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CAIE P2 2013 June Q4
8 marks Moderate -0.3
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
8 marks Moderate -0.5
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
8 marks Standard +0.3
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
10 marks Standard +0.3
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
4 marks Moderate -0.8
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
4 marks Standard +0.3
2 Solve the inequality \(| x - 8 | > | 2 x - 4 |\).
CAIE P2 2013 June Q3
4 marks Moderate -0.8
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 marks Moderate -0.8
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
8 marks Standard +0.3
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
8 marks Moderate -0.3
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
9 marks Standard +0.3
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 \(M\).
CAIE P2 2013 June Q8
9 marks Standard +0.8
8
  1. Prove the identity $$\frac { 1 } { \sin \left( x - 60 ^ { \circ } \right) + \cos \left( x - 30 ^ { \circ } \right) } \equiv \operatorname { cosec } x$$
  2. Hence solve the equation $$\frac { 2 } { \sin \left( x - 60 ^ { \circ } \right) + \cos \left( x - 30 ^ { \circ } \right) } = 3 \cot ^ { 2 } x - 2$$ for \(0 ^ { \circ } < x < 360 ^ { \circ }\).
CAIE P2 2014 June Q1
4 marks Standard +0.3
1 Solve the inequality \(| 3 x - 2 | \geqslant | x + 4 |\).
CAIE P2 2014 June Q2
6 marks Easy -1.2
2 Find the gradient of each of the following curves at the point for which \(x = 0\).
  1. \(y = 3 \sin x + \tan 2 x\)
  2. \(y = \frac { 6 } { 1 + \mathrm { e } ^ { 2 x } }\)
CAIE P2 2014 June Q3
6 marks Standard +0.3
3
  1. Find the quotient when \(6 x ^ { 4 } - x ^ { 3 } - 26 x ^ { 2 } + 4 x + 15\) is divided by ( \(x ^ { 2 } - 4\) ), and confirm that the remainder is 7 .
  2. Hence solve the equation \(6 x ^ { 4 } - x ^ { 3 } - 26 x ^ { 2 } + 4 x + 8 = 0\).
CAIE P2 2014 June Q4
8 marks Moderate -0.3
4
  1. By sketching a suitable pair of graphs, show that the equation $$3 \ln x = 15 - x ^ { 3 }$$ has exactly one real root.
  2. Show by calculation that the root lies between 2.0 and 2.5.
  3. Use the iterative formula \(x _ { n + 1 } = \sqrt [ 3 ] { } \left( 15 - 3 \ln x _ { n } \right)\) to find the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
CAIE P2 2014 June Q6
8 marks Moderate -0.3
6
  1. Show that \(\int _ { 6 } ^ { 16 } \frac { 6 } { 2 x - 7 } \mathrm {~d} x = \ln 125\).
  2. Use the trapezium rule with four intervals to find an approximation to $$\int _ { 1 } ^ { 17 } \log _ { 10 } x d x$$ giving your answer correct to 3 significant figures.
CAIE P2 2014 June Q7
10 marks Standard +0.3
7 The equation of a curve is $$2 x ^ { 2 } + 3 x y + y ^ { 2 } = 3$$
  1. Find the equation of the tangent to the curve at the point \(( 2 , - 1 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
  2. Show that the curve has no stationary points.
CAIE P2 2014 June Q1
4 marks Moderate -0.3
1
  1. Solve the equation \(| x + 2 | = | x - 13 |\).
  2. Hence solve the equation \(\left| 3 ^ { y } + 2 \right| = \left| 3 ^ { y } - 13 \right|\), giving your answer correct to 3 significant figures.
CAIE P2 2014 June Q2
4 marks Standard +0.3
2 Solve the equation \(3 \sin 2 \theta \tan \theta = 2\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P2 2014 June Q3
5 marks Moderate -0.8
3
  1. Find \(\int 4 \cos \left( \frac { 1 } { 3 } x + 2 \right) \mathrm { d } x\).
  2. Use the trapezium rule with three intervals to find an approximation to $$\int _ { 0 } ^ { 12 } \sqrt { } \left( 4 + x ^ { 2 } \right) \mathrm { d } x$$ giving your answer correct to 3 significant figures.
CAIE P2 2014 June Q4
6 marks Moderate -0.3
4 The parametric equations of a curve are $$x = 2 \ln ( t + 1 ) , \quad y = 4 \mathrm { e } ^ { t }$$ Find the equation of the tangent 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 2014 June Q5
6 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{de8af872-9f77-4787-8e66-ed199405ca25-2_583_597_1457_772} The variables \(x\) and \(y\) satisfy the equation \(y = K \left( 2 ^ { p x } \right)\), where \(K\) and \(p\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points ( \(1.35,1.87\) ) and ( \(3.35,3.81\) ), as shown in the diagram. Find the values of \(K\) and \(p\) correct to 2 decimal places.
[0pt] [6]
CAIE P2 2014 June Q6
7 marks Moderate -0.3
6 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = x ^ { 3 } + 2 x + a$$ where \(a\) is a constant.
  1. Given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\), find the value of \(a\).
  2. When \(a\) has this value, find the quotient when \(\mathrm { p } ( x )\) is divided by ( \(x + 2\) ) and hence show that the equation \(\mathrm { p } ( x ) = 0\) has exactly one real root.
CAIE P2 2014 June Q7
9 marks Standard +0.3
7 It is given that \(\int _ { 0 } ^ { a } \left( \frac { 1 } { 2 } \mathrm { e } ^ { 3 x } + x ^ { 2 } \right) \mathrm { d } x = 10\), where \(a\) is a positive constant.
  1. Show that \(a = \frac { 1 } { 3 } \ln \left( 61 - 2 a ^ { 3 } \right)\).
  2. Show by calculation that the value of \(a\) lies between 1.0 and 1.5.
  3. 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.