Questions P3 (1203 questions)

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CAIE P3 2022 June Q9
9 With respect to the origin \(O\), the point \(A\) has position vector given by \(\overrightarrow { O A } = \mathbf { i } + 5 \mathbf { j } + 6 \mathbf { k }\). The line \(l\) has vector equation \(\mathbf { r } = 4 \mathbf { i } + \mathbf { k } + \lambda ( - \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } )\).
  1. Find in degrees the acute angle between the directions of \(O A\) and \(l\).
  2. Find the position vector of the foot of the perpendicular from \(A\) to \(l\).
  3. Hence find the position vector of the reflection of \(A\) in \(l\).
CAIE P3 2022 June Q10
10 The constant \(a\) is such that \(\int _ { 1 } ^ { a } x ^ { 2 } \ln x \mathrm {~d} x = 4\).
  1. Show that \(a = \left( \frac { 35 } { 3 \ln a - 1 } \right) ^ { \frac { 1 } { 3 } }\).
  2. Verify by calculation that \(a\) lies between 2.4 and 2.8.
  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.
    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 June Q1
1 Solve the equation $$3 \mathrm { e } ^ { 2 x } - 4 \mathrm { e } ^ { - 2 x } = 5$$ Give the answer correct to 3 decimal places.
CAIE P3 2023 June Q2
2
  1. Sketch the graph of \(y = | 2 x + 3 |\).
  2. Solve the inequality \(3 x + 8 > | 2 x + 3 |\).
CAIE P3 2023 June Q3
3 Find the coefficient of \(x ^ { 3 }\) in the binomial expansion of \(( 3 + x ) \sqrt { 1 + 4 x }\).
CAIE P3 2023 June Q4
4
  1. Show that the equation \(\sin 2 \theta + \cos 2 \theta = 2 \sin ^ { 2 } \theta\) can be expressed in the form $$\cos ^ { 2 } \theta + 2 \sin \theta \cos \theta - 3 \sin ^ { 2 } \theta = 0$$
  2. Hence solve the equation \(\sin 2 \theta + \cos 2 \theta = 2 \sin ^ { 2 } \theta\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P3 2023 June Q5
4 marks
5 The equation of a curve is \(x ^ { 2 } y - a y ^ { 2 } = 4 a ^ { 3 }\), where \(a\) is a non-zero constant.
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 x y } { 2 a y - x ^ { 2 } }\).
  2. Hence find the coordinates of the points where the tangent to the curve is parallel to the \(y\)-axis. [4]
CAIE P3 2023 June Q6
6 Relative to the origin \(O\), the points \(A , B\) and \(C\) have position vectors given by $$\overrightarrow { O A } = \left( \begin{array} { l } 2
1
3 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { l } 4
3
2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 3
- 2
- 4 \end{array} \right) .$$ The quadrilateral \(A B C D\) is a parallelogram.
  1. Find the position vector of \(D\).
  2. The angle between \(B A\) and \(B C\) is \(\theta\). Find the exact value of \(\cos \theta\).
  3. Hence find the area of \(A B C D\), giving your answer in the form \(p \sqrt { q }\), where \(p\) and \(q\) are integers.
CAIE P3 2023 June Q7
7 The variables \(x\) and \(y\) satisfy the differential equation $$\cos 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { 4 \tan 2 x } { \sin ^ { 2 } 3 y }$$ where \(0 \leqslant x < \frac { 1 } { 4 } \pi\). It is given that \(y = 0\) when \(x = \frac { 1 } { 6 } \pi\).
Solve the differential equation to obtain the value of \(x\) when \(y = \frac { 1 } { 6 } \pi\). Give your answer correct to 3 decimal places.
CAIE P3 2023 June Q8
8 Let \(\mathrm { f } ( x ) = \frac { 3 - 3 x ^ { 2 } } { ( 2 x + 1 ) ( x + 2 ) ^ { 2 } }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence find the exact value of \(\int _ { 0 } ^ { 4 } \mathrm { f } ( x ) \mathrm { d } x\), giving your answer in the form \(a + b \ln c\), where \(a , b\) and \(c\) are integers.
CAIE P3 2023 June Q9
9 The constant \(a\) is such that \(\int _ { 0 } ^ { a } x \mathrm { e } ^ { - 2 x } \mathrm {~d} x = \frac { 1 } { 8 }\).
