Questions — CAIE P3 (1070 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
CAIE P3 2012 November Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{adbef77f-e2ac-40ce-a56b-cf6776534ec1-3_543_1091_1402_529} The diagram shows part of the curve \(y = \sin ^ { 3 } 2 x \cos ^ { 3 } 2 x\). The shaded region shown is bounded by the curve and the \(x\)-axis and its exact area is denoted by \(A\).
  1. Use the substitution \(u = \sin 2 x\) in a suitable integral to find the value of \(A\).
  2. Given that \(\int _ { 0 } ^ { k \pi } \left| \sin ^ { 3 } 2 x \cos ^ { 3 } 2 x \right| \mathrm { d } x = 40 A\), find the value of the constant \(k\).
CAIE P3 2012 November Q8
8 Two lines have equations $$\mathbf { r } = \left( \begin{array} { r } 5
1
- 4 \end{array} \right) + s \left( \begin{array} { r } 1
- 1
3 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r } p
4
- 2 \end{array} \right) + t \left( \begin{array} { r } 2
5
- 4 \end{array} \right) ,$$ where \(p\) is a constant. It is given that the lines intersect.
  1. Find the value of \(p\) and determine the coordinates of the point of intersection.
  2. Find the equation of the plane containing the two lines, giving your answer in the form \(a x + b y + c z = d\), where \(a , b , c\) and \(d\) are integers.
CAIE P3 2012 November Q9
9
  1. Express \(\frac { 9 - 7 x + 8 x ^ { 2 } } { ( 3 - x ) \left( 1 + x ^ { 2 } \right) }\) in partial fractions.
  2. Hence obtain the expansion of \(\frac { 9 - 7 x + 8 x ^ { 2 } } { ( 3 - x ) \left( 1 + x ^ { 2 } \right) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
CAIE P3 2012 November Q10
10
  1. Without using a calculator, solve the equation \(\mathrm { i } w ^ { 2 } = ( 2 - 2 \mathrm { i } ) ^ { 2 }\).
    1. Sketch an Argand diagram showing the region \(R\) consisting of points representing the complex numbers \(z\) where $$| z - 4 - 4 i | \leqslant 2$$
    2. For the complex numbers represented by points in the region \(R\), it is given that $$p \leqslant | z | \leqslant q \quad \text { and } \quad \alpha \leqslant \arg z \leqslant \beta$$ Find the values of \(p , q , \alpha\) and \(\beta\), giving your answers correct to 3 significant figures. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.
      University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE P3 2013 November Q1
1 The equation of a curve is \(y = \frac { 1 + x } { 1 + 2 x }\) for \(x > - \frac { 1 } { 2 }\). Show that the gradient of the curve is always negative.
CAIE P3 2013 November Q2
2 Solve the equation \(2 \left| 3 ^ { x } - 1 \right| = 3 ^ { x }\), giving your answers correct to 3 significant figures.
CAIE P3 2013 November Q3
3 Find the exact value of \(\int _ { 1 } ^ { 4 } \frac { \ln x } { \sqrt { } x } \mathrm {~d} x\).
CAIE P3 2013 November Q4
4 The parametric equations of a curve are $$x = \mathrm { e } ^ { - t } \cos t , \quad y = \mathrm { e } ^ { - t } \sin t$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \tan \left( t - \frac { 1 } { 4 } \pi \right)\).
CAIE P3 2013 November Q5
5
  1. Prove that \(\cot \theta + \tan \theta \equiv 2 \operatorname { cosec } 2 \theta\).
  2. Hence show that \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \operatorname { cosec } 2 \theta \mathrm {~d} \theta = \frac { 1 } { 2 } \ln 3\).
CAIE P3 2013 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{dd7b2aee-4318-48e8-97c0-541e47f2e83a-2_551_567_1416_788} In the diagram, \(A\) is a point on the circumference of a circle with centre \(O\) and radius \(r\). A circular arc with centre \(A\) meets the circumference at \(B\) and \(C\). The angle \(O A B\) is \(\theta\) radians. The shaded region is bounded by the circumference of the circle and the arc with centre \(A\) joining \(B\) and \(C\). The area of the shaded region is equal to half the area of the circle.
  1. Show that \(\cos 2 \theta = \frac { 2 \sin 2 \theta - \pi } { 4 \theta }\).
  2. Use the iterative formula $$\theta _ { n + 1 } = \frac { 1 } { 2 } \cos ^ { - 1 } \left( \frac { 2 \sin 2 \theta _ { n } - \pi } { 4 \theta _ { n } } \right)$$ with initial value \(\theta _ { 1 } = 1\), to determine \(\theta\) correct to 2 decimal places, showing the result of each iteration to 4 decimal places.
