Questions — CAIE P3 (1070 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
CAIE P3 2023 March Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{8c26235b-c78c-40d8-9e8e-213dc1311186-10_627_611_255_767} The diagram shows a circle with centre \(O\) and radius \(r\). The angle of the minor sector \(A O B\) of the circle is \(x\) radians. The area of the major sector of the circle is 3 times the area of the shaded region.
  1. Show that \(x = \frac { 3 } { 4 } \sin x + \frac { 1 } { 2 } \pi\).
  2. Show by calculation that the root of the equation in (a) lies between 2 and 2.5.
  3. Use an iterative formula based on the equation in (a) to calculate this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2023 March Q8
5 marks
8
\includegraphics[max width=\textwidth, alt={}, center]{8c26235b-c78c-40d8-9e8e-213dc1311186-12_437_686_274_719} The diagram shows the curve \(y = x ^ { 3 } \ln x\), for \(x > 0\), and its minimum point \(M\).
  1. Find the exact coordinates of \(M\).
  2. Find the exact area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = \frac { 1 } { 2 }\). [5]
CAIE P3 2023 March Q9
9 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { 3 y } \sin ^ { 2 } 2 x$$ It is given that \(y = 0\) when \(x = 0\).
Solve the differential equation and find the value of \(y\) when \(x = \frac { 1 } { 2 }\).
CAIE P3 2023 March Q10
10 With respect to the origin \(O\), the points \(A , B , C\) and \(D\) have position vectors given by $$\overrightarrow { O A } = \left( \begin{array} { r } 3
- 1
2 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 1
2
- 3 \end{array} \right) , \quad \overrightarrow { O C } = \left( \begin{array} { r } 1
- 2
5 \end{array} \right) \quad \text { and } \quad \overrightarrow { O D } = \left( \begin{array} { r } 5
- 6
CAIE P3 2023 March Q11
11 \end{array} \right)$$
  1. Find the obtuse angle between the vectors \(\overrightarrow { O A }\) and \(\overrightarrow { O B }\).
    The line \(l\) passes through the points \(A\) and \(B\).
  2. Find a vector equation for the line \(l\).
  3. Find the position vector of the point of intersection of the line \(l\) and the line passing through \(C\) and \(D\).
    11 Let \(\mathrm { f } ( x ) = \frac { 5 x ^ { 2 } + x + 11 } { \left( 4 + x ^ { 2 } \right) ( 1 + x ) }\).
  4. Express \(\mathrm { f } ( x )\) in partial fractions.
  5. Hence show that \(\int _ { 0 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = \ln 54 - \frac { 1 } { 8 } \pi\).
    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 March Q1
1 Find the quotient and remainder when \(x ^ { 4 } - 3 x ^ { 3 } + 9 x ^ { 2 } - 12 x + 27\) is divided by \(x ^ { 2 } + 5\).
CAIE P3 2024 March Q2
2
  1. Find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 2 x - 5 ) \sqrt { 4 - x }\).
  2. State the set of values of \(x\) for which the expansion in part (a) is valid.
CAIE P3 2024 March Q3
3 It is given that \(z = - \sqrt { 3 } + \mathrm { i }\).
  1. Express \(z ^ { 2 }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
  2. The complex number \(\omega\) is such that \(z ^ { 2 } \omega\) is real and \(\left| \frac { z ^ { 2 } } { \omega } \right| = 12\). Find the two possible values of \(\omega\), giving your answers in the form \(R \mathrm { e } ^ { \mathrm { i } \alpha }\), where \(R > 0\) and \(- \pi < \alpha \leqslant \pi\).
CAIE P3 2024 March Q4
4 The positive numbers \(p\) and \(q\) are such that $$\ln \left( \frac { p } { q } \right) = a \text { and } \ln \left( q ^ { 2 } p \right) = b .$$ Express \(\ln \left( p ^ { 7 } q \right)\) in terms of \(a\) and \(b\).
CAIE P3 2024 March Q5
5
  1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 4 - 2 i | \leqslant 3\) and \(| z | \geqslant | 10 - z |\).
  2. Find the greatest value of \(\arg z\) for points in this region.
CAIE P3 2024 March Q6
6 The equation of a curve is \(2 y ^ { 2 } + 3 x y + x = x ^ { 2 }\).
  1. Show that \(\frac { \mathrm { dy } } { \mathrm { dx } } = \frac { 2 \mathrm { x } - 3 \mathrm { y } - 1 } { 4 \mathrm { y } + 3 \mathrm { x } }\).
  2. Hence show that the curve does not have a tangent that is parallel to the \(x\)-axis.
CAIE P3 2024 March Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{446573d3-73b1-482a-a3f6-1abddfdd90d0-10_620_517_260_774} The diagram shows the curve \(\mathrm { y } = \mathrm { xe } ^ { 2 \mathrm { x } } - 5 \mathrm { x }\) and its minimum point \(M\), where \(x = \alpha\).
  1. Show that \(\alpha\) satisfies the equation \(\alpha = \frac { 1 } { 2 } \ln \left( \frac { 5 } { 1 + 2 \alpha } \right)\).
