Questions — CAIE P3 (1110 questions)

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CAIE P3 2023 November Q8
7 marks Standard +0.3
8 It is given that \(\frac { 2 + 3 a \mathrm { i } } { a + 2 \mathrm { i } } = \lambda ( 2 - \mathrm { i } )\), where \(a\) and \(\lambda\) are real constants.
  1. Show that \(3 a ^ { 2 } + 4 a - 4 = 0\).
  2. Hence find the possible values of \(a\) and the corresponding values of \(\lambda\).
CAIE P3 2023 November Q9
9 marks Standard +0.8
9 \includegraphics[max width=\textwidth, alt={}, center]{39cf66af-095b-404b-a38c-0aa7684c4a27-14_428_787_274_671} The diagram shows the curve \(y = \sin x \cos 2 x\), for \(0 \leqslant x \leqslant \pi\), and a maximum point \(M\), where \(x = a\). The shaded region between the curve and the \(x\)-axis is denoted by \(R\).
  1. Find the value of \(a\) correct to 2 decimal places.
  2. Find the exact area of the region \(R\), giving your answer in simplified form.
CAIE P3 2023 November Q10
9 marks Standard +0.3
10 The equations of the lines \(l\) and \(m\) are given by $$l : \mathbf { r } = \left( \begin{array} { r } 3 \\ - 2 \\ 1 \end{array} \right) + \lambda \left( \begin{array} { l } 1 \\ 1 \\ 2 \end{array} \right) \quad \text { and } \quad m : \mathbf { r } = \left( \begin{array} { r } 6 \\ - 3 \\ 6 \end{array} \right) + \mu \left( \begin{array} { r } - 2 \\ 4 \\ c \end{array} \right)$$ where \(c\) is a positive constant. It is given that the angle between \(l\) and \(m\) is \(60 ^ { \circ }\).
  1. Find the value of \(c\).
  2. Show that the length of the perpendicular from \(( 6 , - 3,6 )\) to \(l\) is \(\sqrt { 11 }\).
CAIE P3 2023 November Q11
9 marks Standard +0.8
11 The variables \(x\) and \(y\) satisfy the differential equation $$x ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + y ^ { 2 } + y = 0$$ It is given that \(x = 1\) when \(y = 1\).
  1. Solve the differential equation to obtain an expression for \(y\) in terms of \(x\).
  2. State what happens to the value of \(y\) when \(x\) tends to infinity. Give your answer in an exact form.
    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 2023 November Q1
4 marks Standard +0.3
1 Find the set of values of \(x\) satisfying the inequality \(\left| 2 ^ { x + 1 } - 2 \right| < 0.5\), giving your answer to 3 significant figures.
CAIE P3 2023 November Q2
5 marks Moderate -0.3
2 On an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 1 + 2 i | \leqslant | z |\) and \(| z - 2 | \leqslant 1\).
CAIE P3 2023 November Q3
5 marks Moderate -0.8
3 The polynomial \(2 x ^ { 3 } + a x ^ { 2 } + b x + 6\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). When \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\) the remainder is - 38 and when \(\mathrm { p } ( x )\) is divided by \(( 2 x - 1 )\) the remainder is \(\frac { 19 } { 2 }\). Find the values of \(a\) and \(b\).
CAIE P3 2023 November Q4
5 marks Standard +0.3
4 Solve the quadratic equation \(( 3 + \mathrm { i } ) w ^ { 2 } - 2 w + 3 - \mathrm { i } = 0\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
CAIE P3 2023 November Q5
6 marks Challenging +1.2
5 Find the exact coordinates of the stationary points of the curve \(y = \frac { \mathrm { e } ^ { 3 x ^ { 2 } - 1 } } { 1 - x ^ { 2 } }\).
CAIE P3 2023 November Q6
6 marks Standard +0.8
6
  1. Show that the equation \(\cot ^ { 2 } \theta + 2 \cos 2 \theta = 4\) can be written in the form $$4 \sin ^ { 4 } \theta + 3 \sin ^ { 2 } \theta - 1 = 0$$
  2. Hence solve the equation \(\cot ^ { 2 } \theta + 2 \cos 2 \theta = 4\), for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
CAIE P3 2023 November Q7
8 marks Standard +0.3
7 The equation of a curve is \(x ^ { 3 } + y ^ { 2 } + 3 x ^ { 2 } + 3 y = 4\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 3 x ^ { 2 } + 6 x } { 2 y + 3 }\).
