CAIE P1 (Pure Mathematics 1) 2010 November

1

1. Find the first 3 terms in the expansion, in ascending powers of \(x\), of \(\left( 1 - 2 x ^ { 2 } \right) ^ { 8 }\).
2. Find the coefficient of \(x ^ { 4 }\) in the expansion of \(\left( 2 - x ^ { 2 } \right) \left( 1 - 2 x ^ { 2 } \right) ^ { 8 }\).

2 Prove the identity $$\tan ^ { 2 } x - \sin ^ { 2 } x \equiv \tan ^ { 2 } x \sin ^ { 2 } x$$

3 The length, \(x\) metres, of a Green Anaconda snake which is \(t\) years old is given approximately by the formula $$x = 0.7 \sqrt { } ( 2 t - 1 ) ,$$ where \(1 \leqslant t \leqslant 10\). Using this formula, find

1. \(\frac { \mathrm { d } x } { \mathrm {~d} t }\),
2. the rate of growth of a Green Anaconda snake which is 5 years old.

4
\includegraphics[max width=\textwidth, alt={}, center]{ae57d8f1-5a0d-426c-952d-e8b99c6aeaba-2_720_645_1183_751} The diagram shows points \(A , C , B , P\) on the circumference of a circle with centre \(O\) and radius 3 cm . Angle \(A O C =\) angle \(B O C = 2.3\) radians.

1. Find angle \(A O B\) in radians, correct to 4 significant figures.
2. Find the area of the shaded region \(A C B P\), correct to 3 significant figures.

5

1. The first and second terms of an arithmetic progression are 161 and 154 respectively. The sum of the first \(m\) terms is zero. Find the value of \(m\).
2. A geometric progression, in which all the terms are positive, has common ratio \(r\). The sum of the first \(n\) terms is less than \(90 \%\) of the sum to infinity. Show that \(r ^ { n } > 0.1\).

6 A curve has equation \(y = k x ^ { 2 } + 1\) and a line has equation \(y = k x\), where \(k\) is a non-zero constant.

1. Find the set of values of \(k\) for which the curve and the line have no common points.
2. State the value of \(k\) for which the line is a tangent to the curve and, for this case, find the coordinates of the point where the line touches the curve.

7 The function f is defined by $$\mathrm { f } ( x ) = x ^ { 2 } - 4 x + 7 \text { for } x > 2$$

1. Express \(\mathrm { f } ( x )\) in the form \(( x - a ) ^ { 2 } + b\) and hence state the range of f .
2. Obtain an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\). The function g is defined by $$\mathrm { g } ( x ) = x - 2 \text { for } x > 2$$ The function h is such that \(\mathrm { f } = \mathrm { hg }\) and the domain of h is \(x > 0\).
3. Obtain an expression for \(\mathrm { h } ( x )\).

8
\includegraphics[max width=\textwidth, alt={}, center]{ae57d8f1-5a0d-426c-952d-e8b99c6aeaba-3_613_897_1311_623} The diagram shows part of the curve \(y = \frac { 2 } { 1 - x }\) and the line \(y = 3 x + 4\). The curve and the line meet at points \(A\) and \(B\).

1. Find the coordinates of \(A\) and \(B\).
2. Find the length of the line \(A B\) and the coordinates of the mid-point of \(A B\).

9
\includegraphics[max width=\textwidth, alt={}, center]{ae57d8f1-5a0d-426c-952d-e8b99c6aeaba-4_582_1072_255_541} The diagram shows a pyramid \(O A B C P\) in which the horizontal base \(O A B C\) is a square of side 10 cm and the vertex \(P\) is 10 cm vertically above \(O\). The points \(D , E , F , G\) lie on \(O P , A P , B P , C P\) respectively and \(D E F G\) is a horizontal square of side 6 cm . The height of \(D E F G\) above the base is \(a \mathrm {~cm}\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O D\) respectively.

1. Show that \(a = 4\).
2. Express the vector \(\overrightarrow { B G }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
3. Use a scalar product to find angle \(G B A\).

10
\includegraphics[max width=\textwidth, alt={}, center]{ae57d8f1-5a0d-426c-952d-e8b99c6aeaba-4_433_969_1475_587} The diagram shows an open rectangular tank of height \(h\) metres covered with a lid. The base of the tank has sides of length \(x\) metres and \(\frac { 1 } { 2 } x\) metres and the lid is a rectangle with sides of length \(\frac { 5 } { 4 } x\) metres and \(\frac { 4 } { 5 } x\) metres. When full the tank holds \(4 \mathrm {~m} ^ { 3 }\) of water. The material from which the tank is made is of negligible thickness. The external surface area of the tank together with the area of the top of the lid is \(A \mathrm {~m} ^ { 2 }\).

1. Express \(h\) in terms of \(x\) and hence show that \(A = \frac { 3 } { 2 } x ^ { 2 } + \frac { 24 } { x }\).
2. Given that \(x\) can vary, find the value of \(x\) for which \(A\) is a minimum, showing clearly that \(A\) is a minimum and not a maximum.
   [0pt] [5]

11
\includegraphics[max width=\textwidth, alt={}, center]{ae57d8f1-5a0d-426c-952d-e8b99c6aeaba-5_609_897_255_625} The diagram shows part of the curve \(y = \frac { 1 } { ( 3 x + 1 ) ^ { \frac { 1 } { 4 } } }\). The curve cuts the \(y\)-axis at \(A\) and the line \(x = 5\) at \(B\).

1. Show that the equation of the line \(A B\) is \(y = - \frac { 1 } { 10 } x + 1\).
2. Find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.