Questions — CAIE P2 (699 questions)

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CAIE P2 2012 June Q6
6 A curve has parametric equations $$x = \frac { 1 } { ( 2 t + 1 ) ^ { 2 } } , \quad y = \sqrt { } ( t + 2 )$$ The point \(P\) on the curve has parameter \(p\) and it is given that the gradient of the curve at \(P\) is - 1 .
  1. Show that \(p = ( p + 2 ) ^ { \frac { 1 } { 6 } } - \frac { 1 } { 2 }\).
  2. Use an iterative process based on the equation in part (i) to find the value of \(p\) correct to 3 decimal places. Use a starting value of 0.7 and show the result of each iteration to 5 decimal places.
CAIE P2 2012 June Q7
7
  1. Show that \(( 2 \sin x + \cos x ) ^ { 2 }\) can be written in the form \(\frac { 5 } { 2 } + 2 \sin 2 x - \frac { 3 } { 2 } \cos 2 x\).
  2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } ( 2 \sin x + \cos x ) ^ { 2 } \mathrm {~d} x\).
CAIE P2 2012 June Q1
1 Solve the inequality \(| x + 3 | < | 2 x + 1 |\).
CAIE P2 2012 June Q2
2
  1. Given that \(5 ^ { 2 x } + 5 ^ { x } = 12\), find the value of \(5 ^ { x }\).
  2. Hence, using logarithms, solve the equation \(5 ^ { 2 x } + 5 ^ { x } = 12\), giving the value of \(x\) correct to 3 significant figures.
CAIE P2 2012 June Q3
3
  1. Find the quotient when the polynomial $$8 x ^ { 3 } - 4 x ^ { 2 } - 18 x + 13$$ is divided by \(4 x ^ { 2 } + 4 x - 3\), and show that the remainder is 4 .
  2. Hence, or otherwise, factorise the polynomial $$8 x ^ { 3 } - 4 x ^ { 2 } - 18 x + 9$$
CAIE P2 2012 June Q4
4
  1. Express \(9 \sin \theta - 12 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(\alpha\) correct to 2 decimal places. Hence
  2. solve the equation \(9 \sin \theta - 12 \cos \theta = 4\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\),
  3. state the largest value of \(k\) for which the equation \(9 \sin \theta - 12 \cos \theta = k\) has any solutions.
CAIE P2 2012 June Q5
5 The parametric equations of a curve are $$x = \ln ( t + 1 ) , \quad y = \mathrm { e } ^ { 2 t } + 2 t$$
  1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
  2. Find the equation of the normal to the curve at the point for which \(t = 0\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
CAIE P2 2012 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{48ab71ff-c37b-4e0b-b031-d99b0cf517a8-3_421_976_251_580} The diagram shows the curve \(y = \frac { \sin 2 x } { x + 2 }\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). The \(x\)-coordinate of the maximum point \(M\) is denoted by \(\alpha\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show that \(\alpha\) satisfies the equation \(\tan 2 x = 2 x + 4\).
  2. Show by calculation that \(\alpha\) lies between 0.6 and 0.7 .
  3. Use the iterative formula \(x _ { n + 1 } = \frac { 1 } { 2 } \tan ^ { - 1 } \left( 2 x _ { n } + 4 \right)\) to find the value of \(\alpha\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
CAIE P2 2012 June Q7
7
  1. Show that \(\tan ^ { 2 } x + \cos ^ { 2 } x \equiv \sec ^ { 2 } x + \frac { 1 } { 2 } \cos 2 x - \frac { 1 } { 2 }\) and hence find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \tan ^ { 2 } x + \cos ^ { 2 } x \right) d x$$

