Questions — Edexcel C2 (579 questions)

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Edexcel C2 Q26
12 marks Moderate -0.3
\includegraphics{figure_9} 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. 2.
  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 Q27
6 marks Moderate -0.3
  1. Write down the first four terms of the binomial expansion, in ascending powers of \(x\), of \((1 + 3x)^n\), where \(n > 2\). [2]
Given that the coefficient of \(x^3\) in this expansion is ten times the coefficient of \(x^2\),
  1. find the value of \(n\), [2]
  2. find the coefficient of \(x^4\) in the expansion. [2]
Edexcel C2 Q28
10 marks Moderate -0.3
  1. Given that \(3 + 2 \log_2 x = \log_2 y\), show that \(y = 8x^2\). [3]
  2. Hence, or otherwise, find the roots \(\alpha\) and \(\beta\), where \(\alpha < \beta\), of the equation $$3 + 2 \log_2 x = \log_2 (14x - 3).$$ [3]
  3. Show that \(\log_2 \alpha = -2\). [1]
  4. Calculate \(\log_2 \beta\), giving your answer to 3 significant figures. [3]
Edexcel C2 Q29
7 marks Moderate -0.3
$$f(x) = x^3 + ax^2 + bx - 10, \text{ where } a \text{ and } b \text{ are constants.}$$ When \(f(x)\) is divided by \((x - 3)\), the remainder is 14. When \(f(x)\) is divided by \((x + 1)\), the remainder is \(-18\).
  1. Find the value of \(a\) and the value of \(b\). [5]
  2. Show that \((x - 2)\) is a factor of \(f(x)\). [2]
Edexcel C2 Q30
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 Q31
10 marks Moderate -0.8
  1. Differentiate with respect to \(x\) $$2x^3 + \sqrt{x} + \frac{x^2 + 2x}{x^2}.$$ [5]
  2. Evaluate $$\int_1^4 \left(\frac{x}{2} + \frac{1}{x^2}\right) dx.$$ [5]
Edexcel C2 Q32
13 marks Moderate -0.3
  1. An arithmetic series has first term \(a\) and common difference \(d\). Prove that the sum of the first \(n\) terms of the series is $$\frac{1}{2}n[2a + (n - 1)d].$$ [4]
A company made a profit of £54 000 in the year 2001. A model for future performance assumes that yearly profits will increase in an arithmetic sequence with common difference £\(d\). This model predicts total profits of £619 200 for the 9 years 2001 to 2009 inclusive.
  1. Find the value of \(d\). [4]
Using your value of \(d\),
  1. find the predicted profit for the year 2011. [2]
An alternative model assumes that the company's yearly profits will increase in a geometric sequence with common ratio 1.06. Using this alternative model and again taking the profit in 2001 to be £54 000,
  1. find the predicted profit for the year 2011. [3]
Edexcel C2 Q33
13 marks Moderate -0.3
  1. Solve, for \(0° < x < 180°\), the equation $$\sin (2x + 50°) = 0.6,$$ giving your answers to 1 decimal place. [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 Q34
6 marks Moderate -0.8
  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 decimals places. [4]
Edexcel C2 Q35
9 marks Standard +0.3
The sequence \(u_1, u_2, u_3, \ldots, u_n\) is defined by the recurrence relation $$u_{n+1} = pu_n + 5, \quad u_1 = 2, \text{ where } p \text{ is a constant.}$$ 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 Q36
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 Q37
10 marks Standard +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 Q38
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 Q39
11 marks Standard +0.3
\includegraphics{figure_10} Figure 1 shows the cross-section \(ABCD\) of a chocolate bar, where \(AB\), \(CD\) and \(AD\) are straight lines and \(M\) is the mid-point of \(AD\). The length \(AD\) is 28 mm, and \(BC\) is an arc of a circle with centre \(M\). Taking \(A\) as the origin, \(B\), \(C\) and \(D\) have coordinates \((7, 24)\), \((21, 24)\) and \((28, 0)\) respectively.
  1. Show that the length of \(BM\) is 25 mm. [1]
  2. Show that, to 3 significant figures, \(\angle BMC = 0.568\) radians. [3]
  3. Hence calculate, in mm\(^2\), the area of the cross-section of the chocolate bar. [5]
Given that this chocolate bar has length 85 mm,
  1. calculate, to the nearest cm\(^3\), the volume of the bar. [2]
Edexcel C2 Q40
12 marks Moderate -0.3
\includegraphics{figure_11} The curve \(C\), shown in Fig. 2, represents the graph of $$y = \frac{x^2}{25}, \quad x \geq 0.$$ 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 Q41
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 Q42
6 marks Moderate -0.8
\includegraphics{figure_12} 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 Q1
6 marks Moderate -0.8
Given that \(p = \log_q 16\), express in terms of \(p\),
  1. \(\log_q 2\). [2]
  2. \(\log_q (8q)\). [4]
Edexcel C2 Q2
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 Q3
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 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
6 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. [2]
Edexcel C2 Q6
9 marks Standard +0.3
\(f(x) = 6x^3 + px^2 + qx + 8\), where \(p\) and \(q\) are constants. Given that \(f(x)\) is exactly divisible by \((2x - 1)\), and also that when \(f(x)\) is divided by \((x - 1)\) the remainder is \(-7\),
  1. find the value of \(p\) and the value of \(q\). [6]
  2. Hence factorise \(f(x)\) completely. [3]
Edexcel C2 Q7
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 Q8
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]