Questions — CAIE P3 (1070 questions)

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CAIE P3 2023 November Q4
4 Solve the quadratic equation \(( 3 + \mathrm { i } ) w ^ { 2 } - 2 w + 3 - \mathrm { i } = 0\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
CAIE P3 2023 November Q5
5 Find the exact coordinates of the stationary points of the curve \(y = \frac { \mathrm { e } ^ { 3 x ^ { 2 } - 1 } } { 1 - x ^ { 2 } }\).
CAIE P3 2023 November Q6
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
5 marks
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
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
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
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
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
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
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
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
5 marks
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
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
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
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
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.
  4. 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}
  5. 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
1 Expand \(( 9 - 3 x ) ^ { \frac { 1 } { 2 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
CAIE P3 2024 November Q2
2
  1. By sketching a suitable pair of graphs, show that the equation \(\cot 2 x = \sec x\) has exactly one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
  2. Show that if a sequence of real values given by the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \tan ^ { - 1 } \left( \cos x _ { n } \right)$$ converges, then it converges to the root in part (a).
CAIE P3 2024 November Q3
3 The square roots of 6-8i can be expressed in the Cartesian form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact. By first forming a quartic equation in \(x\) or \(y\), find the square roots of \(6 - 8 \mathrm { i }\) in exact Cartesian form.
\includegraphics[max width=\textwidth, alt={}, center]{9da6c2ac-31aa-4063-88b9-e15e38bedd8a-05_2725_35_99_20}
CAIE P3 2024 November Q4
4 Solve the equation \(5 ^ { x } = 5 ^ { x + 2 } - 10\). Give your answer correct to 3 decimal places.
CAIE P3 2024 November Q5
5
  1. The complex number \(u\) is given by $$u = \frac { \left( \cos \frac { 1 } { 7 } \pi + i \sin \frac { 1 } { 7 } \pi \right) ^ { 4 } } { \cos \frac { 1 } { 7 } \pi - i \sin \frac { 1 } { 7 } \pi }$$ Find the exact value of \(\arg u\).
  2. The complex numbers \(u\) and \(u ^ { * }\) are plotted on an Argand diagram. Describe the single geometrical transformation that maps \(u\) onto \(u ^ { * }\) and state the exact value of \(\arg u ^ { * }\).
    \includegraphics[max width=\textwidth, alt={}, center]{9da6c2ac-31aa-4063-88b9-e15e38bedd8a-06_2715_35_110_2012}
    \includegraphics[max width=\textwidth, alt={}, center]{9da6c2ac-31aa-4063-88b9-e15e38bedd8a-07_588_869_255_603} The variables \(x\) and \(y\) satisfy the equation \(a y = b ^ { x }\), where \(a\) and \(b\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points ( \(0.50,2.24\) ) and ( \(3.40,8.27\) ), as shown in the diagram. Find the values of \(a\) and \(b\). Give each value correct to 1 significant figure.
CAIE P3 2024 November Q7
7
  1. Show that the equation \(\tan ^ { 3 } x + 2 \tan 2 x - \tan x = 0\) may be expressed as $$\tan ^ { 4 } x - 2 \tan ^ { 2 } x - 3 = 0$$ for \(\tan x \neq 0\).
    \includegraphics[max width=\textwidth, alt={}, center]{9da6c2ac-31aa-4063-88b9-e15e38bedd8a-09_2725_35_99_20}
  2. Hence solve the equation \(\tan ^ { 3 } 2 \theta + 2 \tan 4 \theta - \tan 2 \theta = 0\) for \(0 < \theta < \pi\). Give your answers in exact form.
CAIE P3 2024 November Q8
8 The parametric equations of a curve are $$x = \tan ^ { 2 } 2 t , \quad y = \cos 2 t$$ for \(0 < t < \frac { 1 } { 4 } \pi\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 1 } { 2 } \cos ^ { 3 } 2 t\).
    \includegraphics[max width=\textwidth, alt={}, center]{9da6c2ac-31aa-4063-88b9-e15e38bedd8a-10_2716_38_109_2012}
    \includegraphics[max width=\textwidth, alt={}, center]{9da6c2ac-31aa-4063-88b9-e15e38bedd8a-11_2725_35_99_20}
  2. Hence find the equation of the normal to the curve at the point where \(t = \frac { 1 } { 8 } \pi\). Give your answer in the form \(y = m x + c\).
CAIE P3 2024 November Q9
9 With respect to the origin \(O\), the points \(A , B\) and \(C\) have position vectors given by $$\overrightarrow { O A } = \left( \begin{array} { r } 2
1
- 3 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { l } 0
4
1 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } - 3
- 2
2 \end{array} \right)$$
  1. The point \(D\) is such that \(A B C D\) is a trapezium with \(\overrightarrow { D C } = 3 \overrightarrow { A B }\). Find the position vector of \(D\).
  2. The diagonals of the trapezium intersect at the point \(P\). Find the position vector of \(P\).
    \includegraphics[max width=\textwidth, alt={}, center]{9da6c2ac-31aa-4063-88b9-e15e38bedd8a-13_2725_35_99_20}
  3. Using a scalar product, calculate angle \(A B C\).
CAIE P3 2024 November Q10
10 A balloon in the shape of a sphere has volume \(V\) and radius \(r\). Air is pumped into the balloon at a constant rate of \(40 \pi\) starting when time \(t = 0\) and \(r = 0\). At the same time, air begins to flow out of the balloon at a rate of \(0.8 \pi r\). The balloon remains a sphere at all times.
  1. Show that \(r\) and \(t\) satisfy the differential equation $$\frac { \mathrm { d } r } { \mathrm {~d} t } = \frac { 50 - r } { 5 r ^ { 2 } }$$
  2. Find the quotient and remainder when \(5 r ^ { 2 }\) is divided by \(50 - r\).
  3. Solve the differential equation in part (a), obtaining an expression for \(t\) in terms of \(r\).
  4. Find the value of \(t\) when the radius of the balloon is 12 .