CAIE P1 (Pure Mathematics 1) 2017 June

1 The coefficients of \(x\) and \(x ^ { 2 }\) in the expansion of \(( 2 + a x ) ^ { 7 }\) are equal. Find the value of the non-zero constant \(a\).

2 The common ratio of a geometric progression is \(r\). The first term of the progression is \(\left( r ^ { 2 } - 3 r + 2 \right)\) and the sum to infinity is \(S\).

1. Show that \(S = 2 - r\).
2. Find the set of possible values that \(S\) can take.

3 Find the coordinates of the points of intersection of the curve \(y = x ^ { \frac { 2 } { 3 } } - 1\) with the curve \(y = x ^ { \frac { 1 } { 3 } } + 1\). [4]

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

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1
3 \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { r } 5
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- 3 \end{array} \right) .$$ The point \(P\) lies on \(A B\) and is such that \(\overrightarrow { A P } = \frac { 1 } { 3 } \overrightarrow { A B }\).

1. Find the position vector of \(P\).
2. Find the distance \(O P\).
3. Determine whether \(O P\) is perpendicular to \(A B\). Justify your answer.
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4. Show that the equation \(\frac { 2 \sin \theta + \cos \theta } { \sin \theta + \cos \theta } = 2 \tan \theta\) may be expressed as \(\cos ^ { 2 } \theta = 2 \sin ^ { 2 } \theta\).
5. Hence solve the equation \(\frac { 2 \sin \theta + \cos \theta } { \sin \theta + \cos \theta } = 2 \tan \theta\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

6 The line \(3 y + x = 25\) is a normal to the curve \(y = x ^ { 2 } - 5 x + k\). Find the value of the constant \(k\).

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\includegraphics[max width=\textwidth, alt={}, center]{4782d612-0ec1-418e-8ef3-c871dce82b44-10_611_732_255_705} The diagram shows two circles with centres \(A\) and \(B\) having radii 8 cm and 10 cm respectively. The two circles intersect at \(C\) and \(D\) where \(C A D\) is a straight line and \(A B\) is perpendicular to \(C D\).

1. Find angle \(A B C\) in radians.
2. Find the area of the shaded region.
   \(8 \quad A ( - 1,1 )\) and \(P ( a , b )\) are two points, where \(a\) and \(b\) are constants. The gradient of \(A P\) is 2 .
3. Find an expression for \(b\) in terms of \(a\).
4. \(B ( 10 , - 1 )\) is a third point such that \(A P = A B\). Calculate the coordinates of the possible positions of \(P\).

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1. Express \(9 x ^ { 2 } - 6 x + 6\) in the form \(( a x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.
   The function f is defined by \(\mathrm { f } ( x ) = 9 x ^ { 2 } - 6 x + 6\) for \(x \geqslant p\), where \(p\) is a constant.
2. State the smallest value of \(p\) for which f is a one-one function.
3. For this value of \(p\), obtain an expression for \(\mathrm { f } ^ { - 1 } ( x )\), and state the domain of \(\mathrm { f } ^ { - 1 }\).
4. State the set of values of \(q\) for which the equation \(\mathrm { f } ( x ) = q\) has no solution.

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1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4782d612-0ec1-418e-8ef3-c871dce82b44-16_451_442_269_888} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} Fig. 1 shows part of the curve \(y = x ^ { 2 } - 1\) and the line \(y = h\), where \(h\) is a constant.
   1. The shaded region is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis. Show that the volume of revolution, \(V\), is given by \(V = \pi \left( \frac { 1 } { 2 } h ^ { 2 } + h \right)\).
   2. Find, showing all necessary working, the area of the shaded region when \(h = 3\).
2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4782d612-0ec1-418e-8ef3-c871dce82b44-17_257_408_1126_904} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} Fig. 2 shows a cross-section of a bowl containing water. When the height of the water level is \(h \mathrm {~cm}\), the volume, \(V \mathrm {~cm} ^ { 3 }\), of water is given by \(V = \pi \left( \frac { 1 } { 2 } h ^ { 2 } + h \right)\). Water is poured into the bowl at a constant rate of \(2 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\). Find the rate, in \(\mathrm { cm } \mathrm { s } ^ { - 1 }\), at which the height of the water level is increasing when the height of the water level is 3 cm .

11 The function f is defined for \(x \geqslant 0\). It is given that f has a minimum value when \(x = 2\) and that \(f ^ { \prime \prime } ( x ) = ( 4 x + 1 ) ^ { - \frac { 1 } { 2 } }\).

1. Find \(\mathrm { f } ^ { \prime } ( x )\).
   It is now given that \(f ^ { \prime \prime } ( 0 ) , f ^ { \prime } ( 0 )\) and \(f ( 0 )\) are the first three terms respectively of an arithmetic progression.
2. Find the value of \(\mathrm { f } ( 0 )\).
3. Find \(\mathrm { f } ( x )\), and hence find the minimum value of f .