OCR C3 (Core Mathematics 3)

Show that $$\int_1^7 \frac{2}{4x-1} \, dx = \ln 3.$$ [4]

Find the set of values of \(x\) such that $$|3x + 1| \leq |x - 2|.$$ [5]

Find all values of \(\theta\) in the interval \(-180 < \theta < 180\) for which $$\tan^2 \theta^\circ + \sec \theta^\circ = 1.$$ [6]

Solve each equation, giving your answers in exact form.

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

1. Prove, by counter-example, that the statement "\(\cosec \theta - \sin \theta > 0\) for all values of \(\theta\) in the interval \(0 < \theta < \pi\)" is false. [2]
2. Find the values of \(\theta\) in the interval \(0 < \theta < \pi\) such that $$\cosec \theta - \sin \theta = 2,$$ giving your answers to 2 decimal places. [5]

The curve \(C\) has the equation \(y = x^2 - 5x + 2\ln \frac{x}{3}\), \(x > 0\).

1. Show that the normal to \(C\) at the point where \(x = 3\) has the equation $$3x + 5y + 21 = 0.$$ [5]
2. Find the \(x\)-coordinates of the stationary points of \(C\). [3]

\includegraphics{figure_7} The diagram shows the curve \(y = \text{f}(x)\) which has a maximum point at \((-45, 7)\) and a minimum point at \((135, -1)\).

1. Showing the coordinates of any stationary points, sketch the curve with equation \(y = 1 + 2\text{f}(x)\). [3]

Given that $$\text{f}(x) = A + 2\sqrt{2} \cos x° - 2\sqrt{2} \sin x°, \quad x \in \mathbb{R}, \quad -180 \leq x \leq 180,$$ where \(A\) is a constant,

1. show that f\((x)\) can be expressed in the form $$\text{f}(x) = A + R \cos (x + \alpha)°,$$ where \(R > 0\) and \(0 < \alpha < 90\), [3]
2. state the value of \(A\), [1]
3. find, to 1 decimal place, the \(x\)-coordinates of the points where the curve \(y = \text{f}(x)\) crosses the \(x\)-axis. [4]

The function f is defined by $$\text{f}(x) \equiv 3 - x^2, \quad x \in \mathbb{R}, \quad x \geq 0.$$

1. State the range of f. [1]
2. Sketch the graphs of \(y = \text{f}(x)\) and \(y = \text{f}^{-1}(x)\) on the same diagram. [3]
3. Find an expression for f\(^{-1}(x)\) and state its domain. [3]

The function g is defined by $$\text{g}(x) \equiv \frac{8}{3-x}, \quad x \in \mathbb{R}, \quad x \neq 3.$$

1. Evaluate fg\((-3)\). [2]
2. Solve the equation $$\text{f}^{-1}(x) = \text{g}(x).$$ [3]

A curve has the equation \(y = (2x + 3)\mathrm{e}^{-x}\).

1. Find the exact coordinates of the stationary point of the curve. [4]

The curve crosses the \(y\)-axis at the point \(P\).

1. Find an equation for the normal to the curve at \(P\). [2]

The normal to the curve at \(P\) meets the curve again at \(Q\).

1. Show that the \(x\)-coordinate of \(Q\) lies between \(-2\) and \(-1\). [3]
2. Use the iterative formula $$x_{n+1} = \frac{3 - 3\mathrm{e}^{x_n}}{\mathrm{e}^{x_n} - 2},$$ with \(x_0 = -1\), to find \(x_1, x_2, x_3\) and \(x_4\). Give the value of \(x_4\) to 2 decimal places. [2]
3. Show that your value for \(x_4\) is the \(x\)-coordinate of \(Q\) correct to 2 decimal places. [2]