  1. Show that \(a = \frac { 1 } { 2 } \ln ( 4 a + 2 )\).
  2. Verify by calculation that \(a\) lies between 0.5 and 1 .
  3. Use an iterative formula based on the equation in (a) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2023 June Q10
10 The polynomial \(x ^ { 3 } + 5 x ^ { 2 } + 31 x + 75\) is denoted by \(\mathrm { p } ( x )\).
  1. Show that \(( x + 3 )\) is a factor of \(\mathrm { p } ( x )\).
  2. Show that \(z = - 1 + 2 \sqrt { 6 } \mathrm { i }\) is a root of \(\mathrm { p } ( z ) = 0\).
  3. Hence find the complex numbers \(z\) which are roots of \(\mathrm { p } \left( z ^ { 2 } \right) = 0\).
    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 June Q1
1 Solve the inequality \(| 5 x - 3 | < 2 | 3 x - 7 |\).
CAIE P3 2023 June Q2
2 Solve the equation \(\ln \left( 2 x ^ { 2 } - 3 \right) = 2 \ln x - \ln 2\), giving your answer in an exact form.
CAIE P3 2023 June Q3
3
  1. On an Argand diagram, sketch the locus of points representing complex numbers \(z\) satisfying \(| z + 3 - 2 \mathrm { i } | = 2\).
  2. Find the least value of \(| z |\) for points on this locus, giving your answer in an exact form.
CAIE P3 2023 June Q4
4 Solve the equation \(2 \cos x - \cos \frac { 1 } { 2 } x = 1\) for \(0 \leqslant x \leqslant 2 \pi\).
CAIE P3 2023 June Q5
5 The complex number \(2 + y \mathrm { i }\) is denoted by \(a\), where \(y\) is a real number and \(y < 0\). It is given that \(\mathrm { f } ( a ) = a ^ { 3 } - a ^ { 2 } - 2 a\).
  1. Find a simplified expression for \(\mathrm { f } ( a )\) in terms of \(y\).
  2. Given that \(\operatorname { Re } ( \mathrm { f } ( a ) ) = - 20\), find \(\arg a\).
CAIE P3 2023 June Q6
6 The equation \(\cot \frac { 1 } { 2 } x = 3 x\) has one root in the interval \(0 < x < \pi\), denoted by \(\alpha\).
  1. Show by calculation that \(\alpha\) lies between 0.5 and 1 .
  2. Show that, if a sequence of positive values given by the iterative formula $$x _ { n + 1 } = \frac { 1 } { 3 } \left( x _ { n } + 4 \tan ^ { - 1 } \left( \frac { 1 } { 3 x _ { n } } \right) \right)$$ converges, then it converges to \(\alpha\).
  3. Use this iterative formula to calculate \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2023 June Q7
7 The equation of a curve is \(3 x ^ { 2 } + 4 x y + 3 y ^ { 2 } = 5\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 3 x + 2 y } { 2 x + 3 y }\).
  2. Hence find the exact coordinates of the two points on the curve at which the tangent is parallel to \(y + 2 x = 0\).
CAIE P3 2023 June Q8
8
  1. The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 + 9 y ^ { 2 } } { \mathrm { e } ^ { 2 x + 1 } } .$$ It is given that \(y = 0\) when \(x = 1\).
    Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).
  2. State what happens to the value of \(y\) as \(x\) tends to infinity. Give your answer in an exact form.
CAIE P3 2023 June Q9
5 marks
9 Let \(\mathrm { f } ( x ) = \frac { 2 x ^ { 2 } + 17 x - 17 } { ( 1 + 2 x ) ( 2 - x ) ^ { 2 } }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence show that \(\int _ { 0 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x = \frac { 5 } { 2 } - \ln 72\).
    \includegraphics[max width=\textwidth, alt={}, center]{60bb482b-fa41-42ea-a112-62851e5a19aa-16_524_725_269_696} The diagram shows the curve \(y = ( x + 5 ) \sqrt { 3 - 2 x }\) and its maximum point \(M\).
CAIE P3 2023 June Q11
11 The points \(A\) and \(B\) have position vectors \(\mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k }\) and \(2 \mathbf { i } - \mathbf { j } + \mathbf { k }\) respectively. The line \(l\) has equation \(\mathbf { r } = \mathbf { i } - \mathbf { j } + 3 \mathbf { k } + \mu ( 2 \mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k } )\).
  1. Show that \(l\) does not intersect the line passing through \(A\) and \(B\).
  2. Find the position vector of the foot of the perpendicular from \(A\) to \(l\).
    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 June Q1
1 Solve the equation \(\ln ( x + 5 ) = 5 + \ln x\). Give your answer correct to 3 decimal places.
CAIE P3 2023 June Q2
2 Find the quotient and remainder when \(2 x ^ { 4 } - 27\) is divided by \(x ^ { 2 } + x + 3\).
CAIE P3 2023 June Q3
3 On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 3 - \mathrm { i } | \leqslant 3\) and \(| z | \geqslant | z - 4 \mathrm { i } |\).