Questions — CAIE P3 (1070 questions)

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CAIE P3 2010 November Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{822f851a-7fae-43b8-9ebc-94588f01e51c-4_597_895_258_625} The diagram shows the curve \(y = x ^ { 3 } \ln x\) and its minimum point \(M\).
  1. Find the exact coordinates of \(M\).
  2. Find the exact area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = 2\).
CAIE P3 2010 November Q1
1 Expand \(( 1 + 2 x ) ^ { - 3 }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
CAIE P3 2010 November Q2
2 The parametric equations of a curve are $$x = \frac { t } { 2 t + 3 } , \quad y = \mathrm { e } ^ { - 2 t }$$ Find the gradient of the curve at the point for which \(t = 0\).
CAIE P3 2010 November Q3
3 The complex number \(w\) is defined by \(w = 2 + \mathrm { i }\).
  1. Showing your working, express \(w ^ { 2 }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real. Find the modulus of \(w ^ { 2 }\).
  2. Shade on an Argand diagram the region whose points represent the complex numbers \(z\) which satisfy $$\left| z - w ^ { 2 } \right| \leqslant \left| w ^ { 2 } \right|$$
CAIE P3 2010 November Q4
4 It is given that \(\mathrm { f } ( x ) = 4 \cos ^ { 2 } 3 x\).
  1. Find the exact value of \(\mathrm { f } ^ { \prime } \left( \frac { 1 } { 9 } \pi \right)\).
  2. Find \(\int \mathrm { f } ( x ) \mathrm { d } x\).
CAIE P3 2010 November Q5
5 Show that \(\int _ { 0 } ^ { 7 } \frac { 2 x + 7 } { ( 2 x + 1 ) ( x + 2 ) } \mathrm { d } x = \ln 50\).
CAIE P3 2010 November Q6
6 The straight line \(l\) passes through the points with coordinates \(( - 5,3,6 )\) and \(( 5,8,1 )\). The plane \(p\) has equation \(2 x - y + 4 z = 9\).
  1. Find the coordinates of the point of intersection of \(l\) and \(p\).
  2. Find the acute angle between \(l\) and \(p\).
CAIE P3 2010 November Q7
7
  1. Given that \(\int _ { 1 } ^ { a } \frac { \ln x } { x ^ { 2 } } \mathrm {~d} x = \frac { 2 } { 5 }\), show that \(a = \frac { 5 } { 3 } ( 1 + \ln a )\).
  2. Use an iteration formula based on the equation \(a = \frac { 5 } { 3 } ( 1 + \ln a )\) to find the value of \(a\) correct to 2 decimal places. Use an initial value of 4 and give the result of each iteration to 4 decimal places.
CAIE P3 2010 November Q8
8
  1. Express \(( \sqrt { } 6 ) \cos \theta + ( \sqrt { } 10 ) \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(\alpha\) correct to 2 decimal places.
  2. Hence, in each of the following cases, find the smallest positive angle \(\theta\) which satisfies the equation
    (a) \(( \sqrt { } 6 ) \cos \theta + ( \sqrt { } 10 ) \sin \theta = - 4\),
    (b) \(( \sqrt { } 6 ) \cos \frac { 1 } { 2 } \theta + ( \sqrt { } 10 ) \sin \frac { 1 } { 2 } \theta = 3\).
CAIE P3 2010 November Q9
9 A biologist is investigating the spread of a weed in a particular region. At time \(t\) weeks after the start of the investigation, the area covered by the weed is \(A \mathrm {~m} ^ { 2 }\). The biologist claims that the rate of increase of \(A\) is proportional to \(\sqrt { } ( 2 A - 5 )\).
  1. Write down a differential equation representing the biologist's claim.
  2. At the start of the investigation, the area covered by the weed was \(7 \mathrm {~m} ^ { 2 }\) and, 10 weeks later, the area covered was \(27 \mathrm {~m} ^ { 2 }\). Assuming that the biologist's claim is correct, find the area covered 20 weeks after the start of the investigation.
CAIE P3 2010 November Q10
10 The polynomial \(\mathrm { p } ( z )\) is defined by $$\mathrm { p } ( z ) = z ^ { 3 } + m z ^ { 2 } + 24 z + 32$$ where \(m\) is a constant. It is given that \(( z + 2 )\) is a factor of \(\mathrm { p } ( z )\).
  1. Find the value of \(m\).
  2. Hence, showing all your working, find
    (a) the three roots of the equation \(\mathrm { p } ( z ) = 0\),
    (b) the six roots of the equation \(\mathrm { p } \left( z ^ { 2 } \right) = 0\).
CAIE P3 2011 November Q1
1 Using the substitution \(u = \mathrm { e } ^ { x }\), or otherwise, solve the equation $$\mathrm { e } ^ { x } = 1 + 6 \mathrm { e } ^ { - x }$$ giving your answer correct to 3 significant figures.
