Questions — CAIE P2 (699 questions)

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CAIE P2 2009 November Q4
4 The parametric equations of a curve are $$x = 1 - \mathrm { e } ^ { - t } , \quad y = \mathrm { e } ^ { t } + \mathrm { e } ^ { - t }$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { 2 t } - 1\).
  2. Hence find the exact value of \(t\) at the point on the curve at which the gradient is 2 .
CAIE P2 2009 November Q5
5 The polynomial \(a x ^ { 3 } + b x ^ { 2 } - 5 x + 2\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x + 1 )\) and \(( x - 2 )\) are factors of \(\mathrm { p } ( x )\).
  1. Find the values of \(a\) and \(b\).
  2. When \(a\) and \(b\) have these values, find the other linear factor of \(\mathrm { p } ( x )\).
CAIE P2 2009 November Q6
6
  1. Express \(3 \cos x + 4 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), stating the exact value of \(R\) and giving the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation $$3 \cos x + 4 \sin x = 4.5$$ giving all solutions in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).
CAIE P2 2009 November Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{729aa2f6-2b62-445f-a2aa-a63b45cb6b64-3_604_971_262_587} The diagram shows the curve \(y = x ^ { 2 } \cos x\), for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\).
  1. Show by differentiation that the \(x\)-coordinate of \(M\) satisfies the equation $$\tan x = \frac { 2 } { x }$$
  2. Verify by calculation that this equation has a root (in radians) between 1 and 1.2.
  3. Use the iterative formula \(x _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 2 } { x _ { n } } \right)\) to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P2 2009 November Q8
8
  1. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \left( \sin 2 x + \sec ^ { 2 } x \right) \mathrm { d } x\).
  2. Show that \(\int _ { 1 } ^ { 4 } \left( \frac { 1 } { 2 x } + \frac { 1 } { x + 1 } \right) \mathrm { d } x = \ln 5\).
CAIE P2 2010 November Q1
1 Solve the inequality \(| x + 1 | > | x - 4 |\).
CAIE P2 2010 November Q2
2 Use logarithms to solve the equation \(5 ^ { x } = 2 ^ { 2 x + 1 }\), giving your answer correct to 3 significant figures.
CAIE P2 2010 November Q3
3 Show that \(\int _ { 0 } ^ { 1 } \left( \mathrm { e } ^ { x } + 1 \right) ^ { 2 } \mathrm {~d} x = \frac { 1 } { 2 } \mathrm { e } ^ { 2 } + 2 \mathrm { e } - \frac { 3 } { 2 }\).
CAIE P2 2010 November Q4
4 The parametric equations of a curve are $$x = 1 + \ln ( t - 2 ) , \quad y = t + \frac { 9 } { t } , \quad \text { for } t > 2$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \left( t ^ { 2 } - 9 \right) ( t - 2 ) } { t ^ { 2 } }\).
  2. Find the coordinates of the only point on the curve at which the gradient is equal to 0 .
CAIE P2 2010 November Q5
5 Solve the equation \(8 + \cot \theta = 2 \operatorname { cosec } ^ { 2 } \theta\), giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P2 2010 November Q6
6 The curve with equation \(y = \frac { 6 } { x ^ { 2 } }\) intersects the line \(y = x + 1\) at the point \(P\).
  1. Verify by calculation that the \(x\)-coordinate of \(P\) lies between 1.4 and 1.6.
  2. Show that the \(x\)-coordinate of \(P\) satisfies the equation $$x = \sqrt { } \left( \frac { 6 } { x + 1 } \right)$$
  3. Use the iterative formula $$x _ { n + 1 } = \sqrt { } \left( \frac { 6 } { x _ { n } + 1 } \right)$$ with initial value \(x _ { 1 } = 1.5\), to determine the \(x\)-coordinate of \(P\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P2 2010 November Q7
7 The polynomial \(3 x ^ { 3 } + 2 x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x - 1 )\) is a factor of \(\mathrm { p } ( x )\), and that when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\) the remainder is 10 .
  1. Find the values of \(a\) and \(b\).
  2. When \(a\) and \(b\) have these values, solve the equation \(\mathrm { p } ( x ) = 0\).
CAIE P2 2010 November Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{dde12c57-5129-43ae-b385-9a8f21f51e49-3_566_787_255_680} The diagram shows the curve \(y = x \sin x\), for \(0 \leqslant x \leqslant \pi\). The point \(Q \left( \frac { 1 } { 2 } \pi , \frac { 1 } { 2 } \pi \right)\) lies on the curve.
