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 2023 March Q9
9 The function f is defined by \(\mathrm { f } ( x ) = - 3 x ^ { 2 } + 2\) for \(x \leqslant - 1\).
  1. State the range of f.
  2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
    The function g is defined by \(\mathrm { g } ( x ) = - x ^ { 2 } - 1\) for \(x \leqslant - 1\).
  3. Solve the equation \(\mathrm { fg } ( x ) - \mathrm { gf } ( x ) + 8 = 0\).
CAIE P1 2023 March Q10
10 At the point \(( 4 , - 1 )\) on a curve, the gradient of the curve is \(- \frac { 3 } { 2 }\). It is given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { - \frac { 1 } { 2 } } + k\), where \(k\) is a constant.
  1. Show that \(k = - 2\).
  2. Find the equation of the curve.
  3. Find the coordinates of the stationary point.
  4. Determine the nature of the stationary point.
CAIE P1 2023 March Q11
11
\includegraphics[max width=\textwidth, alt={}, center]{3bad1d9f-5b9e-4895-aa4e-3e6d9f6c072e-16_599_780_274_671} The diagram shows the curve with equation \(x = y ^ { 2 } + 1\). The points \(A ( 5,2 )\) and \(B ( 2 , - 1 )\) lie on the curve.
  1. Find an equation of the line \(A B\).
  2. Find the volume of revolution when the region between the curve and the line \(A B\) is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE P1 2024 March Q1
1 Find the exact value of \(\int _ { 3 } ^ { \infty } \frac { 2 } { x ^ { 2 } } d x\).
CAIE P1 2024 March Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-03_451_597_255_735} The diagram shows part of the curve with equation \(\mathrm { y } = \mathrm { ksin } \frac { 1 } { 2 } \mathrm { x }\), where \(k\) is a positive constant and \(x\) is measured in radians. The curve has a minimum point \(A\).
  1. State the coordinates of \(A\).
  2. A sequence of transformations is applied to the curve in the following order. Translation of 2 units in the negative \(y\)-direction
    Reflection in the \(x\)-axis
    Find the equation of the new curve and determine the coordinates of the point on the new curve corresponding to \(A\).
CAIE P1 2024 March Q3
3 A curve is such that \(\frac { \mathrm { dy } } { \mathrm { dx } } = 3 ( 4 \mathrm { x } + 5 ) ^ { \frac { 1 } { 2 } }\). It is given that the points \(( 1,9 )\) and \(( 5 , a )\) lie on the curve. Find the value of \(a\).
CAIE P1 2024 March Q4
4
  1. Prove that \(\frac { ( \sin \theta + \cos \theta ) ^ { 2 } - 1 } { \cos ^ { 2 } \theta } \equiv 2 \tan \theta\).
    \includegraphics[max width=\textwidth, alt={}]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_63_1569_333_328} …...........................................................................................................................................
    \includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_65_1570_511_324}
    \includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_62_1570_603_324}
    \includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_72_1570_685_324}
    \includegraphics[max width=\textwidth, alt={}]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_72_1570_776_324} ...................................................................................................................................... .
    \includegraphics[max width=\textwidth, alt={}]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_76_1572_952_322} ........................................................................................................................................
    \includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_72_1570_1137_324}
    \includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_74_1572_1226_322}
    \includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_77_1575_1315_319}
  2. Hence solve the equation \(\frac { ( \sin \theta + \cos \theta ) ^ { 2 } - 1 } { \cos ^ { 2 } \theta } = 5 \tan ^ { 3 } \theta\) for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\).
CAIE P1 2024 March Q5
5 A curve has the equation \(\mathrm { y } = \frac { 3 } { 2 \mathrm { x } ^ { 2 } - 5 }\).
Find the equation of the normal to the curve at the point \(( 2,1 )\), giving your answer in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { c } = 0\), where \(a , b\) and \(c\) are integers.
