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CAIE P2 2024 March Q3
7 marks Moderate -0.3
3 The polynomial \(\mathrm { p } ( x )\) is defined by $$p ( x ) = 6 x ^ { 3 } + a x ^ { 2 } + 3 x - 10$$ where \(a\) is a constant. It is given that \(( 2 x - 1 )\) is a factor of \(\mathrm { p } ( x )\).
  1. Find the value of \(a\) and hence factorise \(\mathrm { p } ( x )\) completely.
  2. Solve the equation \(\mathrm { p } ( \operatorname { cosec } \theta ) = 0\) for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\).
    \includegraphics[max width=\textwidth, alt={}, center]{7b39a2ab-305d-43c5-a1e7-9442d6c13886-06_442_706_278_667} The diagram shows the curve with equation \(\mathrm { y } = \sqrt { 1 + \mathrm { e } ^ { 0.5 \mathrm { x } } }\). The shaded region is bounded by the curve and the straight lines \(x = 0 , x = 6\) and \(y = 0\).
  3. Use the trapezium rule with three intervals to find an approximation to the area of the shaded region. Give your answer correct to 3 significant figures.
  4. The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced.
CAIE P2 2024 March Q5
9 marks Standard +0.3
5
\includegraphics[max width=\textwidth, alt={}, center]{7b39a2ab-305d-43c5-a1e7-9442d6c13886-08_615_469_260_799} The diagram shows part of the curve with equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 3 } } { \mathrm { x } + 2 }\). At the point \(P\), the gradient of the curve is 6 .
  1. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(x = \sqrt [ 3 ] { 12 x + 12 }\).
  2. Show by calculation that the \(x\)-coordinate of \(P\) lies between 3.8 and 4.0 .
  3. Use an iterative formula, based on the equation in part (a), to find the \(x\)-coordinate of \(P\) correct to 3 significant figures. Show the result of each iteration to 5 significant figures.
CAIE P2 2024 March Q6
9 marks Standard +0.3
6
\includegraphics[max width=\textwidth, alt={}, center]{7b39a2ab-305d-43c5-a1e7-9442d6c13886-10_629_620_278_717} The diagram shows the curve with parametric equations $$x = 1 + \sqrt { t } , \quad y = ( \ln t + 2 ) ( \ln t - 3 ) ,$$ for \(0 < t < 25\). The curve crosses the \(x\)-axis at the points \(A\) and \(B\) and has a minimum point \(M\).
  1. Show that \(\frac { \mathrm { dy } } { \mathrm { dx } } = \frac { 4 \mathrm { Int } - 2 } { \sqrt { \mathrm { t } } }\).
  2. Find the exact gradient of the curve at \(B\).
  3. Find the exact coordinates of \(M\).
CAIE P2 2024 March Q7
10 marks Standard +0.3
7
  1. Prove that $$\sin 2 \theta ( a \cot \theta + b \tan \theta ) \equiv a + b + ( a - b ) \cos 2 \theta$$ where \(a\) and \(b\) are constants.
  2. Find the exact value of \(\int _ { \frac { 1 } { 12 } \pi } ^ { \frac { 1 } { 6 } \pi } \sin 2 \theta ( 5 \cot \theta + 3 \tan \theta ) d \theta\).
  3. Solve the equation \(\sin \frac { 2 } { 3 } \alpha \left( 2 \cot \frac { 1 } { 3 } \alpha + 7 \tan \frac { 1 } { 3 } \alpha \right) = 11\) for \(- \pi < \alpha < \pi\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE P2 2020 November Q1
4 marks Moderate -0.5
1 Given that $$\ln ( 2 x + 1 ) - \ln ( x - 3 ) = 2$$ find \(x\) in terms of e.
CAIE P2 2020 November Q2
5 marks Moderate -0.8
2 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = x ^ { 3 } + a x ^ { 2 } + b x + 16$$ where \(a\) and \(b\) are constants. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) and that the remainder is 72 when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\). Find the values of \(a\) and \(b\).
CAIE P2 2020 November Q3
5 marks Moderate -0.3
3
\includegraphics[max width=\textwidth, alt={}, center]{8beee722-7f86-454a-bc36-27e83f1483fd-04_684_455_260_845} The diagram shows the curve \(y = 2 + \mathrm { e } ^ { - 2 x }\). The curve crosses the \(y\)-axis at the point \(A\), and the point \(B\) on the curve has \(x\)-coordinate 1 . The shaded region is bounded by the curve and the line segment \(A B\). Find the exact area of the shaded region.