  2. Verify by calculation that \(\alpha\) lies between 0.4 and 0.5.
  3. Use an iterative formula based on the equation in part (a) to determine \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2024 March Q8
8
  1. Express \(3 \sin x + 2 \sqrt { 2 } \cos \left( x + \frac { 1 } { 4 } \pi \right)\) in the form \(\mathrm { R } \sin ( \mathrm { x } + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). State the exact value of \(R\) and give \(\alpha\) correct to 3 decimal places.
  2. Hence solve the equation $$6 \sin \frac { 1 } { 2 } \theta + 4 \sqrt { 2 } \cos \left( \frac { 1 } { 2 } \theta + \frac { 1 } { 4 } \pi \right) = 3$$ for \(- 4 \pi < \theta < 4 \pi\).
CAIE P3 2024 March Q9
9 Relative to the origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { \mathrm { OA } } = 5 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } , \quad \overrightarrow { \mathrm { OB } } = 8 \mathbf { i } + 2 \mathbf { j } - 6 \mathbf { k } \quad \text { and } \quad \overrightarrow { \mathrm { OC } } = 3 \mathbf { i } + 4 \mathbf { j } - 7 \mathbf { k }$$
  1. Show that \(O A B C\) is a rectangle.
    \includegraphics[max width=\textwidth, alt={}, center]{446573d3-73b1-482a-a3f6-1abddfdd90d0-14_67_1573_557_324}
    \includegraphics[max width=\textwidth, alt={}, center]{446573d3-73b1-482a-a3f6-1abddfdd90d0-14_68_1575_648_322}
    \includegraphics[max width=\textwidth, alt={}]{446573d3-73b1-482a-a3f6-1abddfdd90d0-14_70_1573_737_324} ....................................................................................................................................... .........................................................................................................................................
  2. Use a scalar product to find the acute angle between the diagonals of \(O A B C\).
CAIE P3 2024 March Q10
10 Let \(f ( x ) = \frac { 36 a ^ { 2 } } { ( 2 a + x ) ( 2 a - x ) ( 5 a - 2 x ) }\), where \(a\) is a positive constant.
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence find the exact value of \(\int _ { - a } ^ { a } f ( x ) d x\), giving your answer in the form plnq+rlns where \(p\) and \(r\) are integers and \(q\) and \(s\) are prime numbers.
CAIE P3 2024 March Q11
11 The variables \(y\) and \(\theta\) satisfy the differential equation $$( 1 + y ) ( 1 + \cos 2 \theta ) \frac { d y } { d \theta } = e ^ { 3 y }$$ It is given that \(y = 0\) when \(\theta = \frac { 1 } { 4 } \pi\).
Solve the differential equation and find the exact value of \(\tan \theta\) when \(y = 1\).
If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE P3 2020 November Q1
1 Solve the inequality \(2 - 5 x > 2 | x - 3 |\).
CAIE P3 2020 November Q2
2 On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z | \geqslant 2\) and \(| z - 1 + \mathrm { i } | \leqslant 1\).
CAIE P3 2020 November Q3
3 The parametric equations of a curve are $$x = 3 - \cos 2 \theta , \quad y = 2 \theta + \sin 2 \theta$$ for \(0 < \theta < \frac { 1 } { 2 } \pi\).
Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \cot \theta\).
CAIE P3 2020 November Q4
4 Solve the equation $$\log _ { 10 } ( 2 x + 1 ) = 2 \log _ { 10 } ( x + 1 ) - 1$$ Give your answers correct to 3 decimal places.
CAIE P3 2020 November Q5
5
  1. By sketching a suitable pair of graphs, show that the equation \(\operatorname { cosec } x = 1 + \mathrm { e } ^ { - \frac { 1 } { 2 } x }\) has exactly two roots in the interval \(0 < x < \pi\).
  2. The sequence of values given by the iterative formula $$x _ { n + 1 } = \pi - \sin ^ { - 1 } \left( \frac { 1 } { \mathrm { e } ^ { - \frac { 1 } { 2 } x _ { n } } + 1 } \right)$$ with initial value \(x _ { 1 } = 2\), converges to one of these roots.
    Use the formula to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2020 November Q6
6
  1. Express \(\sqrt { 6 } \cos \theta + 3 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). State the exact value of \(R\) and give \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation \(\sqrt { 6 } \cos \frac { 1 } { 3 } x + 3 \sin \frac { 1 } { 3 } x = 2.5\), for \(0 ^ { \circ } < x < 360 ^ { \circ }\).
CAIE P3 2020 November Q7
7
  1. Verify that \(- 1 + \sqrt { 5 } \mathrm { i }\) is a root of the equation \(2 x ^ { 3 } + x ^ { 2 } + 6 x - 18 = 0\).
  2. Find the other roots of this equation.
CAIE P3 2020 November Q8
8 The coordinates \(( x , y )\) of a general point of a curve satisfy the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } = \left( 1 - 2 x ^ { 2 } \right) y$$ for \(x > 0\). It is given that \(y = 1\) when \(x = 1\).
Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).
CAIE P3 2020 November Q9
9 Let \(\mathrm { f } ( x ) = \frac { 8 + 5 x + 12 x ^ { 2 } } { ( 1 - x ) ( 2 + 3 x ) ^ { 2 } }\).
  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 \(x ^ { 2 }\).