Questions — Edexcel (10514 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 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 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 PURE 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 PURE S1 S2 S3 S4 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 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 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
Edexcel C2 Q4
11 marks Standard +0.3
Given that \(\text{f}(x) = (2x^{\frac{1}{3}} - 3x^{-\frac{1}{2}})^2 + 5\), \(x > 0\),
  1. find, to 3 significant figures, the value of x for which f(x) = 5. [3]
  2. Show that f(x) may be written in the form \(Ax^{\frac{2}{3}} + \frac{B}{x} + C\), where A, B and C are constants to be found. [3]
  3. Hence evaluate \(\int_1^2 \text{f}(x) \, \text{dx}\). [5]
Edexcel C2 Q5
11 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows the cross-section ABCD of a chocolate bar, where AB, CD and AD are straight lines and M is the mid-point of AD. The length AD is 28 mm, and BC is an arc of a circle with centre M. Taking A as the origin, B, C and D have coordinates (7, 24), (21, 24) and (28, 0) respectively.
  1. Show that the length of BM is 25 mm. [1]
  2. Show that, to 3 significant figures, \(\angle BMC = 0.568\) radians. [3]
  3. Hence calculate, in mm², the area of the cross-section of the chocolate bar. [5]
Given that this chocolate bar has length 85 mm,
  1. calculate, to the nearest cm³, the volume of the bar. [2]
Edexcel C2 Q6
9 marks Standard +0.3
\includegraphics{figure_2} Figure 1 shows the curve with equation \(y = 5 + 2x - x^2\) and the line with equation \(y = 2\). The curve and the line intersect at the points A and B.
  1. Find the x-coordinates of A and B. [3]
The shaded region R is bounded by the curve and the line.
  1. Find the area of R. [6]
Edexcel C2 Q7
14 marks Standard +0.3
Find all the values of \(\theta\) in the interval \(0 \leq \theta < 360°\) for which
  1. \(\cos(\theta - 10°) = \cos 15°\), [3]
  2. \(\tan 2\theta = 0.4\), [5]
  3. \(2 \sin \theta \tan \theta = 3\). [6]
Edexcel C2 Q1
6 marks Moderate -0.8
  1. Using the factor theorem, show that \((x + 3)\) is a factor of $$x^3 - 3x^2 - 10x + 24.$$ [2]
  2. Factorise \(x^3 - 3x^2 - 10x + 24\) completely. [4]
Edexcel C2 Q2
7 marks Standard +0.3
\(f(n) = n^3 + pn^2 + 11n + 9\), where \(p\) is a constant.
  1. Given that f(n) has a remainder of 3 when it is divided by \((n + 2)\), prove that \(p = 6\). [2]
  2. Show that f(n) can be written in the form \((n + 2)(n + q)(n + r) + 3\), where \(q\) and \(r\) are integers to be found. [3]
  3. Hence show that f(n) is divisible by 3 for all positive integer values of \(n\). [2]
Edexcel C2 Q3
8 marks Standard +0.3
Find the values of \(\theta\), to 1 decimal place, in the interval \(-180 \leq \theta < 180\) for which $$2 \sin^2 \theta° - 2 \sin \theta° = \cos^2 \theta°.$$ [8]
Edexcel C2 Q4
7 marks Moderate -0.8
Every £1 of money invested in a savings scheme continuously gains interest at a rate of 4% per year. Hence, after \(x\) years, the total value of an initial £1 investment is £\(y\), where $$y = 1.04^x.$$
  1. Sketch the graph of \(y = 1.04^x\), \(x \geq 0\). [2]
  2. Calculate, to the nearest £, the total value of an initial £800 investment after 10 years. [2]
  3. Use logarithms to find the number of years it takes to double the total value of any initial investment. [3]
Edexcel C2 Q5
10 marks Standard +0.3
The curve \(C\) with equation \(y = p + qe^x\), where \(p\) and \(q\) are constants, passes through the point \((0, 2)\). At the point \(P\) (ln 2, \(p + 2q\)) on \(C\), the gradient is 5.
  1. Find the value of \(p\) and the value of \(q\). [5]
The normal to \(C\) at \(P\) crosses the \(x\)-axis at \(L\) and the \(y\)-axis at \(M\).
