Questions — CAIE (7646 questions)

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CAIE P3 2023 November Q6
6 marks Standard +0.8
6
  1. Show that the equation \(\cot ^ { 2 } \theta + 2 \cos 2 \theta = 4\) can be written in the form $$4 \sin ^ { 4 } \theta + 3 \sin ^ { 2 } \theta - 1 = 0$$
  2. Hence solve the equation \(\cot ^ { 2 } \theta + 2 \cos 2 \theta = 4\), for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
CAIE P3 2023 November Q7
8 marks Standard +0.3
7 The equation of a curve is \(x ^ { 3 } + y ^ { 2 } + 3 x ^ { 2 } + 3 y = 4\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 3 x ^ { 2 } + 6 x } { 2 y + 3 }\).
  2. Hence find the coordinates of the points on the curve at which the tangent is parallel to the \(x\)-axis. [5]
CAIE P3 2023 November Q8
7 marks Standard +0.3
8 The variables \(x\) and \(y\) satisfy the differential equation $$\mathrm { e } ^ { 4 x } \frac { \mathrm {~d} y } { \mathrm {~d} x } = \cos ^ { 2 } 3 y .$$ It is given that \(y = 0\) when \(x = 2\).
Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).
CAIE P3 2023 November Q9
11 marks Standard +0.8
9 Let \(\mathrm { f } ( x ) = \frac { 17 x ^ { 2 } - 7 x + 16 } { \left( 2 + 3 x ^ { 2 } \right) ( 2 - x ) }\).
  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 \(x ^ { 3 }\).
  3. State the set of values of \(x\) for which the expansion in (b) is valid. Give your answer in an exact form.
CAIE P3 2023 November Q10
9 marks Standard +0.8
10 \includegraphics[max width=\textwidth, alt={}, center]{a49b720b-f8d2-42ff-b147-5d676993aa4c-16_611_689_274_721} The diagram shows the curve \(y = x \cos 2 x\), for \(x \geqslant 0\).
  1. Find the equation of the tangent to the curve at the point where \(x = \frac { 1 } { 2 } \pi\).
  2. Find the exact area of the shaded region shown in the diagram, bounded by the curve and the \(x\)-axis.
CAIE P3 2023 November Q11
9 marks Standard +0.3
11 The line \(l\) has equation \(\mathbf { r } = \mathbf { i } - 2 \mathbf { j } - 3 \mathbf { k } + \lambda ( - \mathbf { i } + \mathbf { j } + 2 \mathbf { k } )\). The points \(A\) and \(B\) have position vectors \(- 2 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }\) and \(3 \mathbf { i } - \mathbf { j } + \mathbf { k }\) respectively.
  1. Find a unit vector in the direction of \(l\).
    The line \(m\) passes through the points \(A\) and \(B\).
  2. Find a vector equation for \(m\).
  3. Determine whether lines \(l\) and \(m\) are parallel, intersect or are skew.
    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 November Q1
5 marks Standard +0.3
1 The polynomial \(4 x ^ { 3 } + a x ^ { 2 } + 5 x + b\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( 2 x + 1 )\) is a factor of \(\mathrm { p } ( x )\). When \(\mathrm { p } ( x )\) is divided by \(( x - 4 )\) the remainder is equal to 3 times the remainder when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\). Find the values of \(a\) and \(b\).
CAIE P3 2024 November Q2
5 marks Moderate -0.3
2 Find the exact value of \(\int _ { 1 } ^ { 3 } x ^ { 2 } \ln 3 x \mathrm {~d} x\). Give your answer in the form \(a \ln b + c\), where \(a\) and \(c\) are rational and \(b\) is an integer. \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-04_2720_38_105_2010}
CAIE P3 2024 November Q3
4 marks Standard +0.3
3 The equation of a curve is \(\ln ( x + y ) = 3 x ^ { 2 } y\).
Find the gradient of the curve at the point \(( 1,0 )\).
CAIE P3 2024 November Q4
8 marks Standard +0.3
4
  1. Show that \(\sec ^ { 4 } \theta - \tan ^ { 4 } \theta \equiv 1 + 2 \tan ^ { 2 } \theta\). \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-07_2723_35_101_20}
  2. Hence or otherwise solve the equation \(\sec ^ { 4 } 2 \alpha - \tan ^ { 4 } 2 \alpha = 2 \tan ^ { 2 } 2 \alpha \sec ^ { 2 } 2 \alpha\) for \(0 ^ { \circ } < \alpha < 180 ^ { \circ }\). [5]
CAIE P3 2024 November Q5
7 marks Moderate -0.3
5
  1. By sketching a suitable pair of graphs, show that the equation \(2 + \mathrm { e } ^ { - 0.2 x } = \ln ( 1 + x )\) has only one root.
