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 June Q2
2 The function f is defined for \(x \in \mathbb { R }\) by \(\mathrm { f } ( x ) = x ^ { 2 } - 6 x + c\), where \(c\) is a constant. It is given that \(\mathrm { f } ( x ) > 2\) for all values of \(x\). Find the set of possible values of \(c\).
CAIE P1 2023 June Q3
3
  1. Give the complete expansion of \(\left( x + \frac { 2 } { x } \right) ^ { 5 }\).
  2. In the expansion of \(\left( a + b x ^ { 2 } \right) \left( x + \frac { 2 } { x } \right) ^ { 5 }\), the coefficient of \(x\) is zero and the coefficient of \(\frac { 1 } { x }\) is 80 . Find the values of the constants \(a\) and \(b\).
CAIE P1 2023 June Q4
4
  1. Show that the equation $$3 \tan ^ { 2 } x - 3 \sin ^ { 2 } x - 4 = 0$$ may be expressed in the form \(a \cos ^ { 4 } x + b \cos ^ { 2 } x + c = 0\), where \(a , b\) and \(c\) are constants to be found.
  2. Hence solve the equation \(3 \tan ^ { 2 } x - 3 \sin ^ { 2 } x - 4 = 0\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
CAIE P1 2023 June Q5
5 A circle has equation \(( x - 1 ) ^ { 2 } + ( y + 4 ) ^ { 2 } = 40\). A line with equation \(y = x - 9\) intersects the circle at points \(A\) and \(B\).
  1. Find the coordinates of the two points of intersection.
  2. Find an equation of the circle with diameter \(A B\).
CAIE P1 2023 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{51bd3ba6-e1d1-4c07-81cd-d99dd77f9306-07_389_552_267_799} The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius \(r \mathrm {~cm}\). Angle \(A O B = \theta\) radians. It is given that the length of the \(\operatorname { arc } A B\) is 9.6 cm and that the area of the sector \(O A B\) is \(76.8 \mathrm {~cm} ^ { 2 }\).
  1. Find the area of the shaded region.
  2. Find the perimeter of the shaded region.
CAIE P1 2023 June Q7
7 The function f is defined by \(\mathrm { f } ( x ) = 2 - \frac { 5 } { x + 2 }\) for \(x > - 2\).
  1. State the range of f.
  2. Obtain an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
    The function g is defined by \(\mathrm { g } ( x ) = x + 3\) for \(x > 0\).
  3. Obtain an expression for \(\operatorname { fg } ( x )\) giving your answer in the form \(\frac { a x + b } { c x + d }\), where \(a , b , c\) and \(d\) are integers.
CAIE P1 2023 June Q8
8 A progression has first term \(a\) and second term \(\frac { a ^ { 2 } } { a + 2 }\), where \(a\) is a positive constant.
  1. For the case where the progression is geometric and the sum to infinity is 264 , find the value of \(a\).
  2. For the case where the progression is arithmetic and \(a = 6\), determine the least value of \(n\) required for the sum of the first \(n\) terms to be less than - 480 .
CAIE P1 2023 June Q9
9 A curve which passes through \(( 0,3 )\) has equation \(y = \mathrm { f } ( x )\). It is given that \(\mathrm { f } ^ { \prime } ( x ) = 1 - \frac { 2 } { ( x - 1 ) ^ { 3 } }\).
  1. Find the equation of the curve.
    The tangent to the curve at \(( 0,3 )\) intersects the curve again at one other point, \(P\).
  2. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(( 2 x + 1 ) ( x - 1 ) ^ { 2 } - 1 = 0\).
  3. Verify that \(x = \frac { 3 } { 2 }\) satisfies this equation and hence find the \(y\)-coordinate of \(P\).
CAIE P1 2023 June Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{51bd3ba6-e1d1-4c07-81cd-d99dd77f9306-14_832_830_276_653} The diagram shows the points \(A \left( 1 \frac { 1 } { 2 } , 5 \frac { 1 } { 2 } \right)\) and \(B \left( 7 \frac { 1 } { 2 } , 3 \frac { 1 } { 2 } \right)\) lying on the curve with equation \(y = 9 x - ( 2 x + 1 ) ^ { \frac { 3 } { 2 } }\).
