Questions — CAIE P2 (709 questions)

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CAIE P2 2005 November Q1
3 marks Moderate -0.8
1 Solve the inequality \(( 0.8 ) ^ { x } < 0.5\).
CAIE P2 2005 November Q2
6 marks Moderate -0.3
2 The polynomial \(x ^ { 3 } + 2 x ^ { 2 } + 2 x + 3\) is denoted by \(\mathrm { p } ( x )\).
  1. Find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 1\).
  2. Find the quotient and remainder when \(\mathrm { p } ( x )\) is divided by \(x ^ { 2 } + x - 1\).
CAIE P2 2005 November Q3
7 marks Standard +0.3
3
  1. Express \(12 \cos \theta - 5 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), 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 $$12 \cos \theta - 5 \sin \theta = 10$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P2 2005 November Q4
7 marks Standard +0.3
4 The equation of a curve is \(x ^ { 3 } + y ^ { 3 } = 9 x y\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 y - x ^ { 2 } } { y ^ { 2 } - 3 x }\).
  2. Find the equation of the tangent to the curve at the point ( 2,4 ), giving your answer in the form \(a x + b y = c\).
CAIE P2 2005 November Q5
8 marks Standard +0.3
5
  1. By sketching a suitable pair of graphs, show that there is only one value of \(x\) that is a root of the equation $$\frac { 1 } { x } = \ln x$$
  2. Verify by calculation that this root lies between 1 and 2 .
  3. Show that this root also satisfies the equation $$x = \mathrm { e } ^ { \frac { 1 } { x } }$$
  4. Use the iterative formula $$x _ { n + 1 } = \mathrm { e } ^ { \frac { 1 } { x _ { n } } }$$ with initial value \(x _ { 1 } = 1.8\), to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P2 2005 November Q6
9 marks Moderate -0.8
6 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { 2 x } - 2 \mathrm { e } ^ { - x }\). The point \(( 0,1 )\) lies on the curve.
  1. Find the equation of the curve.
  2. The curve has one stationary point. Find the \(x\)-coordinate of this point and determine whether it is a maximum or a minimum point.
CAIE P2 2005 November Q7
10 marks Moderate -0.8
7 \includegraphics[max width=\textwidth, alt={}, center]{d527d21f-0ab5-40fa-8cfd-ebfb4aba0a87-3_493_863_264_641} The diagram shows the part of the curve \(y = \sin ^ { 2 } x\) for \(0 \leqslant x \leqslant \pi\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sin 2 x\).
  2. Hence find the \(x\)-coordinates of the points on the curve at which the gradient of the curve is 0.5 . [3]
  3. By expressing \(\sin ^ { 2 } x\) in terms of \(\cos 2 x\), find the area of the region bounded by the curve and the \(x\)-axis between 0 and \(\pi\).
CAIE P2 2006 November Q1
4 marks Standard +0.3
1 Solve the inequality \(| 2 x - 1 | > | x |\).
CAIE P2 2006 November Q2
6 marks Moderate -0.8
2
  1. Express \(4 ^ { x }\) in terms of \(y\), where \(y = 2 ^ { x }\).
  2. Hence find the values of \(x\) that satisfy the equation $$3 \left( 4 ^ { x } \right) - 10 \left( 2 ^ { x } \right) + 3 = 0 ,$$ giving your answers correct to 2 decimal places.
CAIE P2 2006 November Q3
6 marks Moderate -0.8
3 The polynomial \(4 x ^ { 3 } - 7 x + a\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that ( \(2 x - 3\) ) is a factor of \(\mathrm { p } ( x )\).
  1. Show that \(a = - 3\).
  2. Hence, or otherwise, solve the equation \(\mathrm { p } ( x ) = 0\).
CAIE P2 2006 November Q4
7 marks Standard +0.3
4
  1. Prove the identity $$\tan \left( x + 45 ^ { \circ } \right) - \tan \left( 45 ^ { \circ } - x \right) \equiv 2 \tan 2 x .$$
  2. Hence solve the equation $$\tan \left( x + 45 ^ { \circ } \right) - \tan \left( 45 ^ { \circ } - x \right) = 2 ,$$ for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
CAIE P2 2006 November Q5
8 marks Standard +0.8
5 \includegraphics[max width=\textwidth, alt={}, center]{4029c46c-50a1-4d23-bc29-589417a6b7f5-2_396_392_1603_879} The diagram shows a chord joining two points, \(A\) and \(B\), on the circumference of a circle with centre \(O\) and radius \(r\). The angle \(A O B\) is \(\alpha\) radians, where \(0 < \alpha < \pi\). The area of the shaded segment is one sixth of the area of the circle.
  1. Show that \(\alpha\) satisfies the equation $$x = \frac { 1 } { 3 } \pi + \sin x .$$
  2. Verify by calculation that \(\alpha\) lies between \(\frac { 1 } { 2 } \pi\) and \(\frac { 2 } { 3 } \pi\).
  3. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 3 } \pi + \sin x _ { n } ,$$ with initial value \(x _ { 1 } = 2\), to determine \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P2 2006 November Q6
9 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{4029c46c-50a1-4d23-bc29-589417a6b7f5-3_501_497_269_826} The diagram shows the part of the curve \(y = \frac { \mathrm { e } ^ { 2 x } } { x }\) for \(x > 0\), and its minimum point \(M\).
