Questions P3 (1203 questions)

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CAIE P3 2022 June Q4
4 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x y } { 1 + x ^ { 2 } }$$ and \(y = 2\) when \(x = 0\).
Solve the differential equation, obtaining a simplified expression for \(y\) in terms of \(x\).
CAIE P3 2022 June Q5
5 The polynomial \(a x ^ { 3 } - 10 x ^ { 2 } + b x + 8\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x - 2 )\) is a factor of both \(\mathrm { p } ( x )\) and \(\mathrm { p } ^ { \prime } ( x )\).
  1. Find the values of \(a\) and \(b\).
  2. When \(a\) and \(b\) have these values, factorise \(\mathrm { p } ( x )\) completely.
CAIE P3 2022 June Q6
6 Let \(I = \int _ { 0 } ^ { 3 } \frac { 27 } { \left( 9 + x ^ { 2 } \right) ^ { 2 } } \mathrm {~d} x\).
  1. Using the substitution \(x = 3 \tan \theta\), show that \(I = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos ^ { 2 } \theta \mathrm {~d} \theta\).
  2. Hence find the exact value of \(I\).
CAIE P3 2022 June Q7
7 The complex number \(u\) is defined by \(u = \frac { \sqrt { 2 } - a \sqrt { 2 } \mathrm { i } } { 1 + 2 \mathrm { i } }\), where \(a\) is a positive integer.
  1. Express \(u\) in terms of \(a\), in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
    It is now given that \(a = 3\).
  2. Express \(u\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\), giving the exact values of \(r\) and \(\theta\).
  3. Using your answer to part (b), find the two square roots of \(u\). Give your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\), giving the exact values of \(r\) and \(\theta\).
CAIE P3 2022 June Q8
8 The equation of a curve is \(x ^ { 3 } + y ^ { 3 } + 2 x y + 8 = 0\).
  1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
    The tangent to the curve at the point where \(x = 0\) and the tangent at the point where \(y = 0\) intersect at the acute angle \(\alpha\).
  2. Find the exact value of \(\tan \alpha\).
CAIE P3 2022 June Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{c1fbc9ef-2dc6-43c3-bc58-179f683c9acf-16_696_1104_264_518} In the diagram, \(O A B C D E F G\) is a cuboid in which \(O A = 2\) units, \(O C = 4\) units and \(O G = 2\) units. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O G\) respectively. The point \(M\) is the midpoint of \(D F\). The point \(N\) on \(A B\) is such that \(A N = 3 N B\).
  1. Express the vectors \(\overrightarrow { O M }\) and \(\overrightarrow { M N }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
  2. Find a vector equation for the line through \(M\) and \(N\).
  3. Show that the length of the perpendicular from \(O\) to the line through \(M\) and \(N\) is \(\sqrt { \frac { 53 } { 6 } }\).
CAIE P3 2022 June Q10
2 marks
10
\includegraphics[max width=\textwidth, alt={}, center]{c1fbc9ef-2dc6-43c3-bc58-179f683c9acf-18_471_686_276_717} The curve \(y = x \sqrt { \sin x }\) has one stationary point in the interval \(0 < x < \pi\), where \(x = a\) (see diagram).
  1. Show that \(\tan a = - \frac { 1 } { 2 } a\).
  2. Verify by calculation that \(a\) lies between 2 and 2.5.
  3. Show that if a sequence of values in the interval \(0 < x < \pi\) given by the iterative formula \(x _ { n + 1 } = \pi - \tan ^ { - 1 } \left( \frac { 1 } { 2 } x _ { n } \right)\) converges, then it converges to \(a\), the root of the equation in part (a). [2]
  4. Use the iterative formula given in part (c) 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 2022 June Q1
1 Solve the equation \(\ln \left( \mathrm { e } ^ { 2 x } + 3 \right) = 2 x + \ln 3\), giving your answer correct to 3 decimal places.
CAIE P3 2022 June Q2
2 Solve the equation \(3 \cos 2 \theta = 3 \cos \theta + 2\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P3 2022 June Q3
3 The polynomial \(a x ^ { 3 } + x ^ { 2 } + b x + 3\) is denoted by \(\mathrm { p } ( x )\). It is given that \(\mathrm { p } ( x )\) is divisible by ( \(2 x - 1\) ) and that when \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\) the remainder is 5 . Find the values of \(a\) and \(b\).
CAIE P3 2022 June Q4
4 The equation of a curve is \(y = \cos ^ { 3 } x \sqrt { \sin x }\). It is given that the curve has one stationary point in the interval \(0 < x < \frac { 1 } { 2 } \pi\). Find the \(x\)-coordinate of this stationary point, giving your answer correct to 3 significant figures.
CAIE P3 2022 June Q5
5
  1. By sketching a suitable pair of graphs, show that the equation \(\ln x = 3 x - x ^ { 2 }\) has one real root.
  2. Verify by calculation that the root lies between 2 and 2.8.
  3. Use the iterative formula \(x _ { n + 1 } = \sqrt { 3 x _ { n } - \ln x _ { n } }\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2022 June Q6
6 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = x \mathrm { e } ^ { y - x } ,$$ and \(y = 0\) when \(x = 0\).
  1. Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).
  2. Find the value of \(y\) when \(x = 1\), giving your answer in the form \(a - \ln b\), where \(a\) and \(b\) are integers.
