Questions — Edexcel C2 (579 questions)

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Edexcel C2 Q1
4 marks Moderate -0.8
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
6 marks Moderate -0.3
\(f(x) = ax^3 + bx^2 - 7x + 14\), 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]
Given also that \((x + 2)\) is a factor of \(f(x)\),
  1. find the values of \(a\) and \(b\). [4]
Edexcel C2 Q3
8 marks Moderate -0.8
Find all values of \(\theta\) in the interval \(0 \leq \theta < 360\) for which
  1. \(\cos(\theta + 75)^\circ = 0\). [3]
  2. \(\sin 2\theta^\circ = 0.7\), giving your answers to one decimal place. [5]
Edexcel C2 Q4
9 marks Moderate -0.3
\includegraphics{figure_1} Fig. 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 Q5
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 Q6
12 marks Moderate -0.3
\includegraphics{figure_2} Triangle \(ABC\) has \(AB = 9\) cm, \(BC = 10\) cm and \(CA = 5\) cm. A circle, centre \(A\) and radius 3 cm, intersects \(AB\) and \(AC\) at \(P\) and \(Q\) respectively, as shown in Fig. 2.
  1. Show that, to 3 decimal places, \(\angle BAC = 1.504\) radians. [3]
Calculate,
  1. the area, in cm\(^2\), of the sector \(APQ\), [2]
  2. the area, in cm\(^2\), of the shaded region \(BPQC\), [3]
  3. the perimeter, in cm, of the shaded region \(BPQC\). [4]
Edexcel C2 Q7
12 marks Standard +0.3
\includegraphics{figure_3} Fig. 3 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 \(\theta = 1 + \frac{1}{4}\pi\). [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
14 marks Standard +0.3
\includegraphics{figure_3} A manufacturer produces cartons for fruit juice. Each carton is in the shape of a closed cuboid with base dimensions \(2x\) cm by \(x\) cm and height \(h\) cm, as shown in Fig. 3. Given that the capacity of a carton has to be 1030 cm\(^3\),
  1. express \(h\) in terms of \(x\), [2]
  2. show that the surface area, \(A\) cm\(^2\), of a carton is given by \(A = 4x^2 + \frac{3090}{x}\). [3]
The manufacturer needs to minimise the surface area of a carton.
  1. Use calculus to find the value of \(x\) for which \(A\) is a minimum. [5]
  2. Calculate the minimum value of \(A\). [2]
  3. Prove that this value of \(A\) is a minimum. [2]
Edexcel C2 Q1
3 marks Moderate -0.8
Find the remainder when \(f(x) = 4x^3 + 3x^2 - 2x - 6\) is divided by \((2x + 1)\). [3]
Edexcel C2 Q2
6 marks Moderate -0.3
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 Q3
6 marks Moderate -0.3
  1. Using the substitution \(u = 2^x\), show that the equation \(4^x - 2^{(x + 1)} - 15 = 0\) can be written in the form \(u^2 - 2u - 15 = 0\). [2]
  2. Hence solve the equation \(4^x - 2^{(x + 1)} - 15 = 0\), giving your answers to 2 d. p. [4]
Edexcel C2 Q4
7 marks Moderate -0.3
\includegraphics{figure_1} The shape of a badge is a sector \(ABC\) of a circle with centre \(A\) and radius \(AB\), as shown in Fig 1. The triangle \(ABC\) is equilateral and has a perpendicular height 3 cm.
  1. Find, in surd form, the length \(AB\). [2]
  2. Find, in terms of \(\pi\), the area of the badge. [2]
  3. Prove that the perimeter of the badge is \(\frac{2\sqrt{3}}{3}(\pi + 6)\) cm. [3]
Edexcel C2 Q5
9 marks Moderate -0.3
A circle \(C\) has centre \((3, 4)\) and radius \(3\sqrt{2}\). A straight line \(l\) has equation \(y = x + 3\).
