Questions — CAIE P3 (1070 questions)

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CAIE P3 2009 June Q6
6 The parametric equations of a curve are $$x = a \cos ^ { 3 } t , \quad y = a \sin ^ { 3 } t$$ where \(a\) is a positive constant and \(0 < t < \frac { 1 } { 2 } \pi\).
  1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
  2. Show that the equation of the tangent to the curve at the point with parameter \(t\) is $$x \sin t + y \cos t = a \sin t \cos t$$
  3. Hence show that, if this tangent meets the \(x\)-axis at \(X\) and the \(y\)-axis at \(Y\), then the length of \(X Y\) is always equal to \(a\).
CAIE P3 2009 June Q7
7
  1. Solve the equation \(z ^ { 2 } + ( 2 \sqrt { } 3 ) \mathrm { i } z - 4 = 0\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. Sketch an Argand diagram showing the points representing the roots.
  3. Find the modulus and argument of each root.
  4. Show that the origin and the points representing the roots are the vertices of an equilateral triangle.
CAIE P3 2009 June Q8
8
  1. Express \(\frac { 100 } { x ^ { 2 } ( 10 - x ) }\) in partial fractions.
  2. Given that \(x = 1\) when \(t = 0\), solve the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 1 } { 100 } x ^ { 2 } ( 10 - x )$$ obtaining an expression for \(t\) in terms of \(x\).
CAIE P3 2009 June Q9
9 The line \(l\) has equation \(\mathbf { r } = 4 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } + t ( 2 \mathbf { i } - \mathbf { j } - 2 \mathbf { k } )\). It is given that \(l\) lies in the plane with equation \(2 x + b y + c z = 1\), where \(b\) and \(c\) are constants.
  1. Find the values of \(b\) and \(c\).
  2. The point \(P\) has position vector \(2 \mathbf { j } + 4 \mathbf { k }\). Show that the perpendicular distance from \(P\) to \(l\) is \(\sqrt { } 5\).
CAIE P3 2009 June Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{0f73e750-18a0-49ad-b4cb-fd6d14f0789e-4_424_713_262_715} The diagram shows the curve \(y = x ^ { 2 } \sqrt { } \left( 1 - x ^ { 2 } \right)\) for \(x \geqslant 0\) and its maximum point \(M\).
  1. Find the exact value of the \(x\)-coordinate of \(M\).
  2. Show, by means of the substitution \(x = \sin \theta\), that the area \(A\) of the shaded region between the curve and the \(x\)-axis is given by $$A = \frac { 1 } { 4 } \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { 2 } 2 \theta \mathrm {~d} \theta$$
  3. Hence obtain the exact value of \(A\).
CAIE P3 2010 June Q1
1 Solve the inequality \(| x + 3 a | > 2 | x - 2 a |\), where \(a\) is a positive constant.
CAIE P3 2010 June Q2
2 Solve the equation $$\sin \theta = 2 \cos 2 \theta + 1$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P3 2010 June Q3
3 The variables \(x\) and \(y\) satisfy the equation \(x ^ { n } y = C\), where \(n\) and \(C\) are constants. When \(x = 1.10\), \(y = 5.20\), and when \(x = 3.20 , y = 1.05\).
  1. Find the values of \(n\) and \(C\).
  2. Explain why the graph of \(\ln y\) against \(\ln x\) is a straight line.
CAIE P3 2010 June Q4
4
  1. Using the expansions of \(\cos ( 3 x - x )\) and \(\cos ( 3 x + x )\), prove that $$\frac { 1 } { 2 } ( \cos 2 x - \cos 4 x ) \equiv \sin 3 x \sin x$$
  2. Hence show that $$\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \sin 3 x \sin x \mathrm {~d} x = \frac { 1 } { 8 } \sqrt { } 3$$
CAIE P3 2010 June Q5
5 Given that \(y = 0\) when \(x = 1\), solve the differential equation $$x y \frac { \mathrm {~d} y } { \mathrm {~d} x } = y ^ { 2 } + 4 ,$$ obtaining an expression for \(y ^ { 2 }\) in terms of \(x\).