  2. Show by calculation that this root lies between 7 and 9 . \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-08_2716_40_109_2009}
  3. Use the iterative formula $$x _ { n + 1 } = \exp \left( 2 + \mathrm { e } ^ { - 0.2 x _ { n } } \right) - 1$$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places. \(\left[ \exp ( x ) \right.\) is an alternative notation for \(\left. \mathrm { e } ^ { x } .\right]\) \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-10_481_789_262_639} The diagram shows the curve \(y = \sin 2 x ( 1 + \sin 2 x )\), for \(0 \leqslant x \leqslant \frac { 3 } { 4 } \pi\), and its minimum point \(M\). The shaded region bounded by the curve that lies above the \(x\)-axis and the \(x\)-axis itself is denoted by \(R\).
CAIE P3 2024 November Q7
9 marks Standard +0.8
7
Let \(f ( x ) = \frac { 5 x ^ { 2 } + 8 x + 5 } { ( 1 + 2 x ) \left( 2 + x ^ { 2 } \right) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions. \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-13_2726_34_97_21}
  2. Hence find the coefficient of \(x ^ { 3 }\) in the expansion of \(\mathrm { f } ( x )\).
CAIE P3 2024 November Q8
9 marks Standard +0.3
8
  1. Given that \(z = 1 + y \mathrm { i }\) and that \(y\) is a real number, express \(\frac { 1 } { z }\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are functions of \(y\).
  2. Show that \(\left( a - \frac { 1 } { 2 } \right) ^ { 2 } + b ^ { 2 } = \frac { 1 } { 4 }\), where \(a\) and \(b\) are the functions of \(y\) found in part (a). \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-14_2716_35_108_2012}
  3. On a single Argand diagram, sketch the loci given by the equations \(\operatorname { Re } ( z ) = 1\) and \(\left| z - \frac { 1 } { 2 } \right| = \frac { 1 } { 2 }\), where \(z\) is a complex number.
  4. The complex number \(z\) is such that \(\operatorname { Re } ( z ) = 1\). Use your answer to part (b) to give a geometrical description of the locus of \(\frac { 1 } { z }\).
CAIE P3 2024 November Q9
10 marks Moderate -0.3
9 The position vector of point \(A\) relative to the origin \(O\) is \(\overrightarrow { O A } = 8 \mathbf { i } - 5 \mathbf { j } + 6 \mathbf { k }\).
The line \(l\) passes through \(A\) and is parallel to the vector \(2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k }\).
  1. State a vector equation for \(l\).
  2. The position vector of point \(B\) relative to the origin \(O\) is \(\overrightarrow { O B } = - t \mathbf { i } + 4 t \mathbf { j } + 3 t \mathbf { k }\), where \(t\) is a constant. The line \(l\) also passes through \(B\). Find the value of \(t\).
  3. The line \(m\) has vector equation \(\mathbf { r } = 5 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } + \mu ( a \mathbf { i } - \mathbf { j } + 3 \mathbf { k } )\). The acute angle between the directions of \(l\) and \(m\) is \(\theta\), where \(\cos \theta = \frac { 1 } { \sqrt { 6 } }\).
    Find the possible values of \(a\). \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-18_542_559_251_753} A large cylindrical tank is used to store water. The base of the tank is a circle of radius 4 metres. At time \(t\) minutes, the depth of the water in the tank is \(h\) metres. There is a tap at the bottom of the tank. When the tap is open, water flows out of the tank at a rate proportional to the square root of the volume of water in the tank.
    1. Show that \(\frac { \mathrm { d } h } { \mathrm {~d} t } = - \lambda \sqrt { h }\), where \(\lambda\) is a positive constant. \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-18_2718_42_107_2007}
    2. At time \(t = 0\) the tap is opened. It is given that \(h = 4\) when \(t = 0\) and that \(h = 2.25\) when \(t = 20\). Solve the differential equation to obtain an expression for \(t\) in terms of \(h\), and hence find the time taken to empty the tank.
      If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE P3 2024 November Q1
4 marks Moderate -0.3
1 The complex number \(z\) satisfies \(| z | = 2\) and \(0 \leqslant \arg z \leqslant \frac { 1 } { 4 } \pi\).
  1. On the Argand diagram below, sketch the locus of the points representing \(z\).
  2. On the same diagram, sketch the locus of the points representing \(z ^ { 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-02_1074_1363_628_351} \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-02_1002_26_1820_2017}
CAIE P3 2024 November Q2
5 marks Standard +0.3
2 Let \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 5 x ^ { 2 } + 4\).
  1. Show that if a sequence of values given by the iterative formula $$x _ { n + 1 } = \sqrt { \frac { 4 } { 5 - 2 x _ { n } } }$$ converges, then it converges to a root of the equation \(\mathrm { f } ( x ) = 0\).
  2. The equation has a root close to 1.2 . Use the iterative formula from part (a) and an initial value of 1.2 to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
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.