Questions C2 (1550 questions)

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Edexcel C2 Q4
11 marks Moderate -0.3
The first term of a geometric series is 120. The sum to infinity of the series is 480.
  1. Show that the common ratio, \(r\), is \(\frac{3}{4}\). [3]
  2. Find, to 2 decimal places, the difference between the 5th and 6th terms. [2]
  3. Calculate the sum of the first 7 terms. [2]
The sum of the first \(n\) terms of the series is greater than 300.
  1. Calculate the smallest possible value of \(n\). [4]
Edexcel C2 Q5
8 marks Moderate -0.3
\includegraphics{figure_2} In Figure 2 \(OAB\) is a sector of a circle, radius 5 m. The chord \(AB\) is 6 m long.
  1. Show that \(\cos A\hat{O}B = \frac{7}{25}\). [2]
  2. Hence find the angle \(A\hat{O}B\) in radians, giving your answer to 3 decimal places. [1]
  3. Calculate the area of the sector \(OAB\). [2]
  4. Hence calculate the shaded area. [3]
Edexcel C2 Q6
6 marks Moderate -0.8
The speed, \(v\) m s⁻¹, of a train at time \(t\) seconds is given by \(v = \sqrt{(1.2^t - 1)}, \quad 0 \leq t \leq 30.\) The following table shows the speed of the train at 5 second intervals.
\(t\)051015202530
\(v\)01.222.286.11
  1. Complete the table, giving the values of \(v\) to 2 decimal places. [3]
The distance, \(s\) metres, travelled by the train in 30 seconds is given by $$s = \int_0^{30} \sqrt{(1.2^t - 1)} \, dt.$$
  1. Use the trapezium rule, with all the values from your table, to estimate the value of \(s\). [3]
Edexcel C2 Q7
10 marks Moderate -0.8
The curve \(C\) has equation \(y = 2x^3 - 5x^2 - 4x + 2\).
  1. Find \(\frac{dy}{dx}\). [2]
  2. Using the result from part (a), find the coordinates of the turning points of \(C\). [4]
  3. Find \(\frac{d^2y}{dx^2}\). [2]
  4. Hence, or otherwise, determine the nature of the turning points of \(C\). [2]
Edexcel C2 Q8
9 marks Moderate -0.8
  1. Find all the values of \(\theta\), to 1 decimal place, in the interval \(0° \leq \theta \leq 360°\) for which \(5 \sin (\theta + 30°) = 3\). [4]
  2. Find all the values of \(\theta\), to 1 decimal place, in the interval \(0° \leq \theta \leq 360°\) for which \(\tan^2 \theta = 4\). [5]
Edexcel C2 Q9
10 marks Standard +0.3
\includegraphics{figure_3} Figure 3 shows the shaded region \(R\) which is bounded by the curve \(y = -2x^2 + 4x\) and the line \(y = \frac{3}{2}\). The points \(A\) and \(B\) are the points of intersection of the line and the curve. Find
  1. the \(x\)-coordinates of the points \(A\) and \(B\), [4]
  2. the exact area of \(R\). [6]
Edexcel C2 Q1
4 marks Moderate -0.3
Evaluate \(\int_0^1 \frac{1}{\sqrt{x}} \, dx\), giving your answer in the form \(a + b\sqrt{2}\), where \(a\) and \(b\) are integers. [4]
Edexcel C2 Q2
6 marks Moderate -0.8
\(f(x) = 3x^3 - 5x^2 - 16x + 12\).
  1. Find the remainder when \(f(x)\) is divided by \((x - 2)\). [2]
Given that \((x + 2)\) is a factor of \(f(x)\),
  1. factorise \(f(x)\) completely. [4]
Edexcel C2 Q3
6 marks Moderate -0.8
  1. Find the first four terms, in ascending powers of \(x\), in the bionomial expansion of \((1 + kx)^8\), where \(k\) is a non-zero constant. [3]
Given that, in this expansion, the coefficients of \(x\) and \(x^2\) are equal, find
  1. the value of \(k\), [2]
  2. the coefficient of \(x^3\). [1]
Edexcel C2 Q4
5 marks Moderate -0.8
\includegraphics{figure_1} Figure 1 shows the triangle \(ABC\), with \(AB = 6\) cm, \(BC = 4\) cm and \(CA = 5\) cm.
