CAIE P1 (Pure Mathematics 1) 2014 November

1
\includegraphics[max width=\textwidth, alt={}, center]{8952fc09-004a-4fb6-ad80-5312095a7057-2_668_554_260_797} The diagram shows part of the curve \(y = x ^ { 2 } + 1\). Find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis.

2
\includegraphics[max width=\textwidth, alt={}, center]{8952fc09-004a-4fb6-ad80-5312095a7057-2_313_737_1197_703} The diagram shows a triangle \(A O B\) in which \(O A\) is \(12 \mathrm {~cm} , O B\) is 5 cm and angle \(A O B\) is a right angle. Point \(P\) lies on \(A B\) and \(O P\) is an arc of a circle with centre \(A\). Point \(Q\) lies on \(A B\) and \(O Q\) is an arc of a circle with centre \(B\).

1. Show that angle \(B A O\) is 0.3948 radians, correct to 4 decimal places.
2. Calculate the area of the shaded region.

3

1. Find the first 3 terms, in ascending powers of \(x\), in the expansion of \(( 1 + x ) ^ { 5 }\). The coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 1 + \left( p x + x ^ { 2 } \right) \right) ^ { 5 }\) is 95 .
2. Use the answer to part (i) to find the value of the positive constant \(p\).

4 A curve has equation \(y = \frac { 12 } { 3 - 2 x }\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). A point moves along this curve. As the point passes through \(A\), the \(x\)-coordinate is increasing at a rate of 0.15 units per second and the \(y\)-coordinate is increasing at a rate of 0.4 units per second.
2. Find the possible \(x\)-coordinates of \(A\).

1. Show that the equation \(1 + \sin x \tan x = 5 \cos x\) can be expressed as $$6 \cos ^ { 2 } x - \cos x - 1 = 0$$
2. Hence solve the equation \(1 + \sin x \tan x = 5 \cos x\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\). The equation of a curve is \(y = x ^ { 3 } + a x ^ { 2 } + b x\), where \(a\) and \(b\) are constants.
3. In the case where the curve has no stationary point, show that \(a ^ { 2 } < 3 b\).
4. In the case where \(a = - 6\) and \(b = 9\), find the set of values of \(x\) for which \(y\) is a decreasing function of \(x\).
   \includegraphics[max width=\textwidth, alt={}, center]{8952fc09-004a-4fb6-ad80-5312095a7057-3_634_711_952_717} The diagram shows a pyramid \(O A B C X\). The horizontal square base \(O A B C\) has side 8 units and the centre of the base is \(D\). The top of the pyramid, \(X\), is vertically above \(D\) and \(X D = 10\) units. The mid-point of \(O X\) is \(M\). The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are parallel to \(\overrightarrow { O A }\) and \(\overrightarrow { O C }\) respectively and the unit vector \(\mathbf { k }\) is vertically upwards.
5. Express the vectors \(\overrightarrow { A M }\) and \(\overrightarrow { A C }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
6. Use a scalar product to find angle \(M A C\).
   (a) The sum, \(S _ { n }\), of the first \(n\) terms of an arithmetic progression is given by \(S _ { n } = 32 n - n ^ { 2 }\). Find the first term and the common difference.
   (b) A geometric progression in which all the terms are positive has sum to infinity 20 . The sum of the first two terms is 12.8 . Find the first term of the progression.

9
\includegraphics[max width=\textwidth, alt={}, center]{8952fc09-004a-4fb6-ad80-5312095a7057-4_832_775_258_685} The diagram shows a trapezium \(A B C D\) in which \(A B\) is parallel to \(D C\) and angle \(B A D\) is \(90 ^ { \circ }\). The coordinates of \(A , B\) and \(C\) are \(( 2,6 ) , ( 5 , - 3 )\) and \(( 8,3 )\) respectively.

1. Find the equation of \(A D\).
2. Find, by calculation, the coordinates of \(D\). The point \(E\) is such that \(A B C E\) is a parallelogram.
3. Find the length of \(B E\).

10 A curve is such that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { 24 } { x ^ { 3 } } - 4\). The curve has a stationary point at \(P\) where \(x = 2\).

1. State, with a reason, the nature of this stationary point.
2. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
3. Given that the curve passes through the point \(( 1,13 )\), find the coordinates of the stationary point \(P\).

11 The function f : \(x \mapsto 6 - 4 \cos \left( \frac { 1 } { 2 } x \right)\) is defined for \(0 \leqslant x \leqslant 2 \pi\).

1. Find the exact value of \(x\) for which \(\mathrm { f } ( x ) = 4\).
2. State the range of f.
3. Sketch the graph of \(y = \mathrm { f } ( x )\).
4. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).