Questions (30179 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 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 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
Sort by: Default | Easiest first | Hardest first
CAIE P3 2015 November Q4
8 marks Standard +0.3
4 A curve has parametric equations $$x = t ^ { 2 } + 3 t + 1 , \quad y = t ^ { 4 } + 1$$ The point \(P\) on the curve has parameter \(p\). It is given that the gradient of the curve at \(P\) is 4 .
  1. Show that \(p = \sqrt [ 3 ] { } ( 2 p + 3 )\).
  2. Verify by calculation that the value of \(p\) lies between 1.8 and 2.0.
  3. Use an iterative formula based on the equation in part (i) to find the value of \(p\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2015 November Q5
8 marks Standard +0.8
5 Use the substitution \(u = 4 - 3 \cos x\) to find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { 9 \sin 2 x } { \sqrt { ( 4 - 3 \cos x ) } } \mathrm { d } x\).
CAIE P3 2015 November Q6
8 marks Standard +0.8
6 The angles \(A\) and \(B\) are such that $$\sin \left( A + 45 ^ { \circ } \right) = ( 2 \sqrt { } 2 ) \cos A \quad \text { and } \quad 4 \sec ^ { 2 } B + 5 = 12 \tan B$$ Without using a calculator, find the exact value of \(\tan ( A - B )\).
CAIE P3 2015 November Q7
10 marks Standard +0.3
7
  1. Show that ( \(x + 1\) ) is a factor of \(4 x ^ { 3 } - x ^ { 2 } - 11 x - 6\).
  2. Find \(\int \frac { 4 x ^ { 2 } + 9 x - 1 } { 4 x ^ { 3 } - x ^ { 2 } - 11 x - 6 } \mathrm {~d} x\).
CAIE P3 2015 November Q8
10 marks Standard +0.3
8 A plane has equation \(4 x - y + 5 z = 39\). A straight line is parallel to the vector \(\mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k }\) and passes through the point \(A ( 0,2 , - 8 )\). The line meets the plane at the point \(B\).
  1. Find the coordinates of \(B\).
  2. Find the acute angle between the line and the plane.
  3. The point \(C\) lies on the line and is such that the distance between \(C\) and \(B\) is twice the distance between \(A\) and \(B\). Find the coordinates of each of the possible positions of the point \(C\).
CAIE P3 2015 November Q9
10 marks Standard +0.3
9
  1. It is given that \(( 1 + 3 \mathrm { i } ) w = 2 + 4 \mathrm { i }\). Showing all necessary working, prove that the exact value of \(\left| w ^ { 2 } \right|\) is 2 and find \(\arg \left( w ^ { 2 } \right)\) correct to 3 significant figures.
  2. On a single Argand diagram sketch the loci \(| z | = 5\) and \(| z - 5 | = | z |\). Hence determine the complex numbers represented by points common to both loci, giving each answer in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\).
CAIE P3 2015 November Q10
10 marks Moderate -0.8
10 Naturalists are managing a wildlife reserve to increase the number of plants of a rare species. The number of plants at time \(t\) years is denoted by \(N\), where \(N\) is treated as a continuous variable.
  1. It is given that the rate of increase of \(N\) with respect to \(t\) is proportional to ( \(N - 150\) ). Write down a differential equation relating \(N , t\) and a constant of proportionality.
  2. Initially, when \(t = 0\), the number of plants was 650 . It was noted that, at a time when there were 900 plants, the number of plants was increasing at a rate of 60 per year. Express \(N\) in terms of \(t\).
  3. The naturalists had a target of increasing the number of plants from 650 to 2000 within 15 years. Will this target be met?
CAIE P3 2016 November Q1
3 marks Moderate -0.5
1 Solve the equation \(\frac { 3 ^ { x } + 2 } { 3 ^ { x } - 2 } = 8\), giving your answer correct to 3 decimal places.
CAIE P3 2016 November Q2
4 marks Moderate -0.3
2 Expand \(( 2 - x ) ( 1 + 2 x ) ^ { - \frac { 3 } { 2 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
CAIE P3 2016 November Q3
5 marks Standard +0.3
3 Express the equation \(\sec \theta = 3 \cos \theta + \tan \theta\) as a quadratic equation in \(\sin \theta\). Hence solve this equation for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\).
CAIE P3 2016 November Q4
7 marks Standard +0.8
4 The equation of a curve is \(x y ( x - 6 y ) = 9 a ^ { 3 }\), where \(a\) is a non-zero constant. Show that there is only one point on the curve at which the tangent is parallel to the \(x\)-axis, and find the coordinates of this point.
CAIE P3 2016 November Q5
8 marks Standard +0.8
5
  1. Prove the identity \(\tan 2 \theta - \tan \theta \equiv \tan \theta \sec 2 \theta\).
  2. Hence show that \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \tan \theta \sec 2 \theta \mathrm {~d} \theta = \frac { 1 } { 2 } \ln \frac { 3 } { 2 }\).
CAIE P3 2016 November Q6
9 marks Standard +0.3
6
  1. By sketching a suitable pair of graphs, show that the equation $$\operatorname { cosec } \frac { 1 } { 2 } x = \frac { 1 } { 3 } x + 1$$ has one root in the interval \(0 < x \leqslant \pi\).
  2. Show by calculation that this root lies between 1.4 and 1.6.
  3. Show that, if a sequence of values in the interval \(0 < x \leqslant \pi\) given by the iterative formula $$x _ { n + 1 } = 2 \sin ^ { - 1 } \left( \frac { 3 } { x _ { n } + 3 } \right)$$ converges, then it converges to the root of the equation in part (i).
