Questions — CAIE P3 (1110 questions)

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CAIE P3 2024 November Q3
5 marks Moderate -0.5
3 \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-04_527_634_255_717} The number of bacteria in a population, \(P\), at time \(t\) hours is modelled by the equation \(P = a \mathrm { e } ^ { k t }\), where \(a\) and \(k\) are constants. The graph of \(\ln P\) against \(t\), shown in the diagram, has gradient \(\frac { 1 } { 20 }\) and intersects the vertical axis at \(( 0,3 )\).
  1. State the value of \(k\) and find the value of \(a\) correct to 2 significant figures.
  2. Find the time taken for \(P\) to double. Give your answer correct to the nearest hour. \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-05_2723_33_99_22}
CAIE P3 2024 November Q4
5 marks Moderate -0.5
4 Find the complex number \(z\) satisfying the equation $$\frac { z - 3 \mathrm { i } } { z + 3 \mathrm { i } } = \frac { 2 - 9 \mathrm { i } } { 5 }$$ Give your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
CAIE P3 2024 November Q5
6 marks Standard +0.3
5
  1. Show that \(\cos ^ { 4 } \theta - \sin ^ { 4 } \theta - 4 \sin ^ { 2 } \theta \cos ^ { 2 } \theta \equiv \cos ^ { 2 } 2 \theta + \cos 2 \theta - 1\).
  2. Solve the equation \(\cos ^ { 4 } \alpha - \sin ^ { 4 } \alpha = 4 \sin ^ { 2 } \alpha \cos ^ { 2 } \alpha\) for \(0 ^ { \circ } \leqslant \alpha \leqslant 180 ^ { \circ }\).
CAIE P3 2024 November Q6
7 marks Standard +0.3
6 The lines \(l\) and \(m\) have vector equations $$l : \quad \mathbf { r } = 2 \mathbf { i } + \mathbf { j } - 3 \mathbf { k } + \lambda ( - \mathbf { i } + 2 \mathbf { k } ) \quad \text { and } \quad m : \quad \mathbf { r } = 2 \mathbf { i } + \mathbf { j } - 3 \mathbf { k } + \mu ( 2 \mathbf { i } - \mathbf { j } + 5 \mathbf { k } ) .$$ Lines \(l\) and \(m\) intersect at the point \(P\).
  1. State the coordinates of \(P\).
  2. Find the exact value of the cosine of the acute angle between \(l\) and \(m\).
  3. The point \(A\) on line \(I\) has coordinates ( \(0,1,1\) ). The point \(B\) on line \(m\) has coordinates ( \(0,2 , - 8\) ). Find the exact area of triangle \(A P B\).
CAIE P3 2024 November Q7
8 marks Standard +0.3
7 The parametric equations of a curve are $$x = 3 \sin 2 t , \quad y = \tan t + \cot t$$ for \(0 < t < \frac { 1 } { 2 } \pi\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { - 2 } { 3 \sin ^ { 2 } 2 t }\). \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-10_2716_40_109_2009} \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-11_2723_33_99_22}
  2. Find the equation of the normal to the curve at the point where \(t = \frac { 1 } { 4 } \pi\). Give your answer in the form \(p y + q x + r = 0\), where \(p , q\) and \(r\) are integers.
CAIE P3 2024 November Q8
8 marks Standard +0.3
8 Let \(\mathrm { f } ( x ) = \frac { 7 a ^ { 2 } } { ( a - 2 x ) ( 3 a + x ) }\), where \(a\) is a positive constant.
  1. Express \(\mathrm { f } ( x )\) in partial fractions. \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-12_2718_40_107_2009} \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-13_2726_33_97_22}
  2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\). [4]
  3. State the set of values of \(x\) for which the expansion in part (b) is valid.
CAIE P3 2024 November Q9
8 marks Standard +0.3
9
  1. Find the quotient and remainder when \(x ^ { 4 } + 16\) is divided by \(x ^ { 2 } + 4\).
  2. Hence show that \(\int _ { 2 } ^ { 2 \sqrt { 3 } } \frac { x ^ { 4 } + 16 } { x ^ { 2 } + 4 } \mathrm {~d} x = \frac { 4 } { 3 } ( \pi + 4 )\).
CAIE P3 2024 November Q10
8 marks Moderate -0.3
10 A water tank is in the shape of a cuboid with base area \(40000 \mathrm {~cm} ^ { 2 }\). At time \(t\) minutes the depth of water in the tank is \(h \mathrm {~cm}\). Water is pumped into the tank at a rate of \(50000 \mathrm {~cm} ^ { 3 }\) per minute. Water is leaking out of the tank through a hole in the bottom at a rate of \(600 \mathrm {~cm} ^ { 3 }\) per minute.
