Questions P3 (1243 questions)

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CAIE P3 2010 June Q5
6 marks Standard +0.3
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
8 marks Standard +0.3
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
9 marks Challenging +1.2
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
9 marks Standard +0.8
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 marks Standard +0.8
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
12 marks Standard +0.3
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 \(a x + b y + c z = d\).
CAIE P3 2010 June Q1
4 marks Moderate -0.8
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
5 marks Standard +0.3
2 Show that \(\int _ { 0 } ^ { \pi } x ^ { 2 } \sin x \mathrm {~d} x = \pi ^ { 2 } - 4\).
CAIE P3 2010 June Q3
7 marks Standard +0.3
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
7 marks Standard +0.3
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
8 marks Moderate -0.8
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
8 marks Moderate -0.3
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
8 marks Standard +0.3
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
9 marks Standard +0.8
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 marks Standard +0.3
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 marks Standard +0.3
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 $$\int _ { 1 } ^ { 2 } \frac { 2 x ^ { 3 } - 1 } { x ^ { 2 } ( 2 x - 1 ) } \mathrm { d } x = \frac { 3 } { 2 } + \frac { 1 } { 2 } \ln \left( \frac { 16 } { 27 } \right)$$
CAIE P3 2011 June Q1
4 marks Moderate -0.8
1 Expand \(\sqrt [ 3 ] { } ( 1 - 6 x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.
CAIE P3 2011 June Q2
4 marks Moderate -0.8
2 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in each of the following cases:
  1. \(y = \ln ( 1 + \sin 2 x )\),
  2. \(y = \frac { \tan x } { x }\).
CAIE P3 2011 June Q3
7 marks Moderate -0.3
3 Points \(A\) and \(B\) have coordinates \(( - 1,2,5 )\) and \(( 2 , - 2,11 )\) respectively. The plane \(p\) passes through \(B\) and is perpendicular to \(A B\).
  1. Find an equation of \(p\), giving your answer in the form \(a x + b y + c z = d\).
  2. Find the acute angle between \(p\) and the \(y\)-axis.
CAIE P3 2011 June Q4
7 marks Standard +0.3
4 The polynomial \(\mathrm { f } ( x )\) is defined by $$f ( x ) = 12 x ^ { 3 } + 25 x ^ { 2 } - 4 x - 12$$
  1. Show that \(\mathrm { f } ( - 2 ) = 0\) and factorise \(\mathrm { f } ( x )\) completely.
  2. Given that $$12 \times 27 ^ { y } + 25 \times 9 ^ { y } - 4 \times 3 ^ { y } - 12 = 0$$ state the value of \(3 ^ { y }\) and hence find \(y\) correct to 3 significant figures.
CAIE P3 2011 June Q5
7 marks Standard +0.3
5 The curve with equation $$6 \mathrm { e } ^ { 2 x } + k \mathrm { e } ^ { y } + \mathrm { e } ^ { 2 y } = c$$ where \(k\) and \(c\) are constants, passes through the point \(P\) with coordinates \(( \ln 3 , \ln 2 )\).
  1. Show that \(58 + 2 k = c\).
  2. Given also that the gradient of the curve at \(P\) is - 6 , find the values of \(k\) and \(c\).
CAIE P3 2011 June Q6
8 marks Standard +0.8
6 \includegraphics[max width=\textwidth, alt={}, center]{cc85b13a-7f15-4025-a545-373cda454de8-3_456_495_255_824} The diagram shows a circle with centre \(O\) and radius 10 cm . The chord \(A B\) divides the circle into two regions whose areas are in the ratio \(1 : 4\) and it is required to find the length of \(A B\). The angle \(A O B\) is \(\theta\) radians.
  1. Show that \(\theta = \frac { 2 } { 5 } \pi + \sin \theta\).
  2. Showing all your working, use an iterative formula, based on the equation in part (i), with an initial value of 2.1 , to find \(\theta\) correct to 2 decimal places. Hence find the length of \(A B\) in centimetres correct to 1 decimal place.
CAIE P3 2011 June Q7
8 marks Standard +0.8
7 The integral \(I\) is defined by \(I = \int _ { 0 } ^ { 2 } 4 t ^ { 3 } \ln \left( t ^ { 2 } + 1 \right) \mathrm { d } t\).
  1. Use the substitution \(x = t ^ { 2 } + 1\) to show that \(I = \int _ { 1 } ^ { 5 } ( 2 x - 2 ) \ln x \mathrm {~d} x\).
  2. Hence find the exact value of \(I\).
CAIE P3 2011 June Q8
10 marks Challenging +1.2
8 The complex number \(u\) is defined by \(u = \frac { 6 - 3 \mathrm { i } } { 1 + 2 \mathrm { i } }\).
  1. Showing all your working, find the modulus of \(u\) and show that the argument of \(u\) is \(- \frac { 1 } { 2 } \pi\).
  2. For complex numbers \(z\) satisfying \(\arg ( z - u ) = \frac { 1 } { 4 } \pi\), find the least possible value of \(| z |\).
  3. For complex numbers \(z\) satisfying \(| z - ( 1 + \mathrm { i } ) u | = 1\), find the greatest possible value of \(| z |\).
CAIE P3 2011 June Q9
10 marks Standard +0.8
9
  1. Prove the identity \(\cos 4 \theta + 4 \cos 2 \theta \equiv 8 \cos ^ { 4 } \theta - 3\).
  2. Hence
    1. solve the equation \(\cos 4 \theta + 4 \cos 2 \theta = 1\) for \(- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\),
    2. find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos ^ { 4 } \theta \mathrm {~d} \theta\).