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CAIE P2 2010 June Q5
7 marks Moderate -0.8
5 The equation of a curve is \(y = x ^ { 3 } \mathrm { e } ^ { - x }\).
  1. Show that the curve has a stationary point where \(x = 3\).
  2. Find the equation of the tangent to the curve at the point where \(x = 1\).
CAIE P2 2010 June Q7
9 marks Moderate -0.3
7 The polynomial \(2 x ^ { 3 } + a x ^ { 2 } + b x + 6\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that when \(\mathrm { p } ( x )\) is divided by ( \(x - 3\) ) the remainder is 30 , and that when \(\mathrm { p } ( x )\) is divided by ( \(x + 1\) ) the remainder is 18 .
  1. Find the values of \(a\) and \(b\).
  2. When \(a\) and \(b\) have these values, verify that ( \(x - 2\) ) is a factor of \(\mathrm { p } ( x )\) and hence factorise \(\mathrm { p } ( x )\) completely.
CAIE P2 2010 June Q8
9 marks Standard +0.3
8
  1. Prove the identity $$\sin \left( x - 30 ^ { \circ } \right) + \cos \left( x - 60 ^ { \circ } \right) \equiv ( \sqrt { } 3 ) \sin x$$
  2. Hence solve the equation $$\sin \left( x - 30 ^ { \circ } \right) + \cos \left( x - 60 ^ { \circ } \right) = \frac { 1 } { 2 } \sec x$$ for \(0 ^ { \circ } < x < 360 ^ { \circ }\).
CAIE P2 2010 June Q2
4 marks Moderate -0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{af7c2b6e-1293-4744-8ea0-927fea5ab4ec-2_531_949_431_598} The diagram shows part of the curve \(y = x \mathrm { e } ^ { - x }\). The shaded region \(R\) is bounded by the curve and by the lines \(x = 2 , x = 3\) and \(y = 0\).
  1. Use the trapezium rule with two intervals to estimate the area of \(R\), giving your answer correct to 2 decimal places.
  2. State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the area of \(R\).
CAIE P2 2011 June Q1
3 marks Moderate -0.5
1 Solve the equation \(| 3 x + 4 | = | 2 x + 5 |\).
CAIE P2 2011 June Q2
4 marks Moderate -0.5
2 A curve has parametric equations $$x = 3 t + \sin 2 t , \quad y = 4 + 2 \cos 2 t$$ Find the exact gradient of the curve at the point for which \(t = \frac { 1 } { 6 } \pi\).
CAIE P2 2011 June Q3
5 marks Moderate -0.5
3 \includegraphics[max width=\textwidth, alt={}, center]{d90dc270-b304-4b42-8e0e-37641b8a03b8-2_556_1113_680_516} The variables \(x\) and \(y\) satisfy the equation \(y = K x ^ { m }\), where \(K\) and \(m\) are constants. The graph of \(\ln y\) against \(\ln x\) is a straight line passing through the points \(( 0,2.0 )\) and \(( 6,10.2 )\), as shown in the diagram. Find the values of \(K\) and \(m\), correct to 2 decimal places.
CAIE P2 2011 June Q4
5 marks Moderate -0.8
4 The polynomial \(\mathrm { f } ( x )\) is defined by $$f ( x ) = 3 x ^ { 3 } + a x ^ { 2 } + a x + a$$ where \(a\) is a constant.
  1. Given that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\), find the value of \(a\).
  2. When \(a\) has the value found in part (i), find the quotient when \(\mathrm { f } ( x )\) is divided by ( \(x + 2\) ).
CAIE P2 2011 June Q5
7 marks Moderate -0.8
5 Find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = 4\) in each of the following cases:
  1. \(y = x \ln ( x - 3 )\),
  2. \(y = \frac { x - 1 } { x + 1 }\).