  1. Find the coordinates of the maximum point of the curve.
  2. Verify that the line \(A B\) is the normal to the curve at \(A\).
  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 2024 June Q1
1
  1. Express \(3 y ^ { 2 } - 12 y - 15\) in the form \(3 ( y + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
  2. Hence find the exact solutions of the equation \(3 x ^ { 4 } - 12 x ^ { 2 } - 15 = 0\).
CAIE P1 2024 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{cacac880-5b44-4fae-8ed8-88a095db69cd-03_606_1515_260_274} The diagram shows two curves. One curve has equation \(y = \sin x\) and the other curve has equation \(y = f ( x )\).
  1. In order to transform the curve \(y = \sin x\) to the curve \(y = f ( x )\), the curve \(y = \sin x\) is first reflected in the \(x\)-axis. Describe fully a sequence of two further transformations which are required.
  2. Find \(\mathrm { f } ( x )\) in terms of \(\sin x\).
CAIE P1 2024 June Q3
3 The coefficient of \(x ^ { 3 }\) in the expansion of \(( 3 + \mathrm { ax } ) ^ { 6 }\) is 160 .
  1. Find the value of the constant \(a\).
  2. Hence find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 3 + \mathrm { ax } ) ^ { 6 } ( 1 - 2 \mathrm { x } )\).
CAIE P1 2024 June Q4
4 The equation of a curve is \(y = f ( x )\), where \(f ( x ) = ( 2 x - 1 ) \sqrt { 3 x - 2 } - 2\). The following points lie on the curve. Non-exact values have been given correct to 5 decimal places. $$A ( 2,4 ) , B ( 2.0001 , k ) , C ( 2.001,4.00625 ) , D ( 2.01,4.06261 ) , E ( 2.1,4.63566 ) , F ( 3,11.22876 )$$
  1. Find the value of \(k\). Give your answer correct to 5 decimal places.
    The table shows the gradients of the chords \(A B , A C , A D\) and \(A F\).
    Chord\(A B\)\(A C\)\(A D\)\(A E\)\(A F\)
    Gradient of
    chord
    		6.25016.25116.26087.2288
  2. Find the gradient of the chord \(A E\). Give your answer correct to 4 decimal places.
  3. Deduce the value of \(f ^ { \prime } ( 2 )\) using the values in the table.
CAIE P1 2024 June Q5
5
  1. Prove the identity \(\frac { \sin ^ { 2 } x - \cos x - 1 } { 1 + \cos x } \equiv - \cos x\).
  2. Hence solve the equation \(\frac { \sin ^ { 2 } x - \cos x - 1 } { 2 + 2 \cos x } = \frac { 1 } { 4 }\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
    \includegraphics[max width=\textwidth, alt={}, center]{cacac880-5b44-4fae-8ed8-88a095db69cd-07_583_990_306_539} The function f is defined by \(\mathrm { f } ( x ) = \frac { 2 } { x ^ { 2 } } + 4\) for \(x < 0\). The diagram shows the graph of \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\).
  3. On this diagram, sketch the graph of \(y = f ^ { - 1 } ( x )\). Show any relevant mirror line.
  4. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
  5. Solve the equation \(\mathrm { f } ( x ) = 4.5\).
  6. Explain why the equation \(\mathrm { f } ^ { - 1 } ( x ) = \mathrm { f } ( x )\) has no solution.
    \includegraphics[max width=\textwidth, alt={}, center]{cacac880-5b44-4fae-8ed8-88a095db69cd-08_522_1036_296_513} In the diagram, \(A O D\) and \(B C\) are two parallel straight lines. Arc \(A B\) is part of a circle with centre \(O\) and radius 15 cm . Angle \(B O A = \theta\) radians. Arc \(C D\) is part of a circle with centre \(O\) and radius 10 cm . Angle \(C O D = \frac { 1 } { 2 } \pi\) radians.
