Questions — CAIE P2 (699 questions)

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CAIE P2 2015 June Q2
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
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
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
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
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
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
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
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
2 Solve the equation \(5 \tan 2 \theta = 4 \cot \theta\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P2 2016 June Q3
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
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
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
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
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.
CAIE P2 2016 June Q1
1 Given that \(5 ^ { 3 x } = 7 ^ { 4 y }\), use logarithms to find the value of \(\frac { x } { y }\) correct to 4 significant figures.
CAIE P2 2016 June Q2
2
  1. Find the quotient and remainder when \(2 x ^ { 3 } - 7 x ^ { 2 } - 9 x + 3\) is divided by \(x ^ { 2 } - 2 x + 5\).
  2. Hence find the values of the constants \(p\) and \(q\) such that \(x ^ { 2 } - 2 x + 5\) is a factor of \(2 x ^ { 3 } - 7 x ^ { 2 } + p x + q\).
CAIE P2 2016 June Q3
3
  1. Solve the equation \(| 3 u + 1 | = | 2 u - 5 |\).
  2. Hence solve the equation \(| 3 \cot x + 1 | = | 2 \cot x - 5 |\) for \(0 < x < \frac { 1 } { 2 } \pi\), giving your answer correct to 3 significant figures.
CAIE P2 2016 June Q4
4
  1. Show that \(\sin \left( \theta + 60 ^ { \circ } \right) + \sin \left( \theta + 120 ^ { \circ } \right) \equiv ( \sqrt { } 3 ) \cos \theta\).
  2. Hence
    (a) find the exact value of \(\sin 105 ^ { \circ } + \sin 165 ^ { \circ }\),
    (b) solve the equation \(\sin \left( \theta + 60 ^ { \circ } \right) + \sin \left( \theta + 120 ^ { \circ } \right) = \sec \theta\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P2 2016 June Q5
5 The equation of a curve is \(y = 6 x \mathrm { e } ^ { \frac { 1 } { 3 } x }\). At the point on the curve with \(x\)-coordinate \(p\), the gradient of the curve is 40 .
  1. Show that \(p = 3 \ln \left( \frac { 20 } { p + 3 } \right)\).
  2. Show by calculation that \(3.3 < p < 3.5\).
  3. Use an iterative formula based on the equation in part (i) to find the value of \(p\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
CAIE P2 2016 June Q6
6
  1. Find \(\int \frac { 4 + \mathrm { e } ^ { x } } { 2 \mathrm { e } ^ { 2 x } } \mathrm {~d} x\).
  2. Without using a calculator, find \(\int _ { 2 } ^ { 10 } \frac { 1 } { 2 x + 5 } \mathrm {~d} x\), giving your answer in the form \(\ln k\).

  3. \includegraphics[max width=\textwidth, alt={}, center]{a07e6d2f-ded1-4c62-957b-41fb94b46a2d-3_446_755_580_735} The diagram shows the curve \(y = \log _ { 10 } ( x + 2 )\) for \(0 \leqslant x \leqslant 6\). The region bounded by the curve and the lines \(x = 0 , x = 6\) and \(y = 0\) is denoted by \(R\). Use the trapezium rule with 2 strips to find an estimate of the area of \(R\), giving your answer correct to 1 decimal place.
CAIE P2 2016 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{a07e6d2f-ded1-4c62-957b-41fb94b46a2d-3_423_837_1352_651} The diagram shows the curve with parametric equations $$x = 2 - \cos t , \quad y = 1 + 3 \cos 2 t$$ for \(0 < t < \pi\). The minimum point is \(M\) and the curve crosses the \(x\)-axis at points \(P\) and \(Q\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - 12 \cos t\).
  2. Find the coordinates of \(M\).
  3. Find the gradient of the curve at \(P\) and at \(Q\).
CAIE P2 2016 June Q6
6
  1. Find \(\int \frac { 4 + \mathrm { e } ^ { x } } { 2 \mathrm { e } ^ { 2 x } } \mathrm {~d} x\).
  2. Without using a calculator, find \(\int _ { 2 } ^ { 10 } \frac { 1 } { 2 x + 5 } \mathrm {~d} x\), giving your answer in the form \(\ln k\).

  3. \includegraphics[max width=\textwidth, alt={}, center]{f85c4010-17b1-441c-ae8a-e77573d1b0c3-3_446_755_580_735} The diagram shows the curve \(y = \log _ { 10 } ( x + 2 )\) for \(0 \leqslant x \leqslant 6\). The region bounded by the curve and the lines \(x = 0 , x = 6\) and \(y = 0\) is denoted by \(R\). Use the trapezium rule with 2 strips to find an estimate of the area of \(R\), giving your answer correct to 1 decimal place.
CAIE P2 2016 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{f85c4010-17b1-441c-ae8a-e77573d1b0c3-3_423_837_1352_651} The diagram shows the curve with parametric equations $$x = 2 - \cos t , \quad y = 1 + 3 \cos 2 t$$ for \(0 < t < \pi\). The minimum point is \(M\) and the curve crosses the \(x\)-axis at points \(P\) and \(Q\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - 12 \cos t\).
  2. Find the coordinates of \(M\).
  3. Find the gradient of the curve at \(P\) and at \(Q\).
CAIE P2 2017 June Q1
1 Given that \(5 ^ { x } = 3 ^ { 4 y }\), use logarithms to show that \(y = m x\) and find the value of the constant \(m\) correct to 3 significant figures.
CAIE P2 2017 June Q2
2 Solve the inequality \(| 4 - x | \leqslant | 3 - 2 x |\).