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 2021 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{af7aeda9-2ded-4db4-9ff3-ed6adc67859f-07_778_878_255_630} The diagram shows part of the graph of \(y = a \cos ( b x ) + c\).
  1. Find the values of the positive integers \(a , b\) and \(c\).
  2. For these values of \(a\), \(b\) and \(c\), use the given diagram to determine the number of solutions in the interval \(0 \leqslant x \leqslant 2 \pi\) for each of the following equations.
    1. \(a \cos ( b x ) + c = \frac { 6 } { \pi } x\)
    2. \(a \cos ( b x ) + c = 6 - \frac { 6 } { \pi } x\)
      The diagram shows a metal plate \(A B C\) in which the sides are the straight line \(A B\) and the arcs \(A C\) and \(B C\). The line \(A B\) has length 6 cm . The arc \(A C\) is part of a circle with centre \(B\) and radius 6 cm , and the arc \(B C\) is part of a circle with centre \(A\) and radius 6 cm .
CAIE P1 2021 November Q7
7 A circle with centre \(( 5,2 )\) passes through the point \(( 7,5 )\).
  1. Find an equation of the circle.
    The line \(y = 5 x - 10\) intersects the circle at \(A\) and \(B\).
  2. Find the exact length of the chord \(A B\).
CAIE P1 2021 November Q8
8
  1. Express \(- 3 x ^ { 2 } + 12 x + 2\) in the form \(- 3 ( x - a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
    The one-one function f is defined by \(\mathrm { f } : x \mapsto - 3 x ^ { 2 } + 12 x + 2\) for \(x \leqslant k\).
  2. State the largest possible value of the constant \(k\).
    It is now given that \(k = - 1\).
  3. State the range of f.
  4. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
    The result of translating the graph of \(y = \mathrm { f } ( x )\) by \(\binom { - 3 } { 1 }\) is the graph of \(y = \mathrm { g } ( x )\).
  5. Express \(\mathrm { g } ( x )\) in the form \(p x ^ { 2 } + q x + r\), where \(p , q\) and \(r\) are constants.
CAIE P1 2021 November Q9
9 A curve has equation \(y = \mathrm { f } ( x )\), and it is given that \(\mathrm { f } ^ { \prime } ( x ) = 2 x ^ { 2 } - 7 - \frac { 4 } { x ^ { 2 } }\).
  1. Given that \(\mathrm { f } ( 1 ) = - \frac { 1 } { 3 }\), find \(\mathrm { f } ( x )\).
  2. Find the coordinates of the stationary points on the curve.
  3. Find \(\mathrm { f } ^ { \prime \prime } ( x )\).
  4. Hence, or otherwise, determine the nature of each of the stationary points.
CAIE P1 2021 November Q10
10
  1. Find \(\int _ { 1 } ^ { \infty } \frac { 1 } { ( 3 x - 2 ) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x\).
    \includegraphics[max width=\textwidth, alt={}, center]{af7aeda9-2ded-4db4-9ff3-ed6adc67859f-16_499_689_1322_726} The diagram shows the curve with equation \(y = \frac { 1 } { ( 3 x - 2 ) ^ { \frac { 3 } { 2 } } }\). The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\). The shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
  2. Find the volume of revolution.
    The normal to the curve at the point \(( 1,1 )\) crosses the \(y\)-axis at the point \(A\).
  3. Find the \(y\)-coordinate of \(A\).
    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 2021 November Q1
1 Solve the equation \(2 \cos \theta = 7 - \frac { 3 } { \cos \theta }\) for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\).
CAIE P1 2021 November Q2
2 The graph of \(y = \mathrm { f } ( x )\) is transformed to the graph of \(y = \mathrm { f } ( 2 x ) - 3\).
  1. Describe fully the two single transformations that have been combined to give the resulting transformation.
    The point \(P ( 5,6 )\) lies on the transformed curve \(y = \mathrm { f } ( 2 x ) - 3\).
  2. State the coordinates of the corresponding point on the original curve \(y = \mathrm { f } ( x )\).
CAIE P1 2021 November Q3
3 The function f is defined as follows: $$\mathrm { f } ( x ) = \frac { x + 3 } { x - 1 } \text { for } x > 1$$
  1. Find the value of \(\mathrm { ff } ( 5 )\).
  2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
CAIE P1 2021 November Q4
4 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 8 } { ( 3 x + 2 ) ^ { 2 } }\). The curve passes through the point \(\left( 2,5 \frac { 2 } { 3 } \right)\).