CAIE P2 2011 June Q6
8 marks Moderate -0.3
6
  1. Find \(\int 4 \mathrm { e } ^ { x } \left( 3 + \mathrm { e } ^ { 2 x } \right) \mathrm { d } x\).
  2. Show that \(\int _ { - \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 4 } \pi } \left( 3 + 2 \tan ^ { 2 } \theta \right) \mathrm { d } \theta = \frac { 1 } { 2 } ( 8 + \pi )\).
CAIE P2 2011 June Q7
9 marks Standard +0.3
7
  1. By sketching a suitable pair of graphs, show that the equation $$\mathrm { e } ^ { 2 x } = 14 - x ^ { 2 }$$ has exactly two real roots.
  2. Show by calculation that the positive root lies between 1.2 and 1.3.
  3. Show that this root also satisfies the equation $$x = \frac { 1 } { 2 } \ln \left( 14 - x ^ { 2 } \right) .$$
  4. Use an iteration process based on the equation in part (iii), with a suitable starting value, to find the root correct to 2 decimal places. Give the result of each step of the process to 4 decimal places.
  5. Express \(4 \sin \theta - 6 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places.
  6. Solve the equation \(4 \sin \theta - 6 \cos \theta = 3\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
  7. Find the greatest and least possible values of \(( 4 \sin \theta - 6 \cos \theta ) ^ { 2 } + 8\) as \(\theta\) varies.
CAIE P2 2011 June Q1
4 marks Moderate -0.8
1 Use logarithms to solve the equation \(3 ^ { x } = 2 ^ { x + 2 }\), giving your answer correct to 3 significant figures.
CAIE P2 2011 June Q2
5 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{2c27f384-5289-4c1b-9199-6b4c6ac81e38-2_645_750_429_699} The diagram shows the curve \(y = \sqrt { } \left( 1 + x ^ { 3 } \right)\). Region \(A\) is bounded by the curve and the lines \(x = 0\), \(x = 2\) and \(y = 0\). Region \(B\) is bounded by the curve and the lines \(x = 0\) and \(y = 3\).
  1. Use the trapezium rule with two intervals to find an approximation to the area of region \(A\). Give your answer correct to 2 decimal places.
  2. Deduce an approximation to the area of region \(B\) and explain why this approximation underestimates the true area of region \(B\).
CAIE P2 2011 June Q3
5 marks Moderate -0.3
3 The sequence \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\) defined by $$x _ { 1 } = 1 , \quad x _ { n + 1 } = \frac { 1 } { 2 } \sqrt [ 3 ] { } \left( x _ { n } ^ { 2 } + 6 \right)$$ converges to the value \(\alpha\).
  1. Find the value of \(\alpha\) correct to 3 decimal places. Show your working, giving each calculated value of the sequence to 5 decimal places.
  2. Find, in the form \(a x ^ { 3 } + b x ^ { 2 } + c = 0\), an equation of which \(\alpha\) is a root.
CAIE P2 2011 June Q4
6 marks Moderate -0.8
4
  1. Find the value of \(\int _ { 0 } ^ { \frac { 2 } { 3 } \pi } \sin \left( \frac { 1 } { 2 } x \right) \mathrm { d } x\).
  2. Find \(\int \mathrm { e } ^ { - x } \left( 1 + \mathrm { e } ^ { x } \right) \mathrm { d } x\).
CAIE P2 2011 June Q5
6 marks Moderate -0.3
5 A curve has equation \(x ^ { 2 } + 2 y ^ { 2 } + 5 x + 6 y = 10\). Find the equation of the tangent to the curve at the point \(( 2 , - 1 )\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
[0pt] [6]
CAIE P2 2011 June Q6
7 marks Standard +0.3
6 The curve \(y = 4 x ^ { 2 } \ln x\) has one stationary point.
  1. Find the coordinates of this stationary point, giving your answers correct to 3 decimal places.
  2. Determine whether this point is a maximum or a minimum point.
CAIE P2 2011 June Q7
8 marks Moderate -0.3
7 The cubic polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 6 x ^ { 3 } + a x ^ { 2 } + b x + 10$$ where \(a\) and \(b\) are constants. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) and that, when \(\mathrm { p } ( x )\) is divided by ( \(x + 1\) ), the remainder is 24 .
