Questions — CAIE P3 (1070 questions)

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CAIE P3 2004 June Q6
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
  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
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 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
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
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
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
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
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.
CAIE P3 2005 June Q4
4
  1. Use the substitution \(x = \tan \theta\) to show that $$\int \frac { 1 - x ^ { 2 } } { \left( 1 + x ^ { 2 } \right) ^ { 2 } } \mathrm {~d} x = \int \cos 2 \theta \mathrm {~d} \theta$$
  2. Hence find the value of $$\int _ { 0 } ^ { 1 } \frac { 1 - x ^ { 2 } } { \left( 1 + x ^ { 2 } \right) ^ { 2 } } \mathrm {~d} x$$
CAIE P3 2005 June Q5
5 The polynomial \(x ^ { 4 } + 5 x + a\) is denoted by \(\mathrm { p } ( x )\). It is given that \(x ^ { 2 } - x + 3\) is a factor of \(\mathrm { p } ( x )\).
  1. Find the value of \(a\) and factorise \(\mathrm { p } ( x )\) completely.
  2. Hence state the number of real roots of the equation \(\mathrm { p } ( x ) = 0\), justifying your answer.
CAIE P3 2005 June Q6
6
  1. Prove the identity $$\cos 4 \theta + 4 \cos 2 \theta \equiv 8 \cos ^ { 4 } \theta - 3$$
  2. Hence solve the equation $$\cos 4 \theta + 4 \cos 2 \theta = 2$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P3 2005 June Q7
7
  1. By sketching a suitable pair of graphs, show that the equation $$\operatorname { cosec } x = \frac { 1 } { 2 } x + 1$$ where \(x\) is in radians, has a root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
  2. Verify, by calculation, that this root lies between 0.5 and 1 .
  3. Show that this root also satisfies the equation $$x = \sin ^ { - 1 } \left( \frac { 2 } { x + 2 } \right)$$
  4. Use the iterative formula $$x _ { n + 1 } = \sin ^ { - 1 } \left( \frac { 2 } { x _ { n } + 2 } \right)$$ with initial value \(x _ { 1 } = 0.75\), to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2005 June Q8
8
  1. Using partial fractions, find $$\int \frac { 1 } { y ( 4 - y ) } \mathrm { d } y$$
  2. Given that \(y = 1\) when \(x = 0\), solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = y ( 4 - y ) ,$$ obtaining an expression for \(y\) in terms of \(x\).
  3. State what happens to the value of \(y\) if \(x\) becomes very large and positive.
CAIE P3 2005 June Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{208eab3e-a78c-43b4-918f-a9efc9b4f47a-4_429_748_264_699} The diagram shows part of the curve \(y = \frac { x } { x ^ { 2 } + 1 }\) and its maximum point \(M\). The shaded region \(R\) is bounded by the curve and by the lines \(y = 0\) and \(x = p\).
  1. Calculate the \(x\)-coordinate of \(M\).
  2. Find the area of \(R\) in terms of \(p\).
  3. Hence calculate the value of \(p\) for which the area of \(R\) is 1 , giving your answer correct to 3 significant figures.
CAIE P3 2005 June Q10
10 With respect to the origin \(O\), the points \(A\) and \(B\) have position vectors given by $$\overrightarrow { O A } = 2 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = \mathbf { i } + 4 \mathbf { j } + 3 \mathbf { k }$$ The line \(l\) has vector equation \(\mathbf { r } = 4 \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } + s ( \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )\).
  1. Prove that the line \(I\) does not intersect the line through \(A\) and \(B\).
  2. Find the equation of the plane containing \(l\) and the point \(A\), giving your answer in the form \(a x + b y + c z = d\). \footnotetext{Every reasonable effort has been made to trace all copyright holders where the publishers (i.e. UCLES) are aware that third-party material has been reproduced. The publishers would be pleased to hear from anyone whose rights they have unwittingly infringed.
