Questions P1 (1374 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 2022 November Q4
4 A large industrial water tank is such that, when the depth of the water in the tank is \(x\) metres, the volume \(V \mathrm {~m} ^ { 3 }\) of water in the tank is given by \(V = 243 - \frac { 1 } { 3 } ( 9 - x ) ^ { 3 }\). Water is being pumped into the tank at a constant rate of \(3.6 \mathrm {~m} ^ { 3 }\) per hour. Find the rate of increase of the depth of the water when the depth is 4 m , giving your answer in cm per minute.
CAIE P1 2022 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{8eb3d21b-dc45-493c-9e5c-3c0535c505e8-06_743_750_269_687} The diagram shows a curve which has a maximum point at \(( 8,12 )\) and a minimum point at \(( 8,0 )\). The curve is the result of applying a combination of two transformations to a circle. The first transformation applied is a translation of \(\binom { 7 } { - 3 }\). The second transformation applied is a stretch in the \(y\)-direction.
  1. State the scale factor of the stretch.
  2. State the radius of the original circle.
  3. State the coordinates of the centre of the circle after the translation has been completed but before the stretch is applied.
  4. State the coordinates of the centre of the original circle.
CAIE P1 2022 November Q6
6 It is given that \(\alpha = \cos ^ { - 1 } \left( \frac { 8 } { 17 } \right)\).
Find, without using the trigonometric functions on your calculator, the exact value of \(\frac { 1 } { \sin \alpha } + \frac { 1 } { \tan \alpha }\).
CAIE P1 2022 November Q7
7 The curve \(y = \mathrm { f } ( x )\) is such that \(\mathrm { f } ^ { \prime } ( x ) = \frac { - 3 } { ( x + 2 ) ^ { 4 } }\).
  1. The tangent at a point on the curve where \(x = a\) has gradient \(- \frac { 16 } { 27 }\). Find the possible values of \(a\).
  2. Find \(\mathrm { f } ( x )\) given that the curve passes through the point \(( - 1,5 )\).
CAIE P1 2022 November Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{8eb3d21b-dc45-493c-9e5c-3c0535c505e8-10_492_888_255_625} The diagram shows two identical circles intersecting at points \(A\) and \(B\) and with centres at \(P\) and \(Q\). The radius of each circle is \(r\) and the distance \(P Q\) is \(\frac { 5 } { 3 } r\).
  1. Find the perimeter of the shaded region in terms of \(r\).
  2. Find the area of the shaded region in terms of \(r\).
CAIE P1 2022 November Q9
9 The first term of a geometric progression is 216 and the fourth term is 64.
  1. Find the sum to infinity of the progression.
    The second term of the geometric progression is equal to the second term of an arithmetic progression.
    The third term of the geometric progression is equal to the fifth term of the same arithmetic progression.
  2. Find the sum of the first 21 terms of the arithmetic progression.
    \includegraphics[max width=\textwidth, alt={}, center]{8eb3d21b-dc45-493c-9e5c-3c0535c505e8-14_798_786_269_667} The diagram shows the circle \(x ^ { 2 } + y ^ { 2 } = 2\) and the straight line \(y = 2 x - 1\) intersecting at the points \(A\) and \(B\). The point \(D\) on the \(x\)-axis is such that \(A D\) is perpendicular to the \(x\)-axis.
CAIE P1 2022 November Q11
11 The coordinates of points \(A , B\) and \(C\) are \(A ( 5 , - 2 ) , B ( 10,3 )\) and \(C ( 2 p , p )\), where \(p\) is a constant.
  1. Given that \(A C\) and \(B C\) are equal in length, find the value of the fraction \(p\).
  2. It is now given instead that \(A C\) is perpendicular to \(B C\) and that \(p\) is an integer.
    1. Find the value of \(p\).
    2. Find the equation of the circle which passes through \(A , B\) and \(C\), giving your answer in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\), where \(a , b\) and \(c\) are constants.
      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 2023 November Q1
1
  1. Expand \(( 1 + 3 x ) ^ { 6 }\) in ascending powers of \(x\) up to, and including, the term in \(x ^ { 2 }\).
  2. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 1 - 7 x + x ^ { 2 } \right) ( 1 + 3 x ) ^ { 6 }\).
CAIE P1 2023 November Q2
2 A line has equation \(y = 2 c x + 3\) and a curve has equation \(y = c x ^ { 2 } + 3 x - c\), where \(c\) is a constant.
Showing all necessary working, determine which of the following statements is correct.
A The line and curve intersect only for a particular set of values of \(c\).
