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CAIE P2 2014 June Q8
9 marks Standard +0.8
8 \includegraphics[max width=\textwidth, alt={}, center]{de8af872-9f77-4787-8e66-ed199405ca25-3_581_650_1272_744} The diagram shows the curve $$y = \tan x \cos 2 x , \text { for } 0 \leqslant x < \frac { 1 } { 2 } \pi$$ and its maximum point \(M\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 \cos ^ { 2 } x - \sec ^ { 2 } x - 2\).
  2. Hence find the \(x\)-coordinate of \(M\), giving your answer correct to 2 decimal places.
CAIE P2 2014 June Q5
6 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{22ba6cc7-7375-434e-9eaa-d536684dd727-2_583_597_1457_772} The variables \(x\) and \(y\) satisfy the equation \(y = K \left( 2 ^ { p x } \right)\), where \(K\) and \(p\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points ( \(1.35,1.87\) ) and ( \(3.35,3.81\) ), as shown in the diagram. Find the values of \(K\) and \(p\) correct to 2 decimal places.
[0pt] [6]
CAIE P2 2014 June Q8
9 marks Standard +0.8
8 \includegraphics[max width=\textwidth, alt={}, center]{22ba6cc7-7375-434e-9eaa-d536684dd727-3_581_650_1272_744} The diagram shows the curve $$y = \tan x \cos 2 x , \text { for } 0 \leqslant x < \frac { 1 } { 2 } \pi$$ and its maximum point \(M\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 \cos ^ { 2 } x - \sec ^ { 2 } x - 2\).
  2. Hence find the \(x\)-coordinate of \(M\), giving your answer correct to 2 decimal places.
CAIE P2 2015 June Q1
5 marks Moderate -0.3
1
  1. Solve the equation \(| 3 x + 4 | = | 3 x - 11 |\).
  2. Hence, using logarithms, solve the equation \(\left| 3 \times 2 ^ { y } + 4 \right| = \left| 3 \times 2 ^ { y } - 11 \right|\), giving the answer correct to 3 significant figures.
CAIE P2 2015 June Q2
5 marks Moderate -0.5
2 \includegraphics[max width=\textwidth, alt={}, center]{595e38f4-c52e-4509-8b16-f08e30dec96b-2_456_716_529_712} The variables \(x\) and \(y\) satisfy the equation $$y = A \mathrm { e } ^ { p ( x - 1 ) } ,$$ where \(A\) and \(p\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \(( 2,1.60 )\) and \(( 5,2.92 )\), as shown in the diagram. Find the values of \(A\) and \(p\) correct to 2 significant figures.
CAIE P2 2015 June Q3
5 marks Moderate -0.3
3 The equation of a curve is $$y = 6 \sin x - 2 \cos 2 x$$ Find the equation of the tangent to the curve at the point \(\left( \frac { 1 } { 6 } \pi , 2 \right)\). Give the answer in the form \(y = m x + c\), where the values of \(m\) and \(c\) are correct to 3 significant figures.
CAIE P2 2015 June Q4
7 marks Standard +0.3
4 The polynomials \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are defined by $$\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } + b \quad \text { and } \quad \mathrm { g } ( x ) = x ^ { 3 } + b x ^ { 2 } - a$$ where \(a\) and \(b\) are constants. It is given that ( \(x + 2\) ) is a factor of \(\mathrm { f } ( x )\). It is also given that, when \(\mathrm { g } ( x )\) 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, find the greatest possible value of \(\mathrm { g } ( x ) - \mathrm { f } ( x )\) as \(x\) varies.
