Questions P1 (1374 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure 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 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 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 Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
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 2006 November Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{dd2cb0ec-5df9-4d99-9e15-5ae1f1c07b96-4_387_903_799_623} The diagram shows an open container constructed out of \(200 \mathrm {~cm} ^ { 2 }\) of cardboard. The two vertical end pieces are isosceles triangles with sides \(5 x \mathrm {~cm} , 5 x \mathrm {~cm}\) and \(8 x \mathrm {~cm}\), and the two side pieces are rectangles of length \(y \mathrm {~cm}\) and width \(5 x \mathrm {~cm}\), as shown. The open top is a horizontal rectangle.
  1. Show that \(y = \frac { 200 - 24 x ^ { 2 } } { 10 x }\).
  2. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the container is given by \(V = 240 x - 28.8 x ^ { 3 }\). Given that \(x\) can vary,
  3. find the value of \(x\) for which \(V\) has a stationary value,
  4. determine whether it is a maximum or a minimum stationary value.
CAIE P1 2006 November Q10
10 The function f is defined by \(\mathrm { f } : x \mapsto x ^ { 2 } - 3 x\) for \(x \in \mathbb { R }\).
  1. Find the set of values of \(x\) for which \(\mathrm { f } ( x ) > 4\).
  2. Express \(\mathrm { f } ( x )\) in the form \(( x - a ) ^ { 2 } - b\), stating the values of \(a\) and \(b\).
  3. Write down the range of f .
  4. State, with a reason, whether f has an inverse. The function g is defined by \(\mathrm { g } : x \mapsto x - 3 \sqrt { } x\) for \(x \geqslant 0\).
  5. Solve the equation \(\mathrm { g } ( x ) = 10\).
CAIE P1 2007 November Q1
1 Determine the set of values of the constant \(k\) for which the line \(y = 4 x + k\) does not intersect the curve \(y = x ^ { 2 }\).
CAIE P1 2007 November Q2
2 Find the area of the region enclosed by the curve \(y = 2 \sqrt { } x\), the \(x\)-axis and the lines \(x = 1\) and \(x = 4\).
CAIE P1 2007 November Q3
3
  1. Find the first three terms in the expansion of \(( 2 + u ) ^ { 5 }\) in ascending powers of \(u\).
  2. Use the substitution \(u = x + x ^ { 2 }\) in your answer to part (i) to find the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 2 + x + x ^ { 2 } \right) ^ { 5 }\).
CAIE P1 2007 November Q4
4 The 1st term of an arithmetic progression is \(a\) and the common difference is \(d\), where \(d \neq 0\).
  1. Write down expressions, in terms of \(a\) and \(d\), for the 5th term and the 15th term. The 1st term, the 5th term and the 15th term of the arithmetic progression are the first three terms of a geometric progression.
  2. Show that \(3 a = 8 d\).
  3. Find the common ratio of the geometric progression.
CAIE P1 2007 November Q5
5
  1. Show that the equation \(3 \sin x \tan x = 8\) can be written as \(3 \cos ^ { 2 } x + 8 \cos x - 3 = 0\).
  2. Hence solve the equation \(3 \sin x \tan x = 8\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
CAIE P1 2007 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{e753f588-97bc-4c6a-a82b-7b6a6d0cadc4-2_627_748_1685_699} The three points \(A ( 3,8 ) , B ( 6,2 )\) and \(C ( 10,2 )\) are shown in the diagram. The point \(D\) is such that the line \(D A\) is perpendicular to \(A B\) and \(D C\) is parallel to \(A B\). Calculate the coordinates of \(D\).
CAIE P1 2007 November Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{e753f588-97bc-4c6a-a82b-7b6a6d0cadc4-3_579_659_269_744} In the diagram, \(A B\) is an arc of a circle, centre \(O\) and radius \(r \mathrm {~cm}\), and angle \(A O B = \theta\) radians. The point \(X\) lies on \(O B\) and \(A X\) is perpendicular to \(O B\).