  1. Show that \(200 \frac { \mathrm {~d} h } { \mathrm {~d} t } = 250 - 3 h\).
    \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-17_2723_33_99_22}
  2. It is given that when \(t = 0 , h = 50\). Find the time taken for the depth of water in the tank to reach 80 cm . Give your answer correct to 2 significant figures.
CAIE P3 2024 November Q11
11 marks Standard +0.8
11 \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-18_565_634_260_717} The diagram shows the curve \(y = 2 \sin x \sqrt { 2 + \cos x }\), for \(0 \leqslant x \leqslant 2 \pi\), and its minimum point \(M\), where \(x = a\).
  1. Find the value of \(a\) correct to 2 decimal places. \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-18_2716_38_109_2012} \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-19_2726_33_97_22}
  2. Use the substitution \(u = 2 + \cos x\) to find the exact area of the shaded region \(R\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE P3 2012 June Q9
10 marks Standard +0.3
9 The lines \(l\) and \(m\) have equations \(\mathbf { r } = 3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } + \lambda ( - \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )\) and \(\mathbf { r } = 4 \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k } + \mu ( a \mathbf { i } + b \mathbf { j } - \mathbf { k } )\) respectively, where \(a\) and \(b\) are constants.
  1. Given that \(l\) and \(m\) intersect, show that $$2 a - b = 4 .$$
  2. Given also that \(l\) and \(m\) are perpendicular, find the values of \(a\) and \(b\).
  3. When \(a\) and \(b\) have these values, find the position vector of the point of intersection of \(l\) and \(m\).
CAIE P3 2021 November Q9
11 marks Standard +0.3
9 Two lines \(l\) and \(m\) have equations \(\mathbf { r } = 3 \mathbf { i } + 2 \mathbf { j } + 5 \mathbf { k } + s ( 4 \mathbf { i } - \mathbf { j } + 3 \mathbf { k } )\) and \(\mathbf { r } = \mathbf { i } - \mathbf { j } - 2 \mathbf { k } + t ( - \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } )\) respectively.
  1. Show that \(l\) and \(m\) are perpendicular.
  2. Show that \(l\) and \(m\) intersect and state the position vector of the point of intersection.
  3. Show that the length of the perpendicular from the origin to the line \(m\) is \(\frac { 1 } { 3 } \sqrt { 5 }\).
CAIE P3 2020 Specimen Q1
3 marks Moderate -0.5
1 Find the set of values of \(x\) for which \(3 \left( 2 ^ { 3 x + 1 } \right) < 8\). Give your answer in a simplified exact form.
CAIE P3 2020 Specimen Q2
4 marks Moderate -0.8
2
  1. Expand \(( 1 + 3 x ) ^ { - \frac { 1 } { 3 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
  2. State the set of values of \(x\) for which the expansion is valid.
CAIE P3 2020 Specimen Q3
4 marks Easy -1.3
3
  1. Sketch the graph of \(y = | 2 x - 3 |\).
  2. Solve the inequality \(3 x - 1 > | 2 x - 3 |\).
CAIE P3 2020 Specimen Q4
9 marks Standard +0.3
4 The parametric equations of a curve are $$x = \mathrm { e } ^ { 2 t - 3 } , \quad y = 4 \ln t$$ where \(t > 0\). When \(t = a\) the gradient of the curve is 2 .
  1. Show that \(a\) satisfies the equation \(a = \frac { 1 } { 2 } ( 3 - \ln a )\).
  2. Verify by calculation that this equation has a root between 1 and 2 .
  3. Use the iterative formula \(a _ { n + 1 } = \frac { 1 } { 2 } \left( 3 - \ln a _ { n } \right)\) to calculate \(a\) correct to 2 decimal places, showing the result of each iteration to 4 decimal places.
CAIE P3 2020 Specimen Q5
7 marks Standard +0.8
5
  1. Show that \(\frac { \mathrm { d } } { \mathrm { d } x } \left( x - \tan ^ { - 1 } x \right) = \frac { x ^ { 2 } } { 1 + x ^ { 2 } }\).
  2. Show that \(\int _ { 0 } ^ { \sqrt { 3 } } x \tan ^ { - 1 } x \mathrm {~d} x = \frac { 2 } { 3 } \pi - \frac { 1 } { 2 } \sqrt { 3 }\).
CAIE P3 2020 Specimen Q6
8 marks Moderate -0.5
6 The complex numbers \(1 + 3 \mathrm { i }\) and \(4 + 2 \mathrm { i }\) are denoted by \(u\) and \(v\) respectively.
  1. Find \(\frac { u } { v }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. State the argument of \(\frac { u } { v }\).
    In an Argand diagram, with origin \(O\), the points \(A , B\) and \(C\) represent the complex numbers \(u , v\) and \(u - v\) respectively.
