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 2016 June Q3
Moderate -0.8
3 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { 2 } + \frac { k } { x ^ { 3 } }\) and passes through the point \(P ( 1,9 )\). The gradient of the curve at \(P\) is 2 .
  1. Find the value of the constant \(k\).
  2. Find the equation of the curve.
CAIE P1 2016 June Q4
Standard +0.3
4 The 1st, 3rd and 13th terms of an arithmetic progression are also the 1st, 2nd and 3rd terms respectively of a geometric progression. The first term of each progression is 3 . Find the common difference of the arithmetic progression and the common ratio of the geometric progression.
CAIE P1 2016 June Q5
Standard +0.3
5 A curve has equation \(y = 8 x + ( 2 x - 1 ) ^ { - 1 }\). Find the values of \(x\) at which the curve has a stationary point and determine the nature of each stationary point, justifying your answers.
CAIE P1 2016 June Q6
Standard +0.8
6
\includegraphics[max width=\textwidth, alt={}, center]{8c358a10-a3e1-47b5-ae62-30ba6b76c167-3_655_1011_255_566} The diagram shows triangle \(A B C\) where \(A B = 5 \mathrm {~cm} , A C = 4 \mathrm {~cm}\) and \(B C = 3 \mathrm {~cm}\). Three circles with centres at \(A , B\) and \(C\) have radii \(3 \mathrm {~cm} , 2 \mathrm {~cm}\) and 1 cm respectively. The circles touch each other at points \(E , F\) and \(G\), lying on \(A B , A C\) and \(B C\) respectively. Find the area of the shaded region \(E F G\).
CAIE P1 2016 June Q7
Standard +0.8
7 The point \(P ( x , y )\) is moving along the curve \(y = x ^ { 2 } - \frac { 10 } { 3 } x ^ { \frac { 3 } { 2 } } + 5 x\) in such a way that the rate of change of \(y\) is constant. Find the values of \(x\) at the points at which the rate of change of \(x\) is equal to half the rate of change of \(y\).
CAIE P1 2016 June Q8
Standard +0.3
8
  1. Show that \(3 \sin x \tan x - \cos x + 1 = 0\) can be written as a quadratic equation in \(\cos x\) and hence solve the equation \(3 \sin x \tan x - \cos x + 1 = 0\) for \(0 \leqslant x \leqslant \pi\).
  2. Find the solutions to the equation \(3 \sin 2 x \tan 2 x - \cos 2 x + 1 = 0\) for \(0 \leqslant x \leqslant \pi\).
CAIE P1 2016 June Q9
Moderate -0.8
9 The position vectors of \(A , B\) and \(C\) relative to an origin \(O\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 2 \\ 3 \\ - 4 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { c } 1 \\ 5 \\ p \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 5 \\ 0 \\ 2 \end{array} \right) ,$$ where \(p\) is a constant.
  1. Find the value of \(p\) for which the lengths of \(A B\) and \(C B\) are equal.
  2. For the case where \(p = 1\), use a scalar product to find angle \(A B C\).
CAIE P1 2016 June Q10
Standard +0.3
10 The function f is such that \(\mathrm { f } ( x ) = 2 x + 3\) for \(x \geqslant 0\). The function g is such that \(\mathrm { g } ( x ) = a x ^ { 2 } + b\) for \(x \leqslant q\), where \(a , b\) and \(q\) are constants. The function fg is such that \(\operatorname { fg } ( x ) = 6 x ^ { 2 } - 21\) for \(x \leqslant q\).
  1. Find the values of \(a\) and \(b\).
  2. Find the greatest possible value of \(q\). It is now given that \(q = - 3\).
  3. Find the range of fg.
  4. Find an expression for \(( \mathrm { fg } ) ^ { - 1 } ( x )\) and state the domain of \(( \mathrm { fg } ) ^ { - 1 }\).
CAIE P1 2016 June Q11
Standard +0.3
11 Triangle \(A B C\) has vertices at \(A ( - 2 , - 1 ) , B ( 4,6 )\) and \(C ( 6 , - 3 )\).
  1. Show that triangle \(A B C\) is isosceles and find the exact area of this triangle.
  2. The point \(D\) is the point on \(A B\) such that \(C D\) is perpendicular to \(A B\). Calculate the \(x\)-coordinate of \(D\).
