Questions P3 (1203 questions)

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CAIE P3 2020 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{3149080d-ad1a-4d2e-8e20-eb9977ced619-03_515_901_260_623} The variables \(x\) and \(y\) satisfy the equation \(y ^ { 2 } = A \mathrm { e } ^ { k x }\), where \(A\) and \(k\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points (1.5, 1.2) and (5.24, 2.7) as shown in the diagram. Find the values of \(A\) and \(k\) correct to 2 decimal places.
CAIE P3 2020 June Q3
3 Find the exact value of $$\int _ { 1 } ^ { 4 } x ^ { \frac { 3 } { 2 } } \ln x \mathrm {~d} x$$
CAIE P3 2020 June Q4
4 A curve has equation \(y = \cos x \sin 2 x\).
Find the \(x\)-coordinate of the stationary point in the interval \(0 < x < \frac { 1 } { 2 } \pi\), giving your answer correct to 3 significant figures.
CAIE P3 2020 June Q5
5
  1. Express \(\sqrt { 2 } \cos x - \sqrt { 5 } \sin x\) in the form \(R \cos ( x + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the exact value of \(R\) and the value of \(\alpha\) correct to 3 decimal places.
  2. Hence solve the equation \(\sqrt { 2 } \cos 2 \theta - \sqrt { 5 } \sin 2 \theta = 1\), for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P3 2020 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{3149080d-ad1a-4d2e-8e20-eb9977ced619-08_318_750_260_699} The diagram shows the curve \(y = \frac { x } { 1 + 3 x ^ { 4 } }\), for \(x \geqslant 0\), and its maximum point \(M\).
  1. Find the \(x\)-coordinate of \(M\), giving your answer correct to 3 decimal places.
  2. Using the substitution \(u = \sqrt { 3 } x ^ { 2 }\), find by integration the exact area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = 1\).
CAIE P3 2020 June Q7
7 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y - 1 } { ( x + 1 ) ( x + 3 ) }$$ It is given that \(y = 2\) when \(x = 0\).
Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).
CAIE P3 2020 June Q8
8
  1. Solve the equation \(( 1 + 2 \mathrm { i } ) w + \mathrm { i } w ^ { * } = 3 + 5 \mathrm { i }\). Give your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
    1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 2 - 2 \mathrm { i } | \leqslant 1\) and \(\arg ( z - 4 \mathrm { i } ) \geqslant - \frac { 1 } { 4 } \pi\).
    2. Find the least value of \(\operatorname { Im } z\) for points in this region, giving your answer in an exact form.
CAIE P3 2020 June Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{3149080d-ad1a-4d2e-8e20-eb9977ced619-14_558_686_260_726} The diagram shows the curves \(y = \cos x\) and \(y = \frac { k } { 1 + x }\), where \(k\) is a constant, for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). The curves touch at the point where \(x = p\).
  1. Show that \(p\) satisfies the equation \(\tan p = \frac { 1 } { 1 + p }\).
  2. Use the iterative formula \(p _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 1 } { 1 + p _ { n } } \right)\) to determine the value of \(p\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
  3. Hence find the value of \(k\) correct to 2 decimal places.
CAIE P3 2020 June Q10
10 With respect to the origin \(O\), the points \(A\) and \(B\) have position vectors given by \(\overrightarrow { O A } = 6 \mathbf { i } + 2 \mathbf { j }\) and \(\overrightarrow { O B } = 2 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }\). The midpoint of \(O A\) is \(M\). The point \(N\) lying on \(A B\), between \(A\) and \(B\), is such that \(A N = 2 N B\).
  1. Find a vector equation for the line through \(M\) and \(N\).
    The line through \(M\) and \(N\) intersects the line through \(O\) and \(B\) at the point \(P\).