CAIE P3 2022 June Q7
7 The equation of a curve is \(x ^ { 3 } + 3 x ^ { 2 } y - y ^ { 3 } = 3\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 2 } + 2 x y } { y ^ { 2 } - x ^ { 2 } }\).
  2. Find the coordinates of the points on the curve where the tangent is parallel to the \(x\)-axis.
CAIE P3 2022 June Q8
8 Let \(\mathrm { f } ( x ) = \frac { x ^ { 2 } + 9 x } { ( 3 x - 1 ) \left( x ^ { 2 } + 3 \right) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence find \(\int _ { 1 } ^ { 3 } \mathrm { f } ( x ) \mathrm { d } x\), giving your answer in a simplified exact form.
CAIE P3 2022 June Q9
9 The lines \(l\) and \(m\) have vector equations $$\mathbf { r } = - \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k } + \lambda ( 2 \mathbf { i } - \mathbf { j } - \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 5 \mathbf { i } + 4 \mathbf { j } + 3 \mathbf { k } + \mu ( a \mathbf { i } + b \mathbf { j } + \mathbf { k } )$$ respectively, where \(a\) and \(b\) are constants.
  1. Given that \(l\) and \(m\) intersect, show that \(2 b - a = 4\).
  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 2022 June Q10
10 The complex number \(- 1 + \sqrt { 7 } \mathrm { i }\) is denoted by \(u\). It is given that \(u\) is a root of the equation $$2 x ^ { 3 } + 3 x ^ { 2 } + 14 x + k = 0$$ where \(k\) is a real constant.
  1. Find the value of \(k\).
  2. Find the other two roots of the equation.
  3. On an Argand diagram, sketch the locus of points representing complex numbers \(z\) satisfying the equation \(| z - u | = 2\).
  4. Determine the greatest value of \(\arg z\) for points on this locus, giving your answer in radians.
    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 2022 June Q1
1 Find, in terms of \(a\), the set of values of \(x\) satisfying the inequality $$2 | 3 x + a | < | 2 x + 3 a |$$ where \(a\) is a positive constant.
CAIE P3 2022 June Q2
2 Solve the equation \(\cos \left( \theta - 60 ^ { \circ } \right) = 3 \sin \theta\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P3 2022 June Q3
3
  1. Show that the equation \(\log _ { 3 } ( 2 x + 1 ) = 1 + 2 \log _ { 3 } ( x - 1 )\) can be written as a quadratic equation in \(x\).
  2. Hence solve the equation \(\log _ { 3 } ( 4 y + 1 ) = 1 + 2 \log _ { 3 } ( 2 y - 1 )\), giving your answer correct to 2 decimal places.
CAIE P3 2022 June Q4
4 The curve \(y = \mathrm { e } ^ { - 4 x } \tan x\) has two stationary points in the interval \(0 \leqslant x < \frac { 1 } { 2 } \pi\).
  1. Obtain an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show it can be written in the form \(\sec ^ { 2 } x ( a + b \sin 2 x ) \mathrm { e } ^ { - 4 x }\), where \(a\) and \(b\) are constants.
  2. Hence find the exact \(x\)-coordinates of the two stationary points.
CAIE P3 2022 June Q5
5 The complex number \(3 - \mathrm { i }\) is denoted by \(u\).
  1. Show, on an Argand diagram with origin \(O\), the points \(A , B\) and \(C\) representing the complex numbers \(u , u ^ { * }\) and \(u ^ { * } - u\) respectively. State the type of quadrilateral formed by the points \(O , A , B\) and \(C\).
  2. Express \(\frac { u ^ { * } } { u }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  3. By considering the argument of \(\frac { u ^ { * } } { u }\), or otherwise, prove that \(\tan ^ { - 1 } \left( \frac { 3 } { 4 } \right) = 2 \tan ^ { - 1 } \left( \frac { 1 } { 3 } \right)\).
CAIE P3 2022 June Q6
6 The parametric equations of a curve are \(x = \frac { 1 } { \cos t } , y = \ln \tan t\), where \(0 < t < \frac { 1 } { 2 } \pi\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \cos t } { \sin ^ { 2 } t }\).
  2. Find the equation of the tangent to the curve at the point where \(y = 0\).
CAIE P3 2022 June Q7
7 Let \(\mathrm { f } ( x ) = \frac { 5 x ^ { 2 } + 8 x - 3 } { ( x - 2 ) \left( 2 x ^ { 2 } + 3 \right) }\).
  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 \(x ^ { 2 }\).
CAIE P3 2022 June Q8
8 At time \(t\) days after the start of observations, the number of insects in a population is \(N\). The variation in the number of insects is modelled by a differential equation of the form \(\frac { \mathrm { d } N } { \mathrm {~d} t } = k N ^ { \frac { 3 } { 2 } } \cos 0.02 t\), where \(k\) is a constant and \(N\) is a continuous variable. It is given that when \(t = 0 , N = 100\).
  1. Solve the differential equation, obtaining a relation between \(N , k\) and \(t\).
  2. Given also that \(N = 625\) when \(t = 50\), find the value of \(k\).
  3. Obtain an expression for \(N\) in terms of \(t\), and find the greatest value of \(N\) predicted by this model.