  2. \includegraphics[max width=\textwidth, alt={}, center]{48ab71ff-c37b-4e0b-b031-d99b0cf517a8-3_550_785_1573_721} The region enclosed by the curve \(y = \tan x + \cos x\) and the lines \(x = 0 , x = \frac { 1 } { 4 } \pi\) and \(y = 0\) is shown in the diagram. Find the exact volume of the solid produced when this region is rotated completely about the \(x\)-axis.
CAIE P2 2012 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{0a45a806-007f-4840-85e7-16d4c1a2c599-2_453_771_386_685} 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 \(( 0,2.14 )\) and \(( 5,4.49 )\), as shown in the diagram. Find the values of \(A\) and \(b\), correct to 1 decimal place.
CAIE P2 2012 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{0a45a806-007f-4840-85e7-16d4c1a2c599-3_528_757_251_694} The diagram shows the curve \(y = 4 e ^ { \frac { 1 } { 2 } x } - 6 x + 3\) and its minimum point \(M\).
  1. Show that the \(x\)-coordinate of \(M\) can be written in the form \(\ln a\), where the value of \(a\) is to be stated.
  2. Find the exact value of the area of the region enclosed by the curve and the lines \(x = 0 , x = 2\) and \(y = 0\).
CAIE P2 2013 June Q1
1 Solve the equation \(\left| 2 ^ { x } - 7 \right| = 1\), giving answers correct to 2 decimal places where appropriate.
CAIE P2 2013 June Q2
2 Solve the equation \(\ln ( 3 - 2 x ) - 2 \ln x = \ln 5\).
CAIE P2 2013 June Q3
3
  1. Show that \(12 \sin ^ { 2 } x \cos ^ { 2 } x \equiv \frac { 3 } { 2 } ( 1 - \cos 4 x )\).
  2. Hence show that $$\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 3 } \pi } 12 \sin ^ { 2 } x \cos ^ { 2 } x d x = \frac { \pi } { 8 } + \frac { 3 \sqrt { } 3 } { 16 }$$
CAIE P2 2013 June Q4
4 The polynomial \(a x ^ { 3 } - 5 x ^ { 2 } + b x + 9\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( 2 x + 3 )\) is a factor of \(\mathrm { p } ( x )\), and that when \(\mathrm { p } ( x )\) is divided by \(( x + 1 )\) the remainder is 8 .
  1. Find the values of \(a\) and \(b\).
  2. When \(a\) and \(b\) have these values, factorise \(\mathrm { p } ( x )\) completely.
CAIE P2 2013 June Q5
5 The parametric equations of a curve are $$x = \mathrm { e } ^ { 2 t } , \quad y = 4 t \mathrm { e } ^ { t }$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 ( t + 1 ) } { \mathrm { e } ^ { t } }\).
  2. Find the equation of the normal to the curve at the point where \(t = 0\).
CAIE P2 2013 June Q6
6
  1. By sketching a suitable pair of graphs, show that the equation $$\cot x = 4 x - 2$$ where \(x\) is in radians, has only one root for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
  2. Verify by calculation that this root lies between \(x = 0.7\) and \(x = 0.9\).
  3. Show that this root also satisfies the equation $$x = \frac { 1 + 2 \tan x } { 4 \tan x }$$
  4. Use the iterative formula \(x _ { n + 1 } = \frac { 1 + 2 \tan x _ { n } } { 4 \tan x _ { n } }\) to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P2 2013 June Q7
7
  1. Express \(5 \sin 2 \theta + 2 \cos 2 \theta\) in the form \(R \sin ( 2 \theta + \alpha )\), 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. Hence
  2. solve the equation $$5 \sin 2 \theta + 2 \cos 2 \theta = 4$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\),
  3. determine the least value of \(\frac { 1 } { ( 10 \sin 2 \theta + 4 \cos 2 \theta ) ^ { 2 } }\) as \(\theta\) varies.
CAIE P2 2013 June Q1
1 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 } { 7 - 2 x }\). The point \(( 3,2 )\) lies on the curve. Find the equation of the curve.
CAIE P2 2013 June Q2
2 Solve the inequality \(| x - 8 | > | 2 x - 4 |\).
CAIE P2 2013 June Q3
3
  1. The polynomial \(2 x ^ { 3 } + a x ^ { 2 } - a x - 12\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x + 1 )\) is a factor of \(\mathrm { p } ( x )\). Find the value of \(a\).
  2. When \(a\) has this value, find the remainder when \(\mathrm { p } ( x )\) is divided by \(( x + 3 )\).
CAIE P2 2013 June Q4
4 The variables \(x\) and \(y\) satisfy the equation \(5 ^ { y + 1 } = 2 ^ { 3 x }\).
  1. By taking logarithms, show that the graph of \(y\) against \(x\) is a straight line.
  2. Find the exact value of the gradient of this line and state the coordinates of the point at which the line cuts the \(y\)-axis.
CAIE P2 2013 June Q5
5 The equation of a curve is $$x ^ { 2 } - 2 x ^ { 2 } y + 3 y = 9$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 x - 4 x y } { 2 x ^ { 2 } - 3 }\).
  2. Find the equation of the normal to the curve at the point where \(x = 2\), giving your answer in the form \(a x + b y + c = 0\).
CAIE P2 2013 June Q6
6
  1. By sketching a suitable pair of graphs, show that the equation $$3 \mathrm { e } ^ { x } = 8 - 2 x$$ has only one root.
  2. Verify by calculation that this root lies between \(x = 0.7\) and \(x = 0.8\).
  3. Show that this root also satisfies the equation $$x = \ln \left( \frac { 8 - 2 x } { 3 } \right)$$
  4. Use the iterative formula \(x _ { n + 1 } = \ln \left( \frac { 8 - 2 x _ { n } } { 3 } \right)\) to determine this root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
CAIE P2 2013 June Q7
7
  1. Find the exact area of the region bounded by the curve \(y = 1 + \mathrm { e } ^ { 2 x - 1 }\), the \(x\)-axis and the lines \(x = \frac { 1 } { 2 }\) and \(x = 2\).

  2. \includegraphics[max width=\textwidth, alt={}, center]{e3ee4932-8219-4332-9cd2-e7f835522469-3_469_719_397_753} The diagram shows the curve \(y = \frac { \mathrm { e } ^ { 2 x } } { \sin 2 x }\) for \(0 < x < \frac { 1 } { 2 } \pi\), and its minimum point \(M\). Find the exact \(x\)-coordinate of \(M\).