Questions — Edexcel C2 (579 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 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
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Edexcel C2 Q1
7 marks Moderate -0.8
  1. Find the remainder when \(x^3 - 2x^2 - 4x + 8\) is divided by
    1. \(x - 3\),
    2. \(x + 2\). [3]
  2. Hence, or otherwise, find all the solutions to the equation \(x^3 - 2x^2 - 4x + 8 = 0\). [4]
Edexcel C2 Q2
6 marks Moderate -0.3
The fourth term of a geometric series is 10 and the seventh term of the series is 80. For this series, find
  1. the common ratio, [2]
  2. the first term, [2]
  3. the sum of the first 20 terms, giving your answer to the nearest whole number. [2]
Edexcel C2 Q3
7 marks Moderate -0.8
  1. Find the first 4 terms of the expansion of \(\left(1 + \frac{x}{3}\right)^{18}\) in ascending powers of \(x\), giving each term in its simplest form. [4]
  2. Use your expansion to estimate the value of \((1.005)^{18}\), giving your answer to 5 decimal places. [3]
Edexcel C2 Q4
9 marks Moderate -0.8
  1. Show that the equation \(3 \sin^2 \theta - 2 \cos^2 \theta = 1\) can be written as \(5 \sin^2 \theta = 3\). [2]
  2. Hence solve, for \(0° \leq \theta \leq 360°\), the equation \(3 \sin^2 \theta - 2 \cos^2 \theta = 1\), giving your answer to 1 decimal place. [7]
Edexcel C2 Q5
6 marks Standard +0.3
Given that \(a\) and \(b\) are positive constants, solve the simultaneous equations \(a = 3b\), \(\log_3 a + \log_3 b = 2\). Give your answers as exact numbers. [6]
Edexcel C2 Q6
7 marks Moderate -0.3
\includegraphics{figure_1} Figure 1 shows 3 yachts \(A\), \(B\) and \(C\) which are assumed to be in the same horizontal plane. Yacht \(B\) is 500 m due north of yacht \(A\) and yacht \(C\) is 700 m from \(A\). The bearing of \(C\) from \(A\) is 015°.
  1. Calculate the distance between yacht \(B\) and yacht \(C\), in metres to 3 significant figures. [3]
The bearing of yacht \(C\) from yacht \(B\) is \(\theta°\), as shown in Figure 1.
  1. Calculate the value of \(\theta\). [4]
Edexcel C2 2008 January Q1
7 marks Moderate -0.8
  1. Find the remainder when $$x^3 - 2x^2 - 4x + 8$$ is divided by
    1. \(x - 3\),
    2. \(x + 2\).
    [3]
  2. Hence, or otherwise, find all the solutions to the equation $$x^3 - 2x^2 - 4x + 8 = 0.$$ [4]
Edexcel C2 2008 January Q2
6 marks Moderate -0.3
The fourth term of a geometric series is 10 and the seventh term of the series is 80. For this series, find
  1. the common ratio, [2]
  2. the first term, [2]
  3. the sum of the first 20 terms, giving your answer to the nearest whole number. [2]
Edexcel C2 2008 January Q3
7 marks Moderate -0.8
  1. Find the first 4 terms of the expansion of \(\left(1 + \frac{x}{2}\right)^{10}\) in ascending powers of \(x\), giving each term in its simplest form. [4]
  2. Use your expansion to estimate the value of \((1.005)^{10}\), giving your answer to 5 decimal places. [3]
Edexcel C2 2008 January Q4
9 marks Moderate -0.8
  1. Show that the equation $$3 \sin^2 \theta - 2 \cos^2 \theta = 1$$ can be written as $$5 \sin^2 \theta = 3.$$ [2]
  2. Hence solve, for \(0° \leq \theta < 360°\), the equation $$3 \sin^2 \theta - 2 \cos^2 \theta = 1,$$ giving your answers to 1 decimal place. [7]
Edexcel C2 2008 January Q5
6 marks Moderate -0.3
Given that \(a\) and \(b\) are positive constants, solve the simultaneous equations $$a = 3b,$$ $$\log_3 a + \log_3 b = 2.$$ Give your answers as exact numbers. [6]
Edexcel C2 2008 January Q6
7 marks Moderate -0.8
\includegraphics{figure_1} Figure 1 shows 3 yachts \(A\), \(B\) and \(C\) which are assumed to be in the same horizontal plane. Yacht \(B\) is 500 m due north of yacht \(A\) and yacht \(C\) is 700 m from \(A\). The bearing of \(C\) from \(A\) is 015°.
