CAIE P3 (Pure Mathematics 3) 2021 June

1 Solve the inequality \(| 2 x - 1 | < 3 | x + 1 |\).

2 On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z + 1 - i | \leqslant 1\) and \(\arg ( z - 1 ) \leqslant \frac { 3 } { 4 } \pi\).

3 The variables \(x\) and \(y\) satisfy the equation \(x = A \left( 3 ^ { - y } \right)\), where \(A\) is a constant.

1. Explain why the graph of \(y\) against \(\ln x\) is a straight line and state the exact value of the gradient of the line.
   It is given that the line intersects the \(y\)-axis at the point where \(y = 1.3\).
2. Calculate the value of \(A\), giving your answer correct to 2 decimal places.

4 Using integration by parts, find the exact value of \(\int _ { 0 } ^ { 2 } \tan ^ { - 1 } \left( \frac { 1 } { 2 } x \right) \mathrm { d } x\).

5 The complex number \(u\) is given by \(u = 10 - 4 \sqrt { 6 } \mathrm { i }\).
Find the two square roots of \(u\), giving your answers in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real and exact.

6

1. Prove that \(\operatorname { cosec } 2 \theta - \cot 2 \theta \equiv \tan \theta\).
2. Hence show that \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 3 } \pi } ( \operatorname { cosec } 2 \theta - \cot 2 \theta ) \mathrm { d } \theta = \frac { 1 } { 2 } \ln 2\).

7 A curve is such that the gradient at a general point with coordinates \(( x , y )\) is proportional to \(\frac { y } { \sqrt { x + 1 } }\). The curve passes through the points with coordinates \(( 0,1 )\) and \(( 3 , e )\). By setting up and solving a differential equation, find the equation of the curve, expressing \(y\) in terms of \(x\).

8 The equation of a curve is \(y = e ^ { - 5 x } \tan ^ { 2 } x\) for \(- \frac { 1 } { 2 } \pi < x < \frac { 1 } { 2 } \pi\).
Find the \(x\)-coordinates of the stationary points of the curve. Give your answers correct to 3 decimal places where appropriate.

9 Let \(\mathrm { f } ( x ) = \frac { 14 - 3 x + 2 x ^ { 2 } } { ( 2 + x ) \left( 3 + x ^ { 2 } \right) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
   \includegraphics[max width=\textwidth, alt={}, center]{459b8403-a481-4ece-88c0-e7600a47c8e4-14_292_732_264_705} The diagram shows a trapezium \(A B C D\) in which \(A D = B C = r\) and \(A B = 2 r\). The acute angles \(B A D\) and \(A B C\) are both equal to \(x\) radians. Circular arcs of radius \(r\) with centres \(A\) and \(B\) meet at \(M\), the midpoint of \(A B\).

1. Given that the sum of the areas of the shaded sectors is \(90 \%\) of the area of the trapezium, show that \(x\) satisfies the equation \(x = 0.9 ( 2 - \cos x ) \sin x\).
2. Verify by calculation that \(x\) lies between 0.5 and 0.7 .
3. Show that if a sequence of values in the interval \(0 < x < \frac { 1 } { 2 } \pi\) given by the iterative formula $$x _ { n + 1 } = \cos ^ { - 1 } \left( 2 - \frac { x _ { n } } { 0.9 \sin x _ { n } } \right)$$ converges, then it converges to the root of the equation in part (a).
4. Use this iterative formula to determine \(x\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

11 With respect to the origin \(O\), the points \(A\) and \(B\) have position vectors given by \(\overrightarrow { O A } = 2 \mathbf { i } - \mathbf { j }\) and \(\overrightarrow { O B } = \mathbf { j } - 2 \mathbf { k }\).

1. Show that \(O A = O B\) and use a scalar product to calculate angle \(A O B\) in degrees.
   The midpoint of \(A B\) is \(M\). The point \(P\) on the line through \(O\) and \(M\) is such that \(P A : O A = \sqrt { 7 } : 1\).
2. Find the possible position vectors of \(P\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.