Questions — Edexcel C2 (579 questions)

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Edexcel C2 Q4
6 marks Moderate -0.3
  1. Write down the first three terms, in ascending powers of \(x\), of the binomial expansion of \((1 + px)^{12}\), where \(p\) is a non-zero constant. [2]
Given that, in the expansion of \((1 + px)^{12}\), the coefficient of \(x\) is \((-q)\) and the coefficient of \(x^2\) is \(11q\),
  1. find the value of \(p\) and the value of \(q\). [4]
Edexcel C2 Q5
8 marks Moderate -0.8
Solve, for \(0 \leq x \leq 180°\), the equation $$\sin(x + 10°) = \frac{\sqrt{3}}{2}.$$ [4]
  1. \(\cos 2x = -0.9\), giving your answers to 1 decimal place. [4]
Edexcel C2 Q6
8 marks Moderate -0.8
A river, running between parallel banks, is 20 m wide. The depth, \(y\) metres, of the river measured at a point \(x\) metres from one bank is given by the formula $$y = \frac{1}{10}x(20 - x), \quad 0 \leq x \leq 20.$$
  1. Complete the table below, giving values of \(y\) to 3 decimal places.
    \(x\)048121620
    \(y\)02.7710
    [2]
  2. Use the trapezium rule with all the values in the table to estimate the cross-sectional area of the river. [4]
Given that the cross-sectional area is constant and that the river is flowing uniformly at 2 m s⁻¹,
  1. estimate, in m³, the volume of water flowing per minute, giving your answer to 3 significant figures. [2]
Edexcel C2 Q7
6 marks Moderate -0.3
In the triangle \(ABC\), \(AB = 8\) cm, \(AC = 7\) cm, \(\angle ABC = 0.5\) radians and \(\angle ACB = x\) radians.
  1. Use the sine rule to find the value of \(\sin x\), giving your answer to 3 decimal places. [3]
Given that there are two possible values of \(x\),
  1. find these values of \(x\), giving your answers to 2 decimal places. [3]
Edexcel C2 Q8
9 marks Moderate -0.3
The circle \(C\), with centre at the point \(A\), has equation \(x^2 + y^2 - 10x + 9 = 0\). Find
  1. the coordinates of \(A\), [2]
  2. the radius of \(C\), [2]
  3. the coordinates of the points at which \(C\) crosses the \(x\)-axis. [2]
Given that the line \(l\) with gradient \(\frac{7}{T}\) is a tangent to \(C\), and that \(l\) touches \(C\) at the point \(T\),
  1. find an equation of the line which passes through \(A\) and \(T\). [3]
Edexcel C2 Q9
10 marks Moderate -0.3
  1. A geometric series has first term \(a\) and common ratio \(r\). Prove that the sum of the first \(n\) terms of the series is $$\frac{a(1-r^n)}{1-r}.$$ [4]
Mr King will be paid a salary of £35 000 in the year 2005. Mr King's contract promises a 4% increase in salary every year, the first increase being given in 2006, so that his annual salaries form a geometric sequence.
  1. Find, to the nearest £100, Mr King's salary in the year 2008. [2]
Mr King will receive a salary each year from 2005 until he retires at the end of 2024.
  1. Find, to the nearest £1000, the total amount of salary he will receive in the period from 2005 until he retires at the end of 2024. [4]
Edexcel C2 Q1
8 marks Moderate -0.8
\(f(x) = 2x^3 + x^2 - 5x + c\), where \(c\) is a constant. Given that \(f(1) = 0\),
  1. find the value of \(c\), [2]
  2. factorise \(f(x)\) completely, [4]
  3. find the remainder when \(f(x)\) is divided by \((2x - 3)\). [2]
Edexcel C2 Q2
6 marks Moderate -0.8
  1. Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of \((1 + px)^9\), where \(p\) is a constant. [2]
The first 3 terms are 1, 36x and \(qx^2\), where \(q\) is a constant.
  1. Find the value of \(p\) and the value of \(q\). [4]
Edexcel C2 Q3
7 marks Moderate -0.8
\includegraphics{figure_1} In Figure 1, \(A(4, 0)\) and \(B(3, 5)\) are the end points of a diameter of the circle \(C\). Find
  1. the exact length of \(AB\), [2]
  2. the coordinates of the midpoint \(P\) of \(AB\), [2]
  3. an equation for the circle \(C\). [3]
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]