  4. Use this iterative formula to calculate the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
CAIE P3 2016 November Q7
9 marks Standard +0.8
7 \includegraphics[max width=\textwidth, alt={}, center]{f4614578-f5f6-4283-8185-8b5598ad91d5-3_416_679_258_731} The diagram shows part of the curve \(y = \left( 2 x - x ^ { 2 } \right) \mathrm { e } ^ { \frac { 1 } { 2 } x }\) and its maximum point \(M\).
  1. Find the exact \(x\)-coordinate of \(M\).
  2. Find the exact value of the area of the shaded region bounded by the curve and the positive \(x\)-axis.
CAIE P3 2016 November Q8
9 marks Standard +0.3
8 Two planes have equations \(3 x + y - z = 2\) and \(x - y + 2 z = 3\).
  1. Show that the planes are perpendicular.
  2. Find a vector equation for the line of intersection of the two planes.
CAIE P3 2016 November Q10
11 marks Standard +0.8
10 A large field of area \(4 \mathrm {~km} ^ { 2 }\) is becoming infected with a soil disease. At time \(t\) years the area infected is \(x \mathrm {~km} ^ { 2 }\) and the rate of growth of the infected area is given by the differential equation \(\frac { \mathrm { d } x } { \mathrm {~d} t } = k x ( 4 - x )\), where \(k\) is a positive constant. It is given that when \(t = 0 , x = 0.4\) and that when \(t = 2 , x = 2\).
  1. Solve the differential equation and show that \(k = \frac { 1 } { 4 } \ln 3\).
  2. Find the value of \(t\) when \(90 \%\) of the area of the field is infected.
CAIE P3 2016 November Q7
9 marks Standard +0.8
7 \includegraphics[max width=\textwidth, alt={}, center]{84df6b9a-6118-44a2-9c18-512039ded4fd-3_416_677_258_733} The diagram shows part of the curve \(y = \left( 2 x - x ^ { 2 } \right) \mathrm { e } ^ { \frac { 1 } { 2 } x }\) and its maximum point \(M\).
  1. Find the exact \(x\)-coordinate of \(M\).
  2. Find the exact value of the area of the shaded region bounded by the curve and the positive \(x\)-axis.
CAIE P3 2016 November Q1
3 marks Moderate -0.5
1 It is given that \(z = \ln ( y + 2 ) - \ln ( y + 1 )\). Express \(y\) in terms of \(z\).
CAIE P3 2016 November Q2
4 marks Standard +0.3
2 The equation of a curve is \(y = \frac { \sin x } { 1 + \cos x }\), for \(- \pi < x < \pi\). Show that the gradient of the curve is positive for all \(x\) in the given interval.
CAIE P3 2016 November Q3
6 marks Standard +0.3
3 Express the equation \(\cot 2 \theta = 1 + \tan \theta\) as a quadratic equation in \(\tan \theta\). Hence solve this equation for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P3 2016 November Q4
7 marks Standard +0.3
4 The polynomial \(4 x ^ { 4 } + a x ^ { 2 } + 11 x + b\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(\mathrm { p } ( x )\) is divisible by \(x ^ { 2 } - x + 2\).
  1. Find the values of \(a\) and \(b\).
  2. When \(a\) and \(b\) have these values, find the real roots of the equation \(\mathrm { p } ( x ) = 0\).
CAIE P3 2016 November Q5
7 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{ccadf73b-16f5-463a-8f69-1394839d5325-2_346_437_1155_854} The diagram shows a variable point \(P\) with coordinates \(( x , y )\) and the point \(N\) which is the foot of the perpendicular from \(P\) to the \(x\)-axis. \(P\) moves on a curve such that, for all \(x \geqslant 0\), the gradient of the curve is equal in value to the area of the triangle \(O P N\), where \(O\) is the origin.
  1. State a differential equation satisfied by \(x\) and \(y\). The point with coordinates \(( 0,2 )\) lies on the curve.
  2. Solve the differential equation to obtain the equation of the curve, expressing \(y\) in terms of \(x\).
  3. Sketch the curve.
CAIE P3 2016 November Q6
9 marks Standard +0.3
6 Let \(I = \int _ { 1 } ^ { 4 } \frac { ( \sqrt { } x ) - 1 } { 2 ( x + \sqrt { } x ) } \mathrm { d } x\).
  1. Using the substitution \(u = \sqrt { } x\), show that \(I = \int _ { 1 } ^ { 2 } \frac { u - 1 } { u + 1 } \mathrm {~d} u\).
  2. Hence show that \(I = 1 + \ln \frac { 4 } { 9 }\).
CAIE P3 2016 November Q8
10 marks Standard +0.3
8 Let \(\mathrm { f } ( x ) = \frac { 3 x ^ { 2 } + x + 6 } { ( x + 2 ) \left( x ^ { 2 } + 4 \right) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
CAIE P3 2016 November Q9
10 marks Standard +0.8
9 \includegraphics[max width=\textwidth, alt={}, center]{ccadf73b-16f5-463a-8f69-1394839d5325-3_481_483_1434_831} The diagram shows the curves \(y = x \cos x\) and \(y = \frac { k } { x }\), where \(k\) is a constant, for \(0 < x \leqslant \frac { 1 } { 2 } \pi\). The curves touch at the point where \(x = a\).
  1. Show that \(a\) satisfies the equation \(\tan a = \frac { 2 } { a }\).
  2. Use the iterative formula \(a _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 2 } { a _ { n } } \right)\) to determine \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
  3. Hence find the value of \(k\) correct to 2 decimal places.