  1. Show that the area, \(A \mathrm {~cm} ^ { 2 }\), of the shaded region \(A X B\) is given by $$A = \frac { 1 } { 2 } r ^ { 2 } ( \theta - \sin \theta \cos \theta ) .$$
  2. In the case where \(r = 12\) and \(\theta = \frac { 1 } { 6 } \pi\), find the perimeter of the shaded region \(A X B\), leaving your answer in terms of \(\sqrt { } 3\) and \(\pi\).
CAIE P1 2007 November Q8
8 The equation of a curve is \(y = ( 2 x - 3 ) ^ { 3 } - 6 x\).
  1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) in terms of \(x\).
  2. Find the \(x\)-coordinates of the two stationary points and determine the nature of each stationary point.
CAIE P1 2007 November Q9
9 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 - x\) and the point \(P ( 2,9 )\) lies on the curve. The normal to the curve at \(P\) meets the curve again at \(Q\). Find
  1. the equation of the curve,
  2. the equation of the normal to the curve at \(P\),
  3. the coordinates of \(Q\).
CAIE P1 2007 November Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{e753f588-97bc-4c6a-a82b-7b6a6d0cadc4-4_597_693_274_726} The diagram shows a cube \(O A B C D E F G\) in which the length of each side is 4 units. The unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(\overrightarrow { O A } , \overrightarrow { O C }\) and \(\overrightarrow { O D }\) respectively. The mid-points of \(O A\) and \(D G\) are \(P\) and \(Q\) respectively and \(R\) is the centre of the square face \(A B F E\).
  1. Express each of the vectors \(\overrightarrow { P R }\) and \(\overrightarrow { P Q }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
  2. Use a scalar product to find angle \(Q P R\).
  3. Find the perimeter of triangle \(P Q R\), giving your answer correct to 1 decimal place.
CAIE P1 2007 November Q11
11 The function f is defined by \(\mathrm { f } : x \mapsto 2 x ^ { 2 } - 8 x + 11\) for \(x \in \mathbb { R }\).
  1. Express \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a\), \(b\) and \(c\) are constants.
  2. State the range of f .
  3. Explain why f does not have an inverse. The function g is defined by \(\mathrm { g } : x \mapsto 2 x ^ { 2 } - 8 x + 11\) for \(x \leqslant A\), where \(A\) is a constant.
  4. State the largest value of \(A\) for which g has an inverse.
  5. When \(A\) has this value, obtain an expression, in terms of \(x\), for \(\mathrm { g } ^ { - 1 } ( x )\) and state the range of \(\mathrm { g } ^ { - 1 }\).
CAIE P1 2008 November Q1
1 Find the value of the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( \frac { x } { 2 } + \frac { 2 } { x } \right) ^ { 6 }\).
CAIE P1 2008 November Q2
2 Prove the identity $$\frac { 1 + \sin x } { \cos x } + \frac { \cos x } { 1 + \sin x } \equiv \frac { 2 } { \cos x }$$
CAIE P1 2008 November Q3
3 The first term of an arithmetic progression is 6 and the fifth term is 12 . The progression has \(n\) terms and the sum of all the terms is 90 . Find the value of \(n\).
CAIE P1 2008 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{08729aab-586b-4210-94c9-77b1f6b1d873-2_558_1488_863_331} The diagram shows a semicircular prism with a horizontal rectangular base \(A B C D\). The vertical ends \(A E D\) and \(B F C\) are semicircles of radius 6 cm . The length of the prism is 20 cm . The mid-point of \(A D\) is the origin \(O\), the mid-point of \(B C\) is \(M\) and the mid-point of \(D C\) is \(N\). The points \(E\) and \(F\) are the highest points of the semicircular ends of the prism. The point \(P\) lies on \(E F\) such that \(E P = 8 \mathrm {~cm}\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O D , O M\) and \(O E\) respectively.
  1. Express each of the vectors \(\overrightarrow { P A }\) and \(\overrightarrow { P N }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
  2. Use a scalar product to calculate angle \(A P N\).
CAIE P1 2008 November Q5
5 The function f is such that \(\mathrm { f } ( x ) = a - b \cos x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\), where \(a\) and \(b\) are positive constants. The maximum value of \(\mathrm { f } ( x )\) is 10 and the minimum value is - 2 .
