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CAIE P3 2002 June Q10
11 marks Standard +0.8
10 \includegraphics[max width=\textwidth, alt={}, center]{0f081749-4fe0-46e3-96c2-466e69cf49d3-4_620_894_338_687} The function f is defined by \(\mathrm { f } ( x ) = ( \ln x ) ^ { 2 }\) for \(x > 0\). The diagram shows a sketch of the graph of \(y = \mathrm { f } ( x )\). The minimum point of the graph is \(A\). The point \(B\) has \(x\)-coordinate e .
  1. State the \(x\)-coordinate of \(A\).
  2. Show that \(\mathrm { f } ^ { \prime \prime } ( x ) = 0\) at \(B\).
  3. Use the substitution \(x = \mathrm { e } ^ { u }\) to show that the area of the region bounded by the \(x\)-axis, the line \(x = \mathrm { e }\), and the part of the curve between \(A\) and \(B\) is given by $$\int _ { 0 } ^ { 1 } u ^ { 2 } \mathrm { e } ^ { u } \mathrm {~d} u .$$
  4. Hence, or otherwise, find the exact value of this area.
CAIE P3 2003 June Q1
4 marks Moderate -0.3
1
  1. Show that the equation $$\sin \left( x - 60 ^ { \circ } \right) - \cos \left( 30 ^ { \circ } - x \right) = 1$$ can be written in the form \(\cos x = k\), where \(k\) is a constant.
  2. Hence solve the equation, for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
CAIE P3 2003 June Q2
4 marks Moderate -0.5
2 Find the exact value of \(\int _ { 0 } ^ { 1 } x \mathrm { e } ^ { 2 x } \mathrm {~d} x\).
CAIE P3 2003 June Q3
4 marks Standard +0.3
3 Solve the inequality \(| x - 2 | < 3 - 2 x\).
CAIE P3 2003 June Q4
7 marks Standard +0.3
4 The polynomial \(x ^ { 4 } - 2 x ^ { 3 } - 2 x ^ { 2 } + a\) is denoted by \(\mathrm { f } ( x )\). It is given that \(\mathrm { f } ( x )\) is divisible by \(x ^ { 2 } - 4 x + 4\).
  1. Find the value of \(a\).
  2. When \(a\) has this value, show that \(\mathrm { f } ( x )\) is never negative.
CAIE P3 2003 June Q5
8 marks Standard +0.3
5 The complex number 2 i is denoted by \(u\). The complex number with modulus 1 and argument \(\frac { 2 } { 3 } \pi\) is denoted by \(w\).
  1. Find in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real, the complex numbers \(w , u w\) and \(\frac { u } { w }\).
  2. Sketch an Argand diagram showing the points \(U , A\) and \(B\) representing the complex numbers \(u\), \(u w\) and \(\frac { u } { w }\) respectively.
  3. Prove that triangle \(U A B\) is equilateral.
CAIE P3 2003 June Q6
9 marks Standard +0.3
6 Let \(\mathrm { f } ( x ) = \frac { 9 x ^ { 2 } + 4 } { ( 2 x + 1 ) ( x - 2 ) ^ { 2 } }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Show that, when \(x\) is sufficiently small for \(x ^ { 3 }\) and higher powers to be neglected, $$f ( x ) = 1 - x + 5 x ^ { 2 }$$
CAIE P3 2003 June Q7
9 marks Standard +0.3
7 In a chemical reaction a compound \(X\) is formed from a compound \(Y\). The masses in grams of \(X\) and \(Y\) present at time \(t\) seconds after the start of the reaction are \(x\) and \(y\) respectively. The sum of the two masses is equal to 100 grams throughout the reaction. At any time, the rate of formation of \(X\) is proportional to the mass of \(Y\) at that time. When \(t = 0 , x = 5\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 1.9\).
  1. Show that \(x\) satisfies the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = 0.02 ( 100 - x ) .$$
  2. Solve this differential equation, obtaining an expression for \(x\) in terms of \(t\).
  3. State what happens to the value of \(x\) as \(t\) becomes very large.
CAIE P3 2003 June Q8
10 marks Standard +0.3
8 The equation of a curve is \(y = \ln x + \frac { 2 } { x }\), where \(x > 0\).
  1. Find the coordinates of the stationary point of the curve and determine whether it is a maximum or a minimum point.
  2. The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 2 } { 3 - \ln x _ { n } }$$ with initial value \(x _ { 1 } = 1\), converges to \(\alpha\). State an equation satisfied by \(\alpha\), and hence show that \(\alpha\) is the \(x\)-coordinate of a point on the curve where \(y = 3\).
  3. Use this iterative formula to find \(\alpha\) correct to 2 decimal places, showing the result of each iteration.
CAIE P3 2003 June Q9
10 marks Standard +0.3
9 Two planes have equations \(x + 2 y - 2 z = 2\) and \(2 x - 3 y + 6 z = 3\). The planes intersect in the straight line \(l\).
  1. Calculate the acute angle between the two planes.
  2. Find a vector equation for the line \(l\).
CAIE P3 2003 June Q10
10 marks Standard +0.3
10
  1. Prove the identity $$\cot x - \cot 2 x \equiv \operatorname { cosec } 2 x$$
  2. Show that \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 4 } \pi } \cot x \mathrm {~d} x = \frac { 1 } { 2 } \ln 2\).
  3. Find the exact value of \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 4 } \pi } \operatorname { cosec } 2 x \mathrm {~d} x\), giving your answer in the form \(a \ln b\).