CAIE P1 2024 March Q6
6 It is given that the coefficient of \(x ^ { 3 }\) in the expansion of $$( 2 + a x ) ^ { 4 } ( 5 - a x )$$ is 432 .
Find the value of the constant \(a\).
CAIE P1 2024 March Q7
7 The straight line \(\mathrm { y } = \mathrm { x } + 5\) meets the curve \(2 \mathrm { x } ^ { 2 } + 3 \mathrm { y } ^ { 2 } = \mathrm { k }\) at a single point \(P\).
  1. Find the value of the constant \(k\).
  2. Find the coordinates of \(P\).
CAIE P1 2024 March Q8
8
  1. An arithmetic progression is such that its first term is 6 and its tenth term is 19.5 .
    Find the sum of the first 100 terms of this arithmetic progression.
  2. A geometric progression \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is such that \(a _ { 1 } = 24\) and the common ratio is \(\frac { 1 } { 2 }\). The sum to infinity of this geometric progression is denoted by \(S\). The sum to infinity of the even-numbered terms (i.e. \(a _ { 2 } , a _ { 4 } , a _ { 6 } , \ldots\) ) is denoted by \(S _ { E }\). Find the values of \(S\) and \(S _ { E }\).
CAIE P1 2024 March Q9
9 The functions f and g are defined for all real values of \(x\) by $$f ( x ) = ( 3 x - 2 ) ^ { 2 } + k \quad \text { and } \quad g ( x ) = 5 x - 1$$ where \(k\) is a constant.
  1. Given that the range of the function gf is \(\mathrm { gf } ( x ) \geqslant 39\), find the value of \(k\).
  2. For this value of \(k\), determine the range of the function fg .
  3. The function h is defined for all real values of \(x\) and is such that \(\mathrm { gh } ( x ) = 35 x + 19\). Find an expression for \(\mathrm { g } ^ { - 1 } ( x )\) and hence, or otherwise, find an expression for \(\mathrm { h } ( x )\).
    \includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-12_739_625_260_721} The diagram shows the circle with centre \(C ( - 4,5 )\) and radius \(\sqrt { 20 }\) units. The circle intersects the \(y\)-axis at the points \(A\) and \(B\). The size of angle \(A C B\) is \(\theta\) radians.
CAIE P1 2024 March Q11
11
\includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-14_467_757_262_653} The diagram shows the curve with equation \(\mathrm { y } = 2 \mathrm { x } ^ { - \frac { 2 } { 3 } } - 3 \mathrm { x } ^ { - \frac { 1 } { 3 } } + 1\) for \(x > 0\). The curve crosses the \(x\)-axis at points \(A\) and \(B\) and has a minimum point \(M\).
  1. Find the exact coordinates of \(M\).
  2. Find the area of the region bounded by the curve and the line segment \(A B\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE P1 2020 November Q1
1 Find the set of values of \(m\) for which the line with equation \(y = m x - 3\) and the curve with equation \(y = 2 x ^ { 2 } + 5\) do not meet.
CAIE P1 2020 November Q2
2 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { ( x - 3 ) ^ { 2 } } + x\). It is given that the curve passes through the point (2, 7). Find the equation of the curve.
CAIE P1 2020 November Q3
3 Air is being pumped into a balloon in the shape of a sphere so that its volume is increasing at a constant rate of \(50 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\). Find the rate at which the radius of the balloon is increasing when the radius is 10 cm .
CAIE P1 2020 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{fdd6e942-b5bc-4369-8587-6de120459776-05_615_1169_260_488} In the diagram, the lower curve has equation \(y = \cos \theta\). The upper curve shows the result of applying a combination of transformations to \(y = \cos \theta\). Find, in terms of a cosine function, the equation of the upper curve.
CAIE P1 2020 November Q5
5 In the expansion of \(\left( 2 x ^ { 2 } + \frac { a } { x } \right) ^ { 6 }\), the coefficients of \(x ^ { 6 }\) and \(x ^ { 3 }\) are equal.
