Questions P3 (1203 questions)

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CAIE P3 2013 June Q9
9
  1. Express \(4 \cos \theta + 3 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Give the value of \(\alpha\) correct to 4 decimal places.
  2. Hence
    (a) solve the equation \(4 \cos \theta + 3 \sin \theta = 2\) for \(0 < \theta < 2 \pi\),
    (b) find \(\int \frac { 50 } { ( 4 \cos \theta + 3 \sin \theta ) ^ { 2 } } \mathrm {~d} \theta\).
CAIE P3 2013 June Q10
10 Liquid is flowing into a small tank which has a leak. Initially the tank is empty and, \(t\) minutes later, the volume of liquid in the tank is \(V \mathrm {~cm} ^ { 3 }\). The liquid is flowing into the tank at a constant rate of \(80 \mathrm {~cm} ^ { 3 }\) per minute. Because of the leak, liquid is being lost from the tank at a rate which, at any instant, is equal to \(k V \mathrm {~cm} ^ { 3 }\) per minute where \(k\) is a positive constant.
  1. Write down a differential equation describing this situation and solve it to show that $$V = \frac { 1 } { k } \left( 80 - 80 \mathrm { e } ^ { - k t } \right)$$
  2. It is observed that \(V = 500\) when \(t = 15\), so that \(k\) satisfies the equation $$k = \frac { 4 - 4 e ^ { - 15 k } } { 25 }$$ Use an iterative formula, based on this equation, to find the value of \(k\) correct to 2 significant figures. Use an initial value of \(k = 0.1\) and show the result of each iteration to 4 significant figures.
  3. Determine how much liquid there is in the tank 20 minutes after the liquid started flowing, and state what happens to the volume of liquid in the tank after a long time.
CAIE P3 2013 June Q1
1 Solve the equation \(| x - 2 | = \left| \frac { 1 } { 3 } x \right|\).
CAIE P3 2013 June Q2
2 The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { x _ { n } \left( x _ { n } ^ { 3 } + 100 \right) } { 2 \left( x _ { n } ^ { 3 } + 25 \right) }$$ with initial value \(x _ { 1 } = 3.5\), converges to \(\alpha\).
  1. Use this formula to calculate \(\alpha\) correct to 4 decimal places, showing 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 2013 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{7c125770-1ded-4763-8453-b07ef43e83e9-2_392_727_927_708} The variables \(x\) and \(y\) satisfy the equation \(y = A e ^ { - k x ^ { 2 } }\), where \(A\) and \(k\) are constants. The graph of \(\ln y\) against \(x ^ { 2 }\) is a straight line passing through the points \(( 0.64,0.76 )\) and \(( 1.69,0.32 )\), as shown in the diagram. Find the values of \(A\) and \(k\) correct to 2 decimal places.
CAIE P3 2013 June Q4
4 The polynomial \(a x ^ { 3 } - 20 x ^ { 2 } + x + 3\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that \(( 3 x + 1 )\) is a factor of \(\mathrm { p } ( x )\).
  1. Find the value of \(a\).
  2. When \(a\) has this value, factorise \(\mathrm { p } ( x )\) completely.
CAIE P3 2013 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{7c125770-1ded-4763-8453-b07ef43e83e9-2_446_601_1969_772} The diagram shows the curve with equation $$x ^ { 3 } + x y ^ { 2 } + a y ^ { 2 } - 3 a x ^ { 2 } = 0$$ where \(a\) is a positive constant. The maximum point on the curve is \(M\). Find the \(x\)-coordinate of \(M\) in terms of \(a\).
  1. By differentiating \(\frac { 1 } { \cos x }\), show that the derivative of \(\sec x\) is \(\sec x \tan x\). Hence show that if \(y = \ln ( \sec x + \tan x )\) then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec x\).
  2. Using the substitution \(x = ( \sqrt { } 3 ) \tan \theta\), find the exact value of $$\int _ { 1 } ^ { 3 } \frac { 1 } { \sqrt { \left( 3 + x ^ { 2 } \right) } } \mathrm { d } x$$ expressing your answer as a single logarithm.
