CAIE P1 (Pure Mathematics 1) 2011 November

1

1. Find the first 3 terms in the expansion of \(( 2 - y ) ^ { 5 }\) in ascending powers of \(y\).
2. Use the result in part (i) to find the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 2 - \left( 2 x - x ^ { 2 } \right) \right) ^ { 5 }\).

2 The functions f and g are defined for \(x \in \mathbb { R }\) by $$\begin{aligned} & \mathrm { f } : x \mapsto 3 x + a
& \mathrm {~g} : x \mapsto b - 2 x \end{aligned}$$ where \(a\) and \(b\) are constants. Given that \(\mathrm { ff } ( 2 ) = 10\) and \(\mathrm { g } ^ { - 1 } ( 2 ) = 3\), find

1. the values of \(a\) and \(b\),
2. an expression for \(\mathrm { fg } ( x )\).

3 Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow { O A } = 5 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = 2 \mathbf { i } + 7 \mathbf { j } + p \mathbf { k }$$ where \(p\) is a constant.

1. Find the value of \(p\) for which angle \(A O B\) is \(90 ^ { \circ }\).
2. In the case where \(p = 4\), find the vector which has magnitude 28 and is in the same direction as \(\overrightarrow { A B }\).

4 The equation of a curve is \(y ^ { 2 } + 2 x = 13\) and the equation of a line is \(2 y + x = k\), where \(k\) is a constant.

1. In the case where \(k = 8\), find the coordinates of the points of intersection of the line and the curve.
2. Find the value of \(k\) for which the line is a tangent to the curve.

5

1. Sketch, on the same diagram, the graphs of \(y = \sin x\) and \(y = \cos 2 x\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
2. Verify that \(x = 30 ^ { \circ }\) is a root of the equation \(\sin x = \cos 2 x\), and state the other root of this equation for which \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
3. Hence state the set of values of \(x\), for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\), for which \(\sin x < \cos 2 x\).

6
\includegraphics[max width=\textwidth, alt={}, center]{3fd0b68f-41b1-4eee-8018-bcaf3cf22950-3_801_1273_255_434} The diagram shows a circle \(C _ { 1 }\) touching a circle \(C _ { 2 }\) at a point \(X\). Circle \(C _ { 1 }\) has centre \(A\) and radius 6 cm , and circle \(C _ { 2 }\) has centre \(B\) and radius 10 cm . Points \(D\) and \(E\) lie on \(C _ { 1 }\) and \(C _ { 2 }\) respectively and \(D E\) is parallel to \(A B\). Angle \(D A X = \frac { 1 } { 3 } \pi\) radians and angle \(E B X = \theta\) radians.

1. By considering the perpendicular distances of \(D\) and \(E\) from \(A B\), show that the exact value of \(\theta\) is \(\sin ^ { - 1 } \left( \frac { 3 \sqrt { } 3 } { 10 } \right)\).
2. Find the perimeter of the shaded region, correct to 4 significant figures.

7 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 5 - \frac { 8 } { x ^ { 2 } }\). The line \(3 y + x = 17\) is the normal to the curve at the point \(P\) on the curve. Given that the \(x\)-coordinate of \(P\) is positive, find

1. the coordinates of \(P\),
2. the equation of the curve.

8 The equation of a curve is \(y = \sqrt { } \left( 8 x - x ^ { 2 } \right)\). Find

1. an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), and the coordinates of the stationary point on the curve,
2. the volume obtained when the region bounded by the curve and the \(x\)-axis is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

9
\includegraphics[max width=\textwidth, alt={}, center]{3fd0b68f-41b1-4eee-8018-bcaf3cf22950-4_767_993_255_575} The diagram shows a quadrilateral \(A B C D\) in which the point \(A\) is ( \(- 1 , - 1\) ), the point \(B\) is ( 3,6 ) and the point \(C\) is (9,4). The diagonals \(A C\) and \(B D\) intersect at \(M\). Angle \(B M A = 90 ^ { \circ }\) and \(B M = M D\). Calculate

1. the coordinates of \(M\) and \(D\),
2. the ratio \(A M : M C\).

10

1. An arithmetic progression contains 25 terms and the first term is - 15 . The sum of all the terms in the progression is 525. Calculate
   1. the common difference of the progression,
   2. the last term in the progression,
   3. the sum of all the positive terms in the progression.
2. A college agrees a sponsorship deal in which grants will be received each year for sports equipment. This grant will be \(
   ) 4000\( in 2012 and will increase by \)5 \%$ each year. Calculate
   1. the value of the grant in 2022,
   2. the total amount the college will receive in the years 2012 to 2022 inclusive.