CAIE P2 (Pure Mathematics 2) 2024 June

1 Solve the inequality \(| 5 x + 7 | > | 2 x - 3 |\).

2 Use logarithms to solve the equation \(6 ^ { 2 x - 1 } = 5 e ^ { 3 x + 2 }\). Give your answer correct to 4 significant figures. [4]
\includegraphics[max width=\textwidth, alt={}, center]{6ee58f43-831d-402c-9f9a-2b247b2f7ffc-04_778_486_276_769} The diagram shows the curve with equation \(\mathrm { y } = 8 \mathrm { e } ^ { - \mathrm { x } } - \mathrm { e } ^ { 2 \mathrm { x } }\). The curve crosses the \(y\)-axis at the point \(A\) and the \(x\)-axis at the point \(B\). The shaded region is bounded by the curve and the two axes.

1. Find the gradient of the curve at \(A\).
2. Show that the \(x\)-coordinate of \(B\) is \(\ln 2\) and hence find the area of the shaded region.

4 A curve is defined by the parametric equations $$x = 4 \cos ^ { 2 } t , \quad y = \sqrt { 3 } \sin 2 t$$ for values of \(t\) such that \(0 < t < \frac { 1 } { 2 } \pi\).
Find the equation of the normal to the curve at the point for which \(t = \frac { 1 } { 6 } \pi\). Give your answer in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { c } = 0\) where \(a , b\) and \(c\) are integers.

5 The polynomial \(\mathrm { p } ( x )\) is defined by \(\mathrm { p } ( x ) = 9 x ^ { 3 } + 18 x ^ { 2 } + 5 x + 4\).

1. Find the quotient when \(\mathrm { p } ( x )\) is divided by \(( 3 x + 2 )\), and show that the remainder is 6 .
2. Find the value of \(\int _ { 0 } ^ { 2 } \frac { \mathrm { p } ( x ) } { 3 x + 2 } \mathrm {~d} x\), giving your answer in the form \(\mathrm { a } + \operatorname { lnb }\) where \(a\) and \(b\) are integers.
   \includegraphics[max width=\textwidth, alt={}, center]{6ee58f43-831d-402c-9f9a-2b247b2f7ffc-10_414_693_276_687} The diagram shows the curve with equation \(\mathrm { y } = \frac { \ln ( 2 \mathrm { x } + 1 ) } { \mathrm { x } + 3 }\). The curve has a maximum point \(M\).

1. Find an expression for \(\frac { \mathrm { dy } } { \mathrm { dx } }\).
2. Show that the \(x\)-coordinate of \(M\) satisfies the equation \(x = \frac { x + 3 } { \ln ( 2 x + 1 ) } - 0.5\).
3. Show by calculation that the \(x\)-coordinate of \(M\) lies between 2.5 and 3.0 .
4. Use an iterative formula based on the equation in part (b) to find the \(x\)-coordinate of \(M\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

7

1. Prove that \(2 \sin \theta \operatorname { cosec } 2 \theta \equiv \sec \theta\).
2. Solve the equation \(\tan ^ { 2 } \theta + 7 \sin \theta \operatorname { cosec } 2 \theta = 8\) for \(- \pi < \theta < \pi\).
3. Find \(\int 8 \sin ^ { 2 } \frac { 1 } { 2 } x \operatorname { cosec } ^ { 2 } x d x\).
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