CAIE P3 2004 June Q1
3 marks Moderate -0.8
1 Sketch the graph of \(y = \sec x\), for \(0 \leqslant x \leqslant 2 \pi\).
CAIE P3 2004 June Q2
4 marks Standard +0.3
2 Solve the inequality \(| 2 x + 1 | < | x |\).
CAIE P3 2004 June Q3
4 marks Moderate -0.3
3 Find the gradient of the curve with equation $$2 x ^ { 2 } - 4 x y + 3 y ^ { 2 } = 3$$ at the point \(( 2,1 )\).
CAIE P3 2004 June Q4
6 marks Moderate -0.3
4
  1. Show that if \(y = 2 ^ { x }\), then the equation $$2 ^ { x } - 2 ^ { - x } = 1$$ can be written as a quadratic equation in \(y\).
  2. Hence solve the equation $$2 ^ { x } - 2 ^ { - x } = 1$$
CAIE P3 2004 June Q5
6 marks Standard +0.3
5
  1. Prove the identity $$\sin ^ { 2 } \theta \cos ^ { 2 } \theta \equiv \frac { 1 } { 8 } ( 1 - \cos 4 \theta )$$
  2. Hence find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \sin ^ { 2 } \theta \cos ^ { 2 } \theta \mathrm {~d} \theta$$
CAIE P3 2004 June Q6
6 marks Moderate -0.3
6 Given that \(y = 1\) when \(x = 0\), solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y ^ { 3 } + 1 } { y ^ { 2 } }$$ obtaining an expression for \(y\) in terms of \(x\).
CAIE P3 2004 June Q7
7 marks Standard +0.3
7
  1. The equation \(x ^ { 3 } + x + 1 = 0\) has one real root. Show by calculation that this root lies between - 1 and 0 .
  2. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 3 } - 1 } { 3 x _ { n } ^ { 2 } + 1 }$$ converges, then it converges to the root of the equation given in part (i).
  3. Use this iterative formula, with initial value \(x _ { 1 } = - 0.5\), to determine the root correct to 2 decimal places, showing the result of each iteration.
CAIE P3 2004 June Q8
7 marks Moderate -0.3
8
  1. Find the roots of the equation \(z ^ { 2 } - z + 1 = 0\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. Obtain the modulus and argument of each root.
  3. Show that each root also satisfies the equation \(z ^ { 3 } = - 1\).
CAIE P3 2004 June Q9
9 marks Standard +0.3
9 Let \(\mathrm { f } ( x ) = \frac { x ^ { 2 } + 7 x - 6 } { ( x - 1 ) ( x - 2 ) ( x + 1 ) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Show that, when \(x\) is sufficiently small for \(x ^ { 4 }\) and higher powers to be neglected, $$f ( x ) = - 3 + 2 x - \frac { 3 } { 2 } x ^ { 2 } + \frac { 11 } { 4 } x ^ { 3 } .$$
CAIE P3 2004 June Q10
11 marks Standard +0.3
10 \includegraphics[max width=\textwidth, alt={}, center]{2718ebbb-29e3-46f7-8d8d-ec7d526483f8-3_458_920_1144_609} The diagram shows the curve \(y = \frac { \ln x } { x ^ { 2 } }\) and its maximum point \(M\). The curve cuts the \(x\)-axis at \(A\).
  1. Write down the \(x\)-coordinate of \(A\).
  2. Find the exact coordinates of \(M\).
  3. Use integration by parts to find the exact area of the shaded region enclosed by the curve, the \(x\)-axis and the line \(x = \mathrm { e }\).
CAIE P3 2004 June Q11
12 marks Standard +0.3
11 With respect to the origin \(O\), the points \(P , Q , R , S\) have position vectors given by $$\overrightarrow { O P } = \mathbf { i } - \mathbf { k } , \quad \overrightarrow { O Q } = - 2 \mathbf { i } + 4 \mathbf { j } , \quad \overrightarrow { O R } = 4 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } , \quad \overrightarrow { O S } = 3 \mathbf { i } + 5 \mathbf { j } - 6 \mathbf { k } .$$
  1. Find the equation of the plane containing \(P , Q\) and \(R\), giving your answer in the form \(a x + b y + c z = d\).
  2. The point \(N\) is the foot of the perpendicular from \(S\) to this plane. Find the position vector of \(N\) and show that the length of \(S N\) is 7 .
CAIE P3 2005 June Q1
4 marks Moderate -0.8
1 Expand \(( 1 + 4 x ) ^ { - \frac { 1 } { 2 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.
CAIE P3 2005 June Q2
4 marks Moderate -0.8
2 \includegraphics[max width=\textwidth, alt={}, center]{208eab3e-a78c-43b4-918f-a9efc9b4f47a-2_508_586_450_776} The diagram shows a sketch of the curve \(y = \frac { 1 } { 1 + x ^ { 3 } }\) for values of \(x\) from - 0.6 to 0.6 .
  1. Use the trapezium rule, with two intervals, to estimate the value of $$\int _ { - 0.6 } ^ { 0.6 } \frac { 1 } { 1 + x ^ { 3 } } \mathrm {~d} x$$ giving your answer correct to 2 decimal places.
  2. Explain, with reference to the diagram, why the trapezium rule may be expected to give a good approximation to the true value of the integral in this case.
CAIE P3 2005 June Q3
7 marks Moderate -0.3
3
  1. Solve the equation \(z ^ { 2 } - 2 \mathrm { i } z - 5 = 0\), giving your answers in the form \(x + \mathrm { i } y\) where \(x\) and \(y\) are real.
  2. Find the modulus and argument of each root.
  3. Sketch an Argand diagram showing the points representing the roots.