CAIE P1 (Pure Mathematics 1) 2006 November

1 Find the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( x + \frac { 2 } { x } \right) ^ { 6 }\).

2 Given that \(x = \sin ^ { - 1 } \left( \frac { 2 } { 5 } \right)\), find the exact value of

1. \(\cos ^ { 2 } x\),
2. \(\tan ^ { 2 } x\).

3
\includegraphics[max width=\textwidth, alt={}, center]{dd2cb0ec-5df9-4d99-9e15-5ae1f1c07b96-2_536_606_735_772} In the diagram, \(A O B\) is a sector of a circle with centre \(O\) and radius 12 cm . The point \(A\) lies on the side \(C D\) of the rectangle \(O C D B\). Angle \(A O B = \frac { 1 } { 3 } \pi\) radians. Express the area of the shaded region in the form \(a ( \sqrt { } 3 ) - b \pi\), stating the values of the integers \(a\) and \(b\).

4 The position vectors of points \(A\) and \(B\) are \(\left( \begin{array} { r } - 3
6
3 \end{array} \right)\) and \(\left( \begin{array} { r } - 1
2
4 \end{array} \right)\) respectively, relative to an origin \(O\).

1. Calculate angle \(A O B\).
2. The point \(C\) is such that \(\overrightarrow { A C } = 3 \overrightarrow { A B }\). Find the unit vector in the direction of \(\overrightarrow { O C }\).

5
\includegraphics[max width=\textwidth, alt={}, center]{dd2cb0ec-5df9-4d99-9e15-5ae1f1c07b96-3_684_771_260_685} The three points \(A ( 1,3 ) , B ( 13,11 )\) and \(C ( 6,15 )\) are shown in the diagram. The perpendicular from \(C\) to \(A B\) meets \(A B\) at the point \(D\). Find

1. the equation of \(C D\),
2. the coordinates of \(D\).

6

1. Find the sum of all the integers between 100 and 400 that are divisible by 7 .
2. The first three terms in a geometric progression are \(144 , x\) and 64 respectively, where \(x\) is positive. Find
   1. the value of \(x\),
   2. the sum to infinity of the progression.

7
\includegraphics[max width=\textwidth, alt={}, center]{dd2cb0ec-5df9-4d99-9e15-5ae1f1c07b96-3_490_665_1793_740} The diagram shows the curve \(y = x ( x - 1 ) ( x - 2 )\), which crosses the \(x\)-axis at the points \(O ( 0,0 )\), \(A ( 1,0 )\) and \(B ( 2,0 )\).

1. The tangents to the curve at the points \(A\) and \(B\) meet at the point \(C\). Find the \(x\)-coordinate of \(C\).
2. Show by integration that the area of the shaded region \(R _ { 1 }\) is the same as the area of the shaded region \(R _ { 2 }\).

8 The equation of a curve is \(y = \frac { 6 } { 5 - 2 x }\).

1. Calculate the gradient of the curve at the point where \(x = 1\).
2. A point with coordinates \(( x , y )\) moves along the curve in such a way that the rate of increase of \(y\) has a constant value of 0.02 units per second. Find the rate of increase of \(x\) when \(x = 1\).
3. The region between the curve, the \(x\)-axis and the lines \(x = 0\) and \(x = 1\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Show that the volume obtained is \(\frac { 12 } { 5 } \pi\).

9
\includegraphics[max width=\textwidth, alt={}, center]{dd2cb0ec-5df9-4d99-9e15-5ae1f1c07b96-4_387_903_799_623} The diagram shows an open container constructed out of \(200 \mathrm {~cm} ^ { 2 }\) of cardboard. The two vertical end pieces are isosceles triangles with sides \(5 x \mathrm {~cm} , 5 x \mathrm {~cm}\) and \(8 x \mathrm {~cm}\), and the two side pieces are rectangles of length \(y \mathrm {~cm}\) and width \(5 x \mathrm {~cm}\), as shown. The open top is a horizontal rectangle.

1. Show that \(y = \frac { 200 - 24 x ^ { 2 } } { 10 x }\).
2. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the container is given by \(V = 240 x - 28.8 x ^ { 3 }\). Given that \(x\) can vary,
3. find the value of \(x\) for which \(V\) has a stationary value,
4. determine whether it is a maximum or a minimum stationary value.

10 The function f is defined by \(\mathrm { f } : x \mapsto x ^ { 2 } - 3 x\) for \(x \in \mathbb { R }\).

1. Find the set of values of \(x\) for which \(\mathrm { f } ( x ) > 4\).
2. Express \(\mathrm { f } ( x )\) in the form \(( x - a ) ^ { 2 } - b\), stating the values of \(a\) and \(b\).
3. Write down the range of f .
4. State, with a reason, whether f has an inverse. The function g is defined by \(\mathrm { g } : x \mapsto x - 3 \sqrt { } x\) for \(x \geqslant 0\).
5. Solve the equation \(\mathrm { g } ( x ) = 10\).