SPS SPS SM (SPS SM) 2021 November

Find \(\frac{dy}{dx}\) for the following functions, simplifying your answers as far as possible.

1. \(y = \cos x - 2 \sin 2x\) [2]
2. \(y = \frac{1}{2}x^4 + 2x^4 \ln x\) [3]
3. \(y = \frac{2e^{3x} - 1}{3e^{3x} - 1}\) [3]

1. Express \(\frac{5x+7}{(x+3)(x+1)^2}\) in partial fractions. In this question you must show all of your algebraic steps clearly. [3] The function \(f(x) = \frac{2-6x+5x^2}{x^2(1-2x)}\) can be written in the form; $$f(x) = \frac{-2}{x} + \frac{2}{x^2} + \frac{1}{1-2x}$$
2. Hence find the exact value of \(\int_2^3 \frac{2-6x+5x^2}{x^2(1-2x)} dx\) [3]

In this question you must show detailed algebraic reasoning. Find the coordinates of any stationary points on the curve below. $$y = (1 - 3x)(3 - x)^3$$ [5]

Find the equation of the normal to the curve \(y = 4 \ln(2x - 3)\) at the point where the curve crosses the \(x\) axis. Give your answer in the form \(ax + by + k = 0\) where \(a > 0\). [5]

1. Write \(\log_{16} y - \log_{16} x\) as a single logarithm. [1]
2. Solve the simultaneous equations, giving your answers in an exact form. $$\log_3 y = \log_3(9 - 6x) + 1$$ $$\log_{16} y - \log_{16} x = \frac{1}{4}$$ [3]

1. Prove the following trigonometric identities. You must show all of your algebraic steps clearly. $$(\cos x + \sin x)(\cos x - \sec x) \equiv 2 \cot 2x$$ [3]
2. Solve the following equation for \(x\) in the interval \(0 \leq x \leq \pi\). Giving your answers in terms of \(\pi\). $$\sin\left(2x + \frac{\pi}{6}\right) = \frac{1}{2}\sin\left(2x - \frac{\pi}{6}\right)$$ [4]

The diagram below represents the graph of the function \(y = (2x - 5)^4 - 1\) \includegraphics{figure_7}

1. Find the intersections of this graph with the \(x\) axis. [1]
2. Hence find the exact value of the area bounded by the curve and the \(x\) axis. [4]

1. Express \(2\sqrt{3} \cos 2x - 6 \sin 2x\) in the form \(R\cos(2x + \alpha)\) where \(R > 0\) and \(0 < \alpha < \frac{\pi}{2}\) [3]
2. Hence
   1. Solve the equation \(2\sqrt{3} \cos 2x - 6 \sin 2x = 6\) for \(0 \leq x \leq 2\pi\) Giving your answers in terms of \(\pi\). [3]
3. It can be shown that \(y = 9 \sin 2x + 4 \cos 2x\) can be written as \(y = \sqrt{97} \sin(2x + 24.0°)\)
   1. State the transformations in the order of occurrence which transform the curve \(y = 9 \sin 2x + 4 \cos 2x\) to the curve \(y = \sin x\) [3]
   2. Find the exact maximum and minimum values of the function; $$f(x) = \frac{1}{11 - 9 \sin 2x - 4 \cos 2x}$$ [2]

1. 1. Show that \(\cos^2 x \equiv \frac{1}{2} + \frac{1}{2}\cos 2x\) [1]
   2. Hence find \(\int 2\cos^2 4x \, dx\) [3]
2. Find \(\int \sin^3 x \, dx\) [3]

1. The parametric equations of a curve are \(x = \theta \cos \theta\) and \(y = \sin \theta\) Find the gradient of the curve at the point for which \(\theta = \pi\) [3]
2. A curve is defined parametrically by the equations; $$x = \cos \theta \qquad y = \left(\frac{\sin \theta}{2}\right)\left(\sin \frac{\theta}{2}\right)$$ Show that the cartesian equation of the curve can be written as \(y^2 = \frac{1}{8}(1-x)^2(1+x)\) [4]