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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 2024 November Q3
3 The equation of a curve is \(y = 2 x ^ { 2 } - 3\). Two points \(A\) and \(B\) with \(x\)-coordinates 2 and \(( 2 + h )\) respectively lie on the curve.
  1. Find and simplify an expression for the gradient of the chord \(A B\) in terms of \(h\).
  2. Explain how the gradient of the curve at the point \(A\) can be deduced from the answer to part (a), and state the value of this gradient.
CAIE P1 2024 November Q4
4 Find the term independent of \(x\) in the expansion of each of the following:
  1. \(\left( x + \frac { 3 } { x ^ { 2 } } \right) ^ { 6 }\)
  2. \(\left( 4 x ^ { 3 } - 5 \right) \left( x + \frac { 3 } { x ^ { 2 } } \right) ^ { 6 }\) .
CAIE P1 2024 November Q5
5 The function f is defined by \(\mathrm { f } ( x ) = \frac { 2 x + 1 } { 2 x - 1 }\) for \(x < \frac { 1 } { 2 }\).
    1. State the value of \(\mathrm { f } ( - 1 )\).

    2. \includegraphics[max width=\textwidth, alt={}, center]{39393c20-34df-4167-a8bf-e4067371de81-06_898_892_877_712} The diagram shows the graph of \(y = \mathrm { f } ( x )\). Sketch the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\) on this diagram. Show any relevant mirror line.
    3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of the function \(\mathrm { f } ^ { - 1 }\).
      \includegraphics[max width=\textwidth, alt={}, center]{39393c20-34df-4167-a8bf-e4067371de81-07_2726_35_97_20}
      The function g is defined by \(\mathrm { g } ( x ) = 3 x + 2\) for \(x \in \mathbb { R }\).
  2. Solve the equation \(\mathrm { f } ( x ) = \mathrm { gf } \left( \frac { 1 } { 4 } \right)\).
    \includegraphics[max width=\textwidth, alt={}, center]{39393c20-34df-4167-a8bf-e4067371de81-08_712_734_246_667} The diagram shows a metal plate \(O A B C D E F\) consisting of sectors of two circles,each with centre \(O\) . The radii of sectors \(A O B\) and \(E O F\) are \(r \mathrm {~cm}\) and the radius of sector \(C O D\) is \(2 r \mathrm {~cm}\) . Angle \(A O B =\) angle \(E O F = \theta\) radians and angle \(C O D = 2 \theta\) radians. It is given that the perimeter of the plate is 14 cm and the area of the plate is \(10 \mathrm {~cm} ^ { 2 }\) .
    Given that \(r > \frac { 3 } { 2 }\) and \(\theta < \frac { 3 } { 4 }\) ,find the values of \(r\) and \(\theta\) .
    ΝΙ১W SΙΗ ΝΙ ΞΙΙΥΜ ΙΟΝ Ο0ΝΙ১W SIH NI IyM 1ON OOΝΙ১W SΙΗΙ ΝΙ ΞΙΙΨΜ ΙΟΝ Ο0ΝΙ১W SIHI NI IYM 1ON OC\includegraphics[max width=\textwidth, alt={}]{39393c20-34df-4167-a8bf-e4067371de81-08_437_28_2390_2016}
    \includegraphics[max width=\textwidth, alt={}, center]{39393c20-34df-4167-a8bf-e4067371de81-09_2725_35_99_20}
CAIE P1 2024 November Q7
7
  1. By expressing \(- 2 x ^ { 2 } + 8 x + 11\) in the form \(- a ( x - b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are positive integers, find the coordinates of the vertex of the graph with equation \(y = - 2 x ^ { 2 } + 8 x + 11\).

  2. \includegraphics[max width=\textwidth, alt={}, center]{39393c20-34df-4167-a8bf-e4067371de81-10_561_1603_1137_310} The diagram shows part of the curve with equation \(y = - 2 x ^ { 2 } + 8 x + 11\) and the line with equation \(y = 8 x + 9\). Find the area of the shaded region.
    \includegraphics[max width=\textwidth, alt={}, center]{39393c20-34df-4167-a8bf-e4067371de81-10_2722_43_107_2004}
    \includegraphics[max width=\textwidth, alt={}, center]{39393c20-34df-4167-a8bf-e4067371de81-11_2726_35_97_20}
CAIE P1 2024 November Q8
8 The equation of a circle is \(x ^ { 2 } + y ^ { 2 } + p x + 2 y + q = 0\), where \(p\) and \(q\) are constants.
  1. Express the equation in the form \(( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }\), where \(a\) is to be given in terms of \(p\) and \(r ^ { 2 }\) is to be given in terms of \(p\) and \(q\).
