SPS SPS SM Pure (SPS SM Pure) 2021 May

The function f is defined for all non-negative values of \(x\) by $$f(x) = 3 + \sqrt{x}.$$

1. Evaluate \(f(169)\). [2]
2. Find an expression for \(f^{-1}(x)\) in terms of \(x\). [2]
3. On a single diagram sketch the graphs of \(y = f(x)\) and \(y = f^{-1}(x)\), indicating how the two graphs are related. [3]

1. Use the trapezium rule, with four strips each of width \(0.5\), to estimate the value of $$\int_0^2 e^{x^2} dx$$ giving your answer correct to 3 significant figures. [3]
2. Explain how the trapezium rule could be used to obtain a more accurate estimate. [1]

Vector \(\mathbf{v} = a\mathbf{i} + 0.6\mathbf{j}\), where \(a\) is a constant.

1. Given that the direction of \(\mathbf{v}\) is \(45°\), state the value of \(a\). [1]
2. Given instead that \(\mathbf{v}\) is parallel to \(8\mathbf{i} + 3\mathbf{j}\), find the value of \(a\). [2]
3. Given instead that \(\mathbf{v}\) is a unit vector, find the possible values of \(a\). [3]

Prove that \(\sqrt{2}\cos(2\theta + 45°) \equiv \cos^2\theta - 2\sin\theta\cos\theta - \sin^2\theta\), where \(\theta\) is measured in degrees. [3]

1. Show that \(\sqrt{\frac{1-x}{1+x}} \approx 1 - x + \frac{1}{2}x^2\), for \(|x| < 1\). [5]
2. By taking \(x = \frac{2}{7}\), show that \(\sqrt{5} \approx \frac{111}{49}\). [3]

Shona makes the following claim. "\(n\) is an even positive integer greater than \(2 \Rightarrow 2^n - 1\) is not prime" Prove that Shona's claim is true. [4]

A curve has parametric equations $$x = 2\sin t, \quad y = \cos 2t + 2\sin t$$ for \(-\frac{1}{2}\pi \leqslant t \leqslant \frac{1}{2}\pi\).

1. Show that \(\frac{dy}{dx} = 1 - 2\sin t\) and hence find the coordinates of the stationary point. [5]
2. Find the cartesian equation of the curve. [3]
3. State the set of values that \(x\) can take and hence sketch the curve. [3]

In this question you must show detailed reasoning. The \(n\)th term of a geometric progression is denoted by \(g_n\) and the \(n\)th term of an arithmetic progression is denoted by \(a_n\). It is given that \(g_1 = a_1 = 1 + \sqrt{5}\), \(g_2 = a_2\) and \(g_3 + a_3 = 0\). Given also that the geometric progression is convergent, show that its sum to infinity is \(4 + 2\sqrt{5}\). [12]

In this question you must show detailed reasoning. \includegraphics{figure_9} The diagram shows the curve \(y = \frac{4\cos 2x}{3 - \sin 2x}\) for \(x > 0\), and the normal to the curve at the point \((\frac{1}{4}\pi, 0)\). Show that the exact area of the shaded region enclosed by the curve, the normal to the curve and the \(y\)-axis is \(\ln \frac{2}{3} + \frac{1}{128}\pi^2\). [10]