Questions — Edexcel C2 (579 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 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]
Edexcel C2 Q7
10 marks Moderate -0.3
$$f(x) = 5\sin 3x°, \quad 0 \leq x \leq 180.$$
  1. Sketch the graph of \(f(x)\), indicating the value of \(x\) at each point where the graph intersects the \(x\)-axis [3]
  2. Write down the coordinates of all the maximum and minimum points of \(f(x)\). [3]
  3. Calculate the values of \(x\) for which \(f(x) = 2.5\) [4]
Edexcel C2 Q8
13 marks Moderate -0.3
  1. Solve, for \(0° < x < 180°\), the equation $$\sin (2x + 50°) = 0.6,$$ giving your answers to 1 decimal place. [7]
  2. In the triangle \(ABC\), \(AC = 18\) cm, \(\angle ABC = 60°\) and \(\sin A = \frac{1}{3}\).
    1. Use the sine rule to show that \(BC = 4\sqrt{3}\). [4]
    2. Find the exact value of \(\cos A\). [2]
Edexcel C2 Q9
12 marks Standard +0.3
\includegraphics{figure_2} Figure 2 shows the line with equation \(y = 9 - x\) and the curve with equation \(y = x^2 - 2x + 3\). The line and the curve intersect at the points \(A\) and \(B\), and \(O\) is the origin.
  1. Calculate the coordinates of \(A\) and the coordinates of \(B\). [5]
The shaded region \(R\) is bounded by the line and the curve.
  1. Calculate the area of \(R\). [7]
Edexcel C2 Q1
3 marks Moderate -0.8
\(f(x) = 4x^3 + 3x^2 - 2x - 6\). Find the remainder when \(f(x)\) is divided by \((2x + 1)\). [3]
Edexcel C2 Q2
4 marks Moderate -0.5
The point \(A\) has coordinates \((2, 5)\) and the point \(B\) has coordinates \((-2, 8)\). Find, in cartesian form, an equation of the circle with diameter \(AB\). [4]
Edexcel C2 Q3
6 marks Moderate -0.8
\(f(x) = x^3 - 19x - 30\).
  1. Show that \((x + 2)\) is a factor of \(f(x)\). [2]
  2. Factorise \(f(x)\) completely. [4]
Edexcel C2 Q4
7 marks Standard +0.3
Express \(\frac{3}{x^2 + 2x} + \frac{x - 4}{x^2 - 4}\) as a single fraction in its simplest form. [7]
Edexcel C2 Q5
8 marks Standard +0.3
Find, in degrees, the value of \(\theta\) in the interval \(0 \leq \theta < 360°\) for which $$2\cos^2 \theta - \cos \theta - 1 = \sin^2 \theta$$ Give your answers to 1 decimal place where appropriate. [8]
Edexcel C2 Q6
12 marks Moderate -0.3
A geometric series is \(a + ar + ar^2 + \ldots\)
  1. Prove that the sum of the first \(n\) terms of this series is given by $$S_n = \frac{a(1-r^n)}{1-r}$$ [4]
The second and fourth terms of the series are 3 and 1.08 respectively. Given that all terms in the series are positive, find
  1. the value of \(r\) and the value of \(a\), [5]
  2. the sum to infinity of the series. [3]
Edexcel C2 Q7
12 marks Standard +0.3
\includegraphics{figure_2} A rectangular sheet of metal measures 50 cm by 40 cm. Squares of side \(x\) cm are cut from each corner of the sheet and the remainder is folded along the dotted lines to make an open tray, as shown in Fig. 2.
  1. Show that the volume, \(V\) cm\(^3\), of the tray is given by $$V = 4x(x^2 - 45x + 500)$$ [3]
  2. State the range of possible values of \(x\). [1]
  3. Find the value of \(x\) for which \(V\) is a maximum. [4]
  4. Hence find the maximum value of \(V\). [2]
  5. Justify that the value of \(V\) you found in part (d) is a maximum. [2]
Edexcel C2 Q8
7 marks Moderate -0.8
\includegraphics{figure_1} Figure 1 shows the sector \(AOB\) of a circle, with centre \(O\) and radius 6.5 cm, and \(\angle AOB = 0.8\) radians.
  1. Calculate, in cm\(^2\), the area of the sector \(AOB\). [2]
  2. Show that the length of the chord \(AB\) is 5.06 cm, to 3 significant figures. [3]
The segment \(R\), shaded in Fig. 1, is enclosed by the arc \(AB\) and the straight line \(AB\).
  1. Calculate, in cm, the perimeter of \(R\). [2]
Edexcel C2 Q9
12 marks Standard +0.3
\includegraphics{figure_2} Figure 2 shows the line with equation \(y = x + 1\) and the curve with equation \(y = 6x - x^2 - 3\). The line and the curve intersect at the points \(A\) and \(B\), and \(O\) is the origin.
  1. Calculate the coordinates of \(A\) and the coordinates of \(B\). [5]
The shaded region \(R\) is bounded by the line and the curve.
  1. Calculate the area of \(R\). [7]