Questions — CAIE P3 (1110 questions)

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CAIE P3 2007 June Q10
12 marks Moderate -0.8
10 A model for the height, \(h\) metres, of a certain type of tree at time \(t\) years after being planted assumes that, while the tree is growing, the rate of increase in height is proportional to \(( 9 - h ) ^ { \frac { 1 } { 3 } }\). It is given that, when \(t = 0 , h = 1\) and \(\frac { \mathrm { d } h } { \mathrm {~d} t } = 0.2\).
  1. Show that \(h\) and \(t\) satisfy the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = 0.1 ( 9 - h ) ^ { \frac { 1 } { 3 } } .$$
  2. Solve this differential equation, and obtain an expression for \(h\) in terms of \(t\).
  3. Find the maximum height of the tree and the time taken to reach this height after planting.
  4. Calculate the time taken to reach half the maximum height.
CAIE P3 2008 June Q1
4 marks Standard +0.8
1 Solve the inequality \(| x - 2 | > 3 | 2 x + 1 |\).
CAIE P3 2008 June Q2
5 marks Standard +0.3
2 Solve, correct to 3 significant figures, the equation $$\mathrm { e } ^ { x } + \mathrm { e } ^ { 2 x } = \mathrm { e } ^ { 3 x }$$
CAIE P3 2008 June Q3
6 marks Standard +0.8
3 \includegraphics[max width=\textwidth, alt={}, center]{20893bfc-3300-4205-9d2c-729cc3243971-2_337_828_657_657} In the diagram, \(A B C D\) is a rectangle with \(A B = 3 a\) and \(A D = a\). A circular arc, with centre \(A\) and radius \(r\), joins points \(M\) and \(N\) on \(A B\) and \(C D\) respectively. The angle \(M A N\) is \(x\) radians. The perimeter of the sector \(A M N\) is equal to half the perimeter of the rectangle.
  1. Show that \(x\) satisfies the equation $$\sin x = \frac { 1 } { 4 } ( 2 + x ) \text {. }$$
  2. This equation has only one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\). Use the iterative formula $$x _ { n + 1 } = \sin ^ { - 1 } \left( \frac { 2 + x _ { n } } { 4 } \right) ,$$ with initial value \(x _ { 1 } = 0.8\), to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2008 June Q4
7 marks Standard +0.3
4
  1. Show that the equation \(\tan \left( 30 ^ { \circ } + \theta \right) = 2 \tan \left( 60 ^ { \circ } - \theta \right)\) can be written in the form $$\tan ^ { 2 } \theta + ( 6 \sqrt { } 3 ) \tan \theta - 5 = 0$$
  2. Hence, or otherwise, solve the equation $$\tan \left( 30 ^ { \circ } + \theta \right) = 2 \tan \left( 60 ^ { \circ } - \theta \right) ,$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P3 2008 June Q5
7 marks Standard +0.8
5 The variable complex number \(z\) is given by $$z = 2 \cos \theta + \mathrm { i } ( 1 - 2 \sin \theta ) ,$$ where \(\theta\) takes all values in the interval \(- \pi < \theta \leqslant \pi\).
  1. Show that \(| z - \mathrm { i } | = 2\), for all values of \(\theta\). Hence sketch, in an Argand diagram, the locus of the point representing \(z\).
  2. Prove that the real part of \(\frac { 1 } { z + 2 - \mathrm { i } }\) is constant for \(- \pi < \theta < \pi\).
CAIE P3 2008 June Q6
8 marks Standard +0.8
6 The equation of a curve is \(x y ( x + y ) = 2 a ^ { 3 }\), where \(a\) is a non-zero constant. Show that there is only one point on the curve at which the tangent is parallel to the \(x\)-axis, and find the coordinates of this point.
CAIE P3 2008 June Q7
9 marks Standard +0.3
7 Let \(\mathrm { f } ( x ) \equiv \frac { x ^ { 2 } + 3 x + 3 } { ( x + 1 ) ( x + 3 ) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence show that \(\int _ { 0 } ^ { 3 } \mathrm { f } ( x ) \mathrm { d } x = 3 - \frac { 1 } { 2 } \ln 2\).
CAIE P3 2008 June Q8
9 marks Standard +0.8
8 \includegraphics[max width=\textwidth, alt={}, center]{20893bfc-3300-4205-9d2c-729cc3243971-3_597_951_1471_598} In the diagram the tangent to a curve at a general point \(P\) with coordinates \(( x , y )\) meets the \(x\)-axis at \(T\). The point \(N\) on the \(x\)-axis is such that \(P N\) is perpendicular to the \(x\)-axis. The curve is such that, for all values of \(x\) in the interval \(0 < x < \frac { 1 } { 2 } \pi\), the area of triangle \(P T N\) is equal to \(\tan x\), where \(x\) is in radians.
