Questions — CAIE P1 (1202 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 P1 2016 November Q4
4 The function f is such that \(\mathrm { f } ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 9 x + 2\) for \(x > n\), where \(n\) is an integer. It is given that f is an increasing function. Find the least possible value of \(n\).
CAIE P1 2016 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{5fed65b9-a848-4343-858c-3cbac0608b24-2_609_533_938_804} The diagram shows a major arc \(A B\) of a circle with centre \(O\) and radius 6 cm . Points \(C\) and \(D\) on \(O A\) and \(O B\) respectively are such that the line \(A B\) is a tangent at \(E\) to the arc \(C E D\) of a smaller circle also with centre \(O\). Angle \(C O D = 1.8\) radians.
  1. Show that the radius of the \(\operatorname { arc } C E D\) is 3.73 cm , correct to 3 significant figures.
  2. Find the area of the shaded region.
CAIE P1 2016 November Q6
6 Three points, \(A , B\) and \(C\), are such that \(B\) is the mid-point of \(A C\). The coordinates of \(A\) are ( \(2 , m\) ) and the coordinates of \(B\) are \(( n , - 6 )\), where \(m\) and \(n\) are constants.
  1. Find the coordinates of \(C\) in terms of \(m\) and \(n\). The line \(y = x + 1\) passes through \(C\) and is perpendicular to \(A B\).
  2. Find the values of \(m\) and \(n\).
CAIE P1 2016 November Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{5fed65b9-a848-4343-858c-3cbac0608b24-3_736_399_260_872} The diagram shows a triangular pyramid \(A B C D\). It is given that $$\overrightarrow { A B } = 3 \mathbf { i } + \mathbf { j } + \mathbf { k } , \quad \overrightarrow { A C } = \mathbf { i } - 2 \mathbf { j } - \mathbf { k } \quad \text { and } \quad \overrightarrow { A D } = \mathbf { i } + 4 \mathbf { j } - 7 \mathbf { k }$$
  1. Verify, showing all necessary working, that each of the angles \(D A B , D A C\) and \(C A B\) is \(90 ^ { \circ }\).
  2. Find the exact value of the area of the triangle \(A B C\), and hence find the exact value of the volume of the pyramid.
    [0pt] [The volume \(V\) of a pyramid of base area \(A\) and vertical height \(h\) is given by \(V = \frac { 1 } { 3 } A h\).]
CAIE P1 2016 November Q8
8
  1. Express \(4 x ^ { 2 } + 12 x + 10\) in the form \(( a x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.
  2. Functions f and g are both defined for \(x > 0\). It is given that \(\mathrm { f } ( x ) = x ^ { 2 } + 1\) and \(\mathrm { fg } ( x ) = 4 x ^ { 2 } + 12 x + 10\). Find \(\mathrm { g } ( x )\).
  3. Find \(( \mathrm { fg } ) ^ { - 1 } ( x )\) and give the domain of \(( \mathrm { fg } ) ^ { - 1 }\).
    (a) Two convergent geometric progressions, \(P\) and \(Q\), have the same sum to infinity. The first and second terms of \(P\) are 6 and \(6 r\) respectively. The first and second terms of \(Q\) are 12 and \(- 12 r\) respectively. Find the value of the common sum to infinity.
    (b) The first term of an arithmetic progression is \(\cos \theta\) and the second term is \(\cos \theta + \sin ^ { 2 } \theta\), where \(0 \leqslant \theta \leqslant \pi\). The sum of the first 13 terms is 52 . Find the possible values of \(\theta\).
CAIE P1 2016 November Q10
10 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { a } x ^ { - \frac { 1 } { 2 } } + a x ^ { - \frac { 3 } { 2 } }\), where \(a\) is a positive constant. The point \(A \left( a ^ { 2 } , 3 \right)\) lies on the curve. Find, in terms of \(a\),
  1. the equation of the tangent to the curve at \(A\), simplifying your answer,
  2. the equation of the curve. It is now given that \(B ( 16,8 )\) also lies on the curve.
  3. Find the value of \(a\) and, using this value, find the distance \(A B\).
CAIE P1 2016 November Q11
11 A curve has equation \(y = ( k x - 3 ) ^ { - 1 } + ( k x - 3 )\), where \(k\) is a non-zero constant.
  1. Find the \(x\)-coordinates of the stationary points in terms of \(k\), and determine the nature of each stationary point, justifying your answers.