  3. State fully the geometrical relationship between \(O C\) and \(B A\).
  4. Show that angle \(A O B = \frac { 1 } { 4 } \pi\) radians.
CAIE P3 2020 Specimen Q7
9 marks Standard +0.3
7
  1. By first expanding \(\cos \left( x + 45 ^ { \circ } \right)\), express \(\cos \left( x + 45 ^ { \circ } \right) - \sqrt { 2 } \sin x\) in the form \(R \cos ( x + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(R\) correct to 4 significant figures and the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation $$\cos \left( x + 45 ^ { \circ } \right) - \sqrt { 2 } \sin x = 2$$ for \(0 ^ { \circ } < x < 360 ^ { \circ }\).
CAIE P3 2020 Specimen Q8
10 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{c1eee696-3d7f-410a-91a8-fa902309c117-14_485_716_262_676} In the diagram, \(O A B C\) is a pyramid in which \(O A = 2\) units, \(O B = 4\) units and \(O C = 2\) units. The edge \(O C\) is vertical, the base \(O A B\) is horizontal and angle \(A O B = 90 ^ { \circ }\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A\), \(O B\) and \(O C\) respectively. The midpoints of \(A B\) and \(B C\) are \(M\) and \(N\) respectively.
  1. Express the vectors \(\overrightarrow { O N }\) and \(\overrightarrow { C M }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
  2. Calculate the angle between the directions of \(\overrightarrow { O N }\) and \(\overrightarrow { C M }\).
  3. Show that the length of the perpendicular from \(M\) to \(O N\) is \(\frac { 3 } { 5 } \sqrt { 5 }\).
CAIE P3 2020 Specimen Q9
10 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{c1eee696-3d7f-410a-91a8-fa902309c117-16_307_593_269_735} The diagram shows the curve \(y = \sin ^ { 2 } 2 x \cos x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\).
  1. Find the \(x\)-coordinate of \(M\).
  2. Using the substitution \(u = \sin x\), find the area of the shaded region bounded by the curve and the \(x\)-axis.
CAIE P3 2020 Specimen Q10
11 marks Standard +0.5
10 In a chemical reaction, a compound \(X\) is formed from two compounds \(Y\) and \(Z\).
The masses in grams of \(X , Y\) and \(Z\) present at time \(t\) seconds after the start of the reaction are \(x , 10 - x\) and \(20 - x\) respectively. At any time the rate of formation of \(X\) is proportional to the product of the masses of \(Y\) and \(Z\) present at the time. When \(t = 0 , x = 0\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 2\).
  1. Show that \(x\) and \(t\) satisfy the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = 0.01 ( 10 - x ) ( 20 - x ) .$$
  2. Solve this differential equation and obtain an expression for \(x\) in terms of \(t\).
  3. State what happens to the value of \(x\) when \(t\) becomes large.
CAIE P3 2020 June Q7
9 marks Challenging +1.2
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Using your answer to part (a), show that $$( f ( x ) ) ^ { 2 } = \frac { 1 } { ( 2 x - 1 ) ^ { 2 } } - \frac { 1 } { 2 x - 1 } + \frac { 1 } { 2 x + 1 } + \frac { 1 } { ( 2 x + 1 ) ^ { 2 } }$$
  3. Hence show that \(\int _ { 1 } ^ { 2 } ( \mathrm { f } ( x ) ) ^ { 2 } \mathrm {~d} x = \frac { 2 } { 5 } + \frac { 1 } { 2 } \ln \left( \frac { 5 } { 9 } \right)\).
CAIE P3 2021 June Q10
10 marks Standard +0.3
  1. Given that the sum of the areas of the shaded sectors is \(90 \%\) of the area of the trapezium, show that \(x\) satisfies the equation \(x = 0.9 ( 2 - \cos x ) \sin x\).
  2. Verify by calculation that \(x\) lies between 0.5 and 0.7 .
  3. Show that if a sequence of values in the interval \(0 < x < \frac { 1 } { 2 } \pi\) given by the iterative formula $$x _ { n + 1 } = \cos ^ { - 1 } \left( 2 - \frac { x _ { n } } { 0.9 \sin x _ { n } } \right)$$ converges, then it converges to the root of the equation in part (a).
  4. Use this iterative formula to determine \(x\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2023 June Q10
10 marks Standard +0.3
  1. Find the exact coordinates of \(M\).
  2. Using the substitution \(u = 3 - 2 x\), find by integration the area of the shaded region bounded by the curve and the \(x\)-axis. Give your answer in the form \(a \sqrt { 13 }\), where \(a\) is a rational number. [5]
CAIE P3 2021 November Q9
10 marks Standard +0.3
  1. Find the \(x\)-coordinate of the stationary point of the curve with equation \(y = \mathrm { f } ( x )\).
  2. Using the substitution \(u = \sqrt { x }\), show that \(\int _ { 0 } ^ { 4 } \mathrm { f } ( x ) \mathrm { d } x = \frac { 1 } { 3 } \ln 5\).