CAIE P2 2015 June Q5
8 marks Standard +0.3
5
  1. Given that \(\int _ { 0 } ^ { a } \left( 3 \mathrm { e } ^ { \frac { 1 } { 2 } x } + 1 \right) \mathrm { d } x = 10\), show that the positive constant \(a\) satisfies the equation $$a = 2 \ln \left( \frac { 16 - a } { 6 } \right)$$
  2. Use the iterative formula \(a _ { n + 1 } = 2 \ln \left( \frac { 16 - a _ { n } } { 6 } \right)\) with \(a _ { 1 } = 2\) to find the value of \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
CAIE P2 2015 June Q6
10 marks Standard +0.3
6
  1. Prove that \(2 \operatorname { cosec } 2 \theta \tan \theta \equiv \sec ^ { 2 } \theta\).
  2. Hence
    (a) solve the equation \(2 \operatorname { cosec } 2 \theta \tan \theta = 5\) for \(0 < \theta < \pi\),
    (b) find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } 2 \operatorname { cosec } 4 x \tan 2 x \mathrm {~d} x\).
CAIE P2 2015 June Q7
10 marks Standard +0.3
7 The equation of a curve is $$y ^ { 3 } + 4 x y = 16$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 4 y } { 3 y ^ { 2 } + 4 x }\).
  2. Show that the curve has no stationary points.
  3. Find the coordinates of the point on the curve where the tangent is parallel to the \(y\)-axis.
CAIE P2 2015 June Q1
4 marks Moderate -0.8
1
  1. Use logarithms to solve the equation \(2 ^ { x } = 20 ^ { 5 }\), giving the answer correct to 3 significant figures.
  2. Hence determine the number of integers \(n\) satisfying $$20 ^ { - 5 } < 2 ^ { n } < 20 ^ { 5 }$$
CAIE P2 2015 June Q2
6 marks Moderate -0.8
2
  1. Given that \(( x + 2 )\) is a factor of $$4 x ^ { 3 } + a x ^ { 2 } - ( a + 1 ) x - 18$$ find the value of the constant \(a\).
  2. When \(a\) has this value, factorise \(4 x ^ { 3 } + a x ^ { 2 } - ( a + 1 ) x - 18\) completely.
CAIE P2 2015 June Q3
6 marks Standard +0.3
3 It is given that \(\theta\) is an acute angle measured in degrees such that $$2 \sec ^ { 2 } \theta + 3 \tan \theta = 22$$
  1. Find the value of \(\tan \theta\).
  2. Use an appropriate formula to find the exact value of \(\tan \left( \theta + 135 ^ { \circ } \right)\).
CAIE P2 2015 June Q4
7 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{cc051d68-7e21-4dc1-b34d-6fb7f12a52fd-2_524_625_1425_758} The diagram shows the curve \(y = \mathrm { e } ^ { x } + 4 \mathrm { e } ^ { - 2 x }\) and its minimum point \(M\).
  1. Show that the \(x\)-coordinate of \(M\) is \(\ln 2\).
  2. The region shaded in the diagram is enclosed by the curve and the lines \(x = 0 , x = \ln 2\) and \(y = 0\). Use integration to show that the area of the shaded region is \(\frac { 5 } { 2 }\).
CAIE P2 2015 June Q6
9 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{cc051d68-7e21-4dc1-b34d-6fb7f12a52fd-3_401_586_817_778} The diagram shows part of the curve with equation $$y = 4 \sin ^ { 2 } x + 8 \sin x + 3$$ and its point of intersection \(P\) with the \(x\)-axis.
  1. Find the exact \(x\)-coordinate of \(P\).
  2. Show that the equation of the curve can be written $$y = 5 + 8 \sin x - 2 \cos 2 x$$ and use integration to find the exact area of the shaded region enclosed by the curve and the axes.
CAIE P2 2015 June Q7
10 marks Standard +0.3
7
  1. Find the gradient of the curve $$3 \ln x + 4 \ln y + 6 x y = 6$$ at the point \(( 1,1 )\).
  2. The parametric equations of a curve are $$x = \frac { 10 } { t } - t , \quad y = \sqrt { } ( 2 t - 1 ) .$$ Find the gradient of the curve at the point \(( - 3,3 )\).
CAIE P2 2015 June Q4
7 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{3b217eb4-3bd3-4800-a913-749754bf109f-2_524_625_1425_758} The diagram shows the curve \(y = \mathrm { e } ^ { x } + 4 \mathrm { e } ^ { - 2 x }\) and its minimum point \(M\).
