Questions C2 (1550 questions)

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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]
AQA C2 2009 June Q1
5 marks Moderate -0.8
The triangle \(ABC\), shown in the diagram, is such that \(AB = 7\) cm, \(AC = 5\) cm, \(BC = 8\) cm and angle \(ABC = \theta\). \includegraphics{figure_1}
  1. Show that \(\theta = 38.2°\), correct to the nearest \(0.1°\). [3]
  2. Calculate the area of triangle \(ABC\), giving your answer, in cm\(^2\), to three significant figures. [2]
AQA C2 2009 June Q2
8 marks Moderate -0.8
  1. Write down the value of \(n\) given that \(\frac{1}{x^3} = x^n\). [1]
  2. Expand \(\left(1 + \frac{3}{x^2}\right)^2\). [2]
  3. Hence find \(\int \left(1 + \frac{3}{x^2}\right)^2 dx\). [3]
  4. Hence find the exact value of \(\int_1^3 \left(1 + \frac{3}{x^2}\right)^2 dx\). [2]
AQA C2 2009 June Q3
7 marks Moderate -0.3
The \(n\)th term of a sequence is \(u_n\). The sequence is defined by $$u_{n+1} = ku_n + 12$$ where \(k\) is a constant. The first two terms of the sequence are given by $$u_1 = 16 \quad u_2 = 24$$
  1. Show that \(k = 0.75\). [2]
  2. Find the value of \(u_3\) and the value of \(u_4\). [2]
  3. The limit of \(u_n\) as \(n\) tends to infinity is \(L\).
    1. Write down an equation for \(L\). [1]
    2. Hence find the value of \(L\). [2]
AQA C2 2009 June Q4
6 marks Moderate -0.3
  1. Use the trapezium rule with four ordinates (three strips) to find an approximate value for \(\int_0^6 \sqrt{x^3 + 1} dx\), giving your answer to four significant figures. [4]
  2. The curve with equation \(y = \sqrt{x^3 + 1}\) is stretched parallel to the \(x\)-axis with scale factor \(\frac{1}{2}\) to give the curve with equation \(y = f(x)\). Write down an expression for \(f(x)\). [2]
AQA C2 2009 June Q5
13 marks Standard +0.3
The diagram shows part of a curve with a maximum point \(M\). \includegraphics{figure_5} The equation of the curve is $$y = 15x^{\frac{3}{2}} - x^{\frac{5}{2}}$$
  1. Find \(\frac{dy}{dx}\). [3]
  2. Hence find the coordinates of the maximum point \(M\). [4]
  3. The point \(P(1, 14)\) lies on the curve. Show that the equation of the tangent to the curve at \(P\) is \(y = 20x - 6\). [3]
  4. The tangents to the curve at the points \(P\) and \(M\) intersect at the point \(R\). Find the length of \(RM\). [3]
AQA C2 2009 June Q6
6 marks Moderate -0.3
The diagram shows a sector \(OAB\) of a circle with centre \(O\) and radius \(r\) cm. \includegraphics{figure_6} The angle \(AOB\) is \(1.2\) radians. The area of the sector is \(33.75\) cm\(^2\). Find the perimeter of the sector. [6]
AQA C2 2009 June Q7
11 marks Moderate -0.3
A geometric series has second term \(375\) and fifth term \(81\).
    1. Show that the common ratio of the series is \(0.6\). [3]
    2. Find the first term of the series. [2]
  2. Find the sum to infinity of the series. [2]
  3. The \(n\)th term of the series is \(u_n\). Find the value of \(\sum_{n=6}^{\infty} u_n\). [4]
AQA C2 2009 June Q8
9 marks Moderate -0.3
  1. Given that \(\frac{\sin \theta - \cos \theta}{\cos \theta} = 4\), prove that \(\tan \theta = 5\). [2]
    1. Use an appropriate identity to show that the equation $$2 \cos^2 x - \sin x = 1$$ can be written as $$2 \sin^2 x + \sin x - 1 = 0$$ [2]
    2. Hence solve the equation $$2 \cos^2 x - \sin x = 1$$ giving all solutions in the interval \(0° \leq x \leq 360°\). [5]
AQA C2 2009 June Q9
10 marks Moderate -0.8
    1. Find the value of \(p\) for which \(\sqrt{125} = 5^p\). [2]
    2. Hence solve the equation \(5^{2x} = \sqrt{125}\). [1]
  2. Use logarithms to solve the equation \(3^{2x-1} = 0.05\), giving your value of \(x\) to four decimal places. [3]
  3. It is given that $$\log_a x = 2(\log_a 3 + \log_a 2) - 1$$ Express \(x\) in terms of \(a\), giving your answer in a form not involving logarithms. [4]
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]