CAIE P1 (Pure Mathematics 1) 2024 June

1

1. Express \(3 y ^ { 2 } - 12 y - 15\) in the form \(3 ( y + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
2. Hence find the exact solutions of the equation \(3 x ^ { 4 } - 12 x ^ { 2 } - 15 = 0\).

2
\includegraphics[max width=\textwidth, alt={}, center]{cacac880-5b44-4fae-8ed8-88a095db69cd-03_606_1515_260_274} The diagram shows two curves. One curve has equation \(y = \sin x\) and the other curve has equation \(y = f ( x )\).

1. In order to transform the curve \(y = \sin x\) to the curve \(y = f ( x )\), the curve \(y = \sin x\) is first reflected in the \(x\)-axis. Describe fully a sequence of two further transformations which are required.
2. Find \(\mathrm { f } ( x )\) in terms of \(\sin x\).

3 The coefficient of \(x ^ { 3 }\) in the expansion of \(( 3 + \mathrm { ax } ) ^ { 6 }\) is 160 .

1. Find the value of the constant \(a\).
2. Hence find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 3 + \mathrm { ax } ) ^ { 6 } ( 1 - 2 \mathrm { x } )\).

4 The equation of a curve is \(y = f ( x )\), where \(f ( x ) = ( 2 x - 1 ) \sqrt { 3 x - 2 } - 2\). The following points lie on the curve. Non-exact values have been given correct to 5 decimal places. $$A ( 2,4 ) , B ( 2.0001 , k ) , C ( 2.001,4.00625 ) , D ( 2.01,4.06261 ) , E ( 2.1,4.63566 ) , F ( 3,11.22876 )$$

1. Find the value of \(k\). Give your answer correct to 5 decimal places.
   The table shows the gradients of the chords \(A B , A C , A D\) and \(A F\).

   Chord	\(A B\)	\(A C\)	\(A D\)	\(A E\)	\(A F\)
   Gradient of chord	6.2501	6.2511	6.2608		7.2288

2. Find the gradient of the chord \(A E\). Give your answer correct to 4 decimal places.
3. Deduce the value of \(f ^ { \prime } ( 2 )\) using the values in the table.

5

1. Prove the identity \(\frac { \sin ^ { 2 } x - \cos x - 1 } { 1 + \cos x } \equiv - \cos x\).
2. Hence solve the equation \(\frac { \sin ^ { 2 } x - \cos x - 1 } { 2 + 2 \cos x } = \frac { 1 } { 4 }\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
   \includegraphics[max width=\textwidth, alt={}, center]{cacac880-5b44-4fae-8ed8-88a095db69cd-07_583_990_306_539} The function f is defined by \(\mathrm { f } ( x ) = \frac { 2 } { x ^ { 2 } } + 4\) for \(x < 0\). The diagram shows the graph of \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\).
3. On this diagram, sketch the graph of \(y = f ^ { - 1 } ( x )\). Show any relevant mirror line.
4. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
5. Solve the equation \(\mathrm { f } ( x ) = 4.5\).
6. Explain why the equation \(\mathrm { f } ^ { - 1 } ( x ) = \mathrm { f } ( x )\) has no solution.
   \includegraphics[max width=\textwidth, alt={}, center]{cacac880-5b44-4fae-8ed8-88a095db69cd-08_522_1036_296_513} In the diagram, \(A O D\) and \(B C\) are two parallel straight lines. Arc \(A B\) is part of a circle with centre \(O\) and radius 15 cm . Angle \(B O A = \theta\) radians. Arc \(C D\) is part of a circle with centre \(O\) and radius 10 cm . Angle \(C O D = \frac { 1 } { 2 } \pi\) radians.
7. Show that \(\theta = 0.7297\), correct to 4 decimal places.
8. Find the perimeter and the area of the shape \(A B C D\). Give your answers correct to 3 significant figures.

8

1. The first three terms of an arithmetic progression are \(25,4 p - 1\) and \(13 - p\), where \(p\) is a constant. Find the value of the tenth term of the progression.
2. The first three terms of a geometric progression are \(25,4 q - 1\) and \(13 - q\), where \(q\) is a positive constant. Find the sum to infinity of the progression.
   \includegraphics[max width=\textwidth, alt={}, center]{cacac880-5b44-4fae-8ed8-88a095db69cd-12_586_1567_333_248} The diagram shows part of the curve with equation \(y = \frac { 1 } { ( 5 x - 4 ) ^ { \frac { 1 } { 3 } } }\) and the lines \(x = 2.4\) and \(y = 1\). The curve intersects the line \(y = 1\) at the point \(( 1,1 )\). Find the exact volume of the solid generated when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

10 The equation of a circle is \(( x - 3 ) ^ { 2 } + y ^ { 2 } = 18\). The line with equation \(y = m x + c\) passes through the point \(( 0 , - 9 )\) and is a tangent to the circle. Find the two possible values of \(m\) and, for each value of \(m\), find the coordinates of the point at which the tangent touches the circle.
\includegraphics[max width=\textwidth, alt={}, center]{cacac880-5b44-4fae-8ed8-88a095db69cd-16_855_1600_306_233} A function is defined by \(\mathrm { f } ( x ) = \frac { 4 } { x ^ { 3 } } - \frac { 3 } { x } + 2\) for \(x \neq 0\). The graph of \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\) is shown in the diagram.

1. Find the set of values of \(x\) for which \(\mathrm { f } ( x )\) is decreasing.
2. A triangle is bounded by the \(y\)-axis, the normal to the curve at the point where \(x = 1\) and the tangent to the curve at the point where \(x = - 1\). Find the area of the triangle. Give your answer correct to 3 significant figures.
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