CAIE P3 2013 June Q10
10 The points \(A\) and \(B\) have position vectors \(2 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k }\) and \(5 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }\) respectively. The plane \(p\) has equation \(x + y = 5\).
  1. Find the position vector of the point of intersection of the line through \(A\) and \(B\) and the plane \(p\).
  2. A second plane \(q\) has an equation of the form \(x + b y + c z = d\), where \(b , c\) and \(d\) are constants. The plane \(q\) contains the line \(A B\), and the acute angle between the planes \(p\) and \(q\) is \(60 ^ { \circ }\). Find the equation of \(q\).
CAIE P3 2013 June Q1
1 Solve the inequality \(| 4 x + 3 | > | x |\).
CAIE P3 2013 June Q2
2 It is given that \(\ln ( y + 1 ) - \ln y = 1 + 3 \ln x\). Express \(y\) in terms of \(x\), in a form not involving logarithms.
CAIE P3 2013 June Q3
3 Solve the equation \(\tan 2 x = 5 \cot x\), for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
CAIE P3 2013 June Q4
4
  1. Express \(( \sqrt { } 3 ) \cos x + \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), giving the exact values of \(R\) and \(\alpha\).
  2. Hence show that $$\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 2 } \pi } \frac { 1 } { ( ( \sqrt { } 3 ) \cos x + \sin x ) ^ { 2 } } \mathrm {~d} x = \frac { 1 } { 4 } \sqrt { } 3$$
CAIE P3 2013 June Q5
5 The polynomial \(8 x ^ { 3 } + a x ^ { 2 } + b x + 3\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( 2 x + 1 )\) is a factor of \(\mathrm { p } ( x )\) and that when \(\mathrm { p } ( x )\) is divided by ( \(2 x - 1\) ) the remainder is 1 .
  1. Find the values of \(a\) and \(b\).
  2. When \(a\) and \(b\) have these values, find the remainder when \(\mathrm { p } ( x )\) is divided by \(2 x ^ { 2 } - 1\).
CAIE P3 2013 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{436d891d-92ee-4076-8369-db756d413979-2_435_597_1516_776} The diagram shows the curves \(y = \mathrm { e } ^ { 2 x - 3 }\) and \(y = 2 \ln x\). When \(x = a\) the tangents to the curves are parallel.
  1. Show that \(a\) satisfies the equation \(a = \frac { 1 } { 2 } ( 3 - \ln a )\).
  2. Verify by calculation that this equation has a root between 1 and 2 .
  3. Use the iterative formula \(a _ { n + 1 } = \frac { 1 } { 2 } \left( 3 - \ln a _ { n } \right)\) to calculate \(a\) correct to 2 decimal places, showing the result of each iteration to 4 decimal places.
CAIE P3 2013 June Q7
7 The complex number \(z\) is defined by \(z = a + \mathrm { i } b\), where \(a\) and \(b\) are real. The complex conjugate of \(z\) is denoted by \(z ^ { * }\).
  1. Show that \(| z | ^ { 2 } = z z ^ { * }\) and that \(( z - k \mathrm { i } ) ^ { * } = z ^ { * } + k \mathrm { i }\), where \(k\) is real. In an Argand diagram a set of points representing complex numbers \(z\) is defined by the equation \(| z - 10 \mathrm { i } | = 2 | z - 4 \mathrm { i } |\).
  2. Show, by squaring both sides, that $$z z ^ { * } - 2 \mathrm { i } z ^ { * } + 2 \mathrm { i } z - 12 = 0$$ Hence show that \(| z - 2 i | = 4\).
  3. Describe the set of points geometrically.
CAIE P3 2013 June Q8
8 The variables \(x\) and \(t\) satisfy the differential equation $$t \frac { \mathrm {~d} x } { \mathrm {~d} t } = \frac { k - x ^ { 3 } } { 2 x ^ { 2 } }$$ for \(t > 0\), where \(k\) is a constant. When \(t = 1 , x = 1\) and when \(t = 4 , x = 2\).
