Questions — CAIE P3 (1110 questions)

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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.
CAIE P3 2005 June Q4
7 marks Standard +0.3
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
8 marks Standard +0.8
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
8 marks Standard +0.3
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
8 marks Standard +0.3
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
9 marks Standard +0.3
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 marks Standard +0.3
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
11 marks Standard +0.3
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 \(a x + b y + c z = d\).
CAIE P3 2007 June Q1
4 marks Moderate -0.8
1 Expand \(( 2 + 3 x ) ^ { - 2 }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
CAIE P3 2007 June Q2
4 marks Easy -1.2
2 The polynomial \(x ^ { 3 } - 2 x + a\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that ( \(x + 2\) ) is a factor of \(\mathrm { p } ( x )\).
  1. Find the value of \(a\).
  2. When \(a\) has this value, find the quadratic factor of \(\mathrm { p } ( x )\).
CAIE P3 2007 June Q3
4 marks Moderate -0.5
3 The equation of a curve is \(y = x \sin 2 x\), where \(x\) is in radians. Find the equation of the tangent to the curve at the point where \(x = \frac { 1 } { 4 } \pi\).
CAIE P3 2007 June Q4
6 marks Standard +0.3
4 Using the substitution \(u = 3 ^ { x }\), or otherwise, solve, correct to 3 significant figures, the equation $$3 ^ { x } = 2 + 3 ^ { - x }$$
CAIE P3 2007 June Q5
7 marks Standard +0.3
5
  1. Express \(\cos \theta + ( \sqrt { } 3 ) \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), giving the exact values of \(R\) and \(\alpha\).
  2. Hence show that \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { 1 } { ( \cos \theta + ( \sqrt { } 3 ) \sin \theta ) ^ { 2 } } \mathrm {~d} \theta = \frac { 1 } { \sqrt { } 3 }\).
CAIE P3 2007 June Q6
9 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{8580dddb-cc72-4745-9e0f-1ac641c6506d-2_355_601_1562_772} The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius \(r\). The angle \(A O B\) is \(\alpha\) radians, where \(0 < \alpha < \pi\). The area of triangle \(A O B\) is half the area of the sector.
  1. Show that \(\alpha\) satisfies the equation $$x = 2 \sin x$$
  2. Verify by calculation that \(\alpha\) lies between \(\frac { 1 } { 2 } \pi\) and \(\frac { 2 } { 3 } \pi\).
  3. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 1 } { 3 } \left( x _ { n } + 4 \sin x _ { n } \right)$$ converges, then it converges to a root of the equation in part (i).
  4. Use this iterative formula, with initial value \(x _ { 1 } = 1.8\), to find \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2007 June Q7
9 marks Standard +0.3
7 Let \(I = \int _ { 1 } ^ { 4 } \frac { 1 } { x ( 4 - \sqrt { } x ) } \mathrm { d } x\).
  1. Use the substitution \(u = \sqrt { } x\) to show that \(I = \int _ { 1 } ^ { 2 } \frac { 2 } { u ( 4 - u ) } \mathrm { d } u\).
  2. Hence show that \(I = \frac { 1 } { 2 } \ln 3\).
CAIE P3 2007 June Q8
10 marks Standard +0.3
8 The complex number \(\frac { 2 } { - 1 + \mathrm { i } }\) is denoted by \(u\).
  1. Find the modulus and argument of \(u\) and \(u ^ { 2 }\).
  2. Sketch an Argand diagram showing the points representing the complex numbers \(u\) and \(u ^ { 2 }\). Shade the region whose points represent the complex numbers \(z\) which satisfy both the inequalities \(| z | < 2\) and \(\left| z - u ^ { 2 } \right| < | z - u |\).
CAIE P3 2007 June Q9
10 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{8580dddb-cc72-4745-9e0f-1ac641c6506d-3_693_537_1206_804} The diagram shows a set of rectangular axes \(O x , O y\) and \(O z\), and three points \(A , B\) and \(C\) with position vectors \(\overrightarrow { O A } = \left( \begin{array} { l } 2 \\ 0 \\ 0 \end{array} \right) , \overrightarrow { O B } = \left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right)\) and \(\overrightarrow { O C } = \left( \begin{array} { l } 1 \\ 1 \\ 2 \end{array} \right)\).
  1. Find the equation of the plane \(A B C\), giving your answer in the form \(a x + b y + c z = d\).
  2. Calculate the acute angle between the planes \(A B C\) and \(O A B\).