  1. Show that the area of \(\triangle OLM\), where \(O\) is the origin, is approximately 53.8 [5]
Edexcel C2 Q6
11 marks Moderate -0.3
\includegraphics{figure_3} Figure 3 shows part of the curve \(C\) with equation $$y = \frac{3}{5}x^2 - \frac{1}{4}x^3.$$ The curve \(C\) touches the \(x\)-axis at the origin and passes through the point \(A(p, 0)\).
  1. Show that \(p = 6\). [1]
  2. Find an equation of the tangent to \(C\) at \(A\). [4]
The curve \(C\) has a maximum at the point \(P\).
  1. Find the \(x\)-coordinate of \(P\). [2]
The shaded region \(R\), in Fig. 3, is bounded by \(C\) and the \(x\)-axis.
  1. Find the area of \(R\). [4]
Edexcel C2 Q7
12 marks Moderate -0.3
\includegraphics{figure_1} Figure 1 shows the cross-sections of two drawer handles. Shape \(X\) is a rectangle \(ABCD\) joined to a semicircle with \(BC\) as diameter. The length \(AB = d\) cm and \(BC = 2d\) cm. Shape \(Y\) is a sector \(OPQ\) of a circle with centre \(O\) and radius \(2d\) cm. Angle \(POQ\) is \(\theta\) radians. Given that the areas of the shapes \(X\) and \(Y\) are equal,
  1. prove that \(\theta = 1 + \frac{1}{2}\pi\). [5]
Using this value of \(\theta\), and given that \(d = 3\), find in terms of \(\pi\),
  1. the perimeter of shape \(X\), [2]
  2. the perimeter of shape \(Y\). [3]
  3. Hence find the difference, in mm, between the perimeters of shapes \(X\) and \(Y\). [2]
Edexcel C2 Q8
11 marks Standard +0.8
$$f(x) = \left(1 + \frac{x}{k}\right)^n, \quad k, n \in \mathbb{N}, \quad n > 2.$$ Given that the coefficient of \(x^3\) is twice the coefficient of \(x^2\) in the binomial expansion of f(x),
  1. prove that \(n = 6k + 2\). Given also that the coefficients of \(x^4\) and \(x^5\) are equal and non-zero, [3]
  2. form another equation in \(n\) and \(k\) and hence show that \(k = 2\) and \(n = 14\). Using these values of \(k\) and \(n\), [4]
  3. expand f(x) in ascending powers of \(x\), up to and including the term in \(x^5\). Give each coefficient as an exact fraction in its lowest terms [4]
Edexcel C2 Q1
6 marks Moderate -0.8
f(x) = ax³ + bx² - 7x + 14, where a and b are constants. Given that when f(x) is divided by (x - 1) the remainder is 9.
  1. write down an equation connecting a and b. [2 marks] Given also that (x + 2) is a factor of f(x),
  2. find the values of a and b. [4 marks]
Edexcel C2 Q2
10 marks Moderate -0.8
  1. Differentiate with respect to x $$2x^3 + \sqrt{x} + \frac{x^2 + 2x}{x^2}.$$ [5 marks]
  2. Evaluate $$\int_1^4 \left(\frac{x}{2} + \frac{1}{x^2}\right) dx.$$ [5 marks]
Edexcel C2 Q3
13 marks Moderate -0.3
  1. An arithmetic series has first term a and common difference d. Prove that the sum of the first n terms of the series is $$\frac{1}{2}n[2a + (n - 1)d].$$ [4 marks] A company made a profit of £54000 in the year 2001. A model for future performance assumes that yearly profits will increase in an arithmetic sequence with common difference £d. This model predicts total profits of £619200 for the 9 years 2001 to 2009 inclusive.
  2. Find the value of d. [4 marks] Using your value of d,
  3. find the predicted profit for the year 2011. [2 marks] An alternative model assumes that the company's yearly profits will increase in a geometric sequence with common ratio 1.06. Using this alternative model and again taking the profit in 2001 to be £54000,
  4. find the predicted profit for the year 2011. [3 marks]
Edexcel C2 Q4
9 marks Moderate -0.3
  1. Write down formulae for sin (A + B) and sin (A - B). Using X = A + B and Y = A - B, prove that $$\sin X + \sin Y = 2 \sin \frac{X + Y}{2} \cos \frac{X - Y}{2}.$$ [4 marks]
  2. Hence, or otherwise, solve, for 0 ≤ θ < 360, $$\sin 40° + \sin 20° = 0.$$ [5 marks]
Edexcel C2 Q5
10 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows a gardener's design for the shape of a flower bed with perimeter ABCD. AD is an arc of a circle with centre O and radius 5 m. BC is an arc of a circle with centre O and radius 7 m. OAB and ODC are straight lines and the size of ∠AOD is θ radians.