  1. Show that the normal to the curve at \(Q\) passes through the point \(( \pi , 0 )\).
  2. Find \(\frac { \mathrm { d } } { \mathrm { d } x } ( \sin x - x \cos x )\).
  3. Hence evaluate \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x \sin x \mathrm {~d} x\).
CAIE P2 2010 November Q1
1 Solve the inequality \(| x + 1 | > | x - 4 |\).
CAIE P2 2010 November Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{2aceb797-097c-499b-99b6-cce9f287cb51-3_566_787_255_680} The diagram shows the curve \(y = x \sin x\), for \(0 \leqslant x \leqslant \pi\). The point \(Q \left( \frac { 1 } { 2 } \pi , \frac { 1 } { 2 } \pi \right)\) lies on the curve.
  1. Show that the normal to the curve at \(Q\) passes through the point \(( \pi , 0 )\).
  2. Find \(\frac { \mathrm { d } } { \mathrm { d } x } ( \sin x - x \cos x )\).
  3. Hence evaluate \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x \sin x \mathrm {~d} x\).
CAIE P2 2010 November Q1
1 Solve the inequality \(| 3 x + 1 | > 8\).
CAIE P2 2010 November Q2
2 The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 7 x _ { n } } { 8 } + \frac { 5 } { 2 x _ { n } ^ { 4 } }$$ with initial value \(x _ { 1 } = 1.7\), converges to \(\alpha\).
  1. Use this iterative formula to determine \(\alpha\) correct to 2 decimal places, giving the result of each iteration to 4 decimal places.
  2. State an equation that is satisfied by \(\alpha\) and hence show that \(\alpha = \sqrt [ 5 ] { 20 }\).
CAIE P2 2010 November Q3
3 The polynomial \(x ^ { 3 } + 4 x ^ { 2 } + a x + 2\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that the remainder when \(\mathrm { p } ( x )\) is divided by ( \(x + 1\) ) is equal to the remainder when \(\mathrm { p } ( x )\) is divided by ( \(x - 2\) ).
  1. Find the value of \(a\).
  2. When \(a\) has this value, show that \(( x - 1 )\) is a factor of \(\mathrm { p } ( x )\) and find the quotient when \(\mathrm { p } ( x )\) is divided by \(( x - 1 )\).
CAIE P2 2010 November Q4
4
  1. Find \(\int \mathrm { e } ^ { 1 - 2 x } \mathrm {~d} x\).
  2. Express \(\sin ^ { 2 } 3 x\) in terms of \(\cos 6 x\) and hence find \(\int \sin ^ { 2 } 3 x \mathrm {~d} x\).
CAIE P2 2010 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{e814d76c-8757-4cc4-a69c-e3636b4cab16-2_604_887_1667_628} The variables \(x\) and \(y\) satisfy the equation \(y = A \left( b ^ { x } \right)\), where \(A\) and \(b\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points ( \(1.4,0.8\) ) and ( \(2.2,1.2\) ), as shown in the diagram. Find the values of \(A\) and \(b\), correct to 2 decimal places.
CAIE P2 2010 November Q6
6
  1. Express \(2 \sin \theta - \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation $$2 \sin \theta - \cos \theta = - 0.4$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P2 2010 November Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{e814d76c-8757-4cc4-a69c-e3636b4cab16-3_611_1084_648_532} The diagram shows the curve \(y = \frac { \ln x } { x ^ { 2 } }\) and its maximum point \(M\).
  1. Find the exact coordinates of \(M\).
  2. Use the trapezium rule with three intervals to estimate the value of $$\int _ { 1 } ^ { 4 } \frac { \ln x } { x ^ { 2 } } \mathrm {~d} x$$ giving your answer correct to 2 decimal places.
CAIE P2 2010 November Q8
8 The equation of a curve is $$x ^ { 2 } + 2 x y - y ^ { 2 } + 8 = 0$$
  1. Show that the tangent to the curve at the point \(( - 2,2 )\) is parallel to the \(x\)-axis.
  2. Find the equation of the tangent to the curve at the other point on the curve for which \(x = - 2\), giving your answer in the form \(y = m x + c\).
CAIE P2 2011 November Q1
1 Solve the inequality \(| 4 - 5 x | < 3\).
CAIE P2 2011 November Q2
2 Show that \(\int _ { 2 } ^ { 6 } \frac { 2 } { 4 x + 1 } \mathrm {~d} x = \ln \frac { 5 } { 3 }\).