Questions P3 (1203 questions)

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CAIE P3 2021 November Q3
3
  1. Given the complex numbers \(u = a + \mathrm { i } b\) and \(w = c + \mathrm { i } d\), where \(a , b , c\) and \(d\) are real, prove that \(( u + w ) ^ { * } = u ^ { * } + w ^ { * }\).
  2. Solve the equation \(( z + 2 + \mathrm { i } ) ^ { * } + ( 2 + \mathrm { i } ) z = 0\), giving your answer in the form \(x + \mathrm { i } y\) where \(x\) and \(y\) are real.
CAIE P3 2021 November Q4
4 Express \(\frac { 4 x ^ { 2 } - 13 x + 13 } { ( 2 x - 1 ) ( x - 3 ) }\) in partial fractions.
CAIE P3 2021 November Q5
5
  1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 3 - 2 \mathbf { i } | \leqslant 1\) and \(\operatorname { Im } z \geqslant 2\).
  2. Find the greatest value of \(\arg z\) for points in the shaded region, giving your answer in degrees.
CAIE P3 2021 November Q6
6
  1. Using the expansions of \(\sin ( 3 x + 2 x )\) and \(\sin ( 3 x - 2 x )\), show that $$\frac { 1 } { 2 } ( \sin 5 x + \sin x ) \equiv \sin 3 x \cos 2 x$$
  2. Hence show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin 3 x \cos 2 x \mathrm {~d} x = \frac { 1 } { 5 } ( 3 - \sqrt { 2 } )\).
CAIE P3 2021 November Q7
7 The variables \(x\) and \(y\) satisfy the differential equation $$\mathrm { e } ^ { 2 x } \frac { \mathrm {~d} y } { \mathrm {~d} x } = 4 x y ^ { 2 }$$ and it is given that \(y = 1\) when \(x = 0\).
Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).
CAIE P3 2021 November Q8
8
  1. By first expanding \(\left( \cos ^ { 2 } \theta + \sin ^ { 2 } \theta \right) ^ { 2 }\), show that $$\cos ^ { 4 } \theta + \sin ^ { 4 } \theta \equiv 1 - \frac { 1 } { 2 } \sin ^ { 2 } 2 \theta .$$
  2. Hence solve the equation $$\cos ^ { 4 } \theta + \sin ^ { 4 } \theta = \frac { 5 } { 9 } ,$$ for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P3 2021 November Q9
9 The equation of a curve is \(y \mathrm { e } ^ { 2 x } - y ^ { 2 } \mathrm { e } ^ { x } = 2\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 y \mathrm { e } ^ { x } - y ^ { 2 } } { 2 y - \mathrm { e } ^ { x } }\).
  2. Find the exact coordinates of the point on the curve where the tangent is parallel to the \(y\)-axis.
CAIE P3 2021 November Q10
10 With respect to the origin \(O\), the position vectors of the points \(A\) and \(B\) are given by \(\overrightarrow { O A } = \left( \begin{array} { r } 1
2
- 1 \end{array} \right)\) and \(\overrightarrow { O B } = \left( \begin{array} { l } 0
3
1 \end{array} \right)\).
  1. Find a vector equation for the line \(l\) through \(A\) and \(B\).
  2. The point \(C\) lies on \(l\) and is such that \(\overrightarrow { A C } = 3 \overrightarrow { A B }\). Find the position vector of \(C\).
  3. Find the possible position vectors of the point \(P\) on \(l\) such that \(O P = \sqrt { 14 }\).
CAIE P3 2021 November Q11
11 The equation of a curve is \(y = \sqrt { \tan x }\), for \(0 \leqslant x < \frac { 1 } { 2 } \pi\).
  1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\tan x\), and verify that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) when \(x = \frac { 1 } { 4 } \pi\).
    The value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) is also 1 at another point on the curve where \(x = a\), as shown in the diagram.
    \includegraphics[max width=\textwidth, alt={}, center]{87be326f-f638-43e9-a654-b7b53d5141ef-18_605_492_1493_822}
  2. Show that \(t ^ { 3 } + t ^ { 2 } + 3 t - 1 = 0\), where \(t = \tan a\).
  3. Use the iterative formula $$a _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 1 } { 3 } \left( 1 - \tan ^ { 2 } a _ { n } - \tan ^ { 3 } a _ { n } \right) \right)$$ to determine \(a\) correct to 2 decimal places, giving the result of each iteration to 4 decimal places.
    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 November Q1
1 Find the quotient and remainder when \(2 x ^ { 4 } + 1\) is divided by \(x ^ { 2 } - x + 2\).
CAIE P3 2021 November Q2
2
  1. Sketch the graph of \(y = | 2 x - 3 |\).
  2. Solve the inequality \(| 2 x - 3 | < 3 x + 2\).
CAIE P3 2021 November Q3
3 Solve the equation \(4 ^ { x - 2 } = 4 ^ { x } - 4 ^ { 2 }\), giving your answer correct to 3 decimal places.
CAIE P3 2021 November Q4
4 Find the exact value of \(\int _ { \frac { 1 } { 3 } \pi } ^ { \pi } x \sin \frac { 1 } { 2 } x \mathrm {~d} x\).
CAIE P3 2021 November Q5
5 Solve the equation \(\sin \theta = 3 \cos 2 \theta + 2\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P3 2021 November Q6
6
  1. By first expanding \(\cos \left( x - 60 ^ { \circ } \right)\), show that the expression $$2 \cos \left( x - 60 ^ { \circ } \right) + \cos x$$ can be written 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 2 decimal places.