  1. Calculate the distance between yacht \(B\) and yacht \(C\), in metres to 3 significant figures. [3]
The bearing of yacht \(C\) from yacht \(B\) is \(\theta°\), as shown in Figure 1.
  1. Calculate the value of \(\theta\). [4]
Edexcel C2 2008 January Q7
10 marks Moderate -0.8
\includegraphics{figure_2} In Figure 2 the curve \(C\) has equation \(y = 6x - x^2\) and the line \(L\) has equation \(y = 2x\).
  1. Show that the curve \(C\) intersects the \(x\)-axis at \(x = 0\) and \(x = 6\). [1]
  2. Show that the line \(L\) intersects the curve \(C\) at the points \((0, 0)\) and \((4, 8)\). [3]
The region \(R\), bounded by the curve \(C\) and the line \(L\), is shown shaded in Figure 2.
  1. Use calculus to find the area of \(R\). [6]
Edexcel C2 2008 January Q8
11 marks Standard +0.3
A circle \(C\) has centre \(M\) \((6, 4)\) and radius 3.
  1. Write down the equation of the circle in the form $$(x - a)^2 + (y - b)^2 = r^2.$$ [2]
\includegraphics{figure_3} Figure 3 shows the circle \(C\). The point \(T\) lies on the circle and the tangent at \(T\) passes through the point \(P\) \((12, 6)\). The line \(MP\) cuts the circle at \(Q\).
  1. Show that the angle \(TMQ\) is 1.0766 radians to 4 decimal places. [4]
The shaded region \(TPQ\) is bounded by the straight lines \(TP\), \(QP\) and the arc \(TQ\), as shown in Figure 3.
  1. Find the area of the shaded region \(TPQ\). Give your answer to 3 decimal places. [5]
Edexcel C2 2008 January Q9
12 marks Standard +0.3
\includegraphics{figure_4} Figure 4 shows an open-topped water tank, in the shape of a cuboid, which is made of sheet metal. The base of the tank is a rectangle \(x\) metres by \(y\) metres. The height of the tank is \(x\) metres. The capacity of the tank is 100 m³.
  1. Show that the area \(A\) m² of the sheet metal used to make the tank is given by $$A = \frac{300}{x} + 2x^2.$$ [4]
  2. Use calculus to find the value of \(x\) for which \(A\) is stationary. [4]
  3. Prove that this value of \(x\) gives a minimum value of \(A\). [2]
  4. Calculate the minimum area of sheet metal needed to make the tank. [2]
Edexcel C2 Q1
4 marks Moderate -0.8
\(f(x) = 2x^3 - x^2 + px + 6\), where \(p\) is a constant. Given that \((x - 1)\) is a factor of \(f(x)\), find
  1. the value of \(p\), [2]
  2. the remainder when \(f(x)\) is divided by \((2x + 1)\). [2]
Edexcel C2 Q2
5 marks Easy -1.2
  1. Find \(\int \left( 3 + 4x^3 - \frac{2}{x^2} \right) dx\). [3]
  2. Hence evaluate \(\int_1^2 \left( 3 + 4x^3 - \frac{2}{x^2} \right) dx\). [2]
Edexcel C2 Q3
5 marks Moderate -0.8
\includegraphics{figure_1} Figure 1 shows a logo \(ABD\). The logo is formed from triangle \(ABC\). The mid-point of \(AC\) is \(D\) and \(BC = AD = DC = 6\) cm. \(\angle BCA = 0.4\) radians. The curve \(BD\) is an arc of a circle with centre \(C\) and radius 6 cm.