  1. Show that the \(x\)-coordinate of \(M\) is \(\ln 2\).
  2. The region shaded in the diagram is enclosed by the curve and the lines \(x = 0 , x = \ln 2\) and \(y = 0\). Use integration to show that the area of the shaded region is \(\frac { 5 } { 2 }\).
CAIE P2 2015 June Q6
9 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{3b217eb4-3bd3-4800-a913-749754bf109f-3_401_586_817_778} The diagram shows part of the curve with equation $$y = 4 \sin ^ { 2 } x + 8 \sin x + 3$$ and its point of intersection \(P\) with the \(x\)-axis.
  1. Find the exact \(x\)-coordinate of \(P\).
  2. Show that the equation of the curve can be written $$y = 5 + 8 \sin x - 2 \cos 2 x$$ and use integration to find the exact area of the shaded region enclosed by the curve and the axes.
CAIE P2 2016 June Q1
3 marks Easy -1.2
1 Find the gradient of the curve $$y = 3 e ^ { 4 x } - 6 \ln ( 2 x + 3 )$$ at the point for which \(x = 0\).
CAIE P2 2016 June Q2
5 marks Standard +0.3
2 Solve the equation \(5 \tan 2 \theta = 4 \cot \theta\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P2 2016 June Q3
6 marks Moderate -0.3
3 Given that \(3 \mathrm { e } ^ { x } + 8 \mathrm { e } ^ { - x } = 14\), find the possible values of \(\mathrm { e } ^ { x }\) and hence solve the equation \(3 \mathrm { e } ^ { x } + 8 \mathrm { e } ^ { - x } = 14\) correct to 3 significant figures.
CAIE P2 2016 June Q4
7 marks Standard +0.3
4 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 8 x ^ { 3 } + 30 x ^ { 2 } + 13 x - 25$$
  1. Find the quotient when \(\mathrm { p } ( x )\) is divided by ( \(x + 2\) ), and show that the remainder is 5 .
  2. Hence factorise \(\mathrm { p } ( x ) - 5\) completely.
  3. Write down the roots of the equation \(\mathrm { p } ( | x | ) - 5 = 0\).
CAIE P2 2016 June Q5
9 marks Standard +0.3
5 A curve is defined by the parametric equations $$x = 2 \tan \theta , \quad y = 3 \sin 2 \theta$$ for \(0 \leqslant \theta < \frac { 1 } { 2 } \pi\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 \cos ^ { 4 } \theta - 3 \cos ^ { 2 } \theta\).
  2. Find the coordinates of the stationary point.
  3. Find the gradient of the curve at the point \(\left( 2 \sqrt { } 3 , \frac { 3 } { 2 } \sqrt { } 3 \right)\).
CAIE P2 2016 June Q6
10 marks Standard +0.3
6 The equation of a curve is \(y = \frac { 3 x ^ { 2 } } { x ^ { 2 } + 4 }\). At the point on the curve with positive \(x\)-coordinate \(p\), the gradient of the curve is \(\frac { 1 } { 2 }\).
  1. Show that \(p = \sqrt { } \left( \frac { 48 p - 16 } { p ^ { 2 } + 8 } \right)\).
  2. Show by calculation that \(2 < p < 3\).
  3. Use an iterative formula based on the equation in part (i) to find the value of \(p\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
CAIE P2 2016 June Q7
10 marks Standard +0.3
7
  1. Find \(\int \frac { 1 + \cos ^ { 4 } 2 x } { \cos ^ { 2 } 2 x } \mathrm {~d} x\).
  2. Without using a calculator, find the exact value of \(\int _ { 4 } ^ { 14 } \left( 2 + \frac { 6 } { 3 x - 2 } \right) \mathrm { d } x\), giving your answer in the form \(\ln \left( a \mathrm { e } ^ { b } \right)\), where \(a\) and \(b\) are integers.