  1. Write down an equation of the circle \(C\). [2]
  2. Calculate the exact coordinates of the two points where the line \(l\) intersects \(C\), giving your answers in surds. [5]
  3. Find the distance between these two points. [2]
Edexcel C2 Q6
9 marks Moderate -0.3
The sequence \(u_1, u_2, u_3, \ldots, u_n\) is defined by the recurrence relation $$u_{n+1} = pu_n + 5, u_1 = 2, \text{ where } p \text{ is a constant.}$$ Given that \(u_3 = 8\),
  1. show that one possible value of \(p\) is \(\frac{1}{2}\) and find the other value of \(p\). [5]
Using \(p = \frac{1}{2}\),
  1. write down the value of \(\log_2 p\). [1]
Given also that \(\log_2 q = t\),
  1. express \(\log_2 \left(\frac{p^3}{\sqrt{q}}\right)\) in terms of \(t\). [3]
Edexcel C2 Q7
12 marks Standard +0.3
\includegraphics{figure_2} Fig. 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]
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 \(n = 6k + 2\). [3]
Given also that the coefficients of \(x^4\) and \(x^5\) are equal and non-zero,
  1. form another equation in \(n\) and \(k\) and hence show that \(k = 2\) and \(n = 14\). [4]
Using these values of \(k\) and \(n\),
  1. 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 Q9
12 marks Standard +0.3
\includegraphics{figure_3} 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. 3.
  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 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 Moderate -0.8
  1. Expand \((2\sqrt{x} + 3)^2\). [2]
  2. Hence evaluate \(\int_1^2 (2\sqrt{x} + 3)^2 \, dx\), giving your answer in the form \(a + b\sqrt{2}\), where \(a\) and \(b\) are integers. [5]
Edexcel C2 Q3
7 marks Standard +0.8
The first three terms in the expansion, in ascending powers of \(x\), of \((1 + px)^n\), are \(1 - 18x + 36p^2x^2\). Given that \(n\) is a positive integer, find the value of \(n\) and the value of \(p\). [7]
Edexcel C2 Q4
7 marks Moderate -0.3
A circle \(C\) has equation \(x^2 + y^2 - 6x + 8y - 75 = 0\).
  1. Write down the coordinates of the centre of \(C\), and calculate the radius of \(C\). [3]
A second circle has centre at the point \((15, 12)\) and radius \(10\).
  1. Sketch both circles on a single diagram and find the coordinates of the point where they touch. [4]
Edexcel C2 Q5
10 marks Moderate -0.8
  1. Differentiate \(2x^2 + \sqrt{x} + \frac{x^2 + 2x}{x^2}\) with respect to \(x\) [5]
  2. Evaluate \(\int_1^4 \left(\frac{x}{2} + \frac{1}{x^2}\right) dx\). [5]
Edexcel C2 Q6
10 marks Moderate -0.3
A geometric series has first term \(1200\). Its sum to infinity is \(960\).
  1. Show that the common ratio of the series is \(-\frac{1}{4}\). [3]
  2. Find, to 3 decimal places, the difference between the ninth and tenth terms of the series. [3]
  3. Write down an expression for the sum of the first \(n\) terms of the series. [2]
Given that \(n\) is odd,
  1. prove that the sum of the first \(n\) terms of the series is \(960(1 + 0.25^n)\). [2]
Edexcel C2 Q7
10 marks Moderate -0.3
On a journey, the average speed of a car is \(v\) m s\(^{-1}\). For \(v \geq 5\), the cost per kilometre, \(C\) pence, of the journey is modelled by \(C = \frac{160}{v} + \frac{v^2}{100}\). Using this model,
  1. show, by calculus, that there is a value of \(v\) for which \(C\) has a stationary value, and find this value of \(v\). [5]
  2. Justify that this value of \(v\) gives a minimum value of \(C\). [2]
  3. Find the minimum value of \(C\) and hence find the minimum cost of a 250 km car journey. [3]
Edexcel C2 Q8
12 marks Moderate -0.3
\includegraphics{figure_1} The curve \(C\), shown in Fig. 1, represents the graph of \(y = \frac{x^2}{25}\), \(x \geq 0\). The points \(A\) and \(B\) on the curve \(C\) have \(x\)-coordinates \(5\) and \(10\) respectively.
  1. Write down the \(y\)-coordinates of \(A\) and \(B\). [1]
  2. Find an equation of the tangent to \(C\) at \(A\). [4]
The finite region \(R\) is enclosed by \(C\), the \(y\)-axis and the lines through \(A\) and \(B\) parallel to the \(x\)-axis.
  1. For points \((x, y)\) on \(C\), express \(x\) in terms of \(y\). [2]
  2. Use integration to find the area of \(R\). [5]