    University of Cambridge International Examinations is part of the University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE P3 2006 June Q1
1 Given that \(x = 4 \left( 3 ^ { - y } \right)\), express \(y\) in terms of \(x\).
CAIE P3 2006 June Q2
2 Solve the inequality \(2 x > | x - 1 |\).
CAIE P3 2006 June Q3
3 The parametric equations of a curve are $$x = 2 \theta + \sin 2 \theta , \quad y = 1 - \cos 2 \theta$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \tan \theta\).
CAIE P3 2006 June Q4
4
  1. Express \(7 \cos \theta + 24 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation $$7 \cos \theta + 24 \sin \theta = 15$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P3 2006 June Q5
5 In a certain industrial process, a substance is being produced in a container. The mass of the substance in the container \(t\) minutes after the start of the process is \(x\) grams. At any time, the rate of formation of the substance is proportional to its mass. Also, throughout the process, the substance is removed from the container at a constant rate of 25 grams per minute. When \(t = 0 , x = 1000\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 75\).
  1. Show that \(x\) and \(t\) satisfy the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = 0.1 ( x - 250 )$$
  2. Solve this differential equation, obtaining an expression for \(x\) in terms of \(t\).
CAIE P3 2006 June Q6
6
  1. By sketching a suitable pair of graphs, show that the equation $$2 \cot x = 1 + \mathrm { e } ^ { x }$$ where \(x\) is in radians, has only one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
  2. Verify by calculation that this root lies between 0.5 and 1.0 .
  3. Show that this root also satisfies the equation $$x = \tan ^ { - 1 } \left( \frac { 2 } { 1 + \mathrm { e } ^ { x } } \right)$$
  4. Use the iterative formula $$x _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 2 } { 1 + \mathrm { e } ^ { x _ { n } } } \right) ,$$ with initial value \(x _ { 1 } = 0.7\), to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2006 June Q7
7 The complex number \(2 + \mathrm { i }\) is denoted by \(u\). Its complex conjugate is denoted by \(u ^ { * }\).
  1. Show, on a sketch of an Argand diagram with origin \(O\), the points \(A , B\) and \(C\) representing the complex numbers \(u , u ^ { * }\) and \(u + u ^ { * }\) respectively. Describe in geometrical terms the relationship between the four points \(O , A , B\) and \(C\).
  2. Express \(\frac { u } { u ^ { * } }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  3. By considering the argument of \(\frac { u } { u ^ { * } }\), or otherwise, prove that $$\tan ^ { - 1 } \left( \frac { 4 } { 3 } \right) = 2 \tan ^ { - 1 } \left( \frac { 1 } { 2 } \right) .$$
CAIE P3 2006 June Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{88f67166-7b44-4b04-b323-f43827531495-3_558_1047_950_550} The diagram shows a sketch of the curve \(y = x ^ { \frac { 1 } { 2 } } \ln x\) and its minimum point \(M\). The curve cuts the \(x\)-axis at the point \(( 1,0 )\).
  1. Find the exact value of the \(x\)-coordinate of \(M\).
  2. Use integration by parts to find the area of the shaded region enclosed by the curve, the \(x\)-axis and the line \(x = 4\). Give your answer correct to 2 decimal places.
  3. Express \(\frac { 10 } { ( 2 - x ) \left( 1 + x ^ { 2 } \right) }\) in partial fractions.
  4. Hence, given that \(| x | < 1\), obtain the expansion of \(\frac { 10 } { ( 2 - x ) \left( 1 + x ^ { 2 } \right) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.
CAIE P3 2006 June Q10
10 The points \(A\) and \(B\) have position vectors, relative to the origin \(O\), given by $$\overrightarrow { O A } = \left( \begin{array} { r } - 1
3
5 \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { r } 3
- 1
- 4 \end{array} \right) .$$ The line \(l\) passes through \(A\) and is parallel to \(O B\). The point \(N\) is the foot of the perpendicular from \(B\) to \(l\).
  1. State a vector equation for the line \(l\).
  2. Find the position vector of \(N\) and show that \(B N = 3\).
  3. Find the equation of the plane containing \(A , B\) and \(N\), giving your answer in the form \(a x + b y + c z = d\).