  2. Find the position vector of \(P\).
  3. Calculate angle \(O P M\), giving your answer in degrees.
    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 P3 2020 June Q1
1 Solve the inequality \(| 2 x - 1 | > 3 | x + 2 |\).
CAIE P3 2020 June Q2
2 Find the exact value of \(\int _ { 0 } ^ { 1 } ( 2 - x ) \mathrm { e } ^ { - 2 x } \mathrm {~d} x\).
CAIE P3 2020 June Q3
3
  1. Show that the equation $$\ln \left( 1 + \mathrm { e } ^ { - x } \right) + 2 x = 0$$ can be expressed as a quadratic equation in \(\mathrm { e } ^ { x }\).
  2. Hence solve the equation \(\ln \left( 1 + \mathrm { e } ^ { - x } \right) + 2 x = 0\), giving your answer correct to 3 decimal places.
CAIE P3 2020 June Q4
4 The equation of a curve is \(y = x \tan ^ { - 1 } \left( \frac { 1 } { 2 } x \right)\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. The tangent to the curve at the point where \(x = 2\) meets the \(y\)-axis at the point with coordinates \(( 0 , p )\). Find \(p\).
CAIE P3 2020 June Q5
5 By first expressing the equation $$\tan \theta \tan \left( \theta + 45 ^ { \circ } \right) = 2 \cot 2 \theta$$ as a quadratic equation in \(\tan \theta\), solve the equation for \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).
CAIE P3 2020 June Q6
6
  1. By sketching a suitable pair of graphs, show that the equation \(x ^ { 5 } = 2 + x\) has exactly one real root.
  2. Show that if a sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 4 x _ { n } ^ { 5 } + 2 } { 5 x _ { n } ^ { 4 } - 1 }$$ converges, then it converges to the root of the equation in part (a).
  3. Use the iterative formula with initial value \(x _ { 1 } = 1.5\) to calculate the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
    \(7 \quad\) Let \(\mathrm { f } ( x ) = \frac { 2 } { ( 2 x - 1 ) ( 2 x + 1 ) }\).
CAIE P3 2020 June Q8
8 Relative to the origin \(O\), the points \(A , B\) and \(D\) have position vectors given by $$\overrightarrow { O A } = \mathbf { i } + 2 \mathbf { j } + \mathbf { k } , \quad \overrightarrow { O B } = 2 \mathbf { i } + 5 \mathbf { j } + 3 \mathbf { k } \quad \text { and } \quad \overrightarrow { O D } = 3 \mathbf { i } + 2 \mathbf { k }$$ A fourth point \(C\) is such that \(A B C D\) is a parallelogram.
  1. Find the position vector of \(C\) and verify that the parallelogram is not a rhombus.
  2. Find angle \(B A D\), giving your answer in degrees.
  3. Find the area of the parallelogram correct to 3 significant figures.
CAIE P3 2020 June Q9
9
  1. The complex numbers \(u\) and \(w\) are such that $$u - w = 2 \mathrm { i } \quad \text { and } \quad u w = 6$$ Find \(u\) and \(w\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
  2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities $$| z - 2 - 2 \mathbf { i } | \leqslant 2 , \quad 0 \leqslant \arg z \leqslant \frac { 1 } { 4 } \pi \quad \text { and } \quad \operatorname { Re } z \leqslant 3$$
    \includegraphics[max width=\textwidth, alt={}]{c1bd46f8-a33a-4927-af59-718b1c9dd4e1-16_462_709_260_719}
    A tank containing water is in the form of a hemisphere. The axis is vertical, the lowest point is \(A\) and the radius is \(r\), as shown in the diagram. The depth of water at time \(t\) is \(h\). At time \(t = 0\) the tank is full and the depth of the water is \(r\). At this instant a tap at \(A\) is opened and water begins to flow out at a rate proportional to \(\sqrt { h }\). The tank becomes empty at time \(t = 14\). The volume of water in the tank is \(V\) when the depth is \(h\). It is given that \(V = \frac { 1 } { 3 } \pi \left( 3 r h ^ { 2 } - h ^ { 3 } \right)\).