  1. Find the values of \(a\) and \(b\).
  2. When \(a\) and \(b\) have these values, factorise \(\mathrm { p } ( x )\) completely.
CAIE P2 2011 June Q8
9 marks Standard +0.8
8
  1. Prove that \(\sin ^ { 2 } 2 \theta \left( \operatorname { cosec } ^ { 2 } \theta - \sec ^ { 2 } \theta \right) \equiv 4 \cos 2 \theta\).
  2. Hence
    1. solve for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\) the equation \(\sin ^ { 2 } 2 \theta \left( \operatorname { cosec } ^ { 2 } \theta - \sec ^ { 2 } \theta \right) = 3\),
    2. find the exact value of \(\operatorname { cosec } ^ { 2 } 15 ^ { \circ } - \sec ^ { 2 } 15 ^ { \circ }\).
CAIE P2 2011 June Q2
5 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{ee420db2-bef4-4c2b-8dd2-c8f439dd561e-2_645_750_429_699} The diagram shows the curve \(y = \sqrt { } \left( 1 + x ^ { 3 } \right)\). Region \(A\) is bounded by the curve and the lines \(x = 0\), \(x = 2\) and \(y = 0\). Region \(B\) is bounded by the curve and the lines \(x = 0\) and \(y = 3\).
  1. Use the trapezium rule with two intervals to find an approximation to the area of region \(A\). Give your answer correct to 2 decimal places.
  2. Deduce an approximation to the area of region \(B\) and explain why this approximation underestimates the true area of region \(B\).
CAIE P2 2012 June Q1
4 marks Moderate -0.3
1 Solve the equation \(\left| x ^ { 3 } - 14 \right| = 13\), showing all your working.
CAIE P2 2012 June Q2
5 marks Moderate -0.5
2 \includegraphics[max width=\textwidth, alt={}, center]{beb8df77-e091-4248-812b-20e885c42e37-2_453_771_386_685} The variables \(x\) and \(y\) satisfy the equation \(y = A \left( b ^ { x } \right)\), where \(A\) and \(b\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \(( 0,2.14 )\) and \(( 5,4.49 )\), as shown in the diagram. Find the values of \(A\) and \(b\), correct to 1 decimal place.
CAIE P2 2012 June Q3
7 marks Moderate -0.3
3 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } - 3 x ^ { 2 } - 5 x + a + 4 ,$$ where \(a\) is a constant.
  1. Given that \(( x - 2 )\) is a factor of \(\mathrm { p } ( x )\), find the value of \(a\).
  2. When \(a\) has this value,
    1. factorise \(\mathrm { p } ( x )\) completely,
    2. find the remainder when \(\mathrm { p } ( x )\) is divided by \(( x + 1 )\).
CAIE P2 2012 June Q4
7 marks Standard +0.3
4
  1. Given that \(35 + \sec ^ { 2 } \theta = 12 \tan \theta\), find the value of \(\tan \theta\).
  2. Hence, showing the use of an appropriate formula in each case, find the exact value of
    1. \(\tan \left( \theta - 45 ^ { \circ } \right)\),
    2. \(\tan 2 \theta\).
CAIE P2 2012 June Q5
9 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{beb8df77-e091-4248-812b-20e885c42e37-3_528_757_251_694} The diagram shows the curve \(y = 4 e ^ { \frac { 1 } { 2 } x } - 6 x + 3\) and its minimum point \(M\).
  1. Show that the \(x\)-coordinate of \(M\) can be written in the form \(\ln a\), where the value of \(a\) is to be stated.
  2. Find the exact value of the area of the region enclosed by the curve and the lines \(x = 0 , x = 2\) and \(y = 0\).