CAIE P3 2011 November Q2
2 The parametric equations of a curve are $$x = 3 \left( 1 + \sin ^ { 2 } t \right) , \quad y = 2 \cos ^ { 3 } t$$ Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\), simplifying your answer as far as possible.
CAIE P3 2011 November Q3
3 The polynomial \(x ^ { 4 } + 3 x ^ { 3 } + a x + 3\) is denoted by \(\mathrm { p } ( x )\). It is given that \(\mathrm { p } ( x )\) is divisible by \(x ^ { 2 } - x + 1\).
  1. Find the value of \(a\).
  2. When \(a\) has this value, find the real roots of the equation \(\mathrm { p } ( x ) = 0\).
CAIE P3 2011 November Q4
4 The variables \(x\) and \(\theta\) are related by the differential equation $$\sin 2 \theta \frac { \mathrm {~d} x } { \mathrm {~d} \theta } = ( x + 1 ) \cos 2 \theta$$ where \(0 < \theta < \frac { 1 } { 2 } \pi\). When \(\theta = \frac { 1 } { 12 } \pi , x = 0\). Solve the differential equation, obtaining an expression for \(x\) in terms of \(\theta\), and simplifying your answer as far as possible.
CAIE P3 2011 November Q5
5
  1. By sketching a suitable pair of graphs, show that the equation $$\sec x = 3 - \frac { 1 } { 2 } x ^ { 2 }$$ 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 1 and 1.4.
  3. Show that this root also satisfies the equation $$x = \cos ^ { - 1 } \left( \frac { 2 } { 6 - x ^ { 2 } } \right)$$
  4. Use an iterative formula based on the equation in part (iii) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2011 November Q6
6
  1. Express \(\cos x + 3 \sin x\) in the form \(R \cos ( x - \alpha )\), 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 \(\cos 2 \theta + 3 \sin 2 \theta = 2\), for \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).
CAIE P3 2011 November Q7
7 With respect to the origin \(O\), the position vectors of two points \(A\) and \(B\) are given by \(\overrightarrow { O A } = \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k }\) and \(\overrightarrow { O B } = 3 \mathbf { i } + 4 \mathbf { j }\). The point \(P\) lies on the line through \(A\) and \(B\), and \(\overrightarrow { A P } = \lambda \overrightarrow { A B }\).
  1. Show that \(\overrightarrow { O P } = ( 1 + 2 \lambda ) \mathbf { i } + ( 2 + 2 \lambda ) \mathbf { j } + ( 2 - 2 \lambda ) \mathbf { k }\).
  2. By equating expressions for \(\cos A O P\) and \(\cos B O P\) in terms of \(\lambda\), find the value of \(\lambda\) for which \(O P\) bisects the angle \(A O B\).
  3. When \(\lambda\) has this value, verify that \(A P : P B = O A : O B\).
CAIE P3 2011 November Q8
8 Let \(f ( x ) = \frac { 12 + 8 x - x ^ { 2 } } { ( 2 - x ) \left( 4 + x ^ { 2 } \right) }\).
  1. Express \(\mathrm { f } ( x )\) in the form \(\frac { A } { 2 - x } + \frac { B x + C } { 4 + x ^ { 2 } }\).
  2. Show that \(\int _ { 0 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x = \ln \left( \frac { 25 } { 2 } \right)\).
CAIE P3 2011 November Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{f421f03c-57c9-4feb-91b9-a7f9b12f96ce-3_666_956_1231_593} The diagram shows the curve \(y = x ^ { 2 } \ln x\) and its minimum point \(M\).
  1. Find the exact values of the coordinates of \(M\).
  2. Find the exact value of the area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = \mathrm { e }\).
CAIE P3 2011 November Q10
10
  1. Showing your working, find the two square roots of the complex number \(1 - ( 2 \sqrt { } 6 ) \mathrm { i }\). Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are exact.
  2. On a sketch of an Argand diagram, shade the region whose points represent the complex numbers \(z\) which satisfy the inequality \(| z - 3 i | \leqslant 2\). Find the greatest value of \(\arg z\) for points in this region.
CAIE P3 2011 November Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{9e129863-5994-4e17-81f8-e139515998d1-3_666_956_1231_593} The diagram shows the curve \(y = x ^ { 2 } \ln x\) and its minimum point \(M\).
  1. Find the exact values of the coordinates of \(M\).
  2. Find the exact value of the area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = \mathrm { e }\).
CAIE P3 2011 November Q1
1 Expand \(\frac { 16 } { ( 2 + x ) ^ { 2 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
CAIE P3 2011 November Q2
2 The equation of a curve is \(y = \frac { \mathrm { e } ^ { 2 x } } { 1 + \mathrm { e } ^ { 2 x } }\). Show that the gradient of the curve at the point for which \(x = \ln 3\) is \(\frac { 9 } { 50 }\).
CAIE P3 2011 November Q3
3
  1. Express \(8 \cos \theta + 15 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation \(8 \cos \theta + 15 \sin \theta = 12\), giving all solutions in the interval \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).