  1. Using the fact that the gradient of the curve at \(P\) is \(\frac { P N } { T N }\), show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 2 } y ^ { 2 } \cot x .$$
  2. Given that \(y = 2\) when \(x = \frac { 1 } { 6 } \pi\), solve this differential equation to find the equation of the curve, expressing \(y\) in terms of \(x\).
CAIE P3 2008 June Q9
10 marks Challenging +1.2
9 \includegraphics[max width=\textwidth, alt={}, center]{20893bfc-3300-4205-9d2c-729cc3243971-4_547_1401_264_370} The diagram shows the curve \(y = \mathrm { e } ^ { - \frac { 1 } { 2 } x } \sqrt { } ( 1 + 2 x )\) and its maximum point \(M\). The shaded region between the curve and the axes is denoted by \(R\).
  1. Find the \(x\)-coordinate of \(M\).
  2. Find by integration the volume of the solid obtained when \(R\) is rotated completely about the \(x\)-axis. Give your answer in terms of \(\pi\) and e.
CAIE P3 2008 June Q10
10 marks Standard +0.8
10 The points \(A\) and \(B\) have position vectors, relative to the origin \(O\), given by $$\overrightarrow { O A } = \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } .$$ The line \(l\) has vector equation $$\mathbf { r } = ( 1 - 2 t ) \mathbf { i } + ( 5 + t ) \mathbf { j } + ( 2 - t ) \mathbf { k }$$
  1. Show that \(l\) does not intersect the line passing through \(A\) and \(B\).
  2. The point \(P\) lies on \(l\) and is such that angle \(P A B\) is equal to \(60 ^ { \circ }\). Given that the position vector of \(P\) is \(( 1 - 2 t ) \mathbf { i } + ( 5 + t ) \mathbf { j } + ( 2 - t ) \mathbf { k }\), show that \(3 t ^ { 2 } + 7 t + 2 = 0\). Hence find the only possible position vector of \(P\).
CAIE P3 2009 June Q1
4 marks Moderate -0.8
1 Solve the equation \(\ln \left( 2 + \mathrm { e } ^ { - x } \right) = 2\), giving your answer correct to 2 decimal places.
CAIE P3 2009 June Q2
4 marks Moderate -0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{0f73e750-18a0-49ad-b4cb-fd6d14f0789e-2_531_700_395_719} The diagram shows the curve \(y = \sqrt { } \left( 1 + 2 \tan ^ { 2 } x \right)\) for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\).
  1. Use the trapezium rule with three intervals to estimate the value of $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sqrt { } \left( 1 + 2 \tan ^ { 2 } x \right) \mathrm { d } x$$ giving your answer correct to 2 decimal places.
  2. The estimate found in part (i) is denoted by \(E\). Explain, without further calculation, whether another estimate found using the trapezium rule with six intervals would be greater than \(E\) or less than \(E\).
CAIE P3 2009 June Q3
5 marks Standard +0.3
3
  1. Prove the identity \(\operatorname { cosec } 2 \theta + \cot 2 \theta \equiv \cot \theta\).
  2. Hence solve the equation \(\operatorname { cosec } 2 \theta + \cot 2 \theta = 2\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P3 2009 June Q4
7 marks Standard +0.3
4 The equation \(x ^ { 3 } - 2 x - 2 = 0\) has one real root.
  1. Show by calculation that this root lies between \(x = 1\) and \(x = 2\).
  2. Prove that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 3 } + 2 } { 3 x _ { n } ^ { 2 } - 2 }$$ converges, then it converges to this root.
  3. Use this iterative formula to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2009 June Q5
7 marks Standard +0.3
5 When \(( 1 + 2 x ) ( 1 + a x ) ^ { \frac { 2 } { 3 } }\), where \(a\) is a constant, is expanded in ascending powers of \(x\), the coefficient of the term in \(x\) is zero.
  1. Find the value of \(a\).
  2. When \(a\) has this value, find the term in \(x ^ { 3 }\) in the expansion of \(( 1 + 2 x ) ( 1 + a x ) ^ { \frac { 2 } { 3 } }\), simplifying the coefficient.
CAIE P3 2009 June Q6
8 marks Challenging +1.2
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
8 marks Standard +0.3
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
10 marks Challenging +1.2
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
11 marks Standard +0.3
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
11 marks Standard +0.8
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
4 marks Challenging +1.2
1 Solve the inequality \(| x + 3 a | > 2 | x - 2 a |\), where \(a\) is a positive constant.
CAIE P3 2010 June Q2
6 marks Standard +0.3
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
6 marks Moderate -0.8
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
6 marks Standard +0.3
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 $$\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \sin 3 x \sin x \mathrm {~d} x = \frac { 1 } { 8 } \sqrt { } 3$$