CAIE P3 2010 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{a74e4ddf-d254-45f3-bd9a-adf7cd53b3a6-3_380_641_258_751} The diagram shows a semicircle \(A C B\) with centre \(O\) and radius \(r\). The angle \(B O C\) is \(x\) radians. The area of the shaded segment is a quarter of the area of the semicircle.
  1. Show that \(x\) satisfies the equation $$x = \frac { 3 } { 4 } \pi - \sin x$$
  2. This equation has one root. Verify by calculation that the root lies between 1.3 and 1.5.
  3. Use the iterative formula $$x _ { n + 1 } = \frac { 3 } { 4 } \pi - \sin x _ { n }$$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2010 June Q7
7 The complex number \(2 + 2 \mathrm { i }\) is denoted by \(u\).
  1. Find the modulus and argument of \(u\).
  2. Sketch an Argand diagram showing the points representing the complex numbers 1, i and \(u\). Shade the region whose points represent the complex numbers \(z\) which satisfy both the inequalities \(| z - 1 | \leqslant | z - \mathrm { i } |\) and \(| z - u | \leqslant 1\).
  3. Using your diagram, calculate the value of \(| z |\) for the point in this region for which \(\arg z\) is least.
CAIE P3 2010 June Q8
8
  1. Express \(\frac { 2 } { ( x + 1 ) ( x + 3 ) }\) in partial fractions.
  2. Using your answer to part (i), show that $$\left( \frac { 2 } { ( x + 1 ) ( x + 3 ) } \right) ^ { 2 } \equiv \frac { 1 } { ( x + 1 ) ^ { 2 } } - \frac { 1 } { x + 1 } + \frac { 1 } { x + 3 } + \frac { 1 } { ( x + 3 ) ^ { 2 } }$$
  3. Hence show that \(\int _ { 0 } ^ { 1 } \frac { 4 } { ( x + 1 ) ^ { 2 } ( x + 3 ) ^ { 2 } } \mathrm {~d} x = \frac { 7 } { 12 } - \ln \frac { 3 } { 2 }\).
CAIE P3 2010 June Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{a74e4ddf-d254-45f3-bd9a-adf7cd53b3a6-4_611_895_255_625} The diagram shows the curve \(y = \sqrt { } \left( \frac { 1 - x } { 1 + x } \right)\).
  1. By first differentiating \(\frac { 1 - x } { 1 + x }\), obtain an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\). Hence show that the gradient of the normal to the curve at the point \(( x , y )\) is \(( 1 + x ) \sqrt { } \left( 1 - x ^ { 2 } \right)\).
  2. The gradient of the normal to the curve has its maximum value at the point \(P\) shown in the diagram. Find, by differentiation, the \(x\)-coordinate of \(P\).
CAIE P3 2010 June Q10
10 The lines \(l\) and \(m\) have vector equations $$\mathbf { r } = \mathbf { i } + \mathbf { j } + \mathbf { k } + s ( \mathbf { i } - \mathbf { j } + 2 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 4 \mathbf { i } + 6 \mathbf { j } + \mathbf { k } + t ( 2 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )$$ respectively.
  1. Show that \(l\) and \(m\) intersect.
  2. Calculate the acute angle between the lines.
  3. Find the equation of the plane containing \(l\) and \(m\), giving your answer in the form \(a x + b y + c z = d\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.
    University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE P3 2010 June Q1
1 Solve the equation $$\frac { 2 ^ { x } + 1 } { 2 ^ { x } - 1 } = 5$$ giving your answer correct to 3 significant figures.