  2. Hence find the coordinates of the points on the curve at which the tangent is parallel to the \(x\)-axis. [5]
CAIE P3 2023 November Q8
7 marks Standard +0.3
8 The variables \(x\) and \(y\) satisfy the differential equation $$\mathrm { e } ^ { 4 x } \frac { \mathrm {~d} y } { \mathrm {~d} x } = \cos ^ { 2 } 3 y .$$ It is given that \(y = 0\) when \(x = 2\).
Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).
CAIE P3 2023 November Q9
11 marks Standard +0.8
9 Let \(\mathrm { f } ( x ) = \frac { 17 x ^ { 2 } - 7 x + 16 } { \left( 2 + 3 x ^ { 2 } \right) ( 2 - x ) }\).
  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 \(x ^ { 3 }\).
  3. State the set of values of \(x\) for which the expansion in (b) is valid. Give your answer in an exact form.
CAIE P3 2023 November Q10
9 marks Standard +0.8
10 \includegraphics[max width=\textwidth, alt={}, center]{a49b720b-f8d2-42ff-b147-5d676993aa4c-16_611_689_274_721} The diagram shows the curve \(y = x \cos 2 x\), for \(x \geqslant 0\).
  1. Find the equation of the tangent to the curve at the point where \(x = \frac { 1 } { 2 } \pi\).
  2. Find the exact area of the shaded region shown in the diagram, bounded by the curve and the \(x\)-axis.
CAIE P3 2023 November Q11
9 marks Standard +0.3
11 The line \(l\) has equation \(\mathbf { r } = \mathbf { i } - 2 \mathbf { j } - 3 \mathbf { k } + \lambda ( - \mathbf { i } + \mathbf { j } + 2 \mathbf { k } )\). The points \(A\) and \(B\) have position vectors \(- 2 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }\) and \(3 \mathbf { i } - \mathbf { j } + \mathbf { k }\) respectively.
  1. Find a unit vector in the direction of \(l\).
    The line \(m\) passes through the points \(A\) and \(B\).
  2. Find a vector equation for \(m\).
  3. Determine whether lines \(l\) and \(m\) are parallel, intersect or are skew.
    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 2024 November Q1
5 marks Standard +0.3
1 The polynomial \(4 x ^ { 3 } + a x ^ { 2 } + 5 x + b\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( 2 x + 1 )\) is a factor of \(\mathrm { p } ( x )\). When \(\mathrm { p } ( x )\) is divided by \(( x - 4 )\) the remainder is equal to 3 times the remainder when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\). Find the values of \(a\) and \(b\).
CAIE P3 2024 November Q2
5 marks Moderate -0.3
2 Find the exact value of \(\int _ { 1 } ^ { 3 } x ^ { 2 } \ln 3 x \mathrm {~d} x\). Give your answer in the form \(a \ln b + c\), where \(a\) and \(c\) are rational and \(b\) is an integer. \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-04_2720_38_105_2010}
CAIE P3 2024 November Q3
4 marks Standard +0.3
3 The equation of a curve is \(\ln ( x + y ) = 3 x ^ { 2 } y\).
Find the gradient of the curve at the point \(( 1,0 )\).
CAIE P3 2024 November Q4
8 marks Standard +0.3
4
  1. Show that \(\sec ^ { 4 } \theta - \tan ^ { 4 } \theta \equiv 1 + 2 \tan ^ { 2 } \theta\). \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-07_2723_35_101_20}
  2. Hence or otherwise solve the equation \(\sec ^ { 4 } 2 \alpha - \tan ^ { 4 } 2 \alpha = 2 \tan ^ { 2 } 2 \alpha \sec ^ { 2 } 2 \alpha\) for \(0 ^ { \circ } < \alpha < 180 ^ { \circ }\). [5]
CAIE P3 2024 November Q5
7 marks Moderate -0.3
5
  1. By sketching a suitable pair of graphs, show that the equation \(2 + \mathrm { e } ^ { - 0.2 x } = \ln ( 1 + x )\) has only one root.
  2. Show by calculation that this root lies between 7 and 9 . \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-08_2716_40_109_2009}
  3. Use the iterative formula $$x _ { n + 1 } = \exp \left( 2 + \mathrm { e } ^ { - 0.2 x _ { n } } \right) - 1$$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places. \(\left[ \exp ( x ) \right.\) is an alternative notation for \(\left. \mathrm { e } ^ { x } .\right]\) \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-10_481_789_262_639} The diagram shows the curve \(y = \sin 2 x ( 1 + \sin 2 x )\), for \(0 \leqslant x \leqslant \frac { 3 } { 4 } \pi\), and its minimum point \(M\). The shaded region bounded by the curve that lies above the \(x\)-axis and the \(x\)-axis itself is denoted by \(R\).
