CAIE P1 (Pure Mathematics 1) 2023 June

1 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 } { ( x - 3 ) ^ { 3 } }\) for \(x > 3\). The curve passes through the point \(( 4,5 )\). Find the equation of the curve.

2 The coefficient of \(x ^ { 4 }\) in the expansion of \(( x + a ) ^ { 6 }\) is \(p\) and the coefficient of \(x ^ { 2 }\) in the expansion of \(( a x + 3 ) ^ { 4 }\) is \(q\). It is given that \(p + q = 276\). Find the possible values of the constant \(a\).

3

1. Express \(4 x ^ { 2 } - 24 x + p\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a\) and \(b\) are integers and \(c\) is to be given in terms of the constant \(p\).
2. Hence or otherwise find the set of values of \(p\) for which the equation \(4 x ^ { 2 } - 24 x + p = 0\) has no real roots.

4 Solve the equation \(8 x ^ { 6 } + 215 x ^ { 3 } - 27 = 0\).

5
\includegraphics[max width=\textwidth, alt={}, center]{1662cb34-273c-461d-908c-9fe2ffe889b4-06_599_1086_274_518} The diagram shows the curve with equation \(y = 10 x ^ { \frac { 1 } { 2 } } - \frac { 5 } { 2 } x ^ { \frac { 3 } { 2 } }\) for \(x > 0\). The curve meets the \(x\)-axis at the points \(( 0,0 )\) and \(( 4,0 )\). Find the area of the shaded region.
The diagram shows a sector \(O A B\) of a circle with centre \(O\). Angle \(A O B = \theta\) radians and \(O P = A P = x\).

1. Show that the arc length \(A B\) is \(2 x \theta \cos \theta\).
2. Find the area of the shaded region \(A P B\) in terms of \(x\) and \(\theta\).

7

1. 1. By first expanding \(( \cos \theta + \sin \theta ) ^ { 2 }\), find the three solutions of the equation $$( \cos \theta + \sin \theta ) ^ { 2 } = 1$$ for \(0 \leqslant \theta \leqslant \pi\).
   2. Hence verify that the only solutions of the equation \(\cos \theta + \sin \theta = 1\) for \(0 \leqslant \theta \leqslant \pi\) are 0 and \(\frac { 1 } { 2 } \pi\).
2. Prove the identity \(\frac { \sin \theta } { \cos \theta + \sin \theta } + \frac { 1 - \cos \theta } { \cos \theta - \sin \theta } \equiv \frac { \cos \theta + \sin \theta - 1 } { 1 - 2 \sin ^ { 2 } \theta }\).
3. Using the results of (a) (ii) and (b), solve the equation $$\frac { \sin \theta } { \cos \theta + \sin \theta } + \frac { 1 - \cos \theta } { \cos \theta - \sin \theta } = 2 ( \cos \theta + \sin \theta - 1 )$$ for \(0 \leqslant \theta \leqslant \pi\).

8
\includegraphics[max width=\textwidth, alt={}, center]{1662cb34-273c-461d-908c-9fe2ffe889b4-10_784_913_274_607} The diagram shows the graph of \(y = \mathrm { f } ( x )\) where the function f is defined by $$f ( x ) = 3 + 2 \sin \frac { 1 } { 4 } x \text { for } 0 \leqslant x \leqslant 2 \pi$$

1. On the diagram above, sketch the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\).
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
3. 
   \includegraphics[max width=\textwidth, alt={}, center]{1662cb34-273c-461d-908c-9fe2ffe889b4-11_759_1545_276_331} The diagram above shows part of the graph of the function \(\mathrm { g } ( x ) = 3 + 2 \sin \frac { 1 } { 4 } x\) for \(- 2 \pi \leqslant x \leqslant 2 \pi\).
   Complete the sketch of the graph of \(\mathrm { g } ( x )\) on the diagram above and hence explain whether the function \(g\) has an inverse.
4. Describe fully a sequence of three transformations which can be combined to transform the graph of \(y = \sin x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\) to the graph of \(y = \mathrm { f } ( x )\), making clear the order in which the transformations are applied.

9 The second term of a geometric progression is 16 and the sum to infinity is 100 .

1. Find the two possible values of the first term.
2. Show that the \(n\)th term of one of the two possible geometric progressions is equal to \(4 ^ { n - 2 }\) multiplied by the \(n\)th term of the other geometric progression.

10 The equation of a circle is \(( x - a ) ^ { 2 } + ( y - 3 ) ^ { 2 } = 20\). The line \(y = \frac { 1 } { 2 } x + 6\) is a tangent to the circle at the point \(P\).

1. Show that one possible value of \(a\) is 4 and find the other possible value.
2. For \(a = 4\), find the equation of the normal to the circle at \(P\).
3. For \(a = 4\), find the equations of the two tangents to the circle which are parallel to the normal found in (b).

11 The equation of a curve is $$y = k \sqrt { 4 x + 1 } - x + 5$$ where \(k\) is a positive constant.

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Find the \(x\)-coordinate of the stationary point in terms of \(k\).
3. Given that \(k = 10.5\), find the equation of the normal to the curve at the point where the tangent to the curve makes an angle of \(\tan ^ { - 1 } ( 2 )\) with the positive \(x\)-axis.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.