Questions P2 (867 questions)

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CAIE P2 2010 November Q7
9 marks Moderate -0.8
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
10 marks Standard +0.3
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 Q8
10 marks Standard +0.3
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
3 marks Easy -1.2
1 Solve the inequality \(| 3 x + 1 | > 8\).
CAIE P2 2010 November Q2
5 marks Standard +0.3
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
6 marks Moderate -0.8
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
6 marks Moderate -0.3
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
6 marks Moderate -0.3
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
7 marks Moderate -0.3
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
8 marks Standard +0.3
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
9 marks Standard +0.3
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
3 marks Easy -1.2
1 Solve the inequality \(| 4 - 5 x | < 3\).
CAIE P2 2011 November Q2
5 marks Moderate -0.8
2 Show that \(\int _ { 2 } ^ { 6 } \frac { 2 } { 4 x + 1 } \mathrm {~d} x = \ln \frac { 5 } { 3 }\).
CAIE P2 2011 November Q3
5 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{55794ceb-2d52-459c-8724-6a6a29ab159a-2_705_737_591_703} The diagram shows the part of the curve \(y = \frac { 1 } { 2 } \tan 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). Find the \(x\)-coordinates of the points on this part of the curve at which the gradient is 4 .
CAIE P2 2011 November Q4
5 marks Moderate -0.3
4 Solve the equation \(3 ^ { 2 x } - 7 \left( 3 ^ { x } \right) + 10 = 0\), giving your answers correct to 3 significant figures.
CAIE P2 2011 November Q5
7 marks Moderate -0.8
5 The polynomial \(4 x ^ { 3 } + a x ^ { 2 } + 9 x + 9\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that when \(\mathrm { p } ( x )\) is divided by \(( 2 x - 1 )\) the remainder is 10 .
  1. Find the value of \(a\) and hence verify that ( \(x - 3\) ) is a factor of \(\mathrm { p } ( x )\).
  2. When \(a\) has this value, solve the equation \(\mathrm { p } ( x ) = 0\).
CAIE P2 2011 November Q6
7 marks Moderate -0.3
6
  1. Verify by calculation that the cubic equation $$x ^ { 3 } - 2 x ^ { 2 } + 5 x - 3 = 0$$ has a root that lies between \(x = 0.7\) and \(x = 0.8\).
  2. Show that this root also satisfies an equation of the form $$x = \frac { a x ^ { 2 } + 3 } { x ^ { 2 } + b }$$ where the values of \(a\) and \(b\) are to be found.
  3. With these values of \(a\) and \(b\), use the iterative formula $$x _ { n + 1 } = \frac { a x _ { n } ^ { 2 } + 3 } { x _ { n } ^ { 2 } + b }$$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P2 2011 November Q7
8 marks Standard +0.3
7 The parametric equations of a curve are $$x = \mathrm { e } ^ { 3 t } , \quad y = t ^ { 2 } \mathrm { e } ^ { t } + 3$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { t ( t + 2 ) } { 3 \mathrm { e } ^ { 2 t } }\).
  2. Show that the tangent to the curve at the point \(( 1,3 )\) is parallel to the \(x\)-axis.
  3. Find the exact coordinates of the other point on the curve at which the tangent is parallel to the \(x\)-axis.
CAIE P2 2011 November Q8
10 marks Standard +0.3
8
  1. By first expanding \(\cos ( 2 x + x )\), show that $$\cos 3 x \equiv 4 \cos ^ { 3 } x - 3 \cos x$$
  2. Hence show that $$\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \left( 2 \cos ^ { 3 } x - \cos x \right) d x = \frac { 5 } { 12 }$$
CAIE P2 2011 November Q1
4 marks Standard +0.3
1 Solve the inequality \(| x + 2 | > \left| \frac { 1 } { 2 } x - 2 \right|\).
CAIE P2 2011 November Q2
4 marks Moderate -0.8
2 Use logarithms to solve the equation \(4 ^ { x + 1 } = 5 ^ { 2 x - 3 }\), giving your answer correct to 3 significant figures.
CAIE P2 2011 November Q3
6 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{322eb555-d40a-460c-8c71-5780f5772bcd-2_535_1041_573_552} The diagram shows the curve \(y = x - 2 \ln x\) and its minimum point \(M\).
  1. Find the \(x\)-coordinate of \(M\).
  2. Use the trapezium rule with three intervals to estimate the value of $$\int _ { 2 } ^ { 5 } ( x - 2 \ln x ) \mathrm { d } x$$ giving your answer correct to 2 decimal places.
  3. State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the integral in part (ii).
CAIE P2 2011 November Q4
6 marks Moderate -0.3
4 Find the exact value of the positive constant \(k\) for which $$\int _ { 0 } ^ { k } e ^ { 4 x } d x = \int _ { 0 } ^ { 2 k } e ^ { x } d x$$
CAIE P2 2011 November Q5
7 marks Moderate -0.3
5
  1. By sketching a suitable pair of graphs, show that the equation $$\frac { 1 } { x } = \sin x$$ where \(x\) is in radians, has only one root for \(0 < x \leqslant \frac { 1 } { 2 } \pi\).
  2. Verify by calculation that this root lies between \(x = 1.1\) and \(x = 1.2\).
  3. Use the iterative formula \(x _ { n + 1 } = \frac { 1 } { \sin x _ { n } }\) to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P2 2011 November Q6
7 marks Standard +0.3
6 The parametric equations of a curve are $$x = 1 + 2 \sin ^ { 2 } \theta , \quad y = 4 \tan \theta$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sin \theta \cos ^ { 3 } \theta }\).
  2. Find the equation of the tangent to the curve at the point where \(\theta = \frac { 1 } { 4 } \pi\), giving your answer in the form \(y = m x + c\).