  2. \includegraphics[max width=\textwidth, alt={}, center]{5fed65b9-a848-4343-858c-3cbac0608b24-4_556_855_1032_685} The diagram shows part of the curve for the case when \(k = 1\). Showing all necessary working, find the volume obtained when the region between the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = 2\), shown shaded in the diagram, is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. \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.
    To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at \href{http://www.cie.org.uk}{www.cie.org.uk} after the live examination series. 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 P1 2017 November Q1
1 A curve has equation \(y = 2 x ^ { \frac { 3 } { 2 } } - 3 x - 4 x ^ { \frac { 1 } { 2 } } + 4\). Find the equation of the tangent to the curve at the point \(( 4,0 )\).
CAIE P1 2017 November Q2
2 A function f is defined by \(\mathrm { f } : x \mapsto x ^ { 3 } - x ^ { 2 } - 8 x + 5\) for \(x < a\). It is given that f is an increasing function. Find the largest possible value of the constant \(a\).
CAIE P1 2017 November Q3
3
  1. A geometric progression has first term \(3 a\) and common ratio \(r\). A second geometric progression has first term \(a\) and common ratio \(- 2 r\). The two progressions have the same sum to infinity. Find the value of \(r\).
  2. The first two terms of an arithmetic progression are 15 and 19 respectively. The first two terms of a second arithmetic progression are 420 and 415 respectively. The two progressions have the same sum of the first \(n\) terms. Find the value of \(n\).
CAIE P1 2017 November Q4
4 Machines in a factory make cardboard cones of base radius \(r \mathrm {~cm}\) and vertical height \(h \mathrm {~cm}\). The volume, \(V \mathrm {~cm} ^ { 3 }\), of such a cone is given by \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\). The machines produce cones for which \(h + r = 18\).
  1. Show that \(V = 6 \pi r ^ { 2 } - \frac { 1 } { 3 } \pi r ^ { 3 }\).
  2. Given that \(r\) can vary, find the non-zero value of \(r\) for which \(V\) has a stationary value and show that the stationary value is a maximum.
  3. Find the maximum volume of a cone that can be made by these machines.
CAIE P1 2017 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{5201a3d5-7733-4d10-9de5-0c2255e3ad60-08_446_844_260_648} The diagram shows an isosceles triangle \(A B C\) in which \(A C = 16 \mathrm {~cm}\) and \(A B = B C = 10 \mathrm {~cm}\). The circular arcs \(B E\) and \(B D\) have centres at \(A\) and \(C\) respectively, where \(D\) and \(E\) lie on \(A C\).
  1. Show that angle \(B A C = 0.6435\) radians, correct to 4 decimal places.
  2. Find the area of the shaded region.
CAIE P1 2017 November Q6
6 The points \(A ( 1,1 )\) and \(B ( 5,9 )\) lie on the curve \(6 y = 5 x ^ { 2 } - 18 x + 19\).
  1. Show that the equation of the perpendicular bisector of \(A B\) is \(2 y = 13 - x\).
    The perpendicular bisector of \(A B\) meets the curve at \(C\) and \(D\).
  2. Find, by calculation, the distance \(C D\), giving your answer in the form \(\sqrt { } \left( \frac { p } { q } \right)\), where \(p\) and \(q\) are integers.
CAIE P1 2017 November Q7
7

  1. \includegraphics[max width=\textwidth, alt={}, center]{5201a3d5-7733-4d10-9de5-0c2255e3ad60-12_499_568_267_826} The diagram shows part of the graph of \(y = a + b \sin x\). Find the values of the constants \(a\) and \(b\).
    1. Show that the equation $$( \sin \theta + 2 \cos \theta ) ( 1 + \sin \theta - \cos \theta ) = \sin \theta ( 1 + \cos \theta )$$ may be expressed as \(3 \cos ^ { 2 } \theta - 2 \cos \theta - 1 = 0\).
    2. Hence solve the equation $$( \sin \theta + 2 \cos \theta ) ( 1 + \sin \theta - \cos \theta ) = \sin \theta ( 1 + \cos \theta )$$ for \(- 180 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P1 2017 November Q8
8
  1. Relative to an origin \(O\), the position vectors of two points \(P\) and \(Q\) are \(\mathbf { p }\) and \(\mathbf { q }\) respectively. The point \(R\) is such that \(P Q R\) is a straight line with \(Q\) the mid-point of \(P R\). Find the position vector of \(R\) in terms of \(\mathbf { p }\) and \(\mathbf { q }\), simplifying your answer.
  2. The vector \(6 \mathbf { i } + a \mathbf { j } + b \mathbf { k }\) has magnitude 21 and is perpendicular to \(3 \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k }\). Find the possible values of \(a\) and \(b\), showing all necessary working.
