Questions P3 (1243 questions)

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CAIE P3 2016 March Q3
6 marks Standard +0.3
3 The equation \(x ^ { 5 } - 3 x ^ { 3 } + x ^ { 2 } - 4 = 0\) has one positive root.
  1. Verify by calculation that this root lies between 1 and 2 .
  2. Show that the equation can be rearranged in the form $$\left. x = \sqrt [ 3 ] { ( } 3 x + \frac { 4 } { x ^ { 2 } } - 1 \right)$$
  3. Use an iterative formula based on this rearrangement to determine the positive root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2016 March Q4
7 marks Moderate -0.5
4 The polynomial \(4 x ^ { 3 } + a x + 2\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that ( \(2 x + 1\) ) is a factor of \(\mathrm { p } ( x )\).
  1. Find the value of \(a\).
  2. When \(a\) has this value,
    1. factorise \(\mathrm { p } ( x )\),
    2. solve the inequality \(\mathrm { p } ( x ) > 0\), justifying your answer.
CAIE P3 2016 March Q5
7 marks Standard +0.8
5 Let \(I = \int _ { 0 } ^ { 1 } \frac { 9 } { \left( 3 + x ^ { 2 } \right) ^ { 2 } } \mathrm {~d} x\).
  1. Using the substitution \(x = ( \sqrt { } 3 ) \tan \theta\), show that \(I = \sqrt { } 3 \int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \cos ^ { 2 } \theta \mathrm {~d} \theta\).
  2. Hence find the exact value of \(I\).
CAIE P3 2016 March Q6
8 marks Challenging +1.2
6 A curve has equation $$\sin y \ln x = x - 2 \sin y$$ for \(- \frac { 1 } { 2 } \pi \leqslant y \leqslant \frac { 1 } { 2 } \pi\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
  2. Hence find the exact \(x\)-coordinate of the point on the curve at which the tangent is parallel to the \(x\)-axis.
CAIE P3 2016 March Q7
8 marks Standard +0.3
7 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = x \mathrm { e } ^ { x + y }$$ and it is given that \(y = 0\) when \(x = 0\).
  1. Solve the differential equation and obtain an expression for \(y\) in terms of \(x\).
  2. Explain briefly why \(x\) can only take values less than 1 .
CAIE P3 2016 March Q8
9 marks Standard +0.3
8 The line \(l\) has equation \(\mathbf { r } = \left( \begin{array} { r } 1 \\ 2 \\ - 1 \end{array} \right) + \lambda \left( \begin{array} { l } 2 \\ 1 \\ 3 \end{array} \right)\). The plane \(p\) has equation \(\mathbf { r } \cdot \left( \begin{array} { r } 2 \\ - 1 \\ - 1 \end{array} \right) = 6\).
  1. Show that \(l\) is parallel to \(p\).
  2. A line \(m\) lies in the plane \(p\) and is perpendicular to \(l\). The line \(m\) passes through the point with coordinates (5, 3, 1). Find a vector equation for \(m\).
CAIE P3 2016 March Q9
10 marks Standard +0.3
9 Let \(\mathrm { f } ( x ) = \frac { 3 x ^ { 3 } + 6 x - 8 } { x \left( x ^ { 2 } + 2 \right) }\).
  1. Express \(\mathrm { f } ( x )\) in the form \(A + \frac { B } { x } + \frac { C x + D } { x ^ { 2 } + 2 }\).
  2. Show that \(\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = 3 - \ln 4\).
CAIE P3 2016 March Q10
11 marks Standard +0.3
10
  1. Find the complex number \(z\) satisfying the equation \(z ^ { * } + 1 = 2 \mathrm { i } z\), where \(z ^ { * }\) denotes the complex conjugate of \(z\). Give your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
    1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers satisfying the inequalities \(| z + 1 - 3 \mathrm { i } | \leqslant 1\) and \(\operatorname { Im } z \geqslant 3\), where \(\operatorname { Im } z\) denotes the imaginary part of \(z\).
    2. Determine the difference between the greatest and least values of \(\arg z\) for points lying in this region.
CAIE P3 2017 March Q1
3 marks Moderate -0.5
1 Solve the equation \(\ln \left( 1 + 2 ^ { x } \right) = 2\), giving your answer correct to 3 decimal places.
CAIE P3 2017 March Q2
4 marks Standard +0.8
2 Solve the inequality \(| x - 4 | < 2 | 3 x + 1 |\).
CAIE P3 2017 March Q3
7 marks Moderate -0.3
3
  1. By sketching suitable graphs, show that the equation \(\mathrm { e } ^ { - \frac { 1 } { 2 } x } = 4 - x ^ { 2 }\) has one positive root and one negative root.
  2. Verify by calculation that the negative root lies between - 1 and - 1.5 .
  3. Use the iterative formula \(x _ { n + 1 } = - \sqrt { } \left( 4 - e ^ { - \frac { 1 } { 2 } x _ { n } } \right)\) to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2017 March Q4
7 marks Standard +0.3
4
  1. Express \(8 \cos \theta - 15 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), stating the exact value of \(R\) and giving the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation $$8 \cos 2 x - 15 \sin 2 x = 4$$ for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
CAIE P3 2017 March Q5
7 marks Challenging +1.2
5 The curve with equation \(y = \mathrm { e } ^ { - a x } \tan x\), where \(a\) is a positive constant, has only one point in the interval \(0 < x < \frac { 1 } { 2 } \pi\) at which the tangent is parallel to the \(x\)-axis. Find the value of \(a\) and state the exact value of the \(x\)-coordinate of this point.
CAIE P3 2017 March Q6
8 marks Standard +0.8
6 The line \(l\) has equation \(\mathbf { r } = \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k } + \lambda ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } )\). The plane \(p\) has equation \(3 x + y - 5 z = 20\).
