Questions P3 (1243 questions)

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CAIE P3 2004 November Q4
7 marks Standard +0.3
4
  1. Show that the equation $$\tan \left( 45 ^ { \circ } + x \right) = 2 \tan \left( 45 ^ { \circ } - x \right)$$ can be written in the form $$\tan ^ { 2 } x - 6 \tan x + 1 = 0$$
  2. Hence solve the equation \(\tan \left( 45 ^ { \circ } + x \right) = 2 \tan \left( 45 ^ { \circ } - x \right)\), for \(0 ^ { \circ } < x < 90 ^ { \circ }\).
CAIE P3 2004 November Q5
8 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{8c533469-393c-4e4c-a6ec-eab1303741e7-2_385_476_1653_836} The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius \(r\). The angle \(A O B\) is \(\alpha\) radians, where \(0 < \alpha < \frac { 1 } { 2 } \pi\). The point \(N\) on \(O A\) is such that \(B N\) is perpendicular to \(O A\). The area of the triangle \(O N B\) is half the area of the sector \(O A B\).
  1. Show that \(\alpha\) satisfies the equation \(\sin 2 x = x\).
  2. By sketching a suitable pair of graphs, show that this equation has exactly one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
  3. Use the iterative formula $$x _ { n + 1 } = \sin \left( 2 x _ { n } \right)$$ with initial value \(x _ { 1 } = 1\), to find \(\alpha\) correct to 2 decimal places, showing the result of each iteration.
CAIE P3 2004 November Q6
8 marks Standard +0.3
6 The complex numbers \(1 + 3 \mathrm { i }\) and \(4 + 2 \mathrm { i }\) are denoted by \(u\) and \(v\) respectively.
  1. Find, in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real, the complex numbers \(u - v\) and \(\frac { u } { v }\).
  2. State the argument of \(\frac { u } { v }\). In an Argand diagram, with origin \(O\), the points \(A , B\) and \(C\) represent the numbers \(u , v\) and \(u - v\) respectively.
  3. State fully the geometrical relationship between \(O C\) and \(B A\).
  4. Prove that angle \(A O B = \frac { 1 } { 4 } \pi\) radians.
CAIE P3 2004 November Q7
9 marks Standard +0.8
7 \includegraphics[max width=\textwidth, alt={}, center]{8c533469-393c-4e4c-a6ec-eab1303741e7-3_480_901_973_621} The diagram shows the curve \(y = x ^ { 2 } e ^ { - \frac { 1 } { 2 } x }\).
  1. Find the \(x\)-coordinate of \(M\), the maximum point of the curve.
  2. Find the area of the shaded region enclosed by the curve, the \(x\)-axis and the line \(x = 1\), giving your answer in terms of e.
CAIE P3 2004 November Q8
9 marks Moderate -0.8
8 An appropriate form for expressing \(\frac { 3 x } { ( x + 1 ) ( x - 2 ) }\) in partial fractions is $$\frac { A } { x + 1 } + \frac { B } { x - 2 }$$ where \(A\) and \(B\) are constants.
  1. Without evaluating any constants, state appropriate forms for expressing the following in partial fractions:
    1. \(\frac { 4 x } { ( x + 4 ) \left( x ^ { 2 } + 3 \right) }\),
    2. \(\frac { 2 x + 1 } { ( x - 2 ) ( x + 2 ) ^ { 2 } }\).
  2. Show that \(\int _ { 3 } ^ { 4 } \frac { 3 x } { ( x + 1 ) ( x - 2 ) } \mathrm { d } x = \ln 5\).
CAIE P3 2004 November Q9
10 marks Challenging +1.3
9 The lines \(l\) and \(m\) have vector equations $$\mathbf { r } = 2 \mathbf { i } - \mathbf { j } + 4 \mathbf { k } + s ( \mathbf { i } + \mathbf { j } - \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = - 2 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + t ( - 2 \mathbf { i } + \mathbf { j } + \mathbf { k } )$$ respectively.
  1. Show that \(l\) and \(m\) do not intersect. The point \(P\) lies on \(l\) and the point \(Q\) has position vector \(2 \mathbf { i } - \mathbf { k }\).
