Questions P3 (1203 questions)

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CAIE P3 2016 June Q5
5 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { - 2 y } \tan ^ { 2 } x$$ for \(0 \leqslant x < \frac { 1 } { 2 } \pi\), and it is given that \(y = 0\) when \(x = 0\). Solve the differential equation and calculate the value of \(y\) when \(x = \frac { 1 } { 4 } \pi\).
CAIE P3 2016 June Q6
6 The curve with equation \(y = x ^ { 2 } \cos \frac { 1 } { 2 } x\) has a stationary point at \(x = p\) in the interval \(0 < x < \pi\).
  1. Show that \(p\) satisfies the equation \(\tan \frac { 1 } { 2 } p = \frac { 4 } { p }\).
  2. Verify by calculation that \(p\) lies between 2 and 2.5.
  3. Use the iterative formula \(p _ { n + 1 } = 2 \tan ^ { - 1 } \left( \frac { 4 } { p _ { n } } \right)\) to determine the value of \(p\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2016 June Q7
7 Let \(I = \int _ { 0 } ^ { 1 } \frac { x ^ { 5 } } { \left( 1 + x ^ { 2 } \right) ^ { 3 } } \mathrm {~d} x\).
  1. Using the substitution \(u = 1 + x ^ { 2 }\), show that \(I = \int _ { 1 } ^ { 2 } \frac { ( u - 1 ) ^ { 2 } } { 2 u ^ { 3 } } \mathrm {~d} u\).
  2. Hence find the exact value of \(I\).
CAIE P3 2016 June Q8
8 The points \(A\) and \(B\) have position vectors, relative to the origin \(O\), given by \(\overrightarrow { O A } = \mathbf { i } + \mathbf { j } + \mathbf { k }\) and \(\overrightarrow { O B } = 2 \mathbf { i } + 3 \mathbf { k }\). The line \(l\) has vector equation \(\mathbf { r } = 2 \mathbf { i } - 2 \mathbf { j } - \mathbf { k } + \mu ( - \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )\).
  1. Show that the line passing through \(A\) and \(B\) does not intersect \(l\).
  2. Show that the length of the perpendicular from \(A\) to \(l\) is \(\frac { 1 } { \sqrt { 2 } }\).
CAIE P3 2016 June Q10
10 Let \(\mathrm { f } ( x ) = \frac { 10 x - 2 x ^ { 2 } } { ( x + 3 ) ( x - 1 ) ^ { 2 } }\).
  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 \(x ^ { 2 }\).
CAIE P3 2017 June Q1
1 Solve the inequality \(| 2 x + 1 | < 3 | x - 2 |\).
CAIE P3 2017 June Q2
2 Expand \(\frac { 1 } { \sqrt [ 3 ] { } ( 1 + 6 x ) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.
CAIE P3 2017 June Q3
3 It is given that \(x = \ln ( 1 - y ) - \ln y\), where \(0 < y < 1\).
  1. Show that \(y = \frac { \mathrm { e } ^ { - x } } { 1 + \mathrm { e } ^ { - x } }\).
  2. Hence show that \(\int _ { 0 } ^ { 1 } y \mathrm {~d} x = \ln \left( \frac { 2 \mathrm { e } } { \mathrm { e } + 1 } \right)\).
CAIE P3 2017 June Q4
4 The parametric equations of a curve are $$x = \ln \cos \theta , \quad y = 3 \theta - \tan \theta ,$$ where \(0 \leqslant \theta < \frac { 1 } { 2 } \pi\).
  1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\tan \theta\).
  2. Find the exact \(y\)-coordinate of the point on the curve at which the gradient of the normal is equal to 1 .
    \includegraphics[max width=\textwidth, alt={}, center]{b00cefad-7c3c-4672-b309-f19aafab8b01-08_378_689_260_726} The diagram shows a semicircle with centre \(O\), radius \(r\) and diameter \(A B\). The point \(P\) on its circumference is such that the area of the minor segment on \(A P\) is equal to half the area of the minor segment on \(B P\). The angle \(A O P\) is \(x\) radians.
CAIE P3 2017 June Q6
6 The plane with equation \(2 x + 2 y - z = 5\) is denoted by \(m\). Relative to the origin \(O\), the points \(A\) and \(B\) have coordinates \(( 3,4,0 )\) and \(( - 1,0,2 )\) respectively.
  1. Show that the plane \(m\) bisects \(A B\) at right angles.
    A second plane \(p\) is parallel to \(m\) and nearer to \(O\). The perpendicular distance between the planes is 1 .
  2. Find the equation of \(p\), giving your answer in the form \(a x + b y + c z = d\).
CAIE P3 2017 June Q7
7 Throughout this question the use of a calculator is not permitted.
The complex numbers \(u\) and \(w\) are defined by \(u = - 1 + 7 \mathrm { i }\) and \(w = 3 + 4 \mathrm { i }\).
  1. Showing all your working, find in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real, the complex numbers \(u - 2 w\) and \(\frac { u } { w }\).
    In an Argand diagram with origin \(O\), the points \(A , B\) and \(C\) represent the complex numbers \(u , w\) and \(u - 2 w\) respectively.
