Questions P1 (1374 questions)

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CAIE P1 2016 March Q2
Easy -1.2
2 A curve for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - \frac { 2 } { x ^ { 3 } }\) passes through \(( - 1,3 )\). Find the equation of the curve.
CAIE P1 2016 March Q3
Moderate -0.5
3 The 12th term of an arithmetic progression is 17 and the sum of the first 31 terms is 1023. Find the 31st term.
CAIE P1 2016 March Q4
Moderate -0.3
4
  1. Solve the equation \(\sin ^ { - 1 } ( 3 x ) = - \frac { 1 } { 3 } \pi\), giving the solution in an exact form.
  2. Solve, by factorising, the equation \(2 \cos \theta \sin \theta - 2 \cos \theta - \sin \theta + 1 = 0\) for \(0 \leqslant \theta \leqslant \pi\).
CAIE P1 2016 March Q5
Moderate -0.3
5 Two points have coordinates \(A ( 5,7 )\) and \(B ( 9 , - 1 )\).
  1. Find the equation of the perpendicular bisector of \(A B\). The line through \(C ( 1,2 )\) parallel to \(A B\) meets the perpendicular bisector of \(A B\) at the point \(X\).
  2. Find, by calculation, the distance \(B X\).
CAIE P1 2016 March Q6
Standard +0.3
6 A vacuum flask (for keeping drinks hot) is modelled as a closed cylinder in which the internal radius is \(r \mathrm {~cm}\) and the internal height is \(h \mathrm {~cm}\). The volume of the flask is \(1000 \mathrm {~cm} ^ { 3 }\). A flask is most efficient when the total internal surface area, \(A \mathrm {~cm} ^ { 2 }\), is a minimum.
  1. Show that \(A = 2 \pi r ^ { 2 } + \frac { 2000 } { r }\).
  2. Given that \(r\) can vary, find the value of \(r\), correct to 1 decimal place, for which \(A\) has a stationary value and verify that the flask is most efficient when \(r\) takes this value.
CAIE P1 2016 March Q7
Standard +0.3
7
\includegraphics[max width=\textwidth, alt={}, center]{0f58de6c-aba7-4a79-a962-c23be3ee0aa9-3_529_698_260_721} The diagram shows a pyramid \(O A B C\) with a horizontal triangular base \(O A B\) and vertical height \(O C\). Angles \(A O B , B O C\) and \(A O C\) are each right angles. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O B\) and \(O C\) respectively, with \(O A = 4\) units, \(O B = 2.4\) units and \(O C = 3\) units. The point \(P\) on \(C A\) is such that \(C P = 3\) units.
  1. Show that \(\overrightarrow { C P } = 2.4 \mathbf { i } - 1.8 \mathbf { k }\).
  2. Express \(\overrightarrow { O P }\) and \(\overrightarrow { B P }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
  3. Use a scalar product to find angle \(B P C\).
CAIE P1 2016 March Q8
Standard +0.3
8 The function f is such that \(\mathrm { f } ( x ) = a ^ { 2 } x ^ { 2 } - a x + 3 b\) for \(x \leqslant \frac { 1 } { 2 a }\), where \(a\) and \(b\) are constants.
  1. For the case where \(\mathrm { f } ( - 2 ) = 4 a ^ { 2 } - b + 8\) and \(\mathrm { f } ( - 3 ) = 7 a ^ { 2 } - b + 14\), find the possible values of \(a\) and \(b\).
  2. For the case where \(a = 1\) and \(b = - 1\), find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and give the domain of \(\mathrm { f } ^ { - 1 }\).
CAIE P1 2016 March Q9
6 marks Standard +0.8
9
  1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0f58de6c-aba7-4a79-a962-c23be3ee0aa9-4_433_476_264_872} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} In Fig. 1, \(O A B\) is a sector of a circle with centre \(O\) and radius \(r\). \(A X\) is the tangent at \(A\) to the arc \(A B\) and angle \(B A X = \alpha\).
    1. Show that angle \(A O B = 2 \alpha\).
    2. Find the area of the shaded segment in terms of \(r\) and \(\alpha\).
  2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0f58de6c-aba7-4a79-a962-c23be3ee0aa9-4_451_503_1162_861} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} In Fig. 2, \(A B C\) is an equilateral triangle of side 4 cm . The lines \(A X , B X\) and \(C X\) are tangents to the equal circular \(\operatorname { arcs } A B , B C\) and \(C A\). Use the results in part (a) to find the area of the shaded region, giving your answer in terms of \(\pi\) and \(\sqrt { } 3\).
    [0pt] [6]
CAIE P1 2016 March Q10
Standard +0.3
10
\includegraphics[max width=\textwidth, alt={}, center]{0f58de6c-aba7-4a79-a962-c23be3ee0aa9-5_650_1038_260_550} The diagram shows part of the curve \(y = \frac { 1 } { 16 } ( 3 x - 1 ) ^ { 2 }\), which touches the \(x\)-axis at the point \(P\). The point \(Q ( 3,4 )\) lies on the curve and the tangent to the curve at \(Q\) crosses the \(x\)-axis at \(R\).