  2. Given that the line \(P Q\) is perpendicular to \(l\), find the position vector of \(P\).
  3. Verify that \(Q\) lies on \(m\) and that \(P Q\) is perpendicular to \(m\).
CAIE P3 2004 November Q10
10 marks Standard +0.3
10 A rectangular reservoir has a horizontal base of area \(1000 \mathrm {~m} ^ { 2 }\). At time \(t = 0\), it is empty and water begins to flow into it at a constant rate of \(30 \mathrm {~m} ^ { 3 } \mathrm {~s} ^ { - 1 }\). At the same time, water begins to flow out at a rate proportional to \(\sqrt { } h\), where \(h \mathrm {~m}\) is the depth of the water at time \(t \mathrm {~s}\). When \(h = 1 , \frac { \mathrm {~d} h } { \mathrm {~d} t } = 0.02\).
  1. Show that \(h\) satisfies the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = 0.01 ( 3 - \sqrt { } h )$$ It is given that, after making the substitution \(x = 3 - \sqrt { } h\), the equation in part (i) becomes $$( x - 3 ) \frac { \mathrm { d } x } { \mathrm {~d} t } = 0.005 x$$
  2. Using the fact that \(x = 3\) when \(t = 0\), solve this differential equation, obtaining an expression for \(t\) in terms of \(x\).
  3. Find the time at which the depth of water reaches 4 m .
CAIE P3 2005 November Q1
4 marks Standard +0.3
1 Given that \(a\) is a positive constant, solve the inequality $$| x - 3 a | > | x - a |$$
CAIE P3 2005 November Q2
5 marks Moderate -0.5
2 \includegraphics[max width=\textwidth, alt={}, center]{9275a3ed-8820-481b-9fc8-28c21b81dbed-2_559_789_513_678} Two variable quantities \(x\) and \(y\) are related by the equation \(y = A x ^ { n }\), where \(A\) and \(n\) are constants. The diagram shows the result of plotting \(\ln y\) against \(\ln x\) for four pairs of values of \(x\) and \(y\). Use the diagram to estimate the values of \(A\) and \(n\).
CAIE P3 2005 November Q3
7 marks Standard +0.3
3 The equation of a curve is \(y = x + \cos 2 x\). Find the \(x\)-coordinates of the stationary points of the curve for which \(0 \leqslant x \leqslant \pi\), and determine the nature of each of these stationary points.
CAIE P3 2005 November Q4
7 marks Standard +0.3
4 The equation \(x ^ { 3 } - x - 3 = 0\) has one real root, \(\alpha\).
  1. Show that \(\alpha\) lies between 1 and 2 . Two iterative formulae derived from this equation are as follows: $$\begin{aligned} & x _ { n + 1 } = x _ { n } ^ { 3 } - 3 \\ & x _ { n + 1 } = \left( x _ { n } + 3 \right) ^ { \frac { 1 } { 3 } } \end{aligned}$$ Each formula is used with initial value \(x _ { 1 } = 1.5\).
  2. Show that one of these formulae produces a sequence which fails to converge, and use the other formula to calculate \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2005 November Q5
7 marks Moderate -0.3
5 By expressing \(8 \sin \theta - 6 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), solve the equation $$8 \sin \theta - 6 \cos \theta = 7$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P3 2005 November Q6
8 marks Standard +0.3
6
  1. Use the substitution \(x = \sin ^ { 2 } \theta\) to show that $$\int \sqrt { } \left( \frac { x } { 1 - x } \right) \mathrm { d } x = \int 2 \sin ^ { 2 } \theta \mathrm {~d} \theta$$
  2. Hence find the exact value of $$\left. \int _ { 0 } ^ { \frac { 1 } { 4 } } \sqrt { ( } \frac { x } { 1 - x } \right) \mathrm { d } x$$
CAIE P3 2005 November Q7
8 marks Moderate -0.3
7 The equation \(2 x ^ { 3 } + x ^ { 2 } + 25 = 0\) has one real root and two complex roots.
  1. Verify that \(1 + 2 \mathrm { i }\) is one of the complex roots.
  2. Write down the other complex root of the equation.
  3. Sketch an Argand diagram showing the point representing the complex number \(1 + 2 \mathrm { i }\). Show on the same diagram the set of points representing the complex numbers \(z\) which satisfy $$| z | = | z - 1 - 2 \mathrm { i } |$$
CAIE P3 2005 November Q8
8 marks Standard +0.3
8 In a certain chemical reaction the amount, \(x\) grams, of a substance present is decreasing. The rate of decrease of \(x\) is proportional to the product of \(x\) and the time, \(t\) seconds, since the start of the reaction. Thus \(x\) and \(t\) satisfy the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = - k x t$$ where \(k\) is a positive constant. At the start of the reaction, when \(t = 0 , x = 100\).
