrm {~cm}\). The sides of the prism are vertical. The height of the prism is \(h \mathrm {~cm}\) and the volume of the prism is \(2000 \mathrm {~cm} ^ { 3 }\).

1. Express \(h\) in terms of \(x\) and show that the total surface area of the prism, \(A \mathrm {~cm} ^ { 2 }\), is given by $$A = \frac { \sqrt { } 3 } { 2 } x ^ { 2 } + \frac { 24000 } { \sqrt { } 3 } x ^ { - 1 }$$
2. Given that \(x\) can vary, find the value of \(x\) for which \(A\) has a stationary value.
3. Determine, showing all necessary working, the nature of this stationary value.

7 A curve for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 7 - x ^ { 2 } - 6 x\) passes through the point \(( 3 , - 10 )\).

1. Find the equation of the curve.
2. Express \(7 - x ^ { 2 } - 6 x\) in the form \(a - ( x + b ) ^ { 2 }\), where \(a\) and \(b\) are constants.
3. Find the set of values of \(x\) for which the gradient of the curve is positive.

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\includegraphics[max width=\textwidth, alt={}, center]{028c7979-6b24-42d0-9857-c616a169b2b2-14_590_691_260_726} In the diagram, \(O A X B\) is a sector of a circle with centre \(O\) and radius 10 cm . The length of the chord \(A B\) is 12 cm . The line \(O X\) passes through \(M\), the mid-point of \(A B\), and \(O X\) is perpendicular to \(A B\). The shaded region is bounded by the chord \(A B\) and by the arc of a circle with centre \(X\) and radius \(X A\).

1. Show that angle \(A X B\) is 2.498 radians, correct to 3 decimal places.
2. Find the perimeter of the shaded region.
3. Find the area of the shaded region.

9 The function f is defined by \(\mathrm { f } : x \mapsto \frac { 2 } { 3 - 2 x }\) for \(x \in \mathbb { R } , x \neq \frac { 3 } { 2 }\).

1. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
   The function g is defined by \(\mathrm { g } : x \mapsto 4 x + a\) for \(x \in \mathbb { R }\), where \(a\) is a constant.
2. Find the value of \(a\) for which \(\operatorname { gf } ( - 1 ) = 3\).
3. Find the possible values of \(a\) given that the equation \(\mathrm { f } ^ { - 1 } ( x ) = \mathrm { g } ^ { - 1 } ( x )\) has two equal roots.

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\includegraphics[max width=\textwidth, alt={}, center]{028c7979-6b24-42d0-9857-c616a169b2b2-18_510_410_260_863} The diagram shows part of the curve \(y = \frac { 4 } { 5 - 3 x }\).

1. Find the equation of the normal to the curve at the point where \(x = 1\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
   The shaded region is bounded by the curve, the coordinate axes and the line \(x = 1\).
2. Find, showing all necessary working, the volume obtained when this shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.