  1. State the \(x\)-coordinate of \(P\). Showing all necessary working, find by calculation
  2. the \(x\)-coordinate of \(R\),
  3. the area of the shaded region \(P Q R\).
CAIE P1 2017 March Q1
Moderate -0.5
1 Find the set of values of \(k\) for which the equation \(2 x ^ { 2 } + 3 k x + k = 0\) has distinct real roots.
CAIE P1 2017 March Q2
Standard +0.3
2 In the expansion of \(\left( \frac { 1 } { a x } + 2 a x ^ { 2 } \right) ^ { 5 }\), the coefficient of \(x\) is 5 . Find the value of the constant \(a\).
CAIE P1 2017 March Q3
Standard +0.3
3
\includegraphics[max width=\textwidth, alt={}, center]{f759ce41-708e-4fe7-80b9-adc2be2972ac-04_489_465_258_840} The diagram shows a water container in the form of an inverted pyramid, which is such that when the height of the water level is \(h \mathrm {~cm}\) the surface of the water is a square of side \(\frac { 1 } { 2 } h \mathrm {~cm}\).
  1. Express the volume of water in the container in terms of \(h\).
    [0pt] [The volume of a pyramid having a base area \(A\) and vertical height \(h\) is \(\frac { 1 } { 3 } A h\).]
    Water is steadily dripping into the container at a constant rate of \(20 \mathrm {~cm} ^ { 3 }\) per minute.
  2. Find the rate, in cm per minute, at which the water level is rising when the height of the water level is 10 cm .
    \includegraphics[max width=\textwidth, alt={}, center]{f759ce41-708e-4fe7-80b9-adc2be2972ac-06_403_773_258_685} In the diagram, \(A B = A C = 8 \mathrm {~cm}\) and angle \(C A B = \frac { 2 } { 7 } \pi\) radians. The circular \(\operatorname { arc } B C\) has centre \(A\), the circular arc \(C D\) has centre \(B\) and \(A B D\) is a straight line.
CAIE P1 2017 March Q5
Standard +0.8
5
\includegraphics[max width=\textwidth, alt={}, center]{f759ce41-708e-4fe7-80b9-adc2be2972ac-08_526_499_258_824} The diagram shows the graphs of \(y = \tan x\) and \(y = \cos x\) for \(0 \leqslant x \leqslant \pi\). The graphs intersect at points \(A\) and \(B\).
  1. Find by calculation the \(x\)-coordinate of \(A\).
  2. Find by calculation the coordinates of \(B\).
CAIE P1 2017 March Q6
Moderate -0.3
6 Relative to an origin \(O\), the position vectors of the points \(A\) and \(B\) are given by $$\overrightarrow { O A } = 2 \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = 7 \mathbf { i } + 4 \mathbf { j } + 3 \mathbf { k }$$
  1. Use a scalar product to find angle \(O A B\).
  2. Find the area of triangle \(O A B\).
CAIE P1 2017 March Q7
Moderate -0.3
7 The function f is defined for \(x \geqslant 0\) by \(\mathrm { f } ( x ) = ( 4 x + 1 ) ^ { \frac { 3 } { 2 } }\).
  1. Find \(\mathrm { f } ^ { \prime } ( x )\) and \(\mathrm { f } ^ { \prime \prime } ( x )\).
    The first, second and third terms of a geometric progression are respectively \(\mathrm { f } ( 2 ) , \mathrm { f } ^ { \prime } ( 2 )\) and \(k \mathrm { f } ^ { \prime \prime } ( 2 )\).
  2. Find the value of the constant \(k\).
CAIE P1 2017 March Q8
Moderate -0.8
8 The functions f and g are defined for \(x \geqslant 0\) by $$\begin{aligned} & \mathrm { f } : x \mapsto 2 x ^ { 2 } + 3 \\ & \mathrm {~g} : x \mapsto 3 x + 2 \end{aligned}$$
  1. Show that \(\operatorname { gf } ( x ) = 6 x ^ { 2 } + 11\) and obtain an unsimplified expression for \(\operatorname { fg } ( x )\).
  2. Find an expression for \(( \mathrm { fg } ) ^ { - 1 } ( x )\) and determine the domain of \(( \mathrm { fg } ) ^ { - 1 }\).
  3. Solve the equation \(\mathrm { gf } ( 2 x ) = \mathrm { fg } ( x )\).
CAIE P1 2017 March Q9
Standard +0.8
9 The point \(A ( 2,2 )\) lies on the curve \(y = x ^ { 2 } - 2 x + 2\).
  1. Find the equation of the tangent to the curve at \(A\).
    The normal to the curve at \(A\) intersects the curve again at \(B\).
  2. Find the coordinates of \(B\).
    The tangents at \(A\) and \(B\) intersect each other at \(C\).