    \(7 \quad\) Let \(\mathrm { f } ( x ) = \frac { 2 x ^ { 2 } - 7 x - 1 } { ( x - 2 ) \left( x ^ { 2 } + 3 \right) }\).
  3. Express \(\mathrm { f } ( x )\) in partial fractions.
  4. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
CAIE P3 2013 November Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{dd7b2aee-4318-48e8-97c0-541e47f2e83a-3_704_714_1272_717} The diagram shows three points \(A , B\) and \(C\) whose position vectors with respect to the origin \(O\) are given by \(\overrightarrow { O A } = \left( \begin{array} { r } 2
- 1
2 \end{array} \right) , \overrightarrow { O B } = \left( \begin{array} { l } 0
3
1 \end{array} \right)\) and \(\overrightarrow { O C } = \left( \begin{array} { l } 3
0
4 \end{array} \right)\). The point \(D\) lies on \(B C\), between \(B\) and \(C\), and is such that \(C D = 2 D B\).
  1. Find the equation of the plane \(A B C\), giving your answer in the form \(a x + b y + c z = d\).
  2. Find the position vector of \(D\).
  3. Show that the length of the perpendicular from \(A\) to \(O D\) is \(\frac { 1 } { 3 } \sqrt { } ( 65 )\).
CAIE P3 2013 November Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{dd7b2aee-4318-48e8-97c0-541e47f2e83a-4_335_875_262_635} A tank containing water is in the form of a cone with vertex \(C\). The axis is vertical and the semivertical angle is \(60 ^ { \circ }\), as shown in the diagram. At time \(t = 0\), the tank is full and the depth of water is \(H\). At this instant, a tap at \(C\) is opened and water begins to flow out. The volume of water in the tank decreases at a rate proportional to \(\sqrt { } h\), where \(h\) is the depth of water at time \(t\). The tank becomes empty when \(t = 60\).
  1. Show that \(h\) and \(t\) satisfy a differential equation of the form $$\frac { \mathrm { d } h } { \mathrm {~d} t } = - A h ^ { - \frac { 3 } { 2 } } ,$$ where \(A\) is a positive constant.
  2. Solve the differential equation given in part (i) and obtain an expression for \(t\) in terms of \(h\) and \(H\).
  3. Find the time at which the depth reaches \(\frac { 1 } { 2 } H\).
    [0pt] [The volume \(V\) of a cone of vertical height \(h\) and base radius \(r\) is given by \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\).]
CAIE P3 2013 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{a5f8f007-5176-4686-93d2-4caabeea182e-2_551_569_1416_788} In the diagram, \(A\) is a point on the circumference of a circle with centre \(O\) and radius \(r\). A circular arc with centre \(A\) meets the circumference at \(B\) and \(C\). The angle \(O A B\) is \(\theta\) radians. The shaded region is bounded by the circumference of the circle and the arc with centre \(A\) joining \(B\) and \(C\). The area of the shaded region is equal to half the area of the circle.
  1. Show that \(\cos 2 \theta = \frac { 2 \sin 2 \theta - \pi } { 4 \theta }\).
  2. Use the iterative formula $$\theta _ { n + 1 } = \frac { 1 } { 2 } \cos ^ { - 1 } \left( \frac { 2 \sin 2 \theta _ { n } - \pi } { 4 \theta _ { n } } \right) ,$$ with initial value \(\theta _ { 1 } = 1\), to determine \(\theta\) correct to 2 decimal places, showing the result of each iteration to 4 decimal places.
CAIE P3 2013 November Q7
7 Let \(\mathrm { f } ( x ) = \frac { 2 x ^ { 2 } - 7 x - 1 } { ( x - 2 ) \left( x ^ { 2 } + 3 \right) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
CAIE P3 2013 November Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{a5f8f007-5176-4686-93d2-4caabeea182e-3_704_716_1272_717} The diagram shows three points \(A , B\) and \(C\) whose position vectors with respect to the origin \(O\) are given by \(\overrightarrow { O A } = \left( \begin{array} { r } 2
- 1
2 \end{array} \right) , \overrightarrow { O B } = \left( \begin{array} { l } 0
3
1 \end{array} \right)\) and \(\overrightarrow { O C } = \left( \begin{array} { l } 3
0
4 \end{array} \right)\). The point \(D\) lies on \(B C\), between \(B\) and \(C\), and is such that \(C D = 2 D B\).
