Questions P3 (1203 questions)

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CAIE P3 2002 June Q1
1 Prove the identity $$\cot \theta - \tan \theta \equiv 2 \cot 2 \theta$$
CAIE P3 2002 June Q2
2 Expand \(( 1 - 3 x ) ^ { - \frac { 1 } { 3 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.
CAIE P3 2002 June Q3
3 The polynomial \(x ^ { 4 } + 4 x ^ { 2 } + x + a\) is denoted by \(\mathrm { p } ( x )\). It is given that ( \(x ^ { 2 } + x + 2\) ) is a factor of \(\mathrm { p } ( x )\).
Find the value of \(a\) and the other quadratic factor of \(p ( x )\).
CAIE P3 2002 June Q4
4 The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 2 } { 3 } \left( x _ { n } + \frac { 1 } { x _ { n } ^ { 2 } } \right)$$ with initial value \(x _ { 1 } = 1\), converges to \(\alpha\).
  1. Use this formula to find \(\alpha\) correct to 2 decimal places, showing the result of each iteration.
  2. State an equation satisfied by \(\alpha\), and hence find the exact value of \(\alpha\).
CAIE P3 2002 June Q5
5 The equation of a curve is \(y = 2 \cos x + \sin 2 x\). Find the \(x\)-coordinates of the stationary points on the curve for which \(0 < x < \pi\), and determine the nature of each of these stationary points.
CAIE P3 2002 June Q6
6 Let \(\mathrm { f } ( x ) = \frac { 4 x } { ( 3 x + 1 ) ( x + 1 ) ^ { 2 } }\).
  1. Express \(f ( x )\) in partial fractions.
  2. Hence show that \(\int _ { 0 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x = 1 - \ln 2\).
CAIE P3 2002 June Q7
7 In a certain chemical process a substance is being formed, and \(t\) minutes after the start of the process there are \(m\) grams of the substance present. In the process the rate of increase of \(m\) is proportional to \(( 50 - m ) ^ { 2 }\). When \(t = 0 , m = 0\) and \(\frac { \mathrm { d } m } { \mathrm {~d} t } = 5\).
  1. Show that \(m\) satisfies the differential equation $$\frac { \mathrm { d } m } { \mathrm {~d} t } = 0.002 ( 50 - m ) ^ { 2 }$$
  2. Solve the differential equation, and show that the solution can be expressed in the form $$m = 50 - \frac { 500 } { t + 10 }$$
  3. Calculate the mass of the substance when \(t = 10\), and find the time taken for the mass to increase from 0 to 45 grams.
  4. State what happens to the mass of the substance as \(t\) becomes very large.
CAIE P3 2002 June Q8
8 The straight line \(l\) passes through the points \(A\) and \(B\) whose position vectors are \(\mathbf { i } + \mathbf { k }\) and \(4 \mathbf { i } - \mathbf { j } + 3 \mathbf { k }\) respectively. The plane \(p\) has equation \(x + 3 y - 2 z = 3\).
  1. Given that \(l\) intersects \(p\), find the position vector of the point of intersection.
  2. Find the equation of the plane which contains \(l\) and is perpendicular to \(p\), giving your answer in the form \(a x + b y + c z = 1\).
CAIE P3 2002 June Q9
9 The complex number \(1 + i \sqrt { } 3\) is denoted by \(u\).
  1. Express \(u\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Hence, or otherwise, find the modulus and argument of \(u ^ { 2 }\) and \(u ^ { 3 }\).
  2. Show that \(u\) is a root of the equation \(z ^ { 2 } - 2 z + 4 = 0\), and state the other root of this equation.
  3. Sketch an Argand diagram showing the points representing the complex numbers \(i\) and \(u\). Shade the region whose points represent every complex number \(z\) satisfying both the inequalities $$| z - \mathrm { i } | \leqslant 1 \quad \text { and } \quad \arg z \geqslant \arg u .$$
CAIE P3 2002 June Q10
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
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
2 Find the exact value of \(\int _ { 0 } ^ { 1 } x \mathrm { e } ^ { 2 x } \mathrm {~d} x\).
CAIE P3 2003 June Q3
3 Solve the inequality \(| x - 2 | < 3 - 2 x\).
CAIE P3 2003 June Q4
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
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
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
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
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
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
  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
1 Sketch the graph of \(y = \sec x\), for \(0 \leqslant x \leqslant 2 \pi\).
CAIE P3 2004 June Q2
2 Solve the inequality \(| 2 x + 1 | < | x |\).
CAIE P3 2004 June Q3
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
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
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 $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \sin ^ { 2 } \theta \cos ^ { 2 } \theta \mathrm {~d} \theta$$