  7. Show that \(\theta = 0.7297\), correct to 4 decimal places.
  8. Find the perimeter and the area of the shape \(A B C D\). Give your answers correct to 3 significant figures.
CAIE P1 2024 June Q8
8
  1. The first three terms of an arithmetic progression are \(25,4 p - 1\) and \(13 - p\), where \(p\) is a constant. Find the value of the tenth term of the progression.
  2. The first three terms of a geometric progression are \(25,4 q - 1\) and \(13 - q\), where \(q\) is a positive constant. Find the sum to infinity of the progression.
    \includegraphics[max width=\textwidth, alt={}, center]{cacac880-5b44-4fae-8ed8-88a095db69cd-12_586_1567_333_248} The diagram shows part of the curve with equation \(y = \frac { 1 } { ( 5 x - 4 ) ^ { \frac { 1 } { 3 } } }\) and the lines \(x = 2.4\) and \(y = 1\). The curve intersects the line \(y = 1\) at the point \(( 1,1 )\). Find the exact volume of the solid generated when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
CAIE P1 2024 June Q10
10 The equation of a circle is \(( x - 3 ) ^ { 2 } + y ^ { 2 } = 18\). The line with equation \(y = m x + c\) passes through the point \(( 0 , - 9 )\) and is a tangent to the circle. Find the two possible values of \(m\) and, for each value of \(m\), find the coordinates of the point at which the tangent touches the circle.
\includegraphics[max width=\textwidth, alt={}, center]{cacac880-5b44-4fae-8ed8-88a095db69cd-16_855_1600_306_233} A function is defined by \(\mathrm { f } ( x ) = \frac { 4 } { x ^ { 3 } } - \frac { 3 } { x } + 2\) for \(x \neq 0\). The graph of \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\) is shown in the diagram.
  1. Find the set of values of \(x\) for which \(\mathrm { f } ( x )\) is decreasing.
  2. A triangle is bounded by the \(y\)-axis, the normal to the curve at the point where \(x = 1\) and the tangent to the curve at the point where \(x = - 1\). Find the area of the triangle. Give your answer correct to 3 significant figures.
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE P1 2024 June Q1
1 The coefficient of \(x ^ { 2 }\) in the expansion of \(( 1 - 4 x ) ^ { 6 }\) is 12 times the coefficient of \(x ^ { 2 }\) in the expansion of \(( 2 + a x ) ^ { 5 }\). Find the value of the positive constant \(a\).
CAIE P1 2024 June Q2
2 The curve \(y = x ^ { 2 }\) is transformed to the curve \(y = 4 ( x - 3 ) ^ { 2 } - 8\).
Describe fully a sequence of transformations that have been combined, making clear the order in which the transformations have been applied.
CAIE P1 2024 June Q3
3
  1. Show that the equation \(\frac { 7 \tan \theta } { \cos \theta } + 12 = 0\) can be expressed as $$12 \sin ^ { 2 } \theta - 7 \sin \theta - 12 = 0$$
  2. Hence solve the equation \(\frac { 7 \tan \theta } { \cos \theta } + 12 = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P1 2024 June Q4
4 The function f is defined as follows: $$\mathrm { f } ( x ) = \sqrt { x } - 1 \text { for } x > 1$$
  1. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
    \includegraphics[max width=\textwidth, alt={}, center]{84b3cd80-faf0-4522-8286-52bf7f86dc8a-06_362_1543_1023_251} The diagram shows the graph of \(\mathrm { y } = \mathrm { g } ( \mathrm { x } )\) where \(\mathrm { g } ( x ) = \frac { 1 } { x ^ { 2 } + 2 }\) for \(x \in \mathbb { R }\).
  2. State the range of \(g\) and explain whether \(g ^ { - 1 }\) exists.
    The function h is defined by \(\mathrm { h } ( x ) = \frac { 1 } { x ^ { 2 } + 2 }\) for \(x \geqslant 0\).
  3. Solve the equation \(\mathrm { hf } ( x ) = \mathrm { f } \left( \frac { 25 } { 16 } \right)\). Give your answer in the form \(\mathrm { a } + \mathrm { b } \sqrt { \mathrm { c } }\), where \(a , b\) and \(c\) are
    integers. integers.