B The line and curve intersect for all values of \(c\).
C The line and curve do not intersect for any values of \(c\).
CAIE P1 2023 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{5e3e5418-7976-4232-8550-1da6420a3fcb-05_424_529_248_815} The diagram shows a cubical closed container made of a thin elastic material which is filled with water and frozen. During the freezing process the length, \(x \mathrm {~cm}\), of each edge of the container increases at the constant rate of 0.01 cm per minute. The volume of the container at time \(t\) minutes is \(V \mathrm {~cm} ^ { 3 }\). Find the rate of increase of \(V\) when \(x = 20\).
CAIE P1 2023 November Q4
4 The transformation R denotes a reflection in the \(x\)-axis and the transformation T denotes a translation of \(\binom { 3 } { - 1 }\).
  1. Find the equation, \(y = \mathrm { g } ( x )\), of the curve with equation \(y = x ^ { 2 }\) after it has been transformed by the sequence of transformations R followed by T .
  2. Find the equation, \(y = \mathrm { h } ( x )\), of the curve with equation \(y = x ^ { 2 }\) after it has been transformed by the sequence of transformations T followed by R .
  3. State fully the transformation that maps the curve \(y = \mathrm { g } ( x )\) onto the curve \(y = \mathrm { h } ( x )\).
CAIE P1 2023 November Q5
5
  1. Show that the equation $$4 \sin x + \frac { 5 } { \tan x } + \frac { 2 } { \sin x } = 0$$ may be expressed in the form \(a \cos ^ { 2 } x + b \cos x + c = 0\), where \(a , b\) and \(c\) are integers to be found.
  2. Hence solve the equation \(4 \sin x + \frac { 5 } { \tan x } + \frac { 2 } { \sin x } = 0\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
CAIE P1 2023 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{5e3e5418-7976-4232-8550-1da6420a3fcb-08_534_506_255_815} The diagram shows a motif formed by the major arc \(A B\) of a circle with radius \(r\) and centre \(O\), and the minor arc \(A O B\) of a circle, also with radius \(r\) but with centre \(C\). The point \(C\) lies on the circle with centre \(O\).
  1. Given that angle \(A C B = k \pi\) radians, state the value of the fraction \(k\).
  2. State the perimeter of the shaded motif in terms of \(\pi\) and \(r\).
  3. Find the area of the shaded motif, giving your answer in terms of \(\pi , r\) and \(\sqrt { 3 }\).
CAIE P1 2023 November Q7
7 The sum of the first two terms of a geometric progression is 15 and the sum to infinity is \(\frac { 125 } { 7 }\). The common ratio of the progression is negative. Find the third term of the progression.
CAIE P1 2023 November Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{5e3e5418-7976-4232-8550-1da6420a3fcb-12_684_776_274_680} The diagram shows the curves with equations \(y = 2 ( 2 x - 3 ) ^ { 4 }\) and \(y = ( 2 x - 3 ) ^ { 2 } + 1\) meeting at points \(A\) and \(B\).
  1. By using the substitution \(u = 2 x - 3\) find, by calculation, the coordinates of \(A\) and \(B\).
  2. Find the exact area of the shaded region.
CAIE P1 2023 November Q9
9
  1. Express \(4 x ^ { 2 } - 12 x + 13\) in the form \(( 2 x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
    The function f is defined by \(\mathrm { f } ( x ) = 4 x ^ { 2 } - 12 x + 13\) for \(p < x < q\), where \(p\) and \(q\) are constants. The function g is defined by \(\mathrm { g } ( x ) = 3 x + 1\) for \(x < 8\).
  2. Given that it is possible to form the composite function gf , find the least possible value of \(p\) and the greatest possible value of \(q\).
  3. Find an expression for \(\operatorname { gf } ( x )\).
    The function h is defined by \(\mathrm { h } ( x ) = 4 x ^ { 2 } - 12 x + 13\) for \(x < 0\).
  4. Find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).
CAIE P1 2023 November Q10
10 A curve has a stationary point at \(( 2 , - 10 )\) and is such that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 6 x\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Find the equation of the curve.
  3. Find the coordinates of the other stationary point and determine its nature.
  4. Find the equation of the tangent to the curve at the point where the curve crosses the \(y\)-axis.
    \includegraphics[max width=\textwidth, alt={}, center]{5e3e5418-7976-4232-8550-1da6420a3fcb-18_689_828_276_646} The diagram shows the circle with equation \(( x - 4 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = 40\). Parallel tangents, each with gradient 1 , touch the circle at points \(A\) and \(B\).
