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 2024 June Q1
1 Find the coefficient of \(x ^ { 2 }\) in the expansion of $$( 2 - 5 x ) ( 1 + 3 x ) ^ { 10 }$$
CAIE P1 2024 June Q2
2

  1. \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-04_582_922_335_575} The diagram shows the curve \(y = k \cos \left( x - \frac { 1 } { 6 } \pi \right)\) where \(k\) is a positive constant and \(x\) is measured in radians. The curve crosses the \(x\)-axis at point \(A\) and \(B\) is a minimum point. Find the coordinates of \(A\) and \(B\).
  2. Find the exact value of \(t\) that satisfies the equation $$3 \sin ^ { - 1 } ( 3 t ) + 2 \cos ^ { - 1 } \left( \frac { 1 } { 2 } \sqrt { 2 } \right) = \pi .$$ \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-04_2718_33_141_2013}
CAIE P1 2024 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-05_483_561_287_753} The diagram shows a sector of a circle with centre \(C\). The radii \(C A\) and \(C B\) each have length \(r \mathrm {~cm}\) and the size of the reflex angle \(A C B\) is \(\theta\) radians. The sector, shaded in the diagram, has a perimeter of 65 cm and an area of \(225 \mathrm {~cm} ^ { 2 }\).
  1. Find the values of \(r\) and \(\theta\).
  2. Find the area of triangle \(A C B\).
CAIE P1 2024 June Q4
4
  1. Show that the equation \(\cos \theta ( 7 \tan \theta - 5 \cos \theta ) = 1\) can be written in the form \(a \sin ^ { 2 } \theta + b \sin \theta + c = 0\), where \(a , b\) and \(c\) are integers to be found.
  2. Hence solve the equation \(\cos 2 x ( 7 \tan 2 x - 5 \cos 2 x ) = 1\) for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
    \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-06_2718_35_141_2012}
    \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-07_2714_33_144_22}
CAIE P1 2024 June Q5
5 The equation of a curve is \(y = 2 x ^ { 2 } - \frac { 1 } { 2 x } + 3\).
  1. Find the coordinates of the stationary point.
  2. Determine the nature of the stationary point.
  3. For positive values of \(x\), determine whether the curve shows a function that is increasing, decreasing or neither. Give a reason for your answer.
CAIE P1 2024 June Q6
6 A curve passes through the point \(\left( \frac { 4 } { 5 } , - 3 \right)\) and is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { - 20 } { ( 5 x - 3 ) ^ { 2 } }\).
  1. Find the equation of the curve.
  2. The curve is transformed by a stretch in the \(x\)-direction with scale factor \(\frac { 1 } { 2 }\) followed by a translation of \(\binom { 2 } { 10 }\). Find the equation of the new curve.
    \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-08_2716_38_143_2009}
CAIE P1 2024 June Q7
7 The first term of an arithmetic progression is 1.5 and the sum of the first ten terms is 127.5 .
  1. Find the common difference.
  2. Find the sum of all the terms of the arithmetic progression whose values are between 25 and 100 .
CAIE P1 2024 June Q8
8 A circle with equation \(x ^ { 2 } + y ^ { 2 } - 6 x + 2 y - 15 = 0\) meets the \(y\)-axis at the points \(A\) and \(B\). The tangents to the circle at \(A\) and \(B\) meet at the point \(P\). Find the coordinates of \(P\).
\includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-10_71_1659_466_244}
\includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-10_2723_37_136_2010}
CAIE P1 2024 June Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-12_764_967_292_555} The diagram shows the curve with equation \(y = \sqrt { 2 x ^ { 3 } + 10 }\).
  1. Find the equation of the tangent to the curve at the point where \(x = 3\). Give your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
    \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-12_2716_35_141_2013}
  2. The region shaded in the diagram is enclosed by the curve and the straight lines \(x = 1 , x = 3\) and \(y = 0\). Find the volume of the solid obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
CAIE P1 2024 June Q10
10 The geometric progression \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) has first term 2 and common ratio \(r\) where \(r > 0\). It is given that \(\frac { 9 } { 2 } a _ { 5 } + 7 a _ { 3 } = 8\).
  1. Find the value of \(r\).
  2. Find the sum of the first 20 terms of the geometric progression. Give your answer correct to 4 significant figures.
    \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-14_2725_42_134_2008}
  3. Find the sum to infinity of the progression \(a _ { 2 } , a _ { 5 } , a _ { 8 } , \ldots\).
CAIE P1 2024 June Q11
11 The function f is defined by \(\mathrm { f } ( x ) = 10 + 6 x - x ^ { 2 }\) for \(x \in \mathbb { R }\).
  1. By completing the square, find the range of f .
    \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-16_2715_37_143_2010} The function g is defined by \(\mathrm { g } ( x ) = 4 x + k\) for \(x \in \mathbb { R }\) where \(k\) is a constant.
  2. It is given that the graph of \(y = \mathrm { g } ^ { - 1 } \mathrm { f } ( x )\) meets the graph of \(y = \mathrm { g } ( x )\) at a single point \(P\). Determine the coordinates of \(P\).
    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]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-18_2715_35_143_2012}
CAIE P1 2020 March Q1
1 The function f is defined by \(\mathrm { f } ( x ) = \frac { 1 } { 3 x + 2 } + x ^ { 2 }\) for \(x < - 1\).
Determine whether f is an increasing function, a decreasing function or neither.