  3. Show that \(h\) and \(t\) satisfy a differential equation of the form $$\frac { \mathrm { d } h } { \mathrm {~d} t } = - \frac { B } { 2 r h ^ { \frac { 1 } { 2 } } - h ^ { \frac { 3 } { 2 } } } ,$$ where \(B\) is a positive constant.
  4. Solve the differential equation and obtain an expression for \(t\) in terms of \(h\) and \(r\).
    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 P3 2021 June Q1
1 Solve the inequality \(2 | 3 x - 1 | < | x + 1 |\).
CAIE P3 2021 June Q2
2 Find the real root of the equation \(\frac { 2 \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } } { 2 + \mathrm { e } ^ { x } } = 3\), giving your answer correct to 3 decimal places. Your working should show clearly that the equation has only one real root.
CAIE P3 2021 June Q3
3
  1. Given that \(\cos \left( x - 30 ^ { \circ } \right) = 2 \sin \left( x + 30 ^ { \circ } \right)\), show that \(\tan x = \frac { 2 - \sqrt { 3 } } { 1 - 2 \sqrt { 3 } }\).
  2. Hence solve the equation $$\cos \left( x - 30 ^ { \circ } \right) = 2 \sin \left( x + 30 ^ { \circ } \right)$$ for \(0 ^ { \circ } < x < 360 ^ { \circ }\).
CAIE P3 2021 June Q4
4
  1. Prove that \(\frac { 1 - \cos 2 \theta } { 1 + \cos 2 \theta } \equiv \tan ^ { 2 } \theta\).
  2. Hence find the exact value of \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \frac { 1 - \cos 2 \theta } { 1 + \cos 2 \theta } \mathrm {~d} \theta\).
CAIE P3 2021 June Q5
5
  1. Solve the equation \(z ^ { 2 } - 2 p \mathrm { i } z - q = 0\), where \(p\) and \(q\) are real constants.
    In an Argand diagram with origin \(O\), the roots of this equation are represented by the distinct points \(A\) and \(B\).
  2. Given that \(A\) and \(B\) lie on the imaginary axis, find a relation between \(p\) and \(q\).
  3. Given instead that triangle \(O A B\) is equilateral, express \(q\) in terms of \(p\).
CAIE P3 2021 June Q6
6 The parametric equations of a curve are $$x = \ln ( 2 + 3 t ) , \quad y = \frac { t } { 2 + 3 t }$$
  1. Show that the gradient of the curve is always positive.
  2. Find the equation of the tangent to the curve at the point where it intersects the \(y\)-axis.
CAIE P3 2021 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{a257f49d-5c8f-4c23-be78-46619b746fde-10_353_689_262_726} The diagram shows the curve \(y = \frac { \tan ^ { - 1 } x } { \sqrt { x } }\) and its maximum point \(M\) where \(x = a\).
  1. Show that \(a\) satisfies the equation $$a = \tan \left( \frac { 2 a } { 1 + a ^ { 2 } } \right)$$
  2. Verify by calculation that \(a\) lies between 1.3 and 1.5.
  3. Use an iterative formula based on the equation in part (a) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2021 June Q8
8 With respect to the origin \(O\), the points \(A\) and \(B\) have position vectors given by \(\overrightarrow { O A } = \left( \begin{array} { l } 1
2
1 \end{array} \right)\) and \(\overrightarrow { O B } = \left( \begin{array} { r } 3
1
- 2 \end{array} \right)\). The line \(l\) has equation \(\mathbf { r } = \left( \begin{array} { l } 2
3
1 \end{array} \right) + \lambda \left( \begin{array} { r } 1
- 2
1 \end{array} \right)\).
  1. Find the acute angle between the directions of \(A B\) and \(l\).
  2. Find the position vector of the point \(P\) on \(l\) such that \(A P = B P\).