CAIE P3 2010 June Q2
2 Show that \(\int _ { 0 } ^ { \pi } x ^ { 2 } \sin x \mathrm {~d} x = \pi ^ { 2 } - 4\).
CAIE P3 2010 June Q3
3 It is given that \(\cos a = \frac { 3 } { 5 }\), where \(0 ^ { \circ } < a < 90 ^ { \circ }\). Showing your working and without using a calculator to evaluate \(a\),
  1. find the exact value of \(\sin \left( a - 30 ^ { \circ } \right)\),
  2. find the exact value of \(\tan 2 a\), and hence find the exact value of \(\tan 3 a\).
CAIE P3 2010 June Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{20de5ba6-9426-4431-99af-6e8e62607f3e-2_513_895_1055_625} The diagram shows the curve \(y = \frac { \sin x } { x }\) for \(0 < x \leqslant 2 \pi\), and its minimum point \(M\).
  1. Show that the \(x\)-coordinate of \(M\) satisfies the equation $$x = \tan x$$
  2. The iterative formula $$x _ { n + 1 } = \tan ^ { - 1 } \left( x _ { n } \right) + \pi$$ can be used to determine the \(x\)-coordinate of \(M\). Use this formula to determine the \(x\)-coordinate of \(M\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2010 June Q5
5 The polynomial \(2 x ^ { 3 } + 5 x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( 2 x + 1 )\) is a factor of \(\mathrm { p } ( x )\) and that when \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\) the remainder is 9 .
  1. Find the values of \(a\) and \(b\).
  2. When \(a\) and \(b\) have these values, factorise \(\mathrm { p } ( x )\) completely.
CAIE P3 2010 June Q6
6 The equation of a curve is $$x \ln y = 2 x + 1$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { y } { x ^ { 2 } }\).
  2. Find the equation of the tangent to the curve at the point where \(y = 1\), giving your answer in the form \(a x + b y + c = 0\).
CAIE P3 2010 June Q7
7 The variables \(x\) and \(t\) are related by the differential equation $$\mathrm { e } ^ { 2 t } \frac { \mathrm {~d} x } { \mathrm {~d} t } = \cos ^ { 2 } x$$ where \(t \geqslant 0\). When \(t = 0 , x = 0\).
  1. Solve the differential equation, obtaining an expression for \(x\) in terms of \(t\).
  2. State what happens to the value of \(x\) when \(t\) becomes very large.
  3. Explain why \(x\) increases as \(t\) increases.
CAIE P3 2010 June Q8
8 The variable complex number \(z\) is given by $$z = 1 + \cos 2 \theta + i \sin 2 \theta$$ where \(\theta\) takes all values in the interval \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\).
  1. Show that the modulus of \(z\) is \(2 \cos \theta\) and the argument of \(z\) is \(\theta\).
  2. Prove that the real part of \(\frac { 1 } { z }\) is constant.
CAIE P3 2010 June Q9
9 The plane \(p\) has equation \(3 x + 2 y + 4 z = 13\). A second plane \(q\) is perpendicular to \(p\) and has equation \(a x + y + z = 4\), where \(a\) is a constant.
  1. Find the value of \(a\).
  2. The line with equation \(\mathbf { r } = \mathbf { j } - \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } )\) meets the plane \(p\) at the point \(A\) and the plane \(q\) at the point \(B\). Find the length of \(A B\).
CAIE P3 2010 June Q10
10
  1. Find the values of the constants \(A , B , C\) and \(D\) such that $$\frac { 2 x ^ { 3 } - 1 } { x ^ { 2 } ( 2 x - 1 ) } \equiv A + \frac { B } { x } + \frac { C } { x ^ { 2 } } + \frac { D } { 2 x - 1 }$$
  2. Hence show that $$\int _ { 1 } ^ { 2 } \frac { 2 x ^ { 3 } - 1 } { x ^ { 2 } ( 2 x - 1 ) } \mathrm { d } x = \frac { 3 } { 2 } + \frac { 1 } { 2 } \ln \left( \frac { 16 } { 27 } \right)$$