Find the equation of the curve.
CAIE P1 2021 November Q5
5 The first, third and fifth terms of an arithmetic progression are \(2 \cos x , - 6 \sqrt { 3 } \sin x\) and \(10 \cos x\) respectively, where \(\frac { 1 } { 2 } \pi < x < \pi\).
  1. Find the exact value of \(x\).
  2. Hence find the exact sum of the first 25 terms of the progression.
CAIE P1 2021 November Q6
6 The second term of a geometric progression is 54 and the sum to infinity of the progression is 243 . The common ratio is greater than \(\frac { 1 } { 2 }\). Find the tenth term, giving your answer in exact form.
CAIE P1 2021 November Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{10b2ec29-adca-4313-ae24-bab8b2d9f8a4-08_556_751_255_696} In the diagram the lengths of \(A B\) and \(A C\) are both 15 cm . The point \(P\) is the foot of the perpendicular from \(C\) to \(A B\). The length \(C P = 9 \mathrm {~cm}\). An arc of a circle with centre \(B\) passes through \(C\) and meets \(A B\) at \(Q\).
  1. Show that angle \(A B C = 1.25\) radians, correct to 3 significant figures.
  2. Calculate the area of the shaded region which is bounded by the \(\operatorname { arc } C Q\) and the lines \(C P\) and \(P Q\).
CAIE P1 2021 November Q8
8
  1. It is given that in the expansion of \(( 4 + 2 x ) ( 2 - a x ) ^ { 5 }\), the coefficient of \(x ^ { 2 }\) is - 15 .
    Find the possible values of \(a\).
  2. It is given instead that in the expansion of \(( 4 + 2 x ) ( 2 - a x ) ^ { 5 }\), the coefficient of \(x ^ { 2 }\) is \(k\). It is also given that there is only one value of \(a\) which leads to this value of \(k\). Find the values of \(k\) and \(a\).
CAIE P1 2021 November Q9
3 marks
9 The volume \(V \mathrm {~m} ^ { 3 }\) of a large circular mound of iron ore of radius \(r \mathrm {~m}\) is modelled by the equation \(V = \frac { 3 } { 2 } \left( r - \frac { 1 } { 2 } \right) ^ { 3 } - 1\) for \(r \geqslant 2\). Iron ore is added to the mound at a constant rate of \(1.5 \mathrm {~m} ^ { 3 }\) per second.
[0pt]
  1. Find the rate at which the radius of the mound is increasing at the instant when the radius is 5.5 m . [3]
  2. Find the volume of the mound at the instant when the radius is increasing at 0.1 m per second.
CAIE P1 2021 November Q10
10 The function f is defined by \(\mathrm { f } ( x ) = x ^ { 2 } + \frac { k } { x } + 2\) for \(x > 0\).
  1. Given that the curve with equation \(y = \mathrm { f } ( x )\) has a stationary point when \(x = 2\), find \(k\).
  2. Determine the nature of the stationary point.
  3. Given that this is the only stationary point of the curve, find the range of f .
CAIE P1 2021 November Q11
11
\includegraphics[max width=\textwidth, alt={}, center]{10b2ec29-adca-4313-ae24-bab8b2d9f8a4-16_505_1166_258_486} The diagram shows the line \(x = \frac { 5 } { 2 }\), part of the curve \(y = \frac { 1 } { 2 } x + \frac { 7 } { 10 } - \frac { 1 } { ( x - 2 ) ^ { \frac { 1 } { 3 } } }\) and the normal to the curve at the point \(A \left( 3 , \frac { 6 } { 5 } \right)\).
  1. Find the \(x\)-coordinate of the point where the normal to the curve meets the \(x\)-axis.
  2. Find the area of the shaded region, giving your answer correct to 2 decimal places.
CAIE P1 2021 November Q12
12
\includegraphics[max width=\textwidth, alt={}, center]{10b2ec29-adca-4313-ae24-bab8b2d9f8a4-18_750_981_258_580} The diagram shows the circle with equation \(x ^ { 2 } + y ^ { 2 } - 6 x + 4 y - 27 = 0\) and the tangent to the circle at the point \(P ( 5,4 )\).