    The line with equation \(x + 2 y = 10\) is the tangent to the circle at the point \(A ( 4,3 )\).
    1. Find the equation of the normal to the circle at the point \(A\).
    2. Find the values of \(p\) and \(q\).
CAIE P1 2024 November Q9
9 The equation of a curve is \(y = \frac { 1 } { 2 } k ^ { 2 } x ^ { 2 } - 2 k x + 2\) and the equation of a line is \(y = k x + p\), where \(k\) and \(p\) are constants with \(0 < k < 1\).
  1. It is given that one of the points of intersection of the curve and the line has coordinates \(\left( \frac { 5 } { 2 } , \frac { 1 } { 2 } \right)\). Find the values of \(k\) and \(p\), and find the coordinates of the other point of intersection.
    \includegraphics[max width=\textwidth, alt={}, center]{39393c20-34df-4167-a8bf-e4067371de81-15_2725_35_99_20}
  2. It is given instead that the line and the curve do not intersect. Find the set of possible values of \(p\).
CAIE P1 2024 November Q10
10 A function f with domain \(x > 0\) is such that \(\mathrm { f } ^ { \prime } ( x ) = 8 ( 2 x - 3 ) ^ { \frac { 1 } { 3 } } - 10 x ^ { \frac { 2 } { 3 } }\). It is given that the curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(( 1,0 )\).
  1. Find the equation of the normal to the curve at the point \(( 1,0 )\).
  2. Find \(\mathrm { f } ( x )\).
    \includegraphics[max width=\textwidth, alt={}, center]{39393c20-34df-4167-a8bf-e4067371de81-16_2715_41_110_2008}
    \includegraphics[max width=\textwidth, alt={}, center]{39393c20-34df-4167-a8bf-e4067371de81-17_2723_35_101_20} It is given that the equation \(\mathrm { f } ^ { \prime } ( x ) = 0\) can be expressed in the form $$125 x ^ { 2 } - 128 x + 192 = 0$$
  3. Determine, making your reasoning clear, whether f is an increasing function, a decreasing function or neither.
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
    \includegraphics[max width=\textwidth, alt={}, center]{39393c20-34df-4167-a8bf-e4067371de81-18_2714_38_109_2010}
CAIE P1 2024 November Q1
1 An arithmetic progression has fourth term 15 and eighth term 25.
Find the 30th term of the progression.
CAIE P1 2024 November Q2
2 Find the exact solution of the equation $$\cos \frac { 1 } { 6 } \pi + \tan 2 x + \frac { \sqrt { 3 } } { 2 } = 0 \text { for } - \frac { 1 } { 4 } \pi < x < \frac { 1 } { 4 } \pi .$$
CAIE P1 2024 November Q3
3
  1. Find the coefficients of \(x ^ { 3 }\) and \(x ^ { 4 }\) in the expansion of \(( 3 - a x ) ^ { 5 }\), where \(a\) is a constant. Give your answers in terms of \(a\).
  2. Given that the coefficient of \(x ^ { 4 }\) in the expansion of \(( a x + 7 ) ( 3 - a x ) ^ { 5 }\) is 240 , find the positive value of \(a\).
    \includegraphics[max width=\textwidth, alt={}, center]{49e137bf-42cc-41af-b5d9-85301d4699b8-05_2723_33_99_21}
CAIE P1 2024 November Q4
4 Solve the equation \(4 \sin ^ { 4 } \theta + 12 \sin ^ { 2 } \theta - 7 = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P1 2024 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{49e137bf-42cc-41af-b5d9-85301d4699b8-06_631_1500_260_285} In the diagram, the graph with equation \(y = \mathrm { f } ( x )\) is shown with solid lines and the graph with equation \(y = \mathrm { g } ( x )\) is shown with broken lines.
  1. Describe fully a sequence of three transformations which transforms the graph of \(y = \mathrm { f } ( x )\) to the graph of \(y = \mathrm { g } ( x )\).
