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CAIE P2 2019 November Q2
5 marks Standard +0.3
2
  1. Solve the equation \(| 4 x + 5 | = | x - 7 |\).
  2. Hence, using logarithms, solve the equation \(\left| 2 ^ { y + 2 } + 5 \right| = \left| 2 ^ { y } - 7 \right|\), giving the answer correct to 3 significant figures.
CAIE P2 2019 November Q3
5 marks Moderate -0.5
3 \includegraphics[max width=\textwidth, alt={}, center]{9c26457d-4b65-4cd4-a9b9-128aba92dbf4-04_586_734_260_701} The variables \(x\) and \(y\) satisfy the equation \(y = k x ^ { a }\), where \(k\) and \(a\) are constants. The graph of \(\ln y\) against \(\ln x\) is a straight line passing through the points ( \(0.22,3.96\) ) and ( \(1.32,2.43\) ), as shown in the diagram. Find the values of \(k\) and \(a\) correct to 3 significant figures.
CAIE P2 2019 November Q4
5 marks Standard +0.3
4 The sequence \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\) defined by $$x _ { 1 } = 1 , \quad x _ { n + 1 } = \frac { x _ { n } } { \ln \left( 2 x _ { n } \right) }$$ converges to the value \(\alpha\).
  1. Use the iterative formula to find the value of \(\alpha\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
  2. State an equation satisfied by \(\alpha\) and hence determine the exact value of \(\alpha\).
CAIE P2 2019 November Q5
5 marks Standard +0.3
5 Find the exact coordinates of the stationary point of the curve with equation \(y = \mathrm { e } ^ { - \frac { 1 } { 2 } x } ( 2 x + 5 )\).
CAIE P2 2019 November Q6
9 marks Standard +0.3
6
  1. Show that \(\int _ { 2 } ^ { 18 } \frac { 3 } { 2 x } \mathrm {~d} x = \ln 27\).
  2. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } 4 \sin ^ { 2 } \left( \frac { 3 } { 2 } x \right) \mathrm { d } x\). Show all necessary working.
CAIE P2 2019 November Q7
8 marks Standard +0.3
7 The parametric equations of a curve are $$x = 3 \sin 2 \theta , \quad y = 1 + 2 \tan 2 \theta$$ for \(0 \leqslant \theta < \frac { 1 } { 4 } \pi\).
  1. Find the exact gradient of the curve at the point for which \(\theta = \frac { 1 } { 6 } \pi\).
  2. Find the value of \(\theta\) at the point where the gradient of the curve is 2 , giving the value correct to 3 significant figures.
CAIE P2 2019 November Q8
10 marks Standard +0.3
8
  1. Express \(0.5 \cos \theta - 1.2 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation \(0.5 \cos \theta - 1.2 \sin \theta = 0.8\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
  3. Determine the greatest and least possible values of \(( 3 - \cos \theta + 2.4 \sin \theta ) ^ { 2 }\) as \(\theta\) varies.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE P2 2019 November Q1
5 marks Moderate -0.8
1 Candidates answer on the Question Paper.
Additional Materials: List of Formulae (MF9) \section*{READ THESE INSTRUCTIONS FIRST} Write your centre number, candidate number and name in the spaces at the top of this page.
Write in dark blue or black pen.
You may use an HB pencil for any diagrams or graphs.
Do not use staples, paper clips, glue or correction fluid.
Answer all the questions in the space provided. If additional space is required, you should use the lined page at the end of this booklet. The question number(s) must be clearly shown.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question.
The use of an electronic calculator is expected, where appropriate.
You are reminded of the need for clear presentation in your answers.
At the end of the examination, fasten all your work securely together.
[0pt] The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 50. 1
  1. Solve the inequality \(| 2 x - 7 | < | 2 x - 9 |\).
  2. Hence find the largest integer \(n\) satisfying the inequality \(| 2 \ln n - 7 | < | 2 \ln n - 9 |\).
CAIE P2 Specimen Q1
4 marks Moderate -0.8
1 Use logarithms to solve the equation $$5 ^ { x + 3 } = 7 ^ { x - 1 }$$ giving the answer correct to 3 significant figures.
CAIE P2 Specimen Q2
5 marks Standard +0.3
2 A curve has equation $$y = \frac { 3 x + 1 } { x - 5 }$$ Find the coordinates of the points on the curve at which the gradient is - 4 .
CAIE P2 Specimen Q3
7 marks Moderate -0.3
3
  1. Express \(8 \sin \theta + 15 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation $$8 \sin \theta + 15 \cos \theta = 6$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P2 Specimen Q4
7 marks Moderate -0.3
4
  1. By sketching a suitable pair of graphs, show that the equation $$\ln x = 4 - \frac { 1 } { 2 } x$$ has exactly one real root, \(\alpha\).
  2. Verify by calculation that \(4.5 < \alpha < 5.0\).
  3. Use the iterative formula \(x _ { n + 1 } = 8 - 2 \ln x _ { n }\) to find \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P2 Specimen Q5
7 marks Moderate -0.3
5
  1. Find \(\int \left( \tan ^ { 2 } x + \sin 2 x \right) \mathrm { d } x\).
  2. Find the exact value of \(\int _ { 0 } ^ { 1 } 3 \mathrm { e } ^ { 1 - 2 x } \mathrm {~d} x\).
CAIE P2 Specimen Q6
9 marks Standard +0.3
6
  1. Find the quotient and remainder when $$x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } + 12 x + 6$$ is divided by ( \(x ^ { 2 } - x + 4\) ).
  2. It is given that, when $$x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } + p x + q$$ is divided by \(\left( x ^ { 2 } - x + 4 \right)\), the remainder is zero. Find the values of the constants \(p\) and \(q\).