CAIE P2 2020 November Q4
5 marks Standard +0.3
4
  1. Solve the equation \(| 2 x - 5 | = | x + 6 |\).
  2. Hence find the value of \(y\) such that \(\left| 2 ^ { 1 - y } - 5 \right| = \left| 2 ^ { - y } + 6 \right|\). Give your answer correct to 3 significant figures.
CAIE P2 2020 November Q5
5 marks Moderate -0.3
5 The sequence of values given by the iterative formula \(x _ { n + 1 } = \frac { 6 + 8 x _ { n } } { 8 + x _ { n } ^ { 2 } }\) with initial value \(x _ { 1 } = 2\) converges to \(\alpha\).
  1. Use the iterative formula to find the value of \(\alpha\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
  2. State an equation satisfied by \(\alpha\) and hence determine the exact value of \(\alpha\).
CAIE P2 2020 November Q6
6 marks Standard +0.3
6 It is given that \(3 \sin 2 \theta = \cos \theta\) where \(\theta\) is an angle such that \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).
  1. Find the exact value of \(\sin \theta\).
  2. Find the exact value of \(\sec \theta\).
  3. Find the exact value of \(\cos 2 \theta\).
CAIE P2 2020 November Q7
10 marks Standard +0.3
7 A curve is defined by the parametric equations $$x = 3 t - 2 \sin t , \quad y = 5 t + 4 \cos t$$ where \(0 \leqslant t \leqslant 2 \pi\). At each of the points \(P\) and \(Q\) on the curve, the gradient of the curve is \(\frac { 5 } { 2 }\).
  1. Show that the values of \(t\) at \(P\) and \(Q\) satisfy the equation \(10 \cos t - 8 \sin t = 5\).
  2. Express \(10 \cos t - 8 \sin t\) in the form \(R \cos ( t + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Give the exact value of \(R\) and the value of \(\alpha\) correct to 3 significant figures.
  3. Hence find the values of \(t\) at the points \(P\) and \(Q\).
CAIE P2 2020 November Q8
10 marks Standard +0.3
8 A curve has equation \(y = f ( x )\) where \(f ( x ) = \frac { 4 x ^ { 3 } + 8 x - 4 } { 2 x - 1 }\).
  1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the coordinates of each of the stationary points of the curve \(y = \mathrm { f } ( x )\).
  2. Divide \(4 x ^ { 3 } + 8 x - 4\) by ( \(2 x - 1\) ), and hence find \(\int \mathrm { f } ( x ) \mathrm { d } x\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE P2 2020 November Q1
3 marks Moderate -0.8
1 Solve the equation \(7 \cot \theta = 3 \operatorname { cosec } \theta\) for \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).
CAIE P2 2020 November Q2
5 marks Moderate -0.3
2 Given that \(\frac { 2 ^ { 3 x + 2 } + 8 } { 2 ^ { 3 x } - 7 } = 5\), find the value of \(2 ^ { 3 x }\) and hence, using logarithms, find the value of \(x\) correct to 4 significant figures.
CAIE P2 2020 November Q3
6 marks Moderate -0.3
3
  1. Sketch, on a single diagram, the graphs of \(y = \left| \frac { 1 } { 2 } x - a \right|\) and \(y = \frac { 3 } { 2 } x - \frac { 1 } { 2 } a\), where \(a\) is a positive constant.
  2. Find the coordinates of the point of intersection of the two graphs.
  3. Deduce the solution of the inequality \(\left| \frac { 1 } { 2 } x - a \right| > \frac { 3 } { 2 } x - \frac { 1 } { 2 } a\).
CAIE P2 2020 November Q4
7 marks Moderate -0.3
4
\includegraphics[max width=\textwidth, alt={}, center]{c473f577-1e96-4d11-a0d5-cdfa4873c295-06_460_1445_260_349} The diagram shows the curve with equation \(y = \frac { x - 2 } { x ^ { 2 } + 8 }\). The shaded region is bounded by the curve and the lines \(x = 14\) and \(y = 0\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence determine the exact \(x\)-coordinates of the stationary points.
  2. Use the trapezium rule with three intervals to find an approximation to the area of the shaded region. Give the answer correct to 2 significant figures.
CAIE P2 2020 November Q5
9 marks Standard +0.3
5 The equation of a curve is \(2 \mathrm { e } ^ { 2 x } y - y ^ { 3 } + 4 = 0\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 \mathrm { e } ^ { 2 x } y } { 3 y ^ { 2 } - 2 \mathrm { e } ^ { 2 x } }\).
  2. The curve passes through the point \(( 0,2 )\). Find the equation of the tangent to the curve at this point, giving your answer in the form \(a x + b y + c = 0\).