  1. Solve this differential equation, obtaining a relation between \(x , k\) and \(t\).
  2. 20 seconds after the start of the reaction the amount of substance present is 90 grams. Find the time after the start of the reaction at which the amount of substance present is 50 grams.
CAIE P3 2005 November Q9
10 marks Standard +0.3
9
  1. Express \(\frac { 3 x ^ { 2 } + x } { ( x + 2 ) \left( x ^ { 2 } + 1 \right) }\) in partial fractions.
  2. Hence obtain the expansion of \(\frac { 3 x ^ { 2 } + x } { ( x + 2 ) \left( x ^ { 2 } + 1 \right) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
CAIE P3 2005 November Q10
11 marks Standard +0.3
10 The straight line \(l\) passes through the points \(A\) and \(B\) with position vectors $$2 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } \quad \text { and } \quad \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }$$ respectively. This line intersects the plane \(p\) with equation \(x - 2 y + 2 z = 6\) at the point \(C\).
  1. Find the position vector of \(C\).
  2. Find the acute angle between \(l\) and \(p\).
  3. Show that the perpendicular distance from \(A\) to \(p\) is equal to 2 .
CAIE P3 2006 November Q1
4 marks Standard +0.3
1 Find the set of values of \(x\) satisfying the inequality \(\left| 3 ^ { x } - 8 \right| < 0.5\), giving 3 significant figures in your answer.
CAIE P3 2006 November Q2
4 marks Standard +0.3
2 Solve the equation $$\tan x \tan 2 x = 1 ,$$ giving all solutions in the interval \(0 ^ { \circ } < x < 180 ^ { \circ }\).
CAIE P3 2006 November Q3
6 marks Moderate -0.8
3 The curve with equation \(y = 6 \mathrm { e } ^ { x } - \mathrm { e } ^ { 3 x }\) has one stationary point.
  1. Find the \(x\)-coordinate of this point.
  2. Determine whether this point is a maximum or a minimum point.
CAIE P3 2006 November Q4
6 marks Moderate -0.3
4 Given that \(y = 2\) when \(x = 0\), solve the differential equation $$y \frac { \mathrm {~d} y } { \mathrm {~d} x } = 1 + y ^ { 2 }$$ obtaining an expression for \(y ^ { 2 }\) in terms of \(x\).
CAIE P3 2006 November Q5
6 marks Standard +0.8
5
  1. Simplify \(( \sqrt { } ( 1 + x ) + \sqrt { } ( 1 - x ) ) ( \sqrt { } ( 1 + x ) - \sqrt { } ( 1 - x ) )\), showing your working, and deduce that $$\frac { 1 } { \sqrt { } ( 1 + x ) + \sqrt { } ( 1 - x ) } = \frac { \sqrt { } ( 1 + x ) - \sqrt { } ( 1 - x ) } { 2 x }$$
  2. Using this result, or otherwise, obtain the expansion of $$\frac { 1 } { \sqrt { } ( 1 + x ) + \sqrt { } ( 1 - x ) }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
CAIE P3 2006 November Q6
9 marks Standard +0.3
6 The equation of a curve is \(x ^ { 3 } + 2 y ^ { 3 } = 3 x y\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y - x ^ { 2 } } { 2 y ^ { 2 } - x }\).
  2. Find the coordinates of the point, other than the origin, where the curve has a tangent which is parallel to the \(x\)-axis.
CAIE P3 2006 November Q7
9 marks Standard +0.3
7 The line \(l\) has equation \(\mathbf { r } = \mathbf { j } + \mathbf { k } + s ( \mathbf { i } - 2 \mathbf { j } + \mathbf { k } )\). The plane \(p\) has equation \(x + 2 y + 3 z = 5\).
  1. Show that the line \(l\) lies in the plane \(p\).
  2. A second plane is perpendicular to the plane \(p\), parallel to the line \(l\) and contains the point with position vector \(2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k }\). Find the equation of this plane, giving your answer in the form \(a x + b y + c z = d\).
CAIE P3 2006 November Q8
10 marks Standard +0.3
8 Let \(\mathrm { f } ( x ) = \frac { 7 x + 4 } { ( 2 x + 1 ) ( x + 1 ) ^ { 2 } }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence show that \(\int _ { 0 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = 2 + \ln \frac { 5 } { 3 }\).