AQA C2 (Core Mathematics 2) 2009 June

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]

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]

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]

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]

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]

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]

A geometric series has second term \(375\) and fifth term \(81\).

1. 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]

1. Given that \(\frac{\sin \theta - \cos \theta}{\cos \theta} = 4\), prove that \(\tan \theta = 5\). [2]
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]

1. 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]