CAIE P1 2024 June Q5
5 The first and second terms of an arithmetic progression are \(\tan \theta\) and \(\sin \theta\) respectively, where \(0 < \theta < \frac { 1 } { 2 } \pi\).
  1. Given that \(\theta = \frac { 1 } { 4 } \pi\), find the exact sum of the first 40 terms of the progression.
    The first and second terms of a geometric progression are \(\tan \theta\) and \(\sin \theta\) respectively, where \(0 < \theta < \frac { 1 } { 2 } \pi\).
    1. Find the sum to infinity of the progression in terms of \(\theta\).
    2. Given that \(\theta = \frac { 1 } { 3 } \pi\), find the sum of the first 10 terms of the progression. Give your answer correct to 3 significant figures.
CAIE P1 2024 June Q6
6 The curve with equation \(y = 2 x - 8 x ^ { \frac { 1 } { 2 } }\) has a minimum point at \(A\) and intersects the positive \(x\)-axis at \(B\).
  1. Find the coordinates of \(A\) and \(B\).

  2. \includegraphics[max width=\textwidth, alt={}, center]{84b3cd80-faf0-4522-8286-52bf7f86dc8a-11_522_1561_296_244} The diagram shows the curve with equation \(\mathrm { y } = 2 \mathrm { x } - 8 \mathrm { x } ^ { \frac { 1 } { 2 } }\) and the line \(A B\). It is given that the equation of \(A B\) is \(y = \frac { 2 x - 32 } { 3 }\). Find the area of the shaded region between the curve and the line.
CAIE P1 2024 June Q7
7 The equation of a circle is \(( x - 6 ) ^ { 2 } + ( y + a ) ^ { 2 } = 18\). The line with equation \(y = 2 a - x\) is a tangent to the circle.
  1. Find the two possible values of the constant \(a\).
  2. For the greater value of \(a\), find the equation of the diameter which is perpendicular to the given tangent.
    \includegraphics[max width=\textwidth, alt={}, center]{84b3cd80-faf0-4522-8286-52bf7f86dc8a-14_280_1358_317_349} The diagram shows a symmetrical plate \(A B C D E F\). The line \(A B C D\) is straight and the length of \(B C\) is 2 cm . Each of the two sectors \(A B F\) and \(D C E\) is of radius \(r \mathrm {~cm}\) and each of the angles \(A B F\) and \(D C E\) is equal to \(\frac { 1 } { 3 } \pi\) radians.
  3. It is given that \(r = 0.4 \mathrm {~cm}\).
    1. Show that the length \(\mathrm { EF } = 2.4 \mathrm {~cm}\).
    2. Find the area of the plate. Give your answer correct to 3 significant figures.
  4. It is given instead that the perimeter of the plate is 6 cm . Find the value of \(r\). Give your answer correct to 3 significant figures.
CAIE P1 2024 June Q9
9 A function f is such that \(\mathrm { f } ^ { \prime } ( x ) = 6 ( 2 x - 3 ) ^ { 2 } - 6 x\) for \(x \in \mathbb { R }\).
  1. Determine the set of values of \(x\) for which \(\mathrm { f } ( x )\) is decreasing.
  2. Given that \(\mathrm { f } ( 1 ) = - 1\), find \(\mathrm { f } ( x )\).
CAIE P1 2024 June Q10
10 The equation of a curve is \(\mathrm { y } = ( 5 - 2 \mathrm { x } ) ^ { \frac { 3 } { 2 } } + 5\) for \(x < \frac { 5 } { 2 }\).
  1. A point \(P\) is moving along the curve in such a way that the \(y\)-coordinate of point \(P\) is decreasing at 5 units per second. Find the rate at which the \(x\)-coordinate of point \(P\) is increasing when \(y = 32\).
  2. Point \(A\) on the curve has \(y\)-coordinate 32. Point \(B\) on the curve is such that the gradient of the curve at \(B\) is - 3 . Find the equation of the perpendicular bisector of \(A B\). Give your answer in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { c } = 0\), where \(a , b\) and \(c\) are integers.
    If you use the following page to complete the answer to any question, the question number must be clearly shown.