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CAIE P2 2012 November Q4
4 The parametric equations of a curve are $$x = \ln ( 1 - 2 t ) , \quad y = \frac { 2 } { t } , \quad \text { for } t < 0$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 - 2 t } { t ^ { 2 } }\).
  2. Find the exact coordinates of the only point on the curve at which the gradient is 3 .
CAIE P2 2012 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{9e1bd528-e7c4-4936-a05a-dde1d1ace7c2-2_512_775_1318_683} The diagram shows the curve \(y = \cos x\), for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). A rectangle \(O A B C\) is drawn, where \(B\) is the point on the curve with \(x\)-coordinate \(\theta\), and \(A\) and \(C\) are on the axes, as shown. The shaded region \(R\) is bounded by the curve and by the lines \(x = \theta\) and \(y = 0\).
  1. Find the area of \(R\) in terms of \(\theta\).
  2. The area of the rectangle \(O A B C\) is equal to the area of \(R\). Show that $$\theta = \frac { 1 - \sin \theta } { \cos \theta }$$
  3. Use the iterative formula \(\theta _ { n + 1 } = \frac { 1 - \sin \theta _ { n } } { \cos \theta _ { n } }\), with initial value \(\theta _ { 1 } = 0.5\), to determine the value of \(\theta\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P2 2012 November Q6
6
  1. Use the trapezium rule with two intervals to estimate the value of $$\int _ { 0 } ^ { 1 } \frac { 1 } { 6 + 2 \mathrm { e } ^ { x } } \mathrm {~d} x$$ giving your answer correct to 2 decimal places.
  2. Find \(\int \frac { \left( \mathrm { e } ^ { x } - 2 \right) ^ { 2 } } { \mathrm { e } ^ { 2 x } } \mathrm {~d} x\).
CAIE P2 2012 November Q7
7 The polynomial \(2 x ^ { 3 } - 4 x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that when \(\mathrm { p } ( x )\) is divided by \(( x + 1 )\) the remainder is 4 , and that when \(\mathrm { p } ( x )\) is divided by \(( x - 3 )\) the remainder is 12 .
  1. Find the values of \(a\) and \(b\).
  2. When \(a\) and \(b\) have these values, find the quotient and remainder when \(\mathrm { p } ( x )\) is divided by ( \(x ^ { 2 } - 2\) ).
CAIE P2 2012 November Q8
8
  1. By differentiating \(\frac { 1 } { \cos \theta }\), show that if \(y = \sec \theta\) then \(\frac { \mathrm { d } y } { \mathrm {~d} \theta } = \tan \theta \sec \theta\).
  2. Hence show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} \theta ^ { 2 } } = a \sec ^ { 3 } \theta + b \sec \theta$$ giving the values of \(a\) and \(b\).
  3. Find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( 1 + \tan ^ { 2 } \theta - 3 \sec \theta \tan \theta \right) d \theta$$
CAIE P2 2012 November Q1
1 Solve the inequality \(| 2 x + 1 | < | 2 x - 5 |\).
CAIE P2 2012 November Q2
2 The curve with equation \(y = \frac { \sin 2 x } { \mathrm { e } ^ { 2 x } }\) has one stationary point in the interval \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). Find the exact \(x\)-coordinate of this point.
CAIE P2 2012 November Q3
3 The polynomial \(x ^ { 4 } - 4 x ^ { 3 } + 3 x ^ { 2 } + 4 x - 4\) is denoted by \(\mathrm { p } ( x )\).
  1. Find the quotient when \(\mathrm { p } ( x )\) is divided by \(x ^ { 2 } - 3 x + 2\).
  2. Hence solve the equation \(\mathrm { p } ( x ) = 0\).
CAIE P2 2012 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{0355f624-3a35-4b9e-8520-af011a0fb6db-2_499_787_922_678} The diagram shows the part of the curve \(y = \sqrt { } ( 2 - \sin x )\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
  1. Use the trapezium rule with 2 intervals to estimate the value of $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sqrt { } ( 2 - \sin x ) \mathrm { d } x$$ giving your answer correct to 2 decimal places.
  2. The line \(y = x\) intersects the curve \(y = \sqrt { } ( 2 - \sin x )\) at the point \(P\). Use the iterative formula $$x _ { n + 1 } = \sqrt { } \left( 2 - \sin x _ { n } \right)$$ to determine the \(x\)-coordinate of \(P\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P2 2012 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{0355f624-3a35-4b9e-8520-af011a0fb6db-3_512_732_251_705} 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 \(( 1,2.9 )\) and \(( 3.5,1.4 )\), as shown in the diagram. Find the values of \(A\) and \(b\), correct to 2 decimal places.
CAIE P2 2012 November Q6
6
  1. Find \(\int 4 \mathrm { e } ^ { - \frac { 1 } { 2 } x } \mathrm {~d} x\).
  2. Show that \(\int _ { 1 } ^ { 3 } \frac { 6 } { 3 x - 1 } \mathrm {~d} x = \ln 16\).
CAIE P2 2012 November Q7
7 The equation of a curve is $$3 x ^ { 2 } - 4 x y + 2 y ^ { 2 } - 6 = 0$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 x - 2 y } { 2 x - 2 y }\).
