Questions — CAIE P3 (1070 questions)

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CAIE P3 2010 June Q1
1 Solve the inequality \(| x - 3 | > 2 | x + 1 |\).
CAIE P3 2010 June Q2
2 The variables \(x\) and \(y\) satisfy the equation \(y ^ { 3 } = A \mathrm { e } ^ { 2 x }\), where \(A\) is a constant. The graph of \(\ln y\) against \(x\) is a straight line.
  1. Find the gradient of this line.
  2. Given that the line intersects the axis of \(\ln y\) at the point where \(\ln y = 0.5\), find the value of \(A\) correct to 2 decimal places.
CAIE P3 2010 June Q3
3 Solve the equation $$\tan \left( 45 ^ { \circ } - x \right) = 2 \tan x$$ giving all solutions in the interval \(0 ^ { \circ } < x < 180 ^ { \circ }\).
CAIE P3 2010 June Q4
4 Given that \(x = 1\) when \(t = 0\), solve the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 1 } { x } - \frac { x } { 4 } ,$$ obtaining an expression for \(x ^ { 2 }\) in terms of \(t\).
CAIE P3 2010 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{c5ec981d-7ff7-4698-82c4-eb0506b635a3-2_515_1031_1384_555} The diagram shows the curve \(y = \mathrm { e } ^ { - x } - \mathrm { e } ^ { - 2 x }\) and its maximum point \(M\). The \(x\)-coordinate of \(M\) is denoted by \(p\).
  1. Find the exact value of \(p\).
  2. Show that the area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = p\) is equal to \(\frac { 1 } { 8 }\).
CAIE P3 2010 June Q6
6 The curve \(y = \frac { \ln x } { x + 1 }\) has one stationary point.
  1. Show that the \(x\)-coordinate of this point satisfies the equation $$x = \frac { x + 1 } { \ln x }$$ and that this \(x\)-coordinate lies between 3 and 4 .
  2. Use the iterative formula $$x _ { n + 1 } = \frac { x _ { n } + 1 } { \ln x _ { n } }$$ to determine the \(x\)-coordinate correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2010 June Q7
7
  1. Prove the identity \(\cos 3 \theta \equiv 4 \cos ^ { 3 } \theta - 3 \cos \theta\).
  2. Using this result, find the exact value of $$\int _ { \frac { 1 } { 3 } \pi } ^ { \frac { 1 } { 2 } \pi } \cos ^ { 3 } \theta \mathrm {~d} \theta$$
CAIE P3 2010 June Q8
8
  1. The equation \(2 x ^ { 3 } - x ^ { 2 } + 2 x + 12 = 0\) has one real root and two complex roots. Showing your working, verify that \(1 + i \sqrt { } 3\) is one of the complex roots. State the other complex root.
  2. On a sketch of an Argand diagram, show the point representing the complex number \(1 + \mathrm { i } \sqrt { } 3\). On the same diagram, shade the region whose points represent the complex numbers \(z\) which satisfy both the inequalities \(| z - 1 - i \sqrt { } 3 | \leqslant 1\) and \(\arg z \leqslant \frac { 1 } { 3 } \pi\).
    1. Express \(\frac { 4 + 5 x - x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }\) in partial fractions.
    2. Hence obtain the expansion of \(\frac { 4 + 5 x - x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
CAIE P3 2010 June Q10
10 The straight line \(l\) has equation \(\mathbf { r } = 2 \mathbf { i } - \mathbf { j } - 4 \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } )\). The plane \(p\) has equation \(3 x - y + 2 z = 9\). The line \(l\) intersects the plane \(p\) at the point \(A\).
  1. Find the position vector of \(A\).
  2. Find the acute angle between \(l\) and \(p\).
  3. Find an equation for the plane which contains \(l\) and is perpendicular to \(p\), giving your answer in the form \(a x + b y + c z = d\).
CAIE P3 2011 June Q1
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
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
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
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
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
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
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
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
9
  1. Prove the identity \(\cos 4 \theta + 4 \cos 2 \theta \equiv 8 \cos ^ { 4 } \theta - 3\).
  2. Hence
    (a) solve the equation \(\cos 4 \theta + 4 \cos 2 \theta = 1\) for \(- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\),
    (b) find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos ^ { 4 } \theta \mathrm {~d} \theta\).
CAIE P3 2011 June Q10
10 The number of birds of a certain species in a forested region is recorded over several years. At time \(t\) years, the number of birds is \(N\), where \(N\) is treated as a continuous variable. The variation in the number of birds is modelled by $$\frac { \mathrm { d } N } { \mathrm {~d} t } = \frac { N ( 1800 - N ) } { 3600 }$$ It is given that \(N = 300\) when \(t = 0\).
  1. Find an expression for \(N\) in terms of \(t\).
  2. According to the model, how many birds will there be after a long time?
CAIE P3 2011 June Q1
1 Solve the inequality \(| x | < | 5 + 2 x |\).
CAIE P3 2011 June Q2
2
  1. Show that the equation $$\log _ { 2 } ( x + 5 ) = 5 - \log _ { 2 } x$$ can be written as a quadratic equation in \(x\).
  2. Hence solve the equation $$\log _ { 2 } ( x + 5 ) = 5 - \log _ { 2 } x$$
CAIE P3 2011 June Q3
3 Solve the equation $$\cos \theta + 4 \cos 2 \theta = 3$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P3 2011 June Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{76371b0f-0145-4cc4-a147-27bcd749816a-2_339_1395_1089_374} The diagram shows a semicircle \(A C B\) with centre \(O\) and radius \(r\). The tangent at \(C\) meets \(A B\) produced at \(T\). The angle \(B O C\) is \(x\) radians. The area of the shaded region is equal to the area of the semicircle.
  1. Show that \(x\) satisfies the equation $$\tan x = x + \pi$$
  2. Use the iterative formula \(x _ { n + 1 } = \tan ^ { - 1 } \left( x _ { n } + \pi \right)\) to determine \(x\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2011 June Q5
5 The parametric equations of a curve are $$x = \ln ( \tan t ) , \quad y = \sin ^ { 2 } t$$ where \(0 < t < \frac { 1 } { 2 } \pi\).
  1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
  2. Find the equation of the tangent to the curve at the point where \(x = 0\).
CAIE P3 2011 June Q6
6 A certain curve is such that its gradient at a point \(( x , y )\) is proportional to \(x y\). At the point \(( 1,2 )\) the gradient is 4 .
  1. By setting up and solving a differential equation, show that the equation of the curve is \(y = 2 \mathrm { e } ^ { x ^ { 2 } - 1 }\).
  2. State the gradient of the curve at the point \(( - 1,2 )\) and sketch the curve.