CAIE P1 2017 November Q9
9 Functions f and g are defined for \(x > 3\) by $$\begin{aligned} & \mathrm { f } : x \mapsto \frac { 1 } { x ^ { 2 } - 9 }
& \mathrm {~g} : x \mapsto 2 x - 3 \end{aligned}$$
  1. Find and simplify an expression for \(\operatorname { gg } ( x )\).
  2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
  3. Solve the equation \(\operatorname { fg } ( x ) = \frac { 1 } { 7 }\).
CAIE P1 2017 November Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{5201a3d5-7733-4d10-9de5-0c2255e3ad60-18_401_584_264_776} The diagram shows part of the curve \(y = \frac { 1 } { 2 } \left( x ^ { 4 } - 1 \right)\), defined for \(x \geqslant 0\).
  1. Find, showing all necessary working, the area of the shaded region.
  2. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
  3. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(y\)-axis.
CAIE P1 2017 November Q1
1 Find the term independent of \(x\) in the expansion of \(\left( 2 x - \frac { 1 } { 4 x ^ { 2 } } \right) ^ { 9 }\).
CAIE P1 2017 November Q2
2 A function f is defined by \(\mathrm { f } : x \mapsto 4 - 5 x\) for \(x \in \mathbb { R }\).
  1. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and find the point of intersection of the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\).
  2. Sketch, on the same diagram, the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), making clear the relationship between the graphs.
CAIE P1 2017 November Q3
3
  1. Each year, the value of a certain rare stamp increases by \(5 \%\) of its value at the beginning of the year. A collector bought the stamp for \(
    ) 10000\( at the beginning of 2005. Find its value at the beginning of 2015 correct to the nearest \)\\( 100\).
  2. The sum of the first \(n\) terms of an arithmetic progression is \(\frac { 1 } { 2 } n ( 3 n + 7 )\). Find the 1 st term and the common difference of the progression.
CAIE P1 2017 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{518bb805-5b14-4b41-94fd-38a31a90c218-06_401_698_255_721} The diagram shows a semicircle with centre \(O\) and radius 6 cm . The radius \(O C\) is perpendicular to the diameter \(A B\). The point \(D\) lies on \(A B\), and \(D C\) is an arc of a circle with centre \(B\).
  1. Calculate the length of the \(\operatorname { arc } D C\).
  2. Find the value of
    \(\frac { \text { area of region } P } { \text { area of region } Q }\),
    giving your answer correct to 3 significant figures.
CAIE P1 2017 November Q5
5
  1. Show that the equation \(\cos 2 x \left( \tan ^ { 2 } 2 x + 3 \right) + 3 = 0\) can be expressed as $$2 \cos ^ { 2 } 2 x + 3 \cos 2 x + 1 = 0$$
  2. Hence solve the equation \(\cos 2 x \left( \tan ^ { 2 } 2 x + 3 \right) + 3 = 0\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
CAIE P1 2017 November Q6
6
  1. The function f , defined by \(\mathrm { f } : x \mapsto a + b \sin x\) for \(x \in \mathbb { R }\), is such that \(\mathrm { f } \left( \frac { 1 } { 6 } \pi \right) = 4\) and \(\mathrm { f } \left( \frac { 1 } { 2 } \pi \right) = 3\).
    1. Find the values of the constants \(a\) and \(b\).
    2. Evaluate \(\mathrm { ff } ( 0 )\).
  2. The function g is defined by \(\mathrm { g } : x \mapsto c + d \sin x\) for \(x \in \mathbb { R }\). The range of g is given by \(- 4 \leqslant \mathrm {~g} ( x ) \leqslant 10\). Find the values of the constants \(c\) and \(d\).
CAIE P1 2017 November Q7
7 Points \(A\) and \(B\) lie on the curve \(y = x ^ { 2 } - 4 x + 7\). Point \(A\) has coordinates \(( 4,7 )\) and \(B\) is the stationary point of the curve. The equation of a line \(L\) is \(y = m x - 2\), where \(m\) is a constant.
  1. In the case where \(L\) passes through the mid-point of \(A B\), find the value of \(m\).
  2. Find the set of values of \(m\) for which \(L\) does not meet the curve.
CAIE P1 2017 November Q8
8 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - x ^ { 2 } + 5 x - 4\).
  1. Find the \(x\)-coordinate of each of the stationary points of the curve.
  2. Obtain an expression for \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and hence or otherwise find the nature of each of the stationary points.
  3. Given that the curve passes through the point \(( 6,2 )\), find the equation of the curve.