CAIE P1 2020 March Q2
2 The graph of \(y = \mathrm { f } ( x )\) is transformed to the graph of \(y = 1 + \mathrm { f } \left( \frac { 1 } { 2 } x \right)\).
Describe fully the two single transformations which have been combined to give the resulting transformation.
CAIE P1 2020 March Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{01b98496-a717-4c68-8489-42d2203b700f-04_700_401_260_870} The diagram shows part of the curve with equation \(y = x ^ { 2 } + 1\). The shaded region enclosed by the curve, the \(y\)-axis and the line \(y = 5\) is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis. Find the volume obtained.
CAIE P1 2020 March Q4
4 A curve has equation \(y = x ^ { 2 } - 2 x - 3\). A point is moving along the curve in such a way that at \(P\) the \(y\)-coordinate is increasing at 4 units per second and the \(x\)-coordinate is increasing at 6 units per second. Find the \(x\)-coordinate of \(P\).
CAIE P1 2020 March Q5
5 Solve the equation $$\frac { \tan \theta + 3 \sin \theta + 2 } { \tan \theta - 3 \sin \theta + 1 } = 2$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 90 ^ { \circ }\).
CAIE P1 2020 March Q6
6 The coefficient of \(\frac { 1 } { x }\) in the expansion of \(\left( 2 x + \frac { a } { x ^ { 2 } } \right) ^ { 5 }\) is 720 .
  1. Find the possible values of the constant \(a\).
  2. Hence find the coefficient of \(\frac { 1 } { x ^ { 7 } }\) in the expansion.
CAIE P1 2020 March Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{01b98496-a717-4c68-8489-42d2203b700f-08_574_689_260_726} The diagram shows a sector \(A O B\) which is part of a circle with centre \(O\) and radius 6 cm and with angle \(A O B = 0.8\) radians. The point \(C\) on \(O B\) is such that \(A C\) is perpendicular to \(O B\). The arc \(C D\) is part of a circle with centre \(O\), where \(D\) lies on \(O A\). Find the area of the shaded region.
CAIE P1 2020 March Q8
8 A woman's basic salary for her first year with a particular company is \(
) 30000\( and at the end of the year she also gets a bonus of \)\\( 600\).
  1. For her first year, express her bonus as a percentage of her basic salary.
    At the end of each complete year, the woman's basic salary will increase by \(3 \%\) and her bonus will increase by \(
    ) 100$.
  2. Express the bonus she will be paid at the end of her 24th year as a percentage of the basic salary paid during that year.
CAIE P1 2020 March Q9
9
  1. Express \(2 x ^ { 2 } + 12 x + 11\) in the form \(2 ( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
    The function f is defined by \(\mathrm { f } ( x ) = 2 x ^ { 2 } + 12 x + 11\) for \(x \leqslant - 4\).
  2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
    The function g is defined by \(\mathrm { g } ( x ) = 2 x - 3\) for \(x \leqslant k\).
  3. For the case where \(k = - 1\), solve the equation \(\operatorname { fg } ( x ) = 193\).
  4. State the largest value of \(k\) possible for the composition fg to be defined.
CAIE P1 2020 March Q10
10 The gradient of a curve at the point \(( x , y )\) is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 ( x + 3 ) ^ { \frac { 1 } { 2 } } - x\). The curve has a stationary point at \(( a , 14 )\), where \(a\) is a positive constant.
  1. Find the value of \(a\).
  2. Determine the nature of the stationary point.
  3. Find the equation of the curve.
CAIE P1 2020 March Q11
11
  1. Solve the equation \(3 \tan ^ { 2 } x - 5 \tan x - 2 = 0\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
  2. Find the set of values of \(k\) for which the equation \(3 \tan ^ { 2 } x - 5 \tan x + k = 0\) has no solutions.
  3. For the equation \(3 \tan ^ { 2 } x - 5 \tan x + k = 0\), state the value of \(k\) for which there are three solutions in the interval \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\), and find these solutions.
CAIE P1 2020 March Q12
12 A diameter of a circle \(C _ { 1 }\) has end-points at \(( - 3 , - 5 )\) and \(( 7,3 )\).
  1. Find an equation of the circle \(C _ { 1 }\).
    \includegraphics[max width=\textwidth, alt={}, center]{01b98496-a717-4c68-8489-42d2203b700f-16_618_846_1062_644} The circle \(C _ { 1 }\) is translated by \(\binom { 8 } { 4 }\) to give circle \(C _ { 2 }\), as shown in the diagram.
  2. Find an equation of the circle \(C _ { 2 }\).
    The two circles intersect at points \(R\) and \(S\).
  3. Show that the equation of the line \(R S\) is \(y = - 2 x + 13\).
  4. Hence show that the \(x\)-coordinates of \(R\) and \(S\) satisfy the equation \(5 x ^ { 2 } - 60 x + 159 = 0\).
    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 March Q1
1
  1. Find the first three terms in the expansion, in ascending powers of \(x\), of \(( 1 + x ) ^ { 5 }\).
  2. Find the first three terms in the expansion, in ascending powers of \(x\), of \(( 1 - 2 x ) ^ { 6 }\).
  3. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 1 + x ) ^ { 5 } ( 1 - 2 x ) ^ { 6 }\).
CAIE P1 2021 March Q2
2 By using a suitable substitution, solve the equation $$( 2 x - 3 ) ^ { 2 } - \frac { 4 } { ( 2 x - 3 ) ^ { 2 } } - 3 = 0$$