CAIE P1 2017 June Q1
Moderate -0.8
1 The coefficients of \(x ^ { 2 }\) and \(x ^ { 3 }\) in the expansion of \(( 3 - 2 x ) ^ { 6 }\) are \(a\) and \(b\) respectively. Find the value of \(\frac { a } { b }\).
CAIE P1 2017 June Q2
Standard +0.3
2 Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r }
CAIE P1 2017 June Q3
Moderate -0.8
3
- 6
p \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { r } 2
- 6
- 7 \end{array} \right)$$ and angle \(A O B = 90 ^ { \circ }\).
  1. Find the value of \(p\).
    The point \(C\) is such that \(\overrightarrow { O C } = \frac { 2 } { 3 } \overrightarrow { O A }\).
  2. Find the unit vector in the direction of \(\overrightarrow { B C }\).
    3
  3. Prove the identity \(\frac { 1 + \cos \theta } { \sin \theta } + \frac { \sin \theta } { 1 + \cos \theta } \equiv \frac { 2 } { \sin \theta }\).
  4. Hence solve the equation \(\frac { 1 + \cos \theta } { \sin \theta } + \frac { \sin \theta } { 1 + \cos \theta } = \frac { 3 } { \cos \theta }\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P1 2017 June Q4
Moderate -0.8
4
  1. An arithmetic progression has a first term of 32, a 5th term of 22 and a last term of - 28 . Find the sum of all the terms in the progression.
  2. Each year a school allocates a sum of money for the library. The amount allocated each year increases by \(2.5 \%\) of the amount allocated the previous year. In 2005 the school allocated \(\\) 2000$. Find the total amount allocated in the years 2005 to 2014 inclusive.
CAIE P1 2017 June Q5
Moderate -0.8
5 The equation of a curve is \(y = 2 \cos x\).
  1. Sketch the graph of \(y = 2 \cos x\) for \(- \pi \leqslant x \leqslant \pi\), stating the coordinates of the point of intersection with the \(y\)-axis. Points \(P\) and \(Q\) lie on the curve and have \(x\)-coordinates of \(\frac { 1 } { 3 } \pi\) and \(\pi\) respectively.
  2. Find the length of \(P Q\) correct to 1 decimal place.
    The line through \(P\) and \(Q\) meets the \(x\)-axis at \(H ( h , 0 )\) and the \(y\)-axis at \(K ( 0 , k )\).
  3. Show that \(h = \frac { 5 } { 9 } \pi\) and find the value of \(k\).
CAIE P1 2017 June Q6
Standard +0.3
6 The horizontal base of a solid prism is an equilateral triangle of side \(x \mathrm {~cm}\). The sides of the prism are vertical. The height of the prism is \(h \mathrm {~cm}\) and the volume of the prism is \(2000 \mathrm {~cm} ^ { 3 }\).
  1. Express \(h\) in terms of \(x\) and show that the total surface area of the prism, \(A \mathrm {~cm} ^ { 2 }\), is given by $$A = \frac { \sqrt { } 3 } { 2 } x ^ { 2 } + \frac { 24000 } { \sqrt { } 3 } x ^ { - 1 }$$
  2. Given that \(x\) can vary, find the value of \(x\) for which \(A\) has a stationary value.
  3. Determine, showing all necessary working, the nature of this stationary value.
CAIE P1 2017 June Q7
Moderate -0.8
7 A curve for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 7 - x ^ { 2 } - 6 x\) passes through the point \(( 3 , - 10 )\).
  1. Find the equation of the curve.
  2. Express \(7 - x ^ { 2 } - 6 x\) in the form \(a - ( x + b ) ^ { 2 }\), where \(a\) and \(b\) are constants.
  3. Find the set of values of \(x\) for which the gradient of the curve is positive.
CAIE P1 2017 June Q8
Standard +0.8
8
\includegraphics[max width=\textwidth, alt={}, center]{028c7979-6b24-42d0-9857-c616a169b2b2-14_590_691_260_726} In the diagram, \(O A X B\) is a sector of a circle with centre \(O\) and radius 10 cm . The length of the chord \(A B\) is 12 cm . The line \(O X\) passes through \(M\), the mid-point of \(A B\), and \(O X\) is perpendicular to \(A B\). The shaded region is bounded by the chord \(A B\) and by the arc of a circle with centre \(X\) and radius \(X A\).