  2. Find the coordinates of each of the points on the curve where the tangent is parallel to the \(x\)-axis.
CAIE P2 2012 November Q8
8
  1. Given that \(\tan A = t\) and \(\tan ( A + B ) = 4\), find \(\tan B\) in terms of \(t\).
  2. Solve the equation $$2 \tan \left( 45 ^ { \circ } - x \right) = 3 \tan x$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
CAIE P2 2012 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{96a4df57-b3c7-4dbf-9bea-bb00ed6a4a16-2_512_775_1318_683} The diagram shows the curve \(y = \cos x\), for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). A rectangle \(O A B C\) is drawn, where \(B\) is the point on the curve with \(x\)-coordinate \(\theta\), and \(A\) and \(C\) are on the axes, as shown. The shaded region \(R\) is bounded by the curve and by the lines \(x = \theta\) and \(y = 0\).
  1. Find the area of \(R\) in terms of \(\theta\).
  2. The area of the rectangle \(O A B C\) is equal to the area of \(R\). Show that $$\theta = \frac { 1 - \sin \theta } { \cos \theta }$$
  3. Use the iterative formula \(\theta _ { n + 1 } = \frac { 1 - \sin \theta _ { n } } { \cos \theta _ { n } }\), with initial value \(\theta _ { 1 } = 0.5\), to determine the value of \(\theta\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P2 2013 November Q1
1 Solve the inequality \(| x + 1 | < | 3 x + 5 |\).
CAIE P2 2013 November Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{faf83d93-40b6-4557-bfd5-f94c67470dfa-2_449_639_388_753} The diagram shows the curve \(y = x ^ { 4 } + 2 x - 9\). The curve cuts the positive \(x\)-axis at the point \(P\).
  1. Verify by calculation that the \(x\)-coordinate of \(P\) lies between 1.5 and 1.6.
  2. Show that the \(x\)-coordinate of \(P\) satisfies the equation $$x = \sqrt [ 3 ] { \left( \frac { 9 } { x } - 2 \right) }$$
  3. Use the iterative formula $$x _ { n + 1 } = \sqrt [ 3 ] { \left( \frac { 9 } { x _ { n } } - 2 \right) }$$ to determine the \(x\)-coordinate of \(P\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P2 2013 November Q3
3 The equation of a curve is \(y = \frac { 1 } { 2 } \mathrm { e } ^ { 2 x } - 5 \mathrm { e } ^ { x } + 4 x\). Find the exact \(x\)-coordinate of each of the stationary points of the curve and determine the nature of each stationary point.
CAIE P2 2013 November Q4
4
  1. The polynomial \(x ^ { 3 } + a x ^ { 2 } + b x + 8\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that when \(\mathrm { p } ( x )\) is divided by \(( x - 3 )\) the remainder is 14 , and that when \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\) the remainder is 24 . Find the values of \(a\) and \(b\).
  2. When \(a\) and \(b\) have these values, find the quotient when \(\mathrm { p } ( x )\) is divided by \(x ^ { 2 } + 2 x - 8\) and hence solve the equation \(\mathrm { p } ( x ) = 0\).
CAIE P2 2013 November Q5
5 The parametric equations of a curve are $$x = \cos 2 \theta - \cos \theta , \quad y = 4 \sin ^ { 2 } \theta$$ for \(0 \leqslant \theta \leqslant \pi\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 8 \cos \theta } { 1 - 4 \cos \theta }\).
  2. Find the coordinates of the point on the curve at which the gradient is - 4 .
CAIE P2 2013 November Q6
6
  1. Find
    1. \(\int \frac { \mathrm { e } ^ { 2 x } + 6 } { \mathrm { e } ^ { 2 x } } \mathrm {~d} x\),
    2. \(\int 3 \cos ^ { 2 } x \mathrm {~d} x\).
  2. Use the trapezium rule with 2 intervals to estimate the value of $$\int _ { 1 } ^ { 2 } \frac { 6 } { \ln ( x + 2 ) } \mathrm { d } x$$ giving your answer correct to 2 decimal places.
    1. Express \(3 \cos \theta + \sin \theta\) 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 $$3 \cos 2 x + \sin 2 x = 2$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
CAIE P2 2013 November Q1
1
  1. Find \(\int \frac { 2 } { 4 x - 1 } \mathrm {~d} x\).
  2. Hence find \(\int _ { 1 } ^ { 7 } \frac { 2 } { 4 x - 1 } \mathrm {~d} x\), expressing your answer in the form \(\ln a\), where \(a\) is an integer.
CAIE P2 2013 November Q2
2 The curve \(y = \frac { \mathrm { e } ^ { 3 x - 1 } } { 2 x }\) has one stationary point. Find the coordinates of this stationary point.
CAIE P2 2013 November Q3
3 Solve the equation \(2 \cot ^ { 2 } \theta - 5 \operatorname { cosec } \theta = 10\), giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P2 2013 November Q4
4
  1. The polynomial \(a x ^ { 3 } + b x ^ { 2 } - 25 x - 6\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x - 3 )\) and \(( x + 2 )\) are factors of \(\mathrm { p } ( x )\). Find the values of \(a\) and \(b\).
  2. When \(a\) and \(b\) have these values, factorise \(\mathrm { p } ( x )\) completely.
CAIE P2 2013 November Q5
5 The parametric equations of a curve are $$x = 1 + \sqrt { } t , \quad y = 3 \ln t$$
  1. Find the exact value of the gradient of the curve at the point \(P\) where \(y = 6\).
  2. Show that the tangent to the curve at \(P\) passes through the point \(( 1,0 )\).