  3. Show that the curve has no stationary points.
CAIE P2 2020 November Q6
10 marks Standard +0.3
6
  1. Find \(\int \left( \frac { 8 } { 4 x + 1 } + \frac { 8 } { \cos ^ { 2 } ( 4 x + 1 ) } \right) \mathrm { d } x\).
  2. It is given that \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \left( 3 + 4 \cos ^ { 2 } \frac { 1 } { 2 } x + k \sin 2 x \right) \mathrm { d } x = 10\). Find the exact value of the constant \(k\).
CAIE P2 2020 November Q7
10 marks Standard +0.3
7
\includegraphics[max width=\textwidth, alt={}, center]{c473f577-1e96-4d11-a0d5-cdfa4873c295-12_650_720_260_708} A curve has equation \(y = \mathrm { f } ( x )\) where \(\mathrm { f } ( x ) = x ^ { 4 } - 5 x ^ { 3 } + 6 x ^ { 2 } + 5 x - 15\). As shown in the diagram, the curve crosses the \(x\)-axis at the points \(A\) and \(B\) with coordinates \(( a , 0 )\) and \(( b , 0 )\) respectively.
  1. Use the factor theorem to show that \(( x - 3 )\) is a factor of \(\mathrm { f } ( x )\).
  2. By first finding the quotient when \(\mathrm { f } ( x )\) is divided by \(( x - 3 )\), show that $$a = - \sqrt { \frac { 5 } { 2 - a } } .$$
  3. Use an iterative formula, based on the equation in part (b), to find the value of \(a\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE P2 2020 November Q3
5 marks Moderate -0.3
3
\includegraphics[max width=\textwidth, alt={}, center]{b4a4082c-f3cd-47c5-8673-680dae9a22bd-04_684_455_260_845} The diagram shows the curve \(y = 2 + \mathrm { e } ^ { - 2 x }\). The curve crosses the \(y\)-axis at the point \(A\), and the point \(B\) on the curve has \(x\)-coordinate 1 . The shaded region is bounded by the curve and the line segment \(A B\). Find the exact area of the shaded region.
CAIE P2 2021 November Q1
4 marks Moderate -0.8
1 Find the exact value of \(\int _ { - 1 } ^ { 2 } \left( 4 \mathrm { e } ^ { 2 x } - 2 \mathrm { e } ^ { - x } \right) \mathrm { d } x\).
CAIE P2 2021 November Q2
6 marks Moderate -0.8
2
  1. Sketch, on the same diagram, the graphs of \(y = 3 x\) and \(y = | x - 3 |\).
  2. Find the coordinates of the point where the two graphs intersect.
  3. Deduce the solution of the inequality \(3 x < | x - 3 |\).
CAIE P2 2021 November Q3
5 marks Moderate -0.3
3
\includegraphics[max width=\textwidth, alt={}, center]{83d0697c-b133-47da-a745-dfdafa7dbf10-05_604_933_258_605} The variables \(x\) and \(y\) satisfy the equation \(a ^ { y } = k x\), where \(a\) and \(k\) are constants. The graph of \(y\) against \(\ln x\) is a straight line passing through the points \(( 1.03,6.36 )\) and \(( 2.58,9.00 )\), as shown in the diagram. Find the values of \(a\) and \(k\), giving each value correct to 2 significant figures.
CAIE P2 2021 November Q4
8 marks Standard +0.3
4 The curve with equation \(y = x \mathrm { e } ^ { 2 x } + 5 \mathrm { e } ^ { - x }\) has a minimum point \(M\).
  1. Show that the \(x\)-coordinate of \(M\) satisfies the equation \(x = \frac { 1 } { 3 } \ln 5 - \frac { 1 } { 3 } \ln ( 1 + 2 x )\).
  2. Use an iterative formula, based on the equation in part (a), to find the \(x\)-coordinate of \(M\) correct to 3 significant figures. Use an initial value of 0.35 and give the result of each iteration to 5 significant figures.
CAIE P2 2021 November Q5
8 marks Moderate -0.3
5
\includegraphics[max width=\textwidth, alt={}, center]{83d0697c-b133-47da-a745-dfdafa7dbf10-08_663_433_260_854} The diagram shows the curve with parametric equations $$x = \ln ( 2 t + 3 ) , \quad y = \frac { 2 t - 3 } { 2 t + 3 } .$$ The curve crosses the \(y\)-axis at the point \(A\) and the \(x\)-axis at the point \(B\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 } { 2 t + 3 }\).
  2. Find the gradient of the curve at \(A\).
  3. Find the gradient of the curve at \(B\).