Questions — CAIE P2 (699 questions)

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CAIE P2 2020 November Q8
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
1 Solve the equation \(7 \cot \theta = 3 \operatorname { cosec } \theta\) for \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).
CAIE P2 2020 November Q2
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
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
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
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
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
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
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
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
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
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
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
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 \(B\).
CAIE P2 2021 November Q6
6 The polynomials \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are defined by $$\mathrm { f } ( x ) = 4 x ^ { 3 } + a x ^ { 2 } + 8 x + 15 \quad \text { and } \quad \mathrm { g } ( x ) = x ^ { 2 } + b x + 18$$ where \(a\) and \(b\) are constants.
  1. Given that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\), find the value of \(a\).
  2. Given that the remainder is 40 when \(\mathrm { g } ( x )\) is divided by \(( x - 2 )\), find the value of \(b\).
  3. When \(a\) and \(b\) have these values, factorise \(\mathrm { f } ( x ) - \mathrm { g } ( x )\) completely.
  4. Hence solve the equation \(\mathrm { f } ( \operatorname { cosec } \theta ) - \mathrm { g } ( \operatorname { cosec } \theta ) = 0\) for \(0 < \theta < 2 \pi\).
CAIE P2 2021 November Q7
7
  1. By first expanding \(\cos ( 2 \theta + \theta )\), show that \(\cos 3 \theta \equiv 4 \cos ^ { 3 } \theta - 3 \cos \theta\).
  2. Find the exact value of \(2 \cos ^ { 3 } \left( \frac { 5 } { 18 } \pi \right) - \frac { 3 } { 2 } \cos \left( \frac { 5 } { 18 } \pi \right)\).
  3. Find \(\int \left( 12 \cos ^ { 3 } x - 4 \cos ^ { 3 } 3 x \right) \mathrm { d } x\).
    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 2021 November Q1
1 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } + b x - 10$$ where \(a\) and \(b\) are constants. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) and that the remainder is - 55 when \(\mathrm { p } ( x )\) is divided by \(( x + 3 )\).
  1. Find the values of \(a\) and \(b\).
  2. Hence factorise \(\mathrm { p } ( x )\) completely.
CAIE P2 2021 November Q2
2
  1. Sketch, on the same diagram, the graphs of \(y = x + 3\) and \(y = | 2 x - 1 |\).
  2. Solve the equation \(x + 3 = | 2 x - 1 |\).
  3. Find the value of \(y\) such that \(5 ^ { \frac { 1 } { 2 } y } + 3 = \left| 2 \times 5 ^ { \frac { 1 } { 2 } y } - 1 \right|\). Give your answer correct to 3 significant figures.
CAIE P2 2021 November Q3
3 The curve with equation $$y = 5 x - 2 \tan 2 x$$ has exactly one stationary point in the interval \(0 \leqslant x < \frac { 1 } { 4 } \pi\).
Find the coordinates of this stationary point, giving each coordinate correct to 3 significant figures.
CAIE P2 2021 November Q4
4 Given that \(\int _ { a } ^ { a + 14 } \frac { 1 } { 3 x } \mathrm {~d} x = \ln 2\), find the value of the positive constant \(a\).
CAIE P2 2021 November Q5
5 A curve has equation \(x ^ { 2 } + 4 x \cos 3 y = 6\).
Find the exact value of the gradient of the normal to the curve at the point \(\left( \sqrt { 2 } , \frac { 1 } { 12 } \pi \right)\).
CAIE P2 2021 November Q6
6
  1. By sketching a suitable pair of graphs on the same diagram, show that the equation $$\ln x = 2 \mathrm { e } ^ { - x }$$ has exactly one root.
  2. Verify by calculation that the root lies between 1.5 and 1.6.
  3. Show that if a sequence of values given by the iterative formula $$x _ { n + 1 } = \mathrm { e } ^ { 2 \mathrm { e } ^ { - x _ { n } } }$$ converges, then it converges to the root of the equation in part (a).
  4. Use the iterative formula in part (c) to determine the root correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
CAIE P2 2021 November Q7
7
  1. Prove that \(4 \sin x \sin \left( x + \frac { 1 } { 6 } \pi \right) \equiv \sqrt { 3 } - \sqrt { 3 } \cos 2 x + \sin 2 x\).
  2. Find the exact value of \(\int _ { 0 } ^ { \frac { 5 } { 6 } \pi } 4 \sin x \sin \left( x + \frac { 1 } { 6 } \pi \right) \mathrm { d } x\).
  3. Find the smallest positive value of \(y\) satisfying the equation $$4 \sin ( 2 y ) \sin \left( 2 y + \frac { 1 } { 6 } \pi \right) = \sqrt { 3 } .$$ Give your answer in an exact form.
    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 2021 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{6294c4f4-70a9-4b81-87e0-20e2cc24dd27-05_606_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 Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{6294c4f4-70a9-4b81-87e0-20e2cc24dd27-08_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 \(B\).