  1. Write down the length of the arc \(BD\). [1]
  2. Find the length of \(AB\). [3]
  3. Write down the perimeter of the logo \(ABD\), giving your answer to 3 significant figures. [1]
Edexcel C2 Q4
6 marks Standard +0.3
Solve $$2 \log_3 x - \log_3 (x - 2) = 2, \quad x > 2.$$ [6]
Edexcel C2 Q5
7 marks Moderate -0.8
The second and fifth terms of a geometric series are 9 and 1.125 respectively. For this series find
  1. the value of the common ratio, [3]
  2. the first term, [2]
  3. the sum to infinity. [2]
Edexcel C2 Q6
7 marks Standard +0.3
The circle \(C\), with centre \(A\), has equation $$x^2 + y^2 - 6x + 4y - 12 = 0.$$
  1. Find the coordinates of \(A\). [2]
  2. Show that the radius of \(C\) is 5. [2]
The points \(P\), \(Q\) and \(R\) lie on \(C\). The length of \(PQ\) is 10 and the length of \(PR\) is 3.
  1. Find the length of \(QR\), giving your answer to 1 decimal place. [3]
Edexcel C2 Q7
8 marks Standard +0.3
The first four terms, in ascending powers of \(x\), of the binomial expansion of \((1 + kx)^n\) are $$1 + Ax + Bx^2 + Bx^3 + \ldots,$$ where \(k\) is a positive constant and \(A\), \(B\) and \(n\) are positive integers.
  1. By considering the coefficients of \(x^2\) and \(x^3\), show that \(3 = (n - 2) k\). [4]
Given that \(A = 4\),
  1. find the value of \(n\) and the value of \(k\). [4]
Edexcel C2 Q8
10 marks Moderate -0.3
  1. Solve, for \(0 \leq x < 360°\), the equation \(\cos (x - 20°) = -0.437\), giving your answers to the nearest degree. [4]
  2. Find the exact values of \(\theta\) in the interval \(0 \leq \theta < 360°\) for which $$3 \tan \theta = 2 \cos \theta.$$ [6]
Edexcel C2 Q9
11 marks Standard +0.3
A pencil holder is in the shape of an open circular cylinder of radius \(r\) cm and height \(h\) cm. The surface area of the cylinder (including the base) is 250 cm\(^2\).
  1. Show that the volume, V cm\(^3\), of the cylinder is given by \(V = 125r - \frac{\pi r^3}{2}\). [4]
  2. Use calculus to find the value of \(r\) for which \(V\) has a stationary value. [3]
  3. Prove that the value of \(r\) you found in part (b) gives a maximum value for \(V\). [2]
  4. Calculate, to the nearest cm\(^3\), the maximum volume of the pencil holder. [2]
Edexcel C2 Q10
12 marks Standard +0.3
\includegraphics{figure_2} Figure 2 shows part of the curve \(C\) with equation $$y = 9 - 2x - \frac{2}{\sqrt{x}}, \quad x > 0.$$ The point \(A(1, 5)\) lies on \(C\) and the curve crosses the \(x\)-axis at \(B(b, 0)\), where \(b\) is a constant and \(b > 0\).
  1. Verify that \(b = 4\). [1]
The tangent to \(C\) at the point \(A\) cuts the \(x\)-axis at the point \(D\), as shown in Fig. 2.
  1. Show that an equation of the tangent to \(C\) at \(A\) is \(y + x = 6\). [4]
  2. Find the coordinates of the point \(D\). [1]
The shaded region \(R\), shown in Fig. 2, is bounded by \(C\), the line \(AD\) and the \(x\)-axis.
  1. Use integration to find the area of \(R\). [6]