Questions — CAIE P3 (1070 questions)

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CAIE P3 2023 November Q1
1 Find the exact coordinates of the points on the curve \(y = \frac { x ^ { 2 } } { 1 - 3 x }\) at which the gradient of the tangent is equal to 8 .
CAIE P3 2023 November Q2
2 On an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 2 \mathrm { i } | \leqslant | z + 2 - \mathrm { i } |\) and \(0 \leqslant \arg ( z + 1 ) \leqslant \frac { 1 } { 4 } \pi\).
CAIE P3 2023 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{ce3c4a9c-bf83-4d28-96e2-ef31c3673dea-04_860_451_264_833} The variables \(x\) and \(y\) are related by the equation \(y = a b ^ { x }\), where \(a\) and \(b\) are constants. The diagram shows the result of plotting \(\ln y\) against \(x\) for two pairs of values of \(x\) and \(y\). The coordinates of these points are \(( 1,3.7 )\) and \(( 2.2,6.46 )\). Use this information to find the values of \(a\) and \(b\).
CAIE P3 2023 November Q4
4 The complex number \(u\) is defined by \(u = \frac { 3 + 2 \mathrm { i } } { a - 5 \mathrm { i } }\), 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. Given that \(\arg u = \frac { 1 } { 4 } \pi\), find the value of \(a\).
CAIE P3 2023 November Q5
5
  1. Given that $$\sin \left( x + \frac { 1 } { 6 } \pi \right) - \sin \left( x - \frac { 1 } { 6 } \pi \right) = \cos \left( x + \frac { 1 } { 3 } \pi \right) - \cos \left( x - \frac { 1 } { 3 } \pi \right)$$ find the exact value of \(\tan x\).
  2. Hence find the exact roots of the equation $$\sin \left( x + \frac { 1 } { 6 } \pi \right) - \sin \left( x - \frac { 1 } { 6 } \pi \right) = \cos \left( x + \frac { 1 } { 3 } \pi \right) - \cos \left( x - \frac { 1 } { 3 } \pi \right)$$ for \(0 \leqslant x \leqslant 2 \pi\).
CAIE P3 2023 November Q6
6 The parametric equations of a curve are $$x = \sqrt { t } + 3 , \quad y = \ln t$$ for \(t > 0\).
  1. Obtain a simplified expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
  2. Hence find the exact coordinates of the point on the curve at which the gradient of the normal is - 2 .
CAIE P3 2023 November Q7
7 The variables \(x\) and \(\theta\) satisfy the differential equation $$\frac { x } { \tan \theta } \frac { \mathrm {~d} x } { \mathrm {~d} \theta } = x ^ { 2 } + 3$$ It is given that \(x = 1\) when \(\theta = 0\).
Solve the differential equation, obtaining an expression for \(x ^ { 2 }\) in terms of \(\theta\).
CAIE P3 2023 November Q8
8
  1. By sketching a suitable pair of graphs, show that the equation $$\sqrt { x } = \mathrm { e } ^ { x } - 3$$ has only one root.
  2. Show by calculation that this root lies between 1 and 2 .
  3. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \ln \left( 3 + \sqrt { x _ { n } } \right)$$ converges, then it converges to the root of the equation in (a).
  4. Use the iterative formula to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2023 November Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{ce3c4a9c-bf83-4d28-96e2-ef31c3673dea-12_375_645_274_742} The diagram shows the curve \(y = x \mathrm { e } ^ { - \frac { 1 } { 4 } x ^ { 2 } }\), for \(x \geqslant 0\), and its maximum point \(M\).
  1. Find the exact coordinates of \(M\).
  2. Using the substitution \(x = \sqrt { u }\), or otherwise, find by integration the exact area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = 3\).
CAIE P3 2023 November Q10
10 Let \(\mathrm { f } ( x ) = \frac { 24 x + 13 } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }\).
  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 \(x ^ { 2 }\).
  3. State the set of values of \(x\) for which the expansion in (b) is valid.
CAIE P3 2023 November Q11
11
\includegraphics[max width=\textwidth, alt={}, center]{ce3c4a9c-bf83-4d28-96e2-ef31c3673dea-16_593_780_264_685} In the diagram, \(O A B C D E F G\) is a cuboid in which \(O A = 3\) units, \(O C = 2\) units and \(O D = 2\) units. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O D\) and \(O C\) respectively. \(M\) is the midpoint of \(E F\).
  1. Find the position vector of \(M\).
    The position vector of \(P\) is \(\mathbf { i } + \mathbf { j } + 2 \mathbf { k }\).
  2. Calculate angle PAM.
  3. Find the exact length of the perpendicular from \(P\) to the line passing through \(O\) and \(M\).
    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
1
  1. Sketch the graph of \(y = | 4 x - 2 |\).
  2. Solve the inequality \(1 + 3 x < | 4 x - 2 |\).
CAIE P3 2023 November Q2
2 The parametric equations of a curve are $$x = ( \ln t ) ^ { 2 } , \quad y = \mathrm { e } ^ { 2 - t ^ { 2 } }$$ for \(t > 0\).
Find the gradient of the curve at the point where \(t = \mathrm { e }\), simplifying your answer.
CAIE P3 2023 November Q3
3 The polynomial \(2 x ^ { 3 } + a x ^ { 2 } - 11 x + b\) is denoted by \(\mathrm { p } ( x )\). It is given that \(\mathrm { p } ( x )\) is divisible by \(( 2 x - 1 )\) and that when \(\mathrm { p } ( x )\) is divided by \(( x + 1 )\) the remainder is 12 . Find the values of \(a\) and \(b\).
CAIE P3 2023 November Q4
4
  1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 4 - 3 \mathrm { i } | \leqslant 2\) and \(\operatorname { Re } z \leqslant 3\).
  2. Find the greatest value of \(\arg z\) for points in this region.
CAIE P3 2023 November Q5
5 Find the exact value of \(\int _ { 0 } ^ { 6 } \frac { x ( x + 1 ) } { x ^ { 2 } + 4 } \mathrm {~d} x\).
CAIE P3 2023 November Q6
6
  1. By sketching a suitable pair of graphs, show that the equation $$\cot x = 2 - \cos x$$ has one root in the interval \(0 < x \leqslant \frac { 1 } { 2 } \pi\).
  2. Show by calculation that this root lies between 0.6 and 0.8 .
  3. Use the iterative formula \(x _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 1 } { 2 - \cos x _ { n } } \right)\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2023 November Q7
7
  1. By expressing \(3 \theta\) as \(2 \theta + \theta\), prove the identity \(\cos 3 \theta \equiv 4 \cos ^ { 3 } \theta - 3 \cos \theta\).
  2. Hence solve the equation $$\cos 3 \theta + \cos \theta \cos 2 \theta = \cos ^ { 2 } \theta$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P3 2023 November Q8
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
\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
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
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
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
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
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 \(b\).