  5. Find the equation of the line \(A B\), giving the answer in the form \(y = m x + c\).
  6. Find the coordinates of \(A\), giving each coordinate in surd form.
  7. Find the equation of the tangent at \(A\), giving the answer in the form \(y = m x + c\), where \(c\) is in surd form.
    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 2023 November Q1
1 The coefficient of \(x ^ { 3 }\) in the expansion of \(( 3 + 2 a x ) ^ { 5 }\) is six times the coefficient of \(x ^ { 2 }\) in the expansion of \(( 2 + a x ) ^ { 6 }\). Find the value of the constant \(a\).
CAIE P1 2023 November Q2
2 Find the exact solution of the equation $$\frac { 1 } { 6 } \pi + \tan ^ { - 1 } ( 4 x ) = - \cos ^ { - 1 } \left( \frac { 1 } { 2 } \sqrt { 3 } \right)$$
CAIE P1 2023 November Q3
3 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 2 } x + \frac { 72 } { x ^ { 4 } }\). The curve passes through the point \(P ( 2,8 )\).
  1. Find the equation of the normal to the curve at \(P\).
  2. Find the equation of the curve.
    \includegraphics[max width=\textwidth, alt={}, center]{e48188bc-3332-4248-971d-ebbbbbfb1280-05_456_488_264_826} The diagram shows the shape of a coin. The three \(\operatorname { arcs } A B , B C\) and \(C A\) are parts of circles with centres \(C , A\) and \(B\) respectively. \(A B C\) is an equilateral triangle with sides of length 2 cm .
  3. Find the perimeter of the coin.
  4. Find the area of the face \(A B C\) of the coin, giving the answer in terms of \(\pi\) and \(\sqrt { 3 }\).
CAIE P1 2023 November Q5
5 The first, second and third terms of a geometric progression are \(\sin \theta , \cos \theta\) and \(2 - \sin \theta\) respectively, where \(\theta\) radians is an acute angle.
  1. Find the value of \(\theta\).
  2. Using this value of \(\theta\), find the sum of the first 10 terms of the progression. Give the answer in the form \(\frac { b } { \sqrt { c } - 1 }\), where \(b\) and \(c\) are integers to be found.
CAIE P1 2023 November Q6
6 The equation of a curve is \(y = x ^ { 2 } - 8 x + 5\).
  1. Find the coordinates of the minimum point of the curve.
    The curve is stretched by a factor of 2 parallel to the \(y\)-axis and then translated by \(\binom { 4 } { 1 }\).
  2. Find the coordinates of the minimum point of the transformed curve.
  3. Find the equation of the transformed curve. Give the answer in the form \(y = a x ^ { 2 } + b x + c\), where \(a , b\) and \(c\) are integers to be found.
CAIE P1 2023 November Q7
7
  1. Verify the identity \(( 2 x - 1 ) \left( 4 x ^ { 2 } + 2 x - 1 \right) \equiv 8 x ^ { 3 } - 4 x + 1\).
  2. Prove the identity \(\frac { \tan ^ { 2 } \theta + 1 } { \tan ^ { 2 } \theta - 1 } \equiv \frac { 1 } { 1 - 2 \cos ^ { 2 } \theta }\).
  3. Using the results of (a) and (b), solve the equation $$\frac { \tan ^ { 2 } \theta + 1 } { \tan ^ { 2 } \theta - 1 } = 4 \cos \theta$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P1 2023 November Q8
8 Functions f and g are defined by $$\begin{aligned} & \mathrm { f } ( x ) = ( x + a ) ^ { 2 } - a \text { for } x \leqslant - a ,
& \mathrm {~g} ( x ) = 2 x - 1 \text { for } x \in \mathbb { R } , \end{aligned}$$ where \(a\) is a positive constant.
  1. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
    1. State the domain of the function \(f ^ { - 1 }\).
    2. State the range of the function \(f ^ { - 1 }\).
  3. Given that \(a = \frac { 7 } { 2 }\), solve the equation \(\mathrm { gf } ( x ) = 0\).
CAIE P1 2023 November Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{e48188bc-3332-4248-971d-ebbbbbfb1280-14_807_1278_274_427} The diagram shows curves with equations \(y = 2 x ^ { \frac { 1 } { 2 } } + 13 x ^ { - \frac { 1 } { 2 } }\) and \(y = 3 x ^ { - \frac { 1 } { 2 } } + 12\). The curves intersect at points \(A\) and \(B\).
  1. Find the coordinates of \(A\) and \(B\).
  2. Hence find the area of the shaded region.