  2. Find an expression for \(\mathrm { g } ( x )\) in the form \(a \mathrm { f } ( b x + c )\), where \(a , b\) and \(c\) are integers.
    □
    \includegraphics[max width=\textwidth, alt={}, center]{49e137bf-42cc-41af-b5d9-85301d4699b8-06_2716_31_106_2016}
    \includegraphics[max width=\textwidth, alt={}, center]{49e137bf-42cc-41af-b5d9-85301d4699b8-07_2723_35_101_20}
CAIE P1 2024 November Q6
6 The first term of a convergent geometric progression is 10 . The sum of the first 4 terms of the progression is \(p\) and the sum of the first 8 terms of the progression is \(q\). It is given that \(\frac { q } { p } = \frac { 17 } { 16 }\). Find the two possible values of the sum to infinity.
\includegraphics[max width=\textwidth, alt={}, center]{49e137bf-42cc-41af-b5d9-85301d4699b8-08_801_730_255_669} The diagram shows a metal plate \(A B C D E F\) consisting of five parts. The parts \(B C D\) and \(D E F\) are semicircles. The part \(B A F O\) is a sector of a circle with centre \(O\) and radius 20 cm , and \(D\) lies on this circle. The parts \(O B D\) and \(O D F\) are triangles. Angles \(B O D\) and \(D O F\) are both \(\theta\) radians.
  1. Given that \(\theta = 1.2\), find the area of the metal plate. Give your answer correct to 3 significant figures.
    \includegraphics[max width=\textwidth, alt={}, center]{49e137bf-42cc-41af-b5d9-85301d4699b8-08_2715_42_110_2006}
  2. Given instead that the area of each semicircle is \(50 \pi \mathrm {~cm} ^ { 2 }\), find the exact perimeter of the metal plate.
CAIE P1 2024 November Q8
8
  1. Express \(3 x ^ { 2 } - 12 x + 14\) in the form \(3 ( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants to be found.
    The function \(\mathrm { f } ( x ) = 3 x ^ { 2 } - 12 x + 14\) is defined for \(x \geqslant k\), where \(k\) is a constant.
  2. Find the least value of \(k\) for which the function \(\mathrm { f } ^ { - 1 }\) exists.
    For the rest of this question, you should assume that \(k\) has the value found in part (b).
  3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
    \includegraphics[max width=\textwidth, alt={}, center]{49e137bf-42cc-41af-b5d9-85301d4699b8-11_2726_35_97_20}
  4. Hence or otherwise solve the equation \(\mathrm { ff } ( x ) = 29\).
CAIE P1 2024 November Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{49e137bf-42cc-41af-b5d9-85301d4699b8-12_883_1703_267_182} The diagram shows the curves with equations \(y = x ^ { 3 } - 3 x + 3\) and \(y = 2 x ^ { 3 } - 4 x ^ { 2 } + 3\).
  1. Find the \(x\)-coordinates of the points of intersection of the curves.
  2. Find the area of the shaded region.
CAIE P1 2024 November Q10
10 Points \(A\) and \(B\) have coordinates \(( 4,3 )\) and \(( 8 , - 5 )\) respectively. A circle with radius 10 passes through the points \(A\) and \(B\).
  1. Show that the centre of the circle lies on the line \(y = \frac { 1 } { 2 } x - 4\).
    \includegraphics[max width=\textwidth, alt={}, center]{49e137bf-42cc-41af-b5d9-85301d4699b8-14_2715_35_109_2010}
  2. Find the two possible equations of the circle.
CAIE P1 2024 November Q11
11 The equation of a curve is \(y = k x ^ { \frac { 1 } { 2 } } - 4 x ^ { 2 } + 2\), where \(k\) is a constant.
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) in terms of \(k\).
  2. It is given that \(k = 2\). Find the coordinates of the stationary point and determine its nature.
    \includegraphics[max width=\textwidth, alt={}, center]{49e137bf-42cc-41af-b5d9-85301d4699b8-16_2717_38_110_2010}
  3. Points \(A\) and \(B\) on the curve have \(x\)-coordinates 0.25 and 1 respectively. For a different value of \(k\), the tangents to the curve at the points \(A\) and \(B\) meet at a point with \(x\)-coordinate 0.6 . Find this value of \(k\).
    \includegraphics[max width=\textwidth, alt={}, center]{49e137bf-42cc-41af-b5d9-85301d4699b8-17_58_1569_429_328}
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
    …………………………………………………………………………………………………………....................