  2. Prove that angle \(A O B = \frac { 1 } { 4 } \pi\).
  3. State fully the geometrical relation between the line segments \(O B\) and \(C A\).
CAIE P3 2017 June Q8
5 marks
8
  1. By first expanding \(2 \sin \left( x - 30 ^ { \circ } \right)\), express \(2 \sin \left( x - 30 ^ { \circ } \right) - \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places. [5]
  2. Hence solve the equation $$2 \sin \left( x - 30 ^ { \circ } \right) - \cos x = 1$$ for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
CAIE P3 2017 June Q9
9
  1. Express \(\frac { 1 } { x ( 2 x + 3 ) }\) in partial fractions.
  2. The variables \(x\) and \(y\) satisfy the differential equation $$x ( 2 x + 3 ) \frac { \mathrm { d } y } { \mathrm {~d} x } = y$$ and it is given that \(y = 1\) when \(x = 1\). Solve the differential equation and calculate the value of \(y\) when \(x = 9\), giving your answer correct to 3 significant figures.
CAIE P3 2017 June Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{b00cefad-7c3c-4672-b309-f19aafab8b01-18_324_677_259_734} The diagram shows the curve \(y = \sin x \cos ^ { 2 } 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\) and its maximum point \(M\).
  1. Using the substitution \(u = \cos x\), find by integration the exact area of the shaded region bounded by the curve and the \(x\)-axis.
  2. Find the \(x\)-coordinate of \(M\). Give your answer correct to 2 decimal places.
CAIE P3 2017 June Q1
1 Solve the equation \(\ln \left( x ^ { 2 } + 1 \right) = 1 + 2 \ln x\), giving your answer correct to 3 significant figures.
CAIE P3 2017 June Q2
2 Solve the inequality \(| x - 3 | < 3 x - 4\).
CAIE P3 2017 June Q3
3
  1. Express the equation \(\cot \theta - 2 \tan \theta = \sin 2 \theta\) in the form \(a \cos ^ { 4 } \theta + b \cos ^ { 2 } \theta + c = 0\), where \(a\), \(b\) and \(c\) are constants to be determined.
  2. Hence solve the equation \(\cot \theta - 2 \tan \theta = \sin 2 \theta\) for \(90 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P3 2017 June Q4
4 The parametric equations of a curve are $$x = t ^ { 2 } + 1 , \quad y = 4 t + \ln ( 2 t - 1 )$$
  1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
  2. Find the equation of the normal to the curve at the point where \(t = 1\). Give your answer in the form \(a x + b y + c = 0\).
CAIE P3 2017 June Q5
5 In a certain chemical process a substance \(A\) reacts with and reduces a substance \(B\). The masses of \(A\) and \(B\) at time \(t\) after the start of the process are \(x\) and \(y\) respectively. It is given that \(\frac { \mathrm { d } y } { \mathrm {~d} t } = - 0.2 x y\) and \(x = \frac { 10 } { ( 1 + t ) ^ { 2 } }\). At the beginning of the process \(y = 100\).
  1. Form a differential equation in \(y\) and \(t\), and solve this differential equation.
  2. Find the exact value approached by the mass of \(B\) as \(t\) becomes large. State what happens to the mass of \(A\) as \(t\) becomes large.
CAIE P3 2017 June Q7
7
  1. Prove that if \(y = \frac { 1 } { \cos \theta }\) then \(\frac { \mathrm { d } y } { \mathrm {~d} \theta } = \sec \theta \tan \theta\).
  2. Prove the identity \(\frac { 1 + \sin \theta } { 1 - \sin \theta } \equiv 2 \sec ^ { 2 } \theta + 2 \sec \theta \tan \theta - 1\).
  3. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { 1 + \sin \theta } { 1 - \sin \theta } \mathrm { d } \theta\).
    \(8 \quad\) Let \(\mathrm { f } ( x ) = \frac { 5 x ^ { 2 } - 7 x + 4 } { ( 3 x + 2 ) \left( x ^ { 2 } + 5 \right) }\).
  4. Express \(\mathrm { f } ( x )\) in partial fractions.
  5. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
CAIE P3 2017 June Q9
9 Relative to the origin \(O\), the point \(A\) has position vector given by \(\overrightarrow { O A } = \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k }\). The line \(l\) has equation \(\mathbf { r } = 9 \mathbf { i } - \mathbf { j } + 8 \mathbf { k } + \mu ( 3 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } )\).
  1. Find the position vector of the foot of the perpendicular from \(A\) to \(l\). Hence find the position vector of the reflection of \(A\) in \(l\).
  2. Find the equation of the plane through the origin which contains \(l\). Give your answer in the form \(a x + b y + c z = d\).
  3. Find the exact value of the perpendicular distance of \(A\) from this plane.
CAIE P3 2017 June Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{83a6d80b-dc74-4936-ac32-858a517a843c-18_353_675_260_735} The diagram shows the curve \(y = x ^ { 2 } \cos 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\). The curve has a maximum point at \(M\) where \(x = p\).
  1. Show that \(p\) satisfies the equation \(p = \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { p } \right)\).
  2. Use the iterative formula \(p _ { n + 1 } = \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { p _ { n } } \right)\) to determine the value of \(p\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
  3. Find, showing all necessary working, the exact area of the region bounded by the curve and the \(x\)-axis.
CAIE P3 2017 June Q1
1 Prove the identity \(\frac { \cot x - \tan x } { \cot x + \tan x } \equiv \cos 2 x\).
CAIE P3 2017 June Q2
2 Expand \(( 3 + 2 x ) ^ { - 3 }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
CAIE P3 2017 June Q3
3 Using the substitution \(u = \mathrm { e } ^ { x }\), solve the equation \(4 \mathrm { e } ^ { - x } = 3 \mathrm { e } ^ { x } + 4\). Give your answer correct to 3 significant figures.