  1. Show that angle \(A X B\) is 2.498 radians, correct to 3 decimal places.
  2. Find the perimeter of the shaded region.
  3. Find the area of the shaded region.
CAIE P1 2017 June Q9
Standard +0.3
9 The function f is defined by \(\mathrm { f } : x \mapsto \frac { 2 } { 3 - 2 x }\) for \(x \in \mathbb { R } , x \neq \frac { 3 } { 2 }\).
  1. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
    The function g is defined by \(\mathrm { g } : x \mapsto 4 x + a\) for \(x \in \mathbb { R }\), where \(a\) is a constant.
  2. Find the value of \(a\) for which \(\operatorname { gf } ( - 1 ) = 3\).
  3. Find the possible values of \(a\) given that the equation \(\mathrm { f } ^ { - 1 } ( x ) = \mathrm { g } ^ { - 1 } ( x )\) has two equal roots.
CAIE P1 2017 June Q10
Standard +0.3
10
\includegraphics[max width=\textwidth, alt={}, center]{028c7979-6b24-42d0-9857-c616a169b2b2-18_510_410_260_863} The diagram shows part of the curve \(y = \frac { 4 } { 5 - 3 x }\).
  1. Find the equation of the normal to the curve at the point where \(x = 1\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
    The shaded region is bounded by the curve, the coordinate axes and the line \(x = 1\).
  2. Find, showing all necessary working, the volume obtained when this shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
CAIE P1 2017 June Q1
Moderate -0.8
1
  1. Find the coefficient of \(x\) in the expansion of \(\left( 2 x - \frac { 1 } { x } \right) ^ { 5 }\).
  2. Hence find the coefficient of \(x\) in the expansion of \(\left( 1 + 3 x ^ { 2 } \right) \left( 2 x - \frac { 1 } { x } \right) ^ { 5 }\).
CAIE P1 2017 June Q2
Standard +0.3
2 The point \(A\) has coordinates ( \(- 2,6\) ). The equation of the perpendicular bisector of the line \(A B\) is \(2 y = 3 x + 5\).
  1. Find the equation of \(A B\).
  2. Find the coordinates of \(B\).
CAIE P1 2017 June Q3
Standard +0.3
3
  1. Prove the identity \(\left( \frac { 1 } { \cos \theta } - \tan \theta \right) ^ { 2 } \equiv \frac { 1 - \sin \theta } { 1 + \sin \theta }\).
  2. Hence solve the equation \(\left( \frac { 1 } { \cos \theta } - \tan \theta \right) ^ { 2 } = \frac { 1 } { 2 }\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
    \includegraphics[max width=\textwidth, alt={}, center]{8a3f8707-67a4-4069-aba5-7e9496cb1748-06_572_460_258_845} The diagram shows a circle with radius \(r \mathrm {~cm}\) and centre \(O\). Points \(A\) and \(B\) lie on the circle and \(A B C D\) is a rectangle. Angle \(A O B = 2 \theta\) radians and \(A D = r \mathrm {~cm}\).
CAIE P1 2017 June Q5
Standard +0.3
5 A curve has equation \(y = 3 + \frac { 12 } { 2 - x }\).
  1. Find the equation of the tangent to the curve at the point where the curve crosses the \(x\)-axis.
  2. A point moves along the curve in such a way that the \(x\)-coordinate is increasing at a constant rate of 0.04 units per second. Find the rate of change of the \(y\)-coordinate when \(x = 4\).
CAIE P1 2017 June Q6
Standard +0.8
6
\includegraphics[max width=\textwidth, alt={}, center]{8a3f8707-67a4-4069-aba5-7e9496cb1748-10_588_583_260_781} The diagram shows the straight line \(x + y = 5\) intersecting the curve \(y = \frac { 4 } { x }\) at the points \(A ( 1,4 )\) and \(B ( 4,1 )\). Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
CAIE P1 2017 June Q7
Standard +0.3
7
  1. The first two terms of an arithmetic progression are 16 and 24. Find the least number of terms of the progression which must be taken for their sum to exceed 20000.
  2. A geometric progression has a first term of 6 and a sum to infinity of 18. A new geometric progression is formed by squaring each of the terms of the original progression. Find the sum to infinity of the new progression.