    \includegraphics[max width=\textwidth, alt={}, center]{49e137bf-42cc-41af-b5d9-85301d4699b8-18_2716_35_109_2012}
    \includegraphics[max width=\textwidth, alt={}, center]{49e137bf-42cc-41af-b5d9-85301d4699b8-19_2717_35_106_20}
CAIE P1 2020 Specimen Q1
1 Th fb lowiற் ns $$A ( \Omega , B ( \mathbb { \phi } , C ( \mathbb { \phi } , D ( \mathbb { \phi } ) \quad \text { ad } \quad E ( \mathbb { \phi } )$$ lie t b cure \(y = \mathrm { f } ( x )\). Th tabeb lw sho s th ad ens 6 th \(\mathrm { ch } \mathrm { s } A E\) ad \(B E\). \begin{center} \begin{tabular}{ | c | c | c | c | c | } \hline Ch d & \(A E\) & \(B E\) & \(C E\) & \(D E\)
\hline \begin{tabular}{ c } Grad en
CAIE P1 2020 Specimen Q2
2 Fn tin f adg re d fie dy $$\begin{aligned} & \mathrm { f } : x \mapsto 3 x + 2 \quad x \in \mathbb { R }
& \operatorname { g } \quad x \mapsto 4 x - 2 \quad x \in \mathbb { R } \end{aligned}$$ Sb th equ tiff \({ } ^ { - 1 } ( x ) = g ( x )\).
CAIE P1 2020 Specimen Q3
3 Ara rith etic p og essich s first term 7 Th \(n\)th erm is \& d (3n)ttl erm is \% Fid b lue \(6 n\).
CAIE P1 2020 Specimen Q4
4 A cu h s eq tin \(y = \mathrm { f } ( x )\).I t is g it h \(\mathrm { f } ^ { \prime } ( x ) = \frac { 1 } { \sqrt { x + 6 } } + \frac { 6 } { x ^ { 2 } }\) ad h \(\mathrm { f } ( \mathcal { \beta } = 1\)
Fif ( \(x\) ).
CAIE P1 2020 Specimen Q5
5
  1. Th cn \(y = x ^ { 2 } + 3 x + 4\) s tras latedy \(\binom { 2 } { 0 }\).
    Find imp ify \(\mathbf { b }\) eq tim the tras lated \(\mathrm { n } \mathbb { E }\).
  2. Th g ad \(y = \mathrm { f } ( x )\) is tras fo med \& b g ap \(6 y = \mathrm { B } ( - x )\). Describ fly ly th two sig le tras fo matin wh ch hav b en cm be d to ge th resh tig tras fo matio
    [0pt] [β
CAIE P1 2020 Specimen Q6
6
  1. Fid \(\mathbf { b }\) co fficien \(\mathrm { s } 6 x ^ { 2 }\) ad \(x ^ { 3 }\) irt \(\mathbf { b }\) e in ( \(\left. 2 x \right) ^ { 6 }\). [
  2. Hen e fid b co fficien \(6 x ^ { 3 }\) in b \(\mathbf { b }\) in \(\left( 3 x + 1 ( 2 \quad x ) ^ { 6 } \right.\).
CAIE P1 2020 Specimen Q7
7
  1. Sw that the equr tin \(\sin x \tan x = 5 \mathrm { co } x\) carb esseds s $$6 \text { св } ^ { 2 } x \in \text { в } x \neq 0$$
  2. Hen e sb e the tinl \(\sin x \tan x = 5 c o x\) fo \(\theta \leqslant x \leqslant \theta\)
CAIE P1 2020 Specimen Q8
8 A cu t h seq tin \(y = \frac { 12 } { 3 - 2 x }\).
  1. Fid \(\frac { \mathrm { dy } } { \mathrm { dx } }\). A \(\dot { p } \cap \mathrm { n } \in \mathrm { s }\) alg thscue. Astb \(\dot { p } \cap \mathbf { p }\) sses th \(\mathbf { g } \quad A\), th \(x\)-co \(\dot { \mathbf { d } } \mathbf { a }\) te is in reasig at a rate \(\boldsymbol { 6 }\) ts p r sech d b \(y\)-co d a te is in reasign ta rate \(\mathbf { 6 }\) th ts p r secd
  2. Fid b s sib e \(x\)-co du tes \(6 A\).