  1. Show that \(\cos A = \frac{3}{4}\). [3]
  2. Hence, or otherwise, find the exact value of \(\sin A\). [2]
Edexcel C2 Q5
9 marks Moderate -0.8
The curve \(C\) has equation \(y = x\sqrt{x^2 + 1}, \quad 0 \leq x \leq 2\).
  1. Copy and complete the table below, giving the values of \(y\) to 3 decimal places at \(x = 1\) and \(x = 1.5\).
    \(x\)00.511.52
    \(y\)00.5306
    [2]
  2. Use the trapezium rule, with all the \(y\) values from your table, to find an approximation for the value of \(\int_0^2 x\sqrt{x^2 + 1} \, dx\), giving your answer to 3 significant figures. [4]
\includegraphics{figure_2} Figure 2 shows the curve \(C\) with equation \(y = x\sqrt{x^2 + 1}\), \(0 \leq x \leq 2\), and the straight line segment \(l\), which joins the origin and the point \((2, 6)\). The finite region \(R\) is bounded by \(C\) and \(l\).
  1. Use your answer to part (b) to find an approximation for the area of \(R\), giving your answer to 3 significant figures. [3]
Edexcel C2 Q6
6 marks Moderate -0.3
  1. Find, to 3 significant figures, the value of \(x\) for which \(8^x = 0.8\). [2]
  2. Solve the equation \(2 \log_3 x - \log_3 7x = 1\). [4]
Edexcel C2 Q7
9 marks Moderate -0.3
\includegraphics{figure_3} The points \(A\) and \(B\) lie on a circle with centre \(P\), as shown in Figure 3. The point \(A\) has coordinates \((1, -2)\) and the mid-point \(M\) of \(AB\) has coordinates \((3, 1)\). The line \(l\) passes through the points \(M\) and \(P\).
  1. Find an equation for \(l\). [4]
Given that the \(x\)-coordinate of \(P\) is 6,
  1. use your answer to part (a) to show that the \(y\)-coordinate of \(P\) is \(-1\). [1]
  2. find an equation for the circle. [4]
Edexcel C2 Q8
9 marks Moderate -0.8
A trading company made a profit of £50 000 in 2006 (Year 1). A model for future trading predicts that profits will increase year by year in a geometric sequence with common ratio \(r, r > 1\). The model therefore predicts that in 2007 (Year 2) a profit of £50 000r will be made.
  1. Write down an expression for the predicted profit in Year \(n\). [1]
The model predicts that in Year \(n\), the profit made will exceed £200 000.
  1. Show that \(n > \frac{\log 4}{\log r} + 1\). [3]
Using the model with \(r = 1.09\),
  1. find the year in which the profit made will first exceed £200 000, [2]
  2. find the total of the profits that will be made by the company over the 10 years from 2006 to 2015 inclusive, giving your answer to the nearest £10 000. [3]
Edexcel C2 Q9
10 marks Moderate -0.8
  1. Sketch, for \(0 \leq x \leq 2\pi\), the graph of \(y = \sin\left(x + \frac{\pi}{6}\right)\). [2]
  2. Write down the exact coordinates of the points where the graph meets the coordinate axes. [3]
  3. Solve, for \(0 \leq x \leq 2\pi\), the equation \(\sin\left(x + \frac{\pi}{6}\right) = 0.65\), giving your answers in radians to 2 decimal places. [5]
Edexcel C2 Q10
11 marks Standard +0.3
\includegraphics{figure_4} Figure 4 shows a solid brick in the shape of a cuboid measuring 2x cm by x cm by y cm. The total surface area of the brick is 600 cm².
  1. Show that the volume, \(V\) cm³, of the brick is given by \(V = 200x - \frac{4x^3}{3}\). [4]
Given that \(x\) can vary,
  1. use calculus to find the maximum value of \(V\), giving your answer to the nearest cm³. [5]
  2. Justify that the value of \(V\) you have found is a maximum. [2]
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]