  1. Find the coordinates of \(M\).
  2. Use the trapezium rule with 2 intervals to estimate the value of $$\int _ { 1 } ^ { 2 } \frac { \mathrm { e } ^ { 2 x } } { x } \mathrm {~d} x$$ giving your answer correct to 1 decimal place.
  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).
  4. Given that \(y = \tan 2 x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  5. Hence, or otherwise, show that $$\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \sec ^ { 2 } 2 x \mathrm {~d} x = \frac { 1 } { 2 } \sqrt { } 3$$ and, by using an appropriate trigonometrical identity, find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \tan ^ { 2 } 2 x \mathrm {~d} x\).
  6. Use the identity \(\cos 4 x \equiv 2 \cos ^ { 2 } 2 x - 1\) to find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \frac { 1 } { 1 + \cos 4 x } \mathrm {~d} x$$
CAIE P2 2007 November Q1
4 marks Moderate -0.8
1 Show that $$\int _ { 1 } ^ { 4 } \frac { 1 } { 2 x + 1 } \mathrm {~d} x = \frac { 1 } { 2 } \ln 3$$
CAIE P2 2007 November Q2
5 marks Standard +0.3
2 The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 2 x _ { n } } { 3 } + \frac { 4 } { x _ { n } ^ { 2 } }$$ with initial value \(x _ { 1 } = 2\), 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 find the exact value of \(\alpha\).
CAIE P2 2007 November Q3
5 marks Standard +0.3
3
  1. Solve the inequality \(| y - 5 | < 1\).
  2. Hence solve the inequality \(\left| 3 ^ { x } - 5 \right| < 1\), giving 3 significant figures in your answer.
CAIE P2 2007 November Q4
5 marks Standard +0.3
4 The equation of a curve is \(y = 2 x - \tan x\), where \(x\) is in radians. Find the coordinates of the stationary points of the curve for which \(- \frac { 1 } { 2 } \pi < x < \frac { 1 } { 2 } \pi\).
CAIE P2 2007 November Q5
6 marks Moderate -0.8
5 The polynomial \(3 x ^ { 3 } + 8 x ^ { 2 } + a x - 2\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\).
  1. Find the value of \(a\).
  2. When \(a\) has this value, solve the equation \(\mathrm { p } ( x ) = 0\).
CAIE P2 2007 November Q6
7 marks Standard +0.3
6
  1. Express \(8 \sin \theta - 15 \cos \theta\) 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 $$8 \sin \theta - 15 \cos \theta = 14$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P2 2007 November Q8
10 marks Moderate -0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{8f815127-61b2-4a7f-8687-747950ea6597-3_693_1061_262_541} The diagram shows the curve \(y = x ^ { 2 } \mathrm { e } ^ { - x }\) and its maximum point \(M\).
  1. Find the \(x\)-coordinate of \(M\).
  2. Show that the tangent to the curve at the point where \(x = 1\) passes through the origin.
  3. Use the trapezium rule, with two intervals, to estimate the value of $$\int _ { 1 } ^ { 3 } x ^ { 2 } \mathrm { e } ^ { - x } \mathrm {~d} x$$ giving your answer correct to 2 decimal places.
CAIE P2 2008 November Q1
4 marks Standard +0.3
1 Solve the inequality \(| x - 3 | > | 2 x |\).
CAIE P2 2008 November Q2
5 marks Moderate -0.8
2 The polynomial \(2 x ^ { 3 } - x ^ { 2 } + a x - 6\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that ( \(x + 2\) ) is a factor of \(\mathrm { p } ( x )\).
  1. Find the value of \(a\).
  2. When \(a\) has this value, factorise \(\mathrm { p } ( x )\) completely.
CAIE P2 2008 November Q3
5 marks Moderate -0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{733c3711-0429-415d-a8f3-8de86097635a-2_550_843_769_651} 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 ( \(0,1.3\) ) and ( \(1.6,0.9\) ), as shown in the diagram. Find the values of \(A\) and \(b\), correct to 2 decimal places.
CAIE P2 2008 November Q4
6 marks Moderate -0.3
4
  1. Show that the equation $$\sin \left( x + 30 ^ { \circ } \right) = 2 \cos \left( x + 60 ^ { \circ } \right)$$ can be written in the form $$( 3 \sqrt { } 3 ) \sin x = \cos x$$
  2. Hence solve the equation $$\sin \left( x + 30 ^ { \circ } \right) = 2 \cos \left( x + 60 ^ { \circ } \right)$$ for \(- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
CAIE P2 2008 November Q5
6 marks Moderate -0.8
5 Show that \(\int _ { 1 } ^ { 2 } \left( \frac { 1 } { x } - \frac { 4 } { 2 x + 1 } \right) \mathrm { d } x = \ln \frac { 18 } { 25 }\).