  2. Hence find the value of \(x\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\) for which \(2 \cos \left( x - 60 ^ { \circ } \right) + \cos x\) takes its least possible value.
CAIE P3 2021 November Q7
3 marks
7 The equation of a curve is \(\ln ( x + y ) = x - 2 y\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x + y - 1 } { 2 ( x + y ) + 1 }\).
  2. Find the coordinates of the point on the curve where the tangent is parallel to the \(x\)-axis. \(\quad\) [3]
CAIE P3 2021 November Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{bbe57fc0-a8a5-4fe5-a637-4f9db00bdc13-10_588_789_260_678} In the diagram, \(O A B C D\) is a pyramid with vertex \(D\). The horizontal base \(O A B C\) is a square of side 4 units. The edge \(O D\) is vertical and \(O D = 4\) units. The unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O D\) respectively. The midpoint of \(A B\) is \(M\) and the point \(N\) on \(C D\) is such that \(D N = 3 N C\).
  1. Find a vector equation for the line through \(M\) and \(N\).
  2. Show that the length of the perpendicular from \(O\) to \(M N\) is \(\frac { 1 } { 3 } \sqrt { 82 }\).
    \(9 \quad\) Let \(\mathrm { f } ( x ) = \frac { 1 } { ( 9 - x ) \sqrt { x } }\).
CAIE P3 2021 November Q10
10 A large plantation of area \(20 \mathrm {~km} ^ { 2 }\) is becoming infected with a plant disease. At time \(t\) years the area infected is \(x \mathrm {~km} ^ { 2 }\) and the rate of increase of \(x\) is proportional to the ratio of the area infected to the area not yet infected. When \(t = 0 , x = 1\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 1\).
  1. Show that \(x\) and \(t\) satisfy the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 19 x } { 20 - x }$$
  2. Solve the differential equation and show that when \(t = 1\) the value of \(x\) satisfies the equation \(x = \mathrm { e } ^ { 0.9 + 0.05 x }\).
  3. Use an iterative formula based on the equation in part (b), with an initial value of 2 , to determine \(x\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
  4. Calculate the value of \(t\) at which the entire plantation becomes infected.
CAIE P3 2021 November Q11
11 The complex number \(- \sqrt { 3 } + \mathrm { i }\) is denoted by \(u\).
  1. Express \(u\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\), giving the exact values of \(r\) and \(\theta\).
  2. Hence show that \(u ^ { 6 }\) is real and state its value.
    1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(0 \leqslant \arg ( z - u ) \leqslant \frac { 1 } { 4 } \pi\) and \(\operatorname { Re } z \leqslant 2\).
    2. Find the greatest value of \(| z |\) for points in the shaded region. Give your answer correct to 3 significant figures.
      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 2022 November Q1
1
  1. Sketch the graph of \(y = | 2 x + 1 |\).
  2. Solve the inequality \(3 x + 5 < | 2 x + 1 |\).
CAIE P3 2022 November Q2
2 On a sketch of an Argand diagram shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z | \leqslant 3 , \operatorname { Re } z \geqslant - 2\) and \(\frac { 1 } { 4 } \pi \leqslant \arg z \leqslant \pi\).
CAIE P3 2022 November Q3
3 Solve the equation \(2 ^ { 3 x - 1 } = 5 \left( 3 ^ { - x } \right)\). Give your answer in the form \(\frac { \ln a } { \ln b }\), where \(a\) and \(b\) are integers.
CAIE P3 2022 November Q4
4 Solve the equation \(\tan \left( x + 45 ^ { \circ } \right) = 2 \cot x\) for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
CAIE P3 2022 November Q5
5 The complex numbers \(u\) and \(w\) are defined by \(u = 2 \mathrm { e } ^ { \frac { 1 } { 4 } \pi \mathrm { i } }\) and \(w = 3 \mathrm { e } ^ { \frac { 1 } { 3 } \pi \mathrm { i } }\).
  1. Find \(\frac { u ^ { 2 } } { w }\), giving your answer in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Give the exact values of \(r\) and \(\theta\).
  2. State the least positive integer \(n\) such that both \(\operatorname { Im } w ^ { n } = 0\) and \(\operatorname { Re } w ^ { n } > 0\).
CAIE P3 2022 November Q6
6
  1. Prove the identity \(\cos 4 \theta + 4 \cos 2 \theta + 3 \equiv 8 \cos ^ { 4 } \theta\).
  2. Hence solve the equation \(\cos 4 \theta + 4 \cos 2 \theta = 4\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).