Questions — CAIE P2 (709 questions)

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CAIE P2 2017 March Q4
6 marks Standard +0.3
4 Find the gradient of the curve $$x ^ { 2 } \sin y + \cos 3 y = 4$$ at the point \(\left( 2 , \frac { 1 } { 2 } \pi \right)\).
CAIE P2 2017 March Q5
8 marks Standard +0.3
5 It is given that \(a\) is a positive constant such that $$\int _ { 0 } ^ { a } \left( 1 + 2 x + 3 \mathrm { e } ^ { 3 x } \right) \mathrm { d } x = 250$$
  1. Show that \(a = \frac { 1 } { 3 } \ln \left( 251 - a - a ^ { 2 } \right)\).
  2. Use an iterative formula based on the equation in part (i) to find the value of \(a\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
CAIE P2 2017 March Q6
10 marks Moderate -0.3
6 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } + b x ^ { 2 } - 17 x - a$$ where \(a\) and \(b\) are constants. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) and that the remainder is 28 when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\).
  1. Find the values of \(a\) and \(b\).
  2. Hence factorise \(\mathrm { p } ( x )\) completely.
  3. State the number of roots of the equation \(\mathrm { p } \left( 2 ^ { y } \right) = 0\), justifying your answer. \includegraphics[max width=\textwidth, alt={}, center]{17025451-6f07-4f35-9dfc-869e084b5ed0-10_508_538_310_799} The diagram shows part of the curve $$y = 2 \cos 2 x \cos \left( 2 x + \frac { 1 } { 6 } \pi \right)$$ The shaded region is bounded by the curve and the two axes.
  4. Show that \(2 \cos 2 x \cos \left( 2 x + \frac { 1 } { 6 } \pi \right)\) can be expressed in the form $$k _ { 1 } ( 1 + \cos 4 x ) + k _ { 2 } \sin 4 x ,$$ where the values of the constants \(k _ { 1 }\) and \(k _ { 2 }\) are to be determined.
  5. Find the exact area of the shaded region.
CAIE P2 2019 March Q1
4 marks Standard +0.3
1 Solve the equation \(\sec ^ { 2 } \theta + \tan ^ { 2 } \theta = 5 \tan \theta + 4\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\). Show all necessary working.
CAIE P2 2019 March Q2
4 marks Standard +0.3
2 Given that \(x\) satisfies the equation \(| 2 x + 3 | = | 2 x - 1 |\), find the value of $$| 4 x - 3 | - | 6 x |$$
CAIE P2 2019 March Q3
5 marks Moderate -0.5
3 \includegraphics[max width=\textwidth, alt={}, center]{772c14a1-f79a-4147-a293-0ff34f930e20-04_577_569_260_788} The variables \(x\) and \(y\) satisfy the equation \(y = A \mathrm { e } ^ { p x + p }\), where \(A\) and \(p\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \(( 1,2.835 )\) and \(( 6,6.585 )\), as shown in the diagram. Find the values of \(A\) and \(p\).
CAIE P2 2019 March Q4
6 marks Standard +0.3
4
  1. Find the quotient when \(4 x ^ { 3 } + 8 x ^ { 2 } + 11 x + 9\) is divided by ( \(2 x + 1\) ), and show that the remainder is 5 .
  2. Show that the equation \(4 x ^ { 3 } + 8 x ^ { 2 } + 11 x + 4 = 0\) has exactly one real root.
CAIE P2 2019 March Q5
9 marks Moderate -0.3
5 The equation of a curve is \(y = \frac { \mathrm { e } ^ { 2 x } } { 4 x + 1 }\) and the point \(P\) on the curve has \(y\)-coordinate 10 .
  1. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(x = \frac { 1 } { 2 } \ln ( 40 x + 10 )\).
  2. Use the iterative formula \(x _ { n + 1 } = \frac { 1 } { 2 } \ln \left( 40 x _ { n } + 10 \right)\) with \(x _ { 1 } = 2.3\) to find the \(x\)-coordinate of \(P\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
  3. Find the gradient of the curve at \(P\), giving the answer correct to 3 significant figures.
CAIE P2 2019 March Q6
11 marks Moderate -0.3
6
  1. Show that \(\int _ { 1 } ^ { 4 } \left( \frac { 2 } { x } + \frac { 2 } { 2 x + 1 } \right) \mathrm { d } x = \ln 48\).