  3. Find the coordinates of \(C\).
CAIE P1 2017 March Q10
Moderate -0.3
10
\includegraphics[max width=\textwidth, alt={}, center]{f759ce41-708e-4fe7-80b9-adc2be2972ac-18_611_531_262_808} The diagram shows the curve \(y = \mathrm { f } ( x )\) defined for \(x > 0\). The curve has a minimum point at \(A\) and crosses the \(x\)-axis at \(B\) and \(C\). It is given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x - \frac { 2 } { x ^ { 3 } }\) and that the curve passes through the point \(\left( 4 , \frac { 189 } { 16 } \right)\).
  1. Find the \(x\)-coordinate of \(A\).
  2. Find \(\mathrm { f } ( x )\).
  3. Find the \(x\)-coordinates of \(B\) and \(C\).
  4. Find, showing all necessary working, the area of the shaded region.
    \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 2019 March Q1
Moderate -0.8
1 The coefficient of \(x ^ { 3 }\) in the expansion of \(( 1 - p x ) ^ { 5 }\) is - 2160 . Find the value of the constant \(p\).
CAIE P1 2019 March Q2
Moderate -0.8
2 A curve with equation \(y = \mathrm { f } ( x )\) passes through the points \(( 0,2 )\) and \(( 3 , - 1 )\). It is given that \(\mathrm { f } ^ { \prime } ( x ) = k x ^ { 2 } - 2 x\), where \(k\) is a constant. Find the value of \(k\).
CAIE P1 2019 March Q3
Standard +0.3
3
\includegraphics[max width=\textwidth, alt={}, center]{c8ac31bc-0f76-4d00-a28b-4a07758f9663-05_652_542_260_799} In the diagram, \(C X D\) is a semicircle of radius 7 cm with centre \(A\) and diameter \(C D\). The straight line \(Y A B X\) is perpendicular to \(C D\), and the arc \(C Y D\) is part of a circle with centre \(B\) and radius 8 cm . Find the total area of the region enclosed by the two arcs.
CAIE P1 2019 March Q4
Moderate -0.3
4 A curve has equation \(y = ( 2 x - 1 ) ^ { - 1 } + 2 x\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
  2. Find the \(x\)-coordinates of the stationary points and, showing all necessary working, determine the nature of each stationary point.
CAIE P1 2019 March Q5
Moderate -0.5
5 Two vectors, \(\mathbf { u }\) and \(\mathbf { v }\), are such that $$\mathbf { u } = \left( \begin{array} { l } q
2
CAIE P1 2019 March Q6
2 marks Moderate -0.8
6 \end{array} \right) \quad \text { and } \quad \mathbf { v } = \left( \begin{array} { c }
CAIE P1 2019 March Q8
2 marks Moderate -0.8
8
q - 1
q ^ { 2 } - 7 \end{array} \right)$$ where \(q\) is a constant.
  1. Find the values of \(q\) for which \(\mathbf { u }\) is perpendicular to \(\mathbf { v }\).
  2. Find the angle between \(\mathbf { u }\) and \(\mathbf { v }\) when \(q = 0\).
    6
  3. The first and second terms of a geometric progression are \(p\) and \(2 p\) respectively, where \(p\) is a positive constant. The sum of the first \(n\) terms is greater than \(1000 p\). Show that \(2 ^ { n } > 1001\). [2]
  4. In another case, \(p\) and \(2 p\) are the first and second terms respectively of an arithmetic progression. The \(n\)th term is 336 and the sum of the first \(n\) terms is 7224 . Write down two equations in \(n\) and \(p\) and hence find the values of \(n\) and \(p\).
    7 (a) Solve the equation \(3 \sin ^ { 2 } 2 \theta + 8 \cos 2 \theta = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
    (b)
    \includegraphics[max width=\textwidth, alt={}, center]{c8ac31bc-0f76-4d00-a28b-4a07758f9663-13_540_750_251_735} The diagram shows part of the graph of \(y = a + \tan b x\), where \(x\) is measured in radians and \(a\) and \(b\) are constants. The curve intersects the \(x\)-axis at \(\left( - \frac { 1 } { 6 } \pi , 0 \right)\) and the \(y\)-axis at \(( 0 , \sqrt { } 3 )\). Find the values of \(a\) and \(b\).
    8
  5. Express \(x ^ { 2 } - 4 x + 7\) in the form \(( x + a ) ^ { 2 } + b\).
    The function f is defined by \(\mathrm { f } ( x ) = x ^ { 2 } - 4 x + 7\) for \(x < k\), where \(k\) is a constant.
  6. State the largest value of \(k\) for which f is a decreasing function.
    The value of \(k\) is now given to be 1 .
  7. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
  8. The function g is defined by \(\mathrm { g } ( x ) = \frac { 2 } { x - 1 }\) for \(x > 1\). Find an expression for \(\mathrm { gf } ( x )\) and state the range of gf.