Questions P2 (867 questions)

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CAIE P2 2012 November Q8
9 marks Standard +0.3
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
6 marks Standard +0.3
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
4 marks Standard +0.3
1 Solve the inequality \(| x + 1 | < | 3 x + 5 |\).
CAIE P2 2013 November Q2
6 marks Moderate -0.8
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
6 marks Standard +0.3
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
9 marks Moderate -0.3
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
8 marks Standard +0.3
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
9 marks Moderate -0.3
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
5 marks Moderate -0.8
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
5 marks Standard +0.3
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
6 marks Standard +0.3
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
7 marks Moderate -0.8
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
8 marks Moderate -0.3
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 \(( 1,0 )\).
CAIE P2 2013 November Q6
9 marks Moderate -0.3
6
  1. Find \(\int ( \sin x - \cos x ) ^ { 2 } \mathrm {~d} x\).
    1. Use the trapezium rule with 2 intervals to estimate the value of $$\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 2 } \pi } \operatorname { cosec } x d x$$ giving your answer correct to 3 decimal places.
    2. Using a sketch of the graph of \(y = \operatorname { cosec } x\) for \(0 < x \leqslant \frac { 1 } { 2 } \pi\), explain whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the integral in part (i).
CAIE P2 2013 November Q7
10 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{0900b607-6136-4bf7-a42e-6824d1a21e43-3_451_451_255_845} The diagram shows part of the curve \(y = 8 x + \frac { 1 } { 2 } \mathrm { e } ^ { x }\). The shaded region \(R\) is bounded by the curve and by the lines \(x = 0 , y = 0\) and \(x = a\), where \(a\) is positive. The area of \(R\) is equal to \(\frac { 1 } { 2 }\).
  1. Find an equation satisfied by \(a\), and show that the equation can be written in the form $$a = \sqrt { } \left( \frac { 2 - \mathrm { e } ^ { a } } { 8 } \right)$$
  2. Verify by calculation that the equation \(a = \sqrt { } \left( \frac { 2 - \mathrm { e } ^ { a } } { 8 } \right)\) has a root between 0.2 and 0.3.
  3. Use the iterative formula \(a _ { n + 1 } = \sqrt { } \left( \frac { 2 - \mathrm { e } ^ { a _ { n } } } { 8 } \right)\) to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P2 2013 November Q2
6 marks Moderate -0.5
2 \includegraphics[max width=\textwidth, alt={}, center]{a3e778cb-9f95-4750-ba49-a57ee22af018-2_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 2014 November Q1
3 marks Moderate -0.5
1 Use the trapezium rule with four intervals to find an approximation to $$\int _ { 1 } ^ { 5 } \left| 2 ^ { x } - 8 \right| \mathrm { d } x$$
CAIE P2 2014 November Q2
5 marks Moderate -0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{72d50061-ead5-466a-96fc-2203438d1407-2_654_693_532_724} 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 ( \(0.75,1.70\) ) and ( \(1.53,2.18\) ), as shown in the diagram. Find the values of \(a\) and \(b\) correct to 2 decimal places.
CAIE P2 2014 November Q3
7 marks Moderate -0.8
3
  1. Find \(\int 4 \cos ^ { 2 } \left( \frac { 1 } { 2 } \theta \right) \mathrm { d } \theta\).
  2. Find the exact value of \(\int _ { - 1 } ^ { 6 } \frac { 1 } { 2 x + 3 } \mathrm {~d} x\).
CAIE P2 2014 November Q4
8 marks Moderate -0.3
4 For each of the following curves, find the exact gradient at the point indicated:
  1. \(y = 3 \cos 2 x - 5 \sin x\) at \(\left( \frac { 1 } { 6 } \pi , - 1 \right)\),
  2. \(x ^ { 3 } + 6 x y + y ^ { 3 } = 21\) at \(( 1,2 )\).
CAIE P2 2014 November Q5
9 marks Standard +0.3
5
  1. Given that ( \(x + 2\) ) and ( \(x + 3\) ) are factors of $$5 x ^ { 3 } + a x ^ { 2 } + b$$ find the values of the constants \(a\) and \(b\).
  2. When \(a\) and \(b\) have these values, factorise $$5 x ^ { 3 } + a x ^ { 2 } + b$$ completely, and hence solve the equation $$5 ^ { 3 y + 1 } + a \times 5 ^ { 2 y } + b = 0$$ giving any answers correct to 3 significant figures.
CAIE P2 2014 November Q6
9 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{72d50061-ead5-466a-96fc-2203438d1407-3_296_675_945_735} The diagram shows part of the curve \(y = \frac { x ^ { 2 } } { 1 + \mathrm { e } ^ { 3 x } }\) and its maximum point \(M\). The \(x\)-coordinate of \(M\) is denoted by \(m\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence show that \(m\) satisfies the equation \(x = \frac { 2 } { 3 } \left( 1 + \mathrm { e } ^ { - 3 x } \right)\).
  2. Show by calculation that \(m\) lies between 0.7 and 0.8 .
  3. Use an iterative formula based on the equation in part (i) to find \(m\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
CAIE P2 2014 November Q7
9 marks Standard +0.8
7 The angle \(\alpha\) lies between \(0 ^ { \circ }\) and \(90 ^ { \circ }\) and is such that $$2 \tan ^ { 2 } \alpha + \sec ^ { 2 } \alpha = 5 - 4 \tan \alpha$$
  1. Show that $$3 \tan ^ { 2 } \alpha + 4 \tan \alpha - 4 = 0$$ and hence find the exact value of \(\tan \alpha\).
  2. It is given that the angle \(\beta\) is such that \(\cot ( \alpha + \beta ) = 6\). Without using a calculator, find the exact value of \(\cot \beta\).
CAIE P2 2014 November Q1
3 marks Moderate -0.8
1 Solve the equation \(| 3 x - 1 | = | 2 x + 5 |\).
CAIE P2 2014 November Q2
5 marks Moderate -0.3
2
  1. Find \(\int _ { 0 } ^ { a } \left( \mathrm { e } ^ { - x } + 6 \mathrm { e } ^ { - 3 x } \right) \mathrm { d } x\), where \(a\) is a positive constant.
  2. Deduce the value of \(\int _ { 0 } ^ { \infty } \left( \mathrm { e } ^ { - x } + 6 \mathrm { e } ^ { - 3 x } \right) \mathrm { d } x\).