  1. Find, in terms of θ, an expression for the area of the flower bed. [3 marks] Given that the area of the flower bed is 15 m²,
  2. show that θ = 1.25. [2 marks]
  3. calculate, in m, the perimeter of the flower bed. [3 marks] The gardener now decides to replace arc AD with the straight line AD.
  4. Find, to the nearest cm, the reduction in the perimeter of the flower bed. [2 marks]
Edexcel C2 Q6
11 marks Standard +0.3
  1. Given that $$(2 + x)^5 + (2 - x)^5 ≡ A + Bx^2 + Cx^4,$$ Find the values of the constants A, B and C. [6 marks]
  2. Using the substitution y = x² and your answers to part (a), solve, $$(2 + x)^5 + (2 - x)^5 = 349.$$ [5 marks]
Edexcel C2 Q7
14 marks Standard +0.3
\includegraphics{figure_2} Figure 2 shows part of the curve C with equation y = f(x), where $$f(x) = x^3 - 6x^2 + 5x.$$ The curve crosses the x-axis at the origin O and at the points A and B.
  1. Factorise f(x) completely [3 marks]
  2. Write down the x-coordinates of the points A and B. [1 marks]
  3. Find the gradient of C at A. [3 marks] The region R is bounded by C and the line OA, and the region S is bounded by C and the line AB.
  4. Use integration to find the area of the combined regions R and S, shown shaded in Fig. 2. [7 marks]
Edexcel C2 Q1
4 marks Easy -1.2
A circle \(C\) has equation $$x^2 + y^2 - 10x + 6y - 15 = 0.$$
  1. Find the coordinates of the centre of \(C\). [2]
  2. Find the radius of \(C\). [2]
Edexcel C2 Q2
5 marks Moderate -0.3
Express \(\frac{y + 3}{(y + 1)(y + 2)} - \frac{y + 1}{(y + 2)(y + 3)}\) as a single fraction in its simplest form. [5]
Edexcel C2 Q3
6 marks Moderate -0.8
Given that \(2 \sin 2\theta = \cos 2\theta\),
  1. show that \(\tan 2\theta = 0.5\). [1]
  2. Hence find the values of \(\theta\), to one decimal place, in the interval \(0 \leq \theta < 360\) for which \(2 \sin 2\theta° = \cos 2\theta°\). [5]
Edexcel C2 Q4
7 marks Moderate -0.8
$$f(x) = x^3 - x^2 - 7x + c, \text{ where } c \text{ is a constant.}$$ Given that \(f(4) = 0\),
  1. find the value of \(c\), [2]
  2. factorise \(f(x)\) as the product of a linear factor and a quadratic factor. [3]
  3. Hence show that, apart from \(x = 4\), there are no real values of \(x\) for which \(f(x) = 0\). [2]
Edexcel C2 Q5
8 marks Moderate -0.3
\includegraphics{figure_1} Figure 1 shows the sector \(OAB\) of a circle of radius \(r\) cm. The area of the sector is 15 cm\(^2\) and \(\angle AOB = 1.5\) radians.
  1. Prove that \(r = 2\sqrt{5}\). [3]
  2. Find, in cm, the perimeter of the sector \(OAB\). [2]
The segment \(R\), shaded in Fig 1, is enclosed by the arc \(AB\) and the straight line \(AB\).
  1. Calculate, to 3 decimal places, the area of \(R\). [3]
Edexcel C2 Q6
10 marks Moderate -0.3
The third and fourth terms of a geometric series are 6.4 and 5.12 respectively. Find
  1. the common ratio of the series, [2]
  2. the first term of the series, [2]
  3. the sum to infinity of the series. [2]
  4. Calculate the difference between the sum to infinity of the series and the sum of the first 25 terms of the series. [4]