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CAIE P2 2003 November Q3
3 The polynomial \(x ^ { 4 } - 6 x ^ { 2 } + x + a\) is denoted by \(\mathrm { f } ( x )\).
  1. It is given that \(( x + 1 )\) is a factor of \(\mathrm { f } ( x )\). Find the value of \(a\).
  2. When \(a\) has this value, verify that ( \(x - 2\) ) is also a factor of \(\mathrm { f } ( x )\) and hence factorise \(\mathrm { f } ( x )\) completely.
CAIE P2 2003 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{f5be66c3-7234-4039-9b66-199b35430c7d-3_543_850_778_644} The diagram shows the curve \(y = ( 4 - x ) \mathrm { e } ^ { x }\) and its maximum point \(M\). The curve cuts the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).
  1. Write down the coordinates of \(A\) and \(B\).
  2. Find the \(x\)-coordinate of \(M\).
  3. The point \(P\) on the curve has \(x\)-coordinate \(p\). The tangent to the curve at \(P\) passes through the origin \(O\). Calculate the value of \(p\).
CAIE P2 2003 November Q7
7
  1. By differentiating \(\frac { \cos x } { \sin x }\), show that if \(y = \cot x\) then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \operatorname { cosec } ^ { 2 } x\).
  2. Hence show that \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 2 } \pi } \operatorname { cosec } ^ { 2 } x \mathrm {~d} x = \sqrt { } 3\). By using appropriate trigonometrical identities, find the exact value of
  3. \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 2 } \pi } \cot ^ { 2 } x \mathrm {~d} x\),
  4. \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 2 } \pi } \frac { 1 } { 1 - \cos 2 x } \mathrm {~d} x\).
CAIE P2 2004 November Q1
1 Solve the inequality \(| x + 1 | > | x |\).
CAIE P2 2004 November Q2
2 Solve the equation \(x ^ { 3.9 } = 11 x ^ { 3.2 }\), where \(x \neq 0\).
CAIE P2 2004 November Q3
3 Find the values of \(x\) satisfying the equation $$3 \sin 2 x = \cos x$$ for \(0 ^ { \circ } \leqslant x \leqslant 90 ^ { \circ }\).
CAIE P2 2004 November Q4
4 The cubic polynomial \(2 x ^ { 3 } - 5 x ^ { 2 } + a x + b\) is denoted by \(\mathrm { f } ( x )\). It is given that ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x )\), and that when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) the remainder is - 6 . Find the values of \(a\) and \(b\).
CAIE P2 2004 November Q5
5 The curve with equation \(y = x ^ { 2 } \ln x\), where \(x > 0\), has one stationary point.
  1. Find the \(x\)-coordinate of this point, giving your answer in terms of e .
  2. Determine whether this point is a maximum or a minimum point.
CAIE P2 2004 November Q6
6
  1. By sketching a suitable pair of graphs, show that there is only one value of \(x\) in the interval \(0 < x < \frac { 1 } { 2 } \pi\) that is a root of the equation $$\cot x = x$$
  2. Verify by calculation that this root lies between 0.8 and 0.9 radians.
  3. Show that this value of \(x\) is also a root of the equation $$x = \tan ^ { - 1 } \left( \frac { 1 } { x } \right)$$
  4. Use the iterative formula $$x _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 1 } { x _ { n } } \right)$$ to determine this root correct to 2 decimal places, showing the result of each iteration.
CAIE P2 2004 November Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{25dffd43-9456-449b-be77-8402109ee603-3_608_672_283_733} The diagram shows the curve \(y = 2 \mathrm { e } ^ { x } + 3 \mathrm { e } ^ { - 2 x }\). The curve cuts the \(y\)-axis at \(A\).
  1. Write down the coordinates of \(A\).
  2. Find the equation of the tangent to the curve at \(A\), and state the coordinates of the point where this tangent meets the \(x\)-axis.
  3. Calculate the area of the region bounded by the curve and by the lines \(x = 0 , y = 0\) and \(x = 1\), giving your answer correct to 2 significant figures.
CAIE P2 2004 November Q8
8
  1. Express \(\cos \theta + \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), giving the exact values of \(R\) and \(\alpha\).
  2. Hence show that $$\frac { 1 } { ( \cos \theta + \sin \theta ) ^ { 2 } } = \frac { 1 } { 2 } \sec ^ { 2 } \left( \theta - \frac { 1 } { 4 } \pi \right)$$
  3. By differentiating \(\frac { \sin x } { \cos x }\), show that if \(y = \tan x\) then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec ^ { 2 } x\).
  4. Using the results of parts (ii) and (iii), show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { 1 } { ( \cos \theta + \sin \theta ) ^ { 2 } } \mathrm {~d} \theta = 1$$
CAIE P2 2005 November Q1
1 Solve the inequality \(( 0.8 ) ^ { x } < 0.5\).
CAIE P2 2005 November Q2
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
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
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
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
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
3 marks
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
1 Solve the inequality \(| 2 x - 1 | > | x |\).
CAIE P2 2006 November Q2
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
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
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
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
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
1 Show that $$\int _ { 1 } ^ { 4 } \frac { 1 } { 2 x + 1 } \mathrm {~d} x = \frac { 1 } { 2 } \ln 3$$