  1. The tangent to the circle at \(P\) meets the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\). Find the area of triangle \(O A B\), where \(O\) is the origin.
  2. Points \(Q\) and \(R\) also lie on the circle, such that \(P Q R\) is an equilateral triangle. Find the exact area of triangle \(P Q R\).
    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 2021 November Q1
1 The graph of \(y = \mathrm { f } ( x )\) is transformed to the graph of \(y = 3 - \mathrm { f } ( x )\).
Describe fully, in the correct order, the two transformations that have been combined.
CAIE P1 2021 November Q2
2
  1. Find the first three terms, in ascending powers of \(x\), in the expansion of \(( 1 + a x ) ^ { 6 }\).
  2. Given that the coefficient of \(x ^ { 2 }\) in the expansion of \(( 1 - 3 x ) ( 1 + a x ) ^ { 6 }\) is - 3 , find the possible values of the constant \(a\).
CAIE P1 2021 November Q3
3
  1. Express \(5 y ^ { 2 } - 30 y + 50\) in the form \(5 ( y + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
  2. The function f is defined by \(\mathrm { f } ( x ) = x ^ { 5 } - 10 x ^ { 3 } + 50 x\) for \(x \in \mathbb { R }\). Determine whether \(f\) is an increasing function, a decreasing function or neither.
CAIE P1 2021 November Q4
4 The first term of an arithmetic progression is 84 and the common difference is - 3 .
  1. Find the smallest value of \(n\) for which the \(n\)th term is negative.
    It is given that the sum of the first \(2 k\) terms of this progression is equal to the sum of the first \(k\) terms.
  2. Find the value of \(k\).
CAIE P1 2021 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{cba7391d-2cf7-483c-9a21-bc9fd49860ee-06_720_686_260_733} In the diagram, \(X\) and \(Y\) are points on the line \(A B\) such that \(B X = 9 \mathrm {~cm}\) and \(A Y = 11 \mathrm {~cm}\). Arc \(B C\) is part of a circle with centre \(X\) and radius 9 cm , where \(C X\) is perpendicular to \(A B\). Arc \(A C\) is part of a circle with centre \(Y\) and radius 11 cm .
  1. Show that angle \(X Y C = 0.9582\) radians, correct to 4 significant figures.
  2. Find the perimeter of \(A B C\).
CAIE P1 2021 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{cba7391d-2cf7-483c-9a21-bc9fd49860ee-08_608_597_258_772} The diagram shows the graph of \(y = \mathrm { f } ( x )\).
  1. On this diagram sketch the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\). It is now given that \(\mathrm { f } ( x ) = - \frac { x } { \sqrt { 4 - x ^ { 2 } } }\) where \(- 2 < x < 2\).
  2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
    The function g is defined by \(\mathrm { g } ( x ) = 2 x\) for \(- a < x < a\), where \(a\) is a constant.
  3. State the maximum possible value of \(a\) for which fg can be formed.
  4. Assuming that fg can be formed, find and simplify an expression for \(\mathrm { fg } ( x )\).
CAIE P1 2021 November Q7
7
  1. Show that the equation \(\frac { \tan x + \cos x } { \tan x - \cos x } = k\), where \(k\) is a constant, can be expressed as $$( k + 1 ) \sin ^ { 2 } x + ( k - 1 ) \sin x - ( k + 1 ) = 0$$
  2. Hence solve the equation \(\frac { \tan x + \cos x } { \tan x - \cos x } = 4\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
CAIE P1 2021 November Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{cba7391d-2cf7-483c-9a21-bc9fd49860ee-12_570_961_260_591} The diagram shows the curves with equations \(y = x ^ { - \frac { 1 } { 2 } }\) and \(y = \frac { 5 } { 2 } - x ^ { \frac { 1 } { 2 } }\). The curves intersect at the points \(A \left( \frac { 1 } { 4 } , 2 \right)\) and \(B \left( 4 , \frac { 1 } { 2 } \right)\).
  1. Find the area of the region between the two curves.
  2. The normal to the curve \(y = x ^ { - \frac { 1 } { 2 } }\) at the point \(( 1,1 )\) intersects the \(y\)-axis at the point \(( 0 , p )\). Find the value of \(p\).