Questions — CAIE P2 (699 questions)

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CAIE P2 2015 November Q6
6
  1. Find the quotient and remainder when $$x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } + 12 x + 6$$ is divided by ( \(x ^ { 2 } - x + 4\) ).
  2. It is given that, when $$x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } + p x + q$$ is divided by ( \(x ^ { 2 } - x + 4\) ), the remainder is zero. Find the values of the constants \(p\) and \(q\).
  3. When \(p\) and \(q\) have these values, show that there is exactly one real value of \(x\) satisfying the equation $$x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } + p x + q = 0$$ and state what that value is.
CAIE P2 2015 November Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{250b4df9-2646-4246-bb6d-2be92bf29598-3_553_689_258_726} The parametric equations of a curve are $$x = 6 \sin ^ { 2 } t , \quad y = 2 \sin 2 t + 3 \cos 2 t$$ for \(0 \leqslant t < \pi\). The curve crosses the \(x\)-axis at points \(B\) and \(D\) and the stationary points are \(A\) and \(C\), as shown in the diagram.
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { 3 } \cot 2 t - 1\).
  2. Find the values of \(t\) at \(A\) and \(C\), giving each answer correct to 3 decimal places.
  3. Find the value of the gradient of the curve at \(B\).
CAIE P2 2015 November Q1
1
  1. Solve the equation \(| 3 x - 2 | = 5\).
  2. Hence, using logarithms, solve the equation \(\left| 3 \times 5 ^ { y } - 2 \right| = 5\), giving the answer correct to 3 significant figures.
CAIE P2 2015 November Q2
2 The sequence of values given by the iterative formula $$x _ { n + 1 } = 2 + \frac { 4 } { x _ { n } ^ { 2 } + 2 x _ { n } + 4 }$$ with initial value \(x _ { 1 } = 2\), converges to \(\alpha\).
  1. Determine the value of \(\alpha\) correct to 3 decimal places, giving the result of each iteration to 5 decimal places.
  2. State an equation satisfied by \(\alpha\) and hence find the exact value of \(\alpha\).
CAIE P2 2015 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{7e100be2-9768-4fcd-b516-c714e53b0665-2_456_725_1082_712} 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 ( \(0.22,3.96\) ) and ( \(1.32,2.43\) ), as shown in the diagram. Find the values of \(K\) and \(m\) correct to 2 significant figures.
CAIE P2 2015 November Q4
4 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 6 x ^ { 3 } + 11 x ^ { 2 } + a x + a$$ where \(a\) is a constant. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\).
  1. Use the factor theorem to show that \(a = - 4\).
  2. When \(a = - 4\),
    (a) factorise \(\mathrm { p } ( x )\) completely,
    (b) solve the equation \(6 \sec ^ { 3 } \theta + 11 \sec ^ { 2 } \theta + a \sec \theta + a = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P2 2015 November Q5
5 Find the \(x\)-coordinates of the stationary points of the following curves:
  1. \(y = 4 x \mathrm { e } ^ { - 3 x }\);
  2. \(y = \frac { 4 x ^ { 2 } } { x + 1 }\).
CAIE P2 2015 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{7e100be2-9768-4fcd-b516-c714e53b0665-3_453_650_258_744} The diagram shows the curve with parametric equations $$x = 3 \cos t , \quad y = 2 \cos \left( t - \frac { 1 } { 6 } \pi \right)$$ for \(0 \leqslant t < 2 \pi\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 3 } ( \sqrt { } 3 - \cot t )\).
  2. Find the equation of the tangent to the curve at the point where the curve crosses the positive \(y\)-axis. Give the answer in the form \(y = m x + c\).
CAIE P2 2015 November Q7
7
  1. Show that the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \left( \cos ^ { 2 } x + \frac { 1 } { \cos ^ { 2 } x } \right) \mathrm { d } x\) is \(\frac { 1 } { 6 } \pi + \frac { 9 } { 8 } \sqrt { } 3\).

  2. \includegraphics[max width=\textwidth, alt={}, center]{7e100be2-9768-4fcd-b516-c714e53b0665-3_444_495_1523_865} The diagram shows the curve \(y = \cos x + \frac { 1 } { \cos x }\) for \(0 \leqslant x \leqslant \frac { 1 } { 3 } \pi\). The shaded region is bounded by the curve and the lines \(x = 0 , x = \frac { 1 } { 3 } \pi\) and \(y = 0\). Find the exact volume of the solid obtained when the shaded region is rotated completely about the \(x\)-axis.
CAIE P2 2015 November Q1
1 Find the exact value of \(\int _ { - 1 } ^ { 35 } \frac { 3 } { 2 x + 5 } \mathrm {~d} x\), giving the answer in the form \(\ln k\).
CAIE P2 2015 November Q2
2
  1. Solve the equation \(| 2 x + 3 | = | x + 8 |\).
  2. Hence, using logarithms, solve the equation \(\left| 2 ^ { y + 1 } + 3 \right| = \left| 2 ^ { y } + 8 \right|\). Give the answer correct to 3 significant figures.
CAIE P2 2015 November Q3
3 The parametric equations of a curve are $$x = ( t + 1 ) \mathrm { e } ^ { t } , \quad y = 6 ( t + 4 ) ^ { \frac { 1 } { 2 } }$$ Find the equation of the tangent to the curve when \(t = 0\), giving the answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
CAIE P2 2015 November Q4
4
  1. Find the quotient when \(3 x ^ { 3 } + 5 x ^ { 2 } - 2 x - 1\) is divided by ( \(x - 2\) ), and show that the remainder is 39 .
