CAIE P1 (Pure Mathematics 1) 2002 November

1 Find the value of the term which is independent of \(x\) in the expansion of \(\left( x + \frac { 3 } { x } \right) ^ { 4 }\).

2 A geometric progression, for which the common ratio is positive, has a second term of 18 and a fourth term of 8 . Find

1. the first term and the common ratio of the progression,
2. the sum to infinity of the progression.

3
\includegraphics[max width=\textwidth, alt={}, center]{a10ad459-6f86-4845-ba28-e4e394bf3d1e-2_330_634_753_758} In the diagram, \(O P Q\) is a sector of a circle, centre \(O\) and radius \(r \mathrm {~cm}\). Angle \(Q O P = \theta\) radians. The tangent to the circle at \(Q\) meets \(O P\) extended at \(R\).

1. Show that the area, \(A \mathrm {~cm} ^ { 2 }\), of the shaded region is given by \(A = \frac { 1 } { 2 } r ^ { 2 } ( \tan \theta - \theta )\).
2. In the case where \(\theta = 0.8\) and \(r = 15\), evaluate the length of the perimeter of the shaded region.

4 The gradient at any point \(( x , y )\) on a curve is \(\sqrt { } ( 1 + 2 x )\). The curve passes through the point \(( 4,11 )\). Find

1. the equation of the curve,
2. the point at which the curve intersects the \(y\)-axis.

5

1. Show that the equation \(3 \tan \theta = 2 \cos \theta\) can be expressed as $$2 \sin ^ { 2 } \theta + 3 \sin \theta - 2 = 0$$
2. Hence solve the equation \(3 \tan \theta = 2 \cos \theta\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

6
\includegraphics[max width=\textwidth, alt={}, center]{a10ad459-6f86-4845-ba28-e4e394bf3d1e-3_602_570_260_790} In the diagram, triangle \(A B C\) is right-angled and \(D\) is the mid-point of \(B C\). Angle \(D A C = 30 ^ { \circ }\) and angle \(B A D = x ^ { \circ }\). Denoting the length of \(A D\) by \(l\),

1. express each of \(A C\) and \(B C\) exactly in terms of \(l\), and show that \(A B = \frac { 1 } { 2 } l \sqrt { } 7\),
2. show that \(x = \tan ^ { - 1 } \left( \frac { 2 } { \sqrt { } 3 } \right) - 30\).

7 Given that \(\mathbf { a } = \left( \begin{array} { r } 2
- 2
1 \end{array} \right) , \mathbf { b } = \left( \begin{array} { l } 2
6
3 \end{array} \right)\) and \(\mathbf { c } = \left( \begin{array} { c } p
p
p + 1 \end{array} \right)\), find

1. the angle between the directions of \(\mathbf { a }\) and \(\mathbf { b }\),
2. the value of \(p\) for which \(\mathbf { b }\) and \(\mathbf { c }\) are perpendicular.

8 A curve has equation \(y = x ^ { 3 } + 3 x ^ { 2 } - 9 x + k\), where \(k\) is a constant.

1. Write down an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Find the \(x\)-coordinates of the two stationary points on the curve.
3. Hence find the two values of \(k\) for which the curve has a stationary point on the \(x\)-axis.

9
\includegraphics[max width=\textwidth, alt={}, center]{a10ad459-6f86-4845-ba28-e4e394bf3d1e-4_719_958_264_589} The diagram shows a rectangle \(A B C D\), where \(A\) is \(( 3,2 )\) and \(B\) is \(( 1,6 )\).

1. Find the equation of \(B C\). Given that the equation of \(A C\) is \(y = x - 1\), find
2. the coordinates of \(C\),
3. the perimeter of the rectangle \(A B C D\).

10
\includegraphics[max width=\textwidth, alt={}, center]{a10ad459-6f86-4845-ba28-e4e394bf3d1e-4_595_800_1548_669} The diagram shows the points \(A ( 1,2 )\) and \(B ( 4,4 )\) on the curve \(y = 2 \sqrt { } x\). The line \(B C\) is the normal to the curve at \(B\), and \(C\) lies on the \(x\)-axis. Lines \(A D\) and \(B E\) are perpendicular to the \(x\)-axis.

1. Find the equation of the normal \(B C\).
2. Find the area of the shaded region.

11

1. Express \(2 x ^ { 2 } + 8 x - 10\) in the form \(a ( x + b ) ^ { 2 } + c\).
2. For the curve \(y = 2 x ^ { 2 } + 8 x - 10\), state the least value of \(y\) and the corresponding value of \(x\).
3. Find the set of values of \(x\) for which \(y \geqslant 14\). Given that \(\mathrm { f } : x \mapsto 2 x ^ { 2 } + 8 x - 10\) for the domain \(x \geqslant k\),
4. find the least value of \(k\) for which f is one-one,
5. express \(\mathrm { f } ^ { - 1 } ( x )\) in terms of \(x\) in this case.