CAIE P3 2024 November Q7
9 marks Standard +0.8
7
Let \(f ( x ) = \frac { 5 x ^ { 2 } + 8 x + 5 } { ( 1 + 2 x ) \left( 2 + x ^ { 2 } \right) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions. \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-13_2726_34_97_21}
  2. Hence find the coefficient of \(x ^ { 3 }\) in the expansion of \(\mathrm { f } ( x )\).
CAIE P3 2024 November Q8
9 marks Standard +0.3
8
  1. Given that \(z = 1 + y \mathrm { i }\) and that \(y\) is a real number, express \(\frac { 1 } { z }\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are functions of \(y\).
  2. Show that \(\left( a - \frac { 1 } { 2 } \right) ^ { 2 } + b ^ { 2 } = \frac { 1 } { 4 }\), where \(a\) and \(b\) are the functions of \(y\) found in part (a). \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-14_2716_35_108_2012}
  3. On a single Argand diagram, sketch the loci given by the equations \(\operatorname { Re } ( z ) = 1\) and \(\left| z - \frac { 1 } { 2 } \right| = \frac { 1 } { 2 }\), where \(z\) is a complex number.
  4. The complex number \(z\) is such that \(\operatorname { Re } ( z ) = 1\). Use your answer to part (b) to give a geometrical description of the locus of \(\frac { 1 } { z }\).
CAIE P3 2024 November Q9
10 marks Moderate -0.3
9 The position vector of point \(A\) relative to the origin \(O\) is \(\overrightarrow { O A } = 8 \mathbf { i } - 5 \mathbf { j } + 6 \mathbf { k }\).
The line \(l\) passes through \(A\) and is parallel to the vector \(2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k }\).
  1. State a vector equation for \(l\).
  2. The position vector of point \(B\) relative to the origin \(O\) is \(\overrightarrow { O B } = - t \mathbf { i } + 4 t \mathbf { j } + 3 t \mathbf { k }\), where \(t\) is a constant. The line \(l\) also passes through \(B\). Find the value of \(t\).
  3. The line \(m\) has vector equation \(\mathbf { r } = 5 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } + \mu ( a \mathbf { i } - \mathbf { j } + 3 \mathbf { k } )\). The acute angle between the directions of \(l\) and \(m\) is \(\theta\), where \(\cos \theta = \frac { 1 } { \sqrt { 6 } }\).
    Find the possible values of \(a\). \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-18_542_559_251_753} A large cylindrical tank is used to store water. The base of the tank is a circle of radius 4 metres. At time \(t\) minutes, the depth of the water in the tank is \(h\) metres. There is a tap at the bottom of the tank. When the tap is open, water flows out of the tank at a rate proportional to the square root of the volume of water in the tank.
    1. Show that \(\frac { \mathrm { d } h } { \mathrm {~d} t } = - \lambda \sqrt { h }\), where \(\lambda\) is a positive constant. \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-18_2718_42_107_2007}
    2. At time \(t = 0\) the tap is opened. It is given that \(h = 4\) when \(t = 0\) and that \(h = 2.25\) when \(t = 20\). Solve the differential equation to obtain an expression for \(t\) in terms of \(h\), and hence find the time taken to empty the tank.
      If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE P3 2024 November Q1
4 marks Moderate -0.3
1 The complex number \(z\) satisfies \(| z | = 2\) and \(0 \leqslant \arg z \leqslant \frac { 1 } { 4 } \pi\).
  1. On the Argand diagram below, sketch the locus of the points representing \(z\).
  2. On the same diagram, sketch the locus of the points representing \(z ^ { 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-02_1074_1363_628_351} \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-02_1002_26_1820_2017}
CAIE P3 2024 November Q2
5 marks Standard +0.3
2 Let \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 5 x ^ { 2 } + 4\).
  1. Show that if a sequence of values given by the iterative formula $$x _ { n + 1 } = \sqrt { \frac { 4 } { 5 - 2 x _ { n } } }$$ converges, then it converges to a root of the equation \(\mathrm { f } ( x ) = 0\).
  2. The equation has a root close to 1.2 . Use the iterative formula from part (a) and an initial value of 1.2 to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.