  1. Find the values of \(a\) and \(b\).
  2. Solve the equation \(\mathrm { f } ( x ) = 0\).
  3. Sketch the graph of \(y = \mathrm { f } ( x )\).
CAIE P1 2008 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{08729aab-586b-4210-94c9-77b1f6b1d873-3_597_417_274_865} In the diagram, the circle has centre \(O\) and radius 5 cm . The points \(P\) and \(Q\) lie on the circle, and the arc length \(P Q\) is 9 cm . The tangents to the circle at \(P\) and \(Q\) meet at the point \(T\). Calculate
  1. angle \(P O Q\) in radians,
  2. the length of \(P T\),
  3. the area of the shaded region.
CAIE P1 2008 November Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{08729aab-586b-4210-94c9-77b1f6b1d873-3_385_360_1379_561}
\includegraphics[max width=\textwidth, alt={}, center]{08729aab-586b-4210-94c9-77b1f6b1d873-3_364_369_1379_1219} A wire, 80 cm long, is cut into two pieces. One piece is bent to form a square of side \(x \mathrm {~cm}\) and the other piece is bent to form a circle of radius \(r \mathrm {~cm}\) (see diagram). The total area of the square and the circle is \(A \mathrm {~cm} ^ { 2 }\).
  1. Show that \(A = \frac { ( \pi + 4 ) x ^ { 2 } - 160 x + 1600 } { \pi }\).
  2. Given that \(x\) and \(r\) can vary, find the value of \(x\) for which \(A\) has a stationary value.
CAIE P1 2008 November Q8
8 The equation of a curve is \(y = 5 - \frac { 8 } { x }\).
  1. Show that the equation of the normal to the curve at the point \(P ( 2,1 )\) is \(2 y + x = 4\). This normal meets the curve again at the point \(Q\).
  2. Find the coordinates of \(Q\).
  3. Find the length of \(P Q\).
CAIE P1 2008 November Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{08729aab-586b-4210-94c9-77b1f6b1d873-4_719_670_264_735} The diagram shows the curve \(y = \sqrt { } ( 3 x + 1 )\) and the points \(P ( 0,1 )\) and \(Q ( 1,2 )\) on the curve. The shaded region is bounded by the curve, the \(y\)-axis and the line \(y = 2\).
  1. Find the area of the shaded region.
  2. Find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Tangents are drawn to the curve at the points \(P\) and \(Q\).
  3. Find the acute angle, in degrees correct to 1 decimal place, between the two tangents.
CAIE P1 2008 November Q10
10 The function f is defined by $$\mathrm { f } : x \mapsto 3 x - 2 \text { for } x \in \mathbb { R } .$$
  1. Sketch, in a single diagram, the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), making clear the relationship between the two graphs. The function g is defined by $$\mathrm { g } : x \mapsto 6 x - x ^ { 2 } \text { for } x \in \mathbb { R }$$
  2. Express \(\operatorname { gf } ( x )\) in terms of \(x\), and hence show that the maximum value of \(\operatorname { gf } ( x )\) is 9 . The function h is defined by $$\mathrm { h } : x \mapsto 6 x - x ^ { 2 } \text { for } x \geqslant 3$$
  3. Express \(6 x - x ^ { 2 }\) in the form \(a - ( x - b ) ^ { 2 }\), where \(a\) and \(b\) are positive constants.
  4. Express \(\mathrm { h } ^ { - 1 } ( x )\) in terms of \(x\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE P1 2009 November Q1
1 Solve the equation \(3 \tan \left( 2 x + 15 ^ { \circ } \right) = 4\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
CAIE P1 2009 November Q2
2 The equation of a curve is \(y = 3 \cos 2 x\). The equation of a line is \(x + 2 y = \pi\). On the same diagram, sketch the curve and the line for \(0 \leqslant x \leqslant \pi\).