4.07f - OCR Spec

17 marks Challenging +1.3

Show that $$\tanh^{-1} x = \frac{1}{2} \ln \left(\frac{1+x}{1-x}\right), \quad \text{where } -1 < x < 1.$$ [4]
Given that $$a \cosh x + b \sinh x \equiv r \cosh(x + \alpha), \quad \text{where } a > b > 0,$$ show that $$\alpha = \frac{1}{2} \ln \left(\frac{a+b}{a-b}\right)$$ and find an expression for $r$ in terms of $a$ and $b$. [7]
Hence solve the equation $$5 \cosh x + 4 \sinh x = 10,$$ giving your answers correct to three significant figures. [6]

10 marks Challenging +1.3

Prove that $$\tanh^{-1}(x) = \frac{1}{2} \ln \left(\frac{1+x}{1-x}\right) \quad -k < x < k$$ stating the value of the constant $k$. [5]
Hence, or otherwise, solve the equation $$2x = \tanh\left(\ln \sqrt{2-3x}\right)$$ [5]

Challenging +1.8

Use a hyperbolic substitution and calculus to show that $$\int \frac{x^2}{\sqrt{x^2 - 1}} dx = \frac{1}{2}\left[x\sqrt{x^2 - 1} + \arcosh x\right] + k$$ where $k$ is an arbitrary constant. (6) \includegraphics{figure_8} Figure 1 shows a sketch of part of the curve $C$ with equation $$y = \frac{4}{15}x \arcosh x \quad x \geqslant 1$$ The finite region $R$, shown shaded in Figure 1, is bounded by the curve $C$, the $x$-axis and the line with equation $x = 3$
Using algebraic integration and the result from part (a), show that the area of $R$ is given by $$\frac{1}{15}\left[17\ln\left(3 + 2\sqrt{2}\right) - 6\sqrt{2}\right]$$ (5) This is the last question on the paper.

9 marks Standard +0.3

Given that $u = \tanh x$, use the definition of $\tanh x$ in terms of exponentials to show that $$x = \frac{1}{2}\ln\left(\frac{1+u}{1-u}\right).$$ [4]
Solve the equation $4\tanh^2 x + \tanh x - 3 = 0$, giving the solution in the form $a\ln b$ where $a$ and $b$ are rational numbers to be determined. [4]
Explain why the equation in part (b) has only one root. [1]

6 marks Standard +0.3

Solve exactly the equation $$5 \cosh x - \sinh x = 7,$$ giving your answers in logarithmic form. [6]

4.07f Inverse hyperbolic: logarithmic forms