Questions — Edexcel (10514 questions)

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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]
Edexcel C2 Q9
13 marks Moderate -0.3
  1. Solve, for \(0° < x < 180°\), the equation \(\sin (2x + 50°) = 0.6\), giving your answers to 1 d. p. [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 Q1
4 marks Moderate -0.8
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 Q2
6 marks Moderate -0.8
\includegraphics{figure_1} The circle \(C\), with centre \((a, b)\) and radius \(5\), touches the \(x\)-axis at \((4, 0)\), as shown in Fig. 1.
  1. Write down the value of \(a\) and the value of \(b\). [1]
  2. Find a cartesian equation of \(C\). [2]
A tangent to the circle, drawn from the point \(P(8, 17)\), touches the circle at \(T\).
  1. Find, to 3 significant figures, the length of \(PT\). [3]
Edexcel C2 Q3
7 marks Standard +0.3
\(\text{f}(n) = n^3 + pn^2 + 11n + 9\), where \(p\) is a constant.
  1. Given that f\((n)\) has a remainder of \(3\) when it is divided by \((n + 2)\), prove that \(p = 6\). [2]
  2. Show that f\((n)\) can be written in the form \((n + 2)(n + q)(n + r) + 3\), where \(q\) and \(r\) are integers to be found. [3]
  3. Hence show that f\((n)\) is divisible by \(3\) for all positive integer values of \(n\). [2]
Edexcel C2 Q4
8 marks Moderate -0.8
  1. Sketch, for \(0 \leq x \leq 360°\), the graph of \(y = \sin (x + 30°)\). [2]
  2. Write down the coordinates of the points at which the graph meets the axes. [3]
  3. Solve, for \(0 \leq x < 360°\), the equation \(\sin (x + 30°) = -\frac{1}{2}\). [3]
Edexcel C2 Q5
7 marks Moderate -0.3
The expansion of \((2 - px)^6\) in ascending powers of \(x\), as far as the term in \(x^2\), is $$64 + Ax + 135x^2.$$ Given that \(p > 0\), find the value of \(p\) and the value of \(A\). [7]
Edexcel C2 Q6
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 Q7
8 marks Moderate -0.3
\includegraphics{figure_2} Fig. 2 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 Q8
12 marks Standard +0.3
\includegraphics{figure_3} Fig. 3 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 Q9
15 marks Moderate -0.3
For the curve \(C\) with equation \(y = x^4 - 8x^2 + 3\),
  1. find \(\frac{dy}{dx}\), [2]
  2. find the coordinates of each of the stationary points, [5]
  3. determine the nature of each stationary point. [3]
The point \(A\), on the curve \(C\), has \(x\)-coordinate \(1\).
  1. Find an equation for the normal to \(C\) at \(A\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [5]
Edexcel C2 Q1
6 marks Moderate -0.8
  1. Show that \((x + 2)\) is a factor of \(f(x) = x^3 - 19x - 30\). [2]
  2. Factorise \(f(x)\) completely. [4]
Edexcel C2 Q2
8 marks Standard +0.3
For the binomial expansion, in descending powers of \(x\), of \(\left( x^3 - \frac{1}{2x} \right)^{12}\),
  1. find the first 4 terms, simplifying each term. [5]
  2. Find, in its simplest form, the term independent of \(x\) in this expansion. [3]
Edexcel C2 Q3
9 marks Moderate -0.8
The curve \(C\) has equation \(y = \cos \left( x + \frac{\pi}{4} \right)\), \(0 \leq x \leq 2\pi\).
  1. Sketch \(C\). [2]
  2. Write down the exact coordinates of the points at which \(C\) meets the coordinate axes. [3]
  3. Solve, for \(x\) in the interval \(0 \leq x \leq 2\pi\), \(\cos \left( x + \frac{\pi}{4} \right) = 0.5\), giving your answers in terms of \(\pi\). [4]