  1. Find the equation of the plane \(A B C\), giving your answer in the form \(a x + b y + c z = d\).
  2. Find the position vector of \(D\).
  3. Show that the length of the perpendicular from \(A\) to \(O D\) is \(\frac { 1 } { 3 } \sqrt { } ( 65 )\).
CAIE P3 2013 November Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{a5f8f007-5176-4686-93d2-4caabeea182e-4_328_867_264_639} A tank containing water is in the form of a cone with vertex \(C\). The axis is vertical and the semivertical angle is \(60 ^ { \circ }\), as shown in the diagram. At time \(t = 0\), the tank is full and the depth of water is \(H\). At this instant, a tap at \(C\) is opened and water begins to flow out. The volume of water in the tank decreases at a rate proportional to \(\sqrt { } h\), where \(h\) is the depth of water at time \(t\). The tank becomes empty when \(t = 60\).
  1. Show that \(h\) and \(t\) satisfy a differential equation of the form $$\frac { \mathrm { d } h } { \mathrm {~d} t } = - A h ^ { - \frac { 3 } { 2 } } ,$$ where \(A\) is a positive constant.
  2. Solve the differential equation given in part (i) and obtain an expression for \(t\) in terms of \(h\) and \(H\).
  3. Find the time at which the depth reaches \(\frac { 1 } { 2 } H\).
    [0pt] [The volume \(V\) of a cone of vertical height \(h\) and base radius \(r\) is given by \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\).]
CAIE P3 2013 November Q1
1 Given that \(2 \ln ( x + 4 ) - \ln x = \ln ( x + a )\), express \(x\) in terms of \(a\).
CAIE P3 2013 November Q2
2 Use the substitution \(u = 3 x + 1\) to find \(\int \frac { 3 x } { 3 x + 1 } \mathrm {~d} x\).
CAIE P3 2013 November Q3
3 The polynomial \(\mathrm { f } ( x )\) is defined by $$f ( x ) = x ^ { 3 } + a x ^ { 2 } - a x + 14$$ where \(a\) is a constant. It is given that ( \(x + 2\) ) is a factor of \(\mathrm { f } ( x )\).
  1. Find the value of \(a\).
  2. Show that, when \(a\) has this value, the equation \(\mathrm { f } ( x ) = 0\) has only one real root.
CAIE P3 2013 November Q4
4 A curve has equation \(3 \mathrm { e } ^ { 2 x } y + \mathrm { e } ^ { x } y ^ { 3 } = 14\). Find the gradient of the curve at the point \(( 0,2 )\).
CAIE P3 2013 November Q5
5 It is given that \(\int _ { 0 } ^ { p } 4 x \mathrm { e } ^ { - \frac { 1 } { 2 } x } \mathrm {~d} x = 9\), where \(p\) is a positive constant.
  1. Show that \(p = 2 \ln \left( \frac { 8 p + 16 } { 7 } \right)\).
  2. Use an iterative process based on the equation in part (i) to find the value of \(p\) correct to 3 significant figures. Use a starting value of 3.5 and give the result of each iteration to 5 significant figures.
CAIE P3 2013 November Q6
6 Two planes have equations \(3 x - y + 2 z = 9\) and \(x + y - 4 z = - 1\).
  1. Find the acute angle between the planes.
  2. Find a vector equation of the line of intersection of the planes.
CAIE P3 2013 November Q7
7
  1. Given that \(\sec \theta + 2 \operatorname { cosec } \theta = 3 \operatorname { cosec } 2 \theta\), show that \(2 \sin \theta + 4 \cos \theta = 3\).
  2. Express \(2 \sin \theta + 4 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\) where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the value of \(\alpha\) correct to 2 decimal places.
  3. Hence solve the equation \(\sec \theta + 2 \operatorname { cosec } \theta = 3 \operatorname { cosec } 2 \theta\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
CAIE P3 2013 November Q8
8
  1. Express \(\frac { 7 x ^ { 2 } + 8 } { ( 1 + x ) ^ { 2 } ( 2 - 3 x ) }\) in partial fractions.
  2. Hence expand \(\frac { 7 x ^ { 2 } + 8 } { ( 1 + x ) ^ { 2 } ( 2 - 3 x ) }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
CAIE P3 2013 November Q9
9
  1. Without using a calculator, use the formula for the solution of a quadratic equation to solve $$( 2 - \mathrm { i } ) z ^ { 2 } + 2 z + 2 + \mathrm { i } = 0$$ Give your answers in the form \(a + b \mathrm { i }\).
  2. The complex number \(w\) is defined by \(w = 2 \mathrm { e } ^ { \frac { 1 } { 4 } \pi \mathrm { i } }\). In an Argand diagram, the points \(A , B\) and \(C\) represent the complex numbers \(w , w ^ { 3 }\) and \(w ^ { * }\) respectively (where \(w ^ { * }\) denotes the complex conjugate of \(w\) ). Draw the Argand diagram showing the points \(A , B\) and \(C\), and calculate the area of triangle \(A B C\).