  1. Find the value of the non-zero constant \(a\).
  2. Find the coefficient of \(x ^ { 6 }\) in the expansion of \(\left( 1 - x ^ { 3 } \right) \left( 2 x ^ { 2 } + \frac { a } { x } \right) ^ { 6 }\).
CAIE P1 2020 November Q6
6 The equation of a curve is \(y = 2 + \sqrt { 25 - x ^ { 2 } }\).
Find the coordinates of the point on the curve at which the gradient is \(\frac { 4 } { 3 }\).
CAIE P1 2020 November Q7
7
  1. Show that \(\frac { \sin \theta } { 1 - \sin \theta } - \frac { \sin \theta } { 1 + \sin \theta } \equiv 2 \tan ^ { 2 } \theta\).
  2. Hence solve the equation \(\frac { \sin \theta } { 1 - \sin \theta } - \frac { \sin \theta } { 1 + \sin \theta } = 8\), for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P1 2020 November Q8
8 A geometric progression has first term \(a\), common ratio \(r\) and sum to infinity \(S\). A second geometric progression has first term \(a\), common ratio \(R\) and sum to infinity \(2 S\).
  1. Show that \(r = 2 R - 1\).
    It is now given that the 3rd term of the first progression is equal to the 2nd term of the second progression.
  2. Express \(S\) in terms of \(a\).
CAIE P1 2020 November Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{fdd6e942-b5bc-4369-8587-6de120459776-12_583_661_260_740} The diagram shows a circle with centre \(A\) passing through the point \(B\). A second circle has centre \(B\) and passes through \(A\). The tangent at \(B\) to the first circle intersects the second circle at \(C\) and \(D\). The coordinates of \(A\) are ( \(- 1,4\) ) and the coordinates of \(B\) are ( 3,2 ).
  1. Find the equation of the tangent CBD.
  2. Find an equation of the circle with centre \(B\).
  3. Find, by calculation, the \(x\)-coordinates of \(C\) and \(D\).
    \includegraphics[max width=\textwidth, alt={}, center]{fdd6e942-b5bc-4369-8587-6de120459776-14_746_652_262_744} The diagram shows a sector \(C A B\) which is part of a circle with centre \(C\). A circle with centre \(O\) and radius \(r\) lies within the sector and touches it at \(D , E\) and \(F\), where \(C O D\) is a straight line and angle \(A C D\) is \(\theta\) radians.
CAIE P1 2020 November Q11
11 The functions \(f\) and \(g\) are defined by $$\begin{array} { l l } f ( x ) = x ^ { 2 } + 3 & \text { for } x > 0
g ( x ) = 2 x + 1 & \text { for } x > - \frac { 1 } { 2 } \end{array}$$
  1. Find an expression for \(\mathrm { fg } ( x )\).
  2. Find an expression for \(( \mathrm { fg } ) ^ { - 1 } ( x )\) and state the domain of \(( \mathrm { fg } ) ^ { - 1 }\).
  3. Solve the equation \(\mathrm { fg } ( x ) - 3 = \mathrm { gf } ( x )\).
CAIE P1 2020 November Q12
12
\includegraphics[max width=\textwidth, alt={}, center]{fdd6e942-b5bc-4369-8587-6de120459776-18_557_677_264_733} The diagram shows a curve with equation \(y = 4 x ^ { \frac { 1 } { 2 } } - 2 x\) for \(x \geqslant 0\), and a straight line with equation \(y = 3 - x\). The curve crosses the \(x\)-axis at \(A ( 4,0 )\) and crosses the straight line at \(B\) and \(C\).
  1. Find, by calculation, the \(x\)-coordinates of \(B\) and \(C\).
  2. Show that \(B\) is a stationary point on the curve.
  3. Find the area of the shaded region.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE P1 2020 November Q1
1 The coefficient of \(x ^ { 3 }\) in the expansion of \(( 1 + k x ) ( 1 - 2 x ) ^ { 5 }\) is 20 .
Find the value of the constant \(k\).