  1. Show that the line \(l\) lies in the plane \(p\).
  2. A second plane is parallel to \(l\), perpendicular to \(p\) and contains the point with position vector \(3 \mathbf { i } - \mathbf { j } + 2 \mathbf { k }\). Find the equation of this plane, giving your answer in the form \(a x + b y + c z = d\). [5]
CAIE P3 2017 March Q7
9 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{e26f21c5-3776-4c86-8440-6959c5e37486-12_444_382_258_886} A water tank has vertical sides and a horizontal rectangular base, as shown in the diagram. The area of the base is \(2 \mathrm {~m} ^ { 2 }\). At time \(t = 0\) the tank is empty and water begins to flow into it at a rate of \(1 \mathrm {~m} ^ { 3 }\) per hour. At the same time water begins to flow out from the base at a rate of \(0.2 \sqrt { } h \mathrm {~m} ^ { 3 }\) per hour, where \(h \mathrm {~m}\) is the depth of water in the tank at time \(t\) hours.
  1. Form a differential equation satisfied by \(h\) and \(t\), and show that the time \(T\) hours taken for the depth of water to reach 4 m is given by $$T = \int _ { 0 } ^ { 4 } \frac { 10 } { 5 - \sqrt { } h } \mathrm {~d} h$$
  2. Using the substitution \(u = 5 - \sqrt { } h\), find the value of \(T\).
CAIE P3 2017 March Q9
10 marks Standard +0.8
9 Let \(\mathrm { f } ( x ) = \frac { x ( 6 - x ) } { ( 2 + x ) \left( 4 + x ^ { 2 } \right) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
CAIE P3 2017 March Q10
10 marks Standard +0.3
10 \includegraphics[max width=\textwidth, alt={}, center]{e26f21c5-3776-4c86-8440-6959c5e37486-18_337_529_260_808} The diagram shows the curve \(y = ( \ln x ) ^ { 2 }\). The \(x\)-coordinate of the point \(P\) is equal to e, and the normal to the curve at \(P\) meets the \(x\)-axis at \(Q\).
  1. Find the \(x\)-coordinate of \(Q\).
  2. Show that \(\int \ln x \mathrm {~d} x = x \ln x - x + c\), where \(c\) is a constant.
  3. Using integration by parts, or otherwise, find the exact value of the area of the shaded region between the curve, the \(x\)-axis and the normal \(P Q\).
CAIE P3 2019 March Q1
5 marks Moderate -0.8
1
  1. Show that the equation \(\log _ { 10 } ( x - 4 ) = 2 - \log _ { 10 } x\) can be written as a quadratic equation in \(x\).
  2. Hence solve the equation \(\log _ { 10 } ( x - 4 ) = 2 - \log _ { 10 } x\), giving your answer correct to 3 significant figures.
CAIE P3 2019 March Q2
5 marks Standard +0.3
2 The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 6 } + 12 x _ { n } } { 3 x _ { n } ^ { 5 } + 8 }$$ with initial value \(x _ { 1 } = 2\), converges to \(\alpha\).
  1. Use the formula to calculate \(\alpha\) correct to 4 decimal places. Give the result of each iteration to 6 decimal places.
  2. State an equation satisfied by \(\alpha\) and hence find the exact value of \(\alpha\).
CAIE P3 2019 March Q3
6 marks Standard +0.3
3
  1. Given that \(\sin \left( \theta + 45 ^ { \circ } \right) + 2 \cos \left( \theta + 60 ^ { \circ } \right) = 3 \cos \theta\), find the exact value of \(\tan \theta\) in a form involving surds. You need not simplify your answer.
  2. Hence solve the equation \(\sin \left( \theta + 45 ^ { \circ } \right) + 2 \cos \left( \theta + 60 ^ { \circ } \right) = 3 \cos \theta\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
CAIE P3 2019 March Q4
5 marks Standard +0.3
4 Show that \(\int _ { 1 } ^ { 4 } x ^ { - \frac { 3 } { 2 } } \ln x \mathrm {~d} x = 2 - \ln 4\).
CAIE P3 2019 March Q5
5 marks Standard +0.8
5 The variables \(x\) and \(y\) satisfy the relation \(\sin y = \tan x\), where \(- \frac { 1 } { 2 } \pi < y < \frac { 1 } { 2 } \pi\). Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \cos x \sqrt { } ( \cos 2 x ) } .$$
CAIE P3 2019 March Q6
7 marks Moderate -0.5
6 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = k y ^ { 3 } \mathrm { e } ^ { - x }$$ where \(k\) is a constant. It is given that \(y = 1\) when \(x = 0\), and that \(y = \sqrt { } \mathrm { e }\) when \(x = 1\). Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).
CAIE P3 2019 March Q7
10 marks Standard +0.3
7
  1. Showing all working and without using a calculator, solve the equation $$( 1 + \mathrm { i } ) z ^ { 2 } - ( 4 + 3 \mathrm { i } ) z + 5 + \mathrm { i } = 0$$ Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. The complex number \(u\) is given by $$u = - 1 - \mathrm { i }$$ On a sketch of an Argand diagram show the point representing \(u\). Shade the region whose points represent complex numbers satisfying the inequalities \(| z | < | z - 2 \mathrm { i } |\) and \(\frac { 1 } { 4 } \pi < \arg ( z - u ) < \frac { 1 } { 2 } \pi\).
CAIE P3 2019 March Q8
10 marks Standard +0.3
8 Let \(\mathrm { f } ( x ) = \frac { 12 + 12 x - 4 x ^ { 2 } } { ( 2 + x ) ( 3 - 2 x ) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).