  1. Solve the differential equation, finding the value of \(k\) and obtaining an expression for \(x\) in terms of \(t\).
  2. State what happens to the value of \(x\) as \(t\) becomes large.
CAIE P3 2013 June Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{436d891d-92ee-4076-8369-db756d413979-3_307_601_1553_772} The diagram shows the curve \(y = \sin ^ { 2 } 2 x \cos x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\).
  1. Find the \(x\)-coordinate of \(M\).
  2. Using the substitution \(u = \sin x\), find by integration the area of the shaded region bounded by the curve and the \(x\)-axis.
CAIE P3 2013 June Q10
10 The line \(l\) has equation \(\mathbf { r } = \mathbf { i } + \mathbf { j } + \mathbf { k } + \lambda ( a \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )\), where \(a\) is a constant. The plane \(p\) has equation \(x + 2 y + 2 z = 6\). Find the value or values of \(a\) in each of the following cases.
  1. The line \(l\) is parallel to the plane \(p\).
  2. The line \(l\) intersects the line passing through the points with position vectors \(3 \mathbf { i } + 2 \mathbf { j } + \mathbf { k }\) and \(\mathbf { i } + \mathbf { j } - \mathbf { k }\).
  3. The acute angle between the line \(l\) and the plane \(p\) is \(\tan ^ { - 1 } 2\).
CAIE P3 2014 June Q1
1
  1. Simplify \(\sin 2 \alpha \sec \alpha\).
  2. Given that \(3 \cos 2 \beta + 7 \cos \beta = 0\), find the exact value of \(\cos \beta\).
CAIE P3 2014 June Q2
2 Use the substitution \(u = 1 + 3 \tan x\) to find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { \sqrt { } ( 1 + 3 \tan x ) } { \cos ^ { 2 } x } d x$$
CAIE P3 2014 June Q3
3 The parametric equations of a curve are $$x = \ln ( 2 t + 3 ) , \quad y = \frac { 3 t + 2 } { 2 t + 3 }$$ Find the gradient of the curve at the point where it crosses the \(y\)-axis.
CAIE P3 2014 June Q4
4 The variables \(x\) and \(y\) are related by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 y \mathrm { e } ^ { 3 x } } { 2 + \mathrm { e } ^ { 3 x } }$$ Given that \(y = 36\) when \(x = 0\), find an expression for \(y\) in terms of \(x\).
CAIE P3 2014 June Q5
5 The complex number \(z\) is defined by \(z = \frac { 9 \sqrt { } 3 + 9 i } { \sqrt { } 3 - i }\). Find, showing all your working,
  1. an expression for \(z\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\),
  2. the two square roots of \(z\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
CAIE P3 2014 June Q6
6 It is given that \(2 \ln ( 4 x - 5 ) + \ln ( x + 1 ) = 3 \ln 3\).
  1. Show that \(16 x ^ { 3 } - 24 x ^ { 2 } - 15 x - 2 = 0\).
  2. By first using the factor theorem, factorise \(16 x ^ { 3 } - 24 x ^ { 2 } - 15 x - 2\) completely.
  3. Hence solve the equation \(2 \ln ( 4 x - 5 ) + \ln ( x + 1 ) = 3 \ln 3\).
CAIE P3 2014 June Q7
7 The straight line \(l\) has equation \(\mathbf { r } = 4 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } + \lambda ( 2 \mathbf { i } - 3 \mathbf { j } + 6 \mathbf { k } )\). The plane \(p\) passes through the point \(( 4 , - 1,2 )\) and is perpendicular to \(l\).
  1. Find the equation of \(p\), giving your answer in the form \(a x + b y + c z = d\).
  2. Find the perpendicular distance from the origin to \(p\).
  3. A second plane \(q\) is parallel to \(p\) and the perpendicular distance between \(p\) and \(q\) is 14 units. Find the possible equations of \(q\).