  2. Hence show that the equation \(3 x ^ { 3 } + 5 x ^ { 2 } - 2 x - 40 = 0\) has exactly one real root.
CAIE P2 2015 November Q5
5 It is given that \(\int _ { 0 } ^ { a } \left( 3 \mathrm { e } ^ { 3 x } + 5 \mathrm { e } ^ { x } \right) \mathrm { d } x = 100\), where \(a\) is a positive constant.
  1. Show that \(a = \frac { 1 } { 3 } \ln \left( 106 - 5 \mathrm { e } ^ { a } \right)\).
  2. Use an iterative formula based on the equation in part (i) to find the value of \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
CAIE P2 2015 November Q6
6
  1. Express \(( \sqrt { } 5 ) \cos \theta - 2 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(\alpha\) correct to 2 decimal places.
  2. Hence
    (a) solve the equation \(( \sqrt { } 5 ) \cos \theta - 2 \sin \theta = 0.9\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\),
    (b) state the greatest and least values of $$10 + ( \sqrt { } 5 ) \cos \theta - 2 \sin \theta$$ as \(\theta\) varies.
CAIE P2 2015 November Q7
7 The equation of a curve is \(y = \frac { \sin 2 x } { \cos x + 1 }\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 \left( \cos ^ { 2 } x + \cos x - 1 \right) } { \cos x + 1 }\).
  2. Find the \(x\)-coordinate of each stationary point of the curve in the interval \(- \pi < x < \pi\). Give each answer correct to 3 significant figures.
CAIE P2 2016 November Q1
1
  1. It is given that \(x\) satisfies the equation \(3 ^ { 2 x } = 5 \left( 3 ^ { x } \right) + 14\). Find the value of \(3 ^ { x }\) and, using logarithms, find the value of \(x\) correct to 3 significant figures.
  2. Hence state the values of \(x\) satisfying the equation \(3 ^ { 2 | x | } = 5 \left( 3 ^ { | x | } \right) + 14\).
CAIE P2 2016 November Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{3edf4fb5-c1f9-4c99-8e23-fa666185e0ee-2_374_728_536_705} The variables \(x\) and \(y\) satisfy the equation \(y = A \mathrm { e } ^ { p x }\), where \(A\) and \(p\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \(( 5,3.17 )\) and \(( 10,4.77 )\), as shown in the diagram. Find the values of \(A\) and \(p\) correct to 2 decimal places.
CAIE P2 2016 November Q3
3 A curve has equation \(y = 2 \sin 2 x - 5 \cos 2 x + 6\) and is defined for \(0 \leqslant x \leqslant \pi\). Find the \(x\)-coordinates of the stationary points of the curve, giving your answers correct to 3 significant figures.
CAIE P2 2016 November Q4
4 It is given that the positive constant \(a\) is such that $$\int _ { - a } ^ { a } \left( 4 \mathrm { e } ^ { 2 x } + 5 \right) \mathrm { d } x = 100$$
  1. Show that \(a = \frac { 1 } { 2 } \ln \left( 50 + \mathrm { e } ^ { - 2 a } - 5 a \right)\).
  2. Use the iterative formula \(a _ { n + 1 } = \frac { 1 } { 2 } \ln \left( 50 + \mathrm { e } ^ { - 2 a _ { n } } - 5 a _ { n } \right)\) to find \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
  3. Show that \(\frac { \cos 2 x + 9 \cos x + 5 } { \cos x + 4 } \equiv 2 \cos x + 1\).
  4. Hence find the exact value of \(\int _ { - \pi } ^ { \pi } \frac { \cos 4 x + 9 \cos 2 x + 5 } { \cos 2 x + 4 } \mathrm {~d} x\).
CAIE P2 2016 November Q6
6 The equation of a curve is \(3 x ^ { 2 } + 4 x y + y ^ { 2 } = 24\). Find the equation of the normal to the curve at the point ( 1,3 ), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
CAIE P2 2016 November Q7
7 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } + 3 x ^ { 2 } + b x + 12$$ where \(a\) and \(b\) are constants. It is given that \(( x + 3 )\) is a factor of \(\mathrm { p } ( x )\). It is also given that the remainder is 18 when \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\).
  1. Find the values of \(a\) and \(b\).
  2. When \(a\) and \(b\) have these values,
    (a) show that the equation \(\mathrm { p } ( x ) = 0\) has exactly one real root,
    (b) solve the equation \(\mathrm { p } ( \sec y ) = 0\) for \(- 180 ^ { \circ } < y < 180 ^ { \circ }\).
CAIE P2 2016 November Q1
1 Solve the equation \(| 0.4 x - 0.8 | = 2\).
CAIE P2 2016 November Q2
2
  1. Given that \(\frac { 1 + 4 ^ { y } } { 3 + 2 ^ { y } } = 5\), find the value of \(2 ^ { y }\).
  2. Use logarithms to find the value of \(y\) correct to 3 significant figures.
CAIE P2 2016 November Q3
3 The definite integral \(I\) is defined by \(I = \int _ { 0 } ^ { 2 } \left( 4 \mathrm { e } ^ { \frac { 1 } { 2 } x } + 3 \right) \mathrm { d } x\).
  1. Show that \(I = 8 \mathrm { e } - 2\).
  2. Sketch the curve \(y = 4 \mathrm { e } ^ { \frac { 1 } { 2 } x } + 3\) for \(0 \leqslant x \leqslant 2\).
  3. State whether an estimate of \(I\) obtained by using the trapezium rule will be more than or less than \(8 \mathrm { e } - 2\). Justify your answer.