  3. When \(p\) and \(q\) have these values, show that there is exactly one real value of \(x\) satisfying the equation $$x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } + p x + q = 0$$ and state what that value is.
CAIE P2 Specimen Q7
11 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{77672e56-a268-47b8-ab8b-cd84b4b3de4f-10_551_689_258_726} The parametric equations of a curve are $$x = 6 \sin ^ { 2 } t , \quad y = 2 \sin 2 t + 3 \cos 2 t$$ for \(0 \leqslant t < \pi\). The curve crosses the \(x\)-axis at points \(B\) and \(D\) and the stationary points are \(A\) and \(C\), as shown in the diagram.
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { 3 } \cot 2 t - 1\).
  2. Find the values of \(t\) at \(A\) and \(C\), giving each answer correct to 3 decimal places.
  3. Find the value of the gradient of the curve at \(B\).
CAIE P3 2020 June Q1
4 marks Standard +0.3
1 Find the set of values of \(x\) for which \(2 \left( 3 ^ { 1 - 2 x } \right) < 5 ^ { x }\). Give your answer in a simplified exact form. [4]
CAIE P3 2020 June Q2
5 marks Moderate -0.8
2
  1. Expand \(( 2 - 3 x ) ^ { - 2 }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
  2. State the set of values of \(x\) for which the expansion is valid.
CAIE P3 2020 June Q3
6 marks Standard +0.3
3 Express the equation \(\tan \left( \theta + 60 ^ { \circ } \right) = 2 + \tan \left( 60 ^ { \circ } - \theta \right)\) as a quadratic equation in \(\tan \theta\), and hence solve the equation for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P3 2020 June Q4
6 marks Standard +0.8
4 The curve with equation \(y = \mathrm { e } ^ { 2 x } ( \sin x + 3 \cos x )\) has a stationary point in the interval \(0 \leqslant x \leqslant \pi\).
  1. Find the \(x\)-coordinate of this point, giving your answer correct to 2 decimal places.
  2. Determine whether the stationary point is a maximum or a minimum.
CAIE P3 2020 June Q5
8 marks Standard +0.3
5
  1. Find the quotient and remainder when \(2 x ^ { 3 } - x ^ { 2 } + 6 x + 3\) is divided by \(x ^ { 2 } + 3\).
  2. Using your answer to part (a), find the exact value of \(\int _ { 1 } ^ { 3 } \frac { 2 x ^ { 3 } - x ^ { 2 } + 6 x + 3 } { x ^ { 2 } + 3 } \mathrm {~d} x\).
CAIE P3 2020 June Q6
8 marks Challenging +1.2
6 \includegraphics[max width=\textwidth, alt={}, center]{da8dedae-4714-408e-a983-90ece63d9e91-08_501_1086_262_525} The diagram shows a circle with centre \(O\) and radius \(r\). The tangents to the circle at the points \(A\) and \(B\) meet at \(T\), and angle \(A O B\) is \(2 x\) radians. The shaded region is bounded by the tangents \(A T\) and \(B T\), and by the minor \(\operatorname { arc } A B\). The area of the shaded region is equal to the area of the circle.
  1. Show that \(x\) satisfies the equation \(\tan x = \pi + x\).
  2. This equation has one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\). Verify by calculation that this root lies between 1 and 1.4.
  3. Use the iterative formula $$x _ { n + 1 } = \tan ^ { - 1 } \left( \pi + x _ { n } \right)$$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2020 June Q7
8 marks Standard +0.3
7 Let \(\mathrm { f } ( x ) = \frac { \cos x } { 1 + \sin x }\).
  1. Show that \(\mathrm { f } ^ { \prime } ( x ) < 0\) for all \(x\) in the interval \(- \frac { 1 } { 2 } \pi < x < \frac { 3 } { 2 } \pi\).
  2. Find \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 2 } \pi } \mathrm { f } ( x ) \mathrm { d } x\). Give your answer in a simplified exact form.
CAIE P3 2020 June Q8
9 marks Standard +0.3
8 A certain curve is such that its gradient at a point \(( x , y )\) is proportional to \(\frac { y } { x \sqrt { x } }\). The curve passes through the points with coordinates \(( 1,1 )\) and \(( 4 , \mathrm { e } )\).
  1. By setting up and solving a differential equation, find the equation of the curve, expressing \(y\) in terms of \(x\).
  2. Describe what happens to \(y\) as \(x\) tends to infinity.
CAIE P3 2020 June Q9
9 marks Standard +0.3
9 With respect to the origin \(O\), the vertices of a triangle \(A B C\) have position vectors $$\overrightarrow { O A } = 2 \mathbf { i } + 5 \mathbf { k } , \quad \overrightarrow { O B } = 3 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } \quad \text { and } \quad \overrightarrow { O C } = \mathbf { i } + \mathbf { j } + \mathbf { k }$$
  1. Using a scalar product, show that angle \(A B C\) is a right angle.
  2. Show that triangle \(A B C\) is isosceles.
  3. Find the exact length of the perpendicular from \(O\) to the line through \(B\) and \(C\).
CAIE P3 2020 June Q10
12 marks Standard +0.3
10
  1. The complex number \(u\) is defined by \(u = \frac { 3 \mathrm { i } } { a + 2 \mathrm { i } }\), where \(a\) is real.
    1. Express \(u\) in the Cartesian form \(x + \mathrm { i } y\), where \(x\) and \(y\) are in terms of \(a\).
    2. Find the exact value of \(a\) for which \(\arg u ^ { * } = \frac { 1 } { 3 } \pi\).
    1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 2 \mathbf { i } | \leqslant | z - 1 - \mathbf { i } |\) and \(| z - 2 - \mathbf { i } | \leqslant 2\).
    2. Calculate the least value of \(\arg z\) for points in this region.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.