  2. Find \(\int \sin 2 x ( \cot x + 2 \operatorname { cosec } x ) \mathrm { d } x\).
CAIE P2 2019 March Q7
11 marks Standard +0.3
7 The parametric equations of a curve are $$x = 2 t - \sin 2 t , \quad y = 5 t + \cos 2 t$$ for \(0 \leqslant t \leqslant \frac { 1 } { 2 } \pi\). At the point \(P\) on the curve, the gradient of the curve is 2 .
  1. Show that the value of the parameter at \(P\) satisfies the equation \(2 \sin 2 t - 4 \cos 2 t = 1\).
  2. By first expressing \(2 \sin 2 t - 4 \cos 2 t\) in the form \(R \sin ( 2 t - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), find the coordinates of \(P\). Give each coordinate correct to 3 significant figures.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE P2 2002 November Q1
4 marks Standard +0.3
1 Solve the inequality \(| 2 x - 1 | < | 3 x |\).
CAIE P2 2002 November Q2
5 marks Moderate -0.8
2 The cubic polynomial \(2 x ^ { 3 } + a x ^ { 2 } + b\) is denoted by \(\mathrm { f } ( x )\). It is given that ( \(x + 1\) ) is a factor of \(\mathrm { f } ( x )\), and that when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\) the remainder is - 5 . Find the values of \(a\) and \(b\).
CAIE P2 2002 November Q3
6 marks Moderate -0.8
3
  1. Express \(9 ^ { x }\) in terms of \(y\), where \(y = 3 ^ { x }\).
  2. Hence solve the equation $$2 \left( 9 ^ { x } \right) - 7 \left( 3 ^ { x } \right) + 3 = 0 ,$$ expressing your answers for \(x\) in terms of logarithms where appropriate.
CAIE P2 2002 November Q4
8 marks Standard +0.3
4
  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 $$\sin x = \frac { 1 } { x ^ { 2 } }$$
  2. Verify by calculation that this root lies between 1 and 1.5.
  3. Show that this value of \(x\) is also a root of the equation $$x = \sqrt { } ( \operatorname { cosec } x )$$
  4. Use the iterative formula $$x _ { n + 1 } = \sqrt { } \left( \operatorname { cosec } x _ { n } \right)$$ to determine this root correct to 3 significant figures, showing the value of each approximation that you calculate.
CAIE P2 2002 November Q5
9 marks Standard +0.3
5 The angle \(x\), measured in degrees, satisfies the equation $$\cos \left( x - 30 ^ { \circ } \right) = 3 \sin \left( x - 60 ^ { \circ } \right)$$
  1. By expanding each side, show that the equation may be simplified to $$( 2 \sqrt { } 3 ) \cos x = \sin x$$
  2. Find the two possible values of \(x\) lying between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).
  3. Find the exact value of \(\cos 2 x\), giving your answer as a fraction.
CAIE P2 2002 November Q6
9 marks Moderate -0.8
6
  1. Find the value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } ( \sin 2 x + \cos x ) \mathrm { d } x\).
  2. \includegraphics[max width=\textwidth, alt={}, center]{9894d97f-3b7b-4dbe-b94a-2c8415442038-3_517_880_422_669} The diagram shows part of the curve \(y = \frac { 1 } { x + 1 }\). The shaded region \(R\) is bounded by the curve and by the lines \(x = 1 , y = 0\) and \(x = p\).
    1. Find, in terms of \(p\), the area of \(R\).
    2. Hence find, correct to 1 decimal place, the value of \(p\) for which the area of \(R\) is equal to 2 .
CAIE P2 2002 November Q7
9 marks Standard +0.3
7 The equation of a curve is $$2 x ^ { 2 } + 3 y ^ { 2 } - 2 x y = 10 .$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y - 2 x } { 3 y - x }\).
  2. Find the coordinates of the points on the curve where the tangent is parallel to the \(x\)-axis.
CAIE P2 2004 November Q1
3 marks Moderate -0.5
1 Solve the inequality \(| x + 1 | > | x |\).
CAIE P2 2004 November Q2
3 marks Moderate -0.8
2 Solve the equation \(x ^ { 3.9 } = 11 x ^ { 3.2 }\), where \(x \neq 0\).
CAIE P2 2004 November Q3
4 marks Moderate -0.3
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
5 marks Moderate -0.8
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
6 marks Moderate -0.3
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
8 marks Moderate -0.3
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
11 marks Moderate -0.3
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
10 marks Standard +0.3
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 $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { 1 } { ( \cos \theta + \sin \theta ) ^ { 2 } } \mathrm {~d} \theta = 1$$