OCR C3 (Core Mathematics 3)

Evaluate $$\int_2^6 \sqrt{3x-2} \, dx.$$ [4]

Differentiate each of the following with respect to \(x\) and simplify your answers.

1. \(\frac{6}{\sqrt{2x-7}}\) [2]
2. \(x^2 e^{-x}\) [3]

1. Prove the identity $$\sqrt{2} \cos (x + 45)° + 2 \cos (x - 30)° \equiv (1 + \sqrt{3}) \cos x°.$$ [4]
2. Hence, find the exact value of \(\cos 75°\) in terms of surds. [3]

$$\text{f}(x) = x^2 + 5x - 2 \sec x, \quad x \in \mathbb{R}, \quad -\frac{\pi}{2} < x < \frac{\pi}{2}.$$

1. Show that the equation \(\text{f}(x) = 0\) has a root, \(\alpha\), such that \(1 < \alpha < 1.5\) [2]
2. Show that a suitable rearrangement of the equation \(\text{f}(x) = 0\) leads to the iterative formula $$x_{n+1} = \cos^{-1} \left( \frac{2}{x_n^2 + 5x_n} \right).$$ [3]
3. Use the iterative formula in part (ii) with a starting value of 1.25 to find \(\alpha\) correct to 3 decimal places. You should show the result of each iteration. [3]

The function f is defined by $$\text{f}(x) \equiv 2 + \ln (3x - 2), \quad x \in \mathbb{R}, \quad x > \frac{2}{3}.$$

1. Find the exact value of \(\text{f}(1)\). [2]
2. Find an equation for the tangent to the curve \(y = \text{f}(x)\) at the point where \(x = 1\). [4]
3. Find an expression for \(\text{f}^{-1}(x)\). [2]

1. Sketch on the same diagram the graphs of \(y = |x| - a\) and \(y = |3x + 5a|\), where \(a\) is a positive constant. Show on your diagram the coordinates of any points where each graph meets the coordinate axes. [5]
2. Solve the equation $$|x| - a = |3x + 5a|.$$ [4]

\includegraphics{figure_7} The diagram shows the curve with equation \(y = 2x - e^{\frac{1}{2}x}\). The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = 2\) and \(x = 4\).

1. Find the area of the shaded region, giving your answer in terms of e. [4]

The shaded region is rotated through four right angles about the \(x\)-axis.

1. Using Simpson's rule with two strips, estimate the volume of the solid formed. [5]

1. Sketch on the same diagram the graphs of $$y = \sin^{-1} x, \quad -1 \leq x \leq 1$$ and $$y = \cos^{-1} (2x), \quad -\frac{1}{2} \leq x \leq \frac{1}{2}.$$ [3]

Given that the graphs intersect at the point with coordinates \((a, b)\),

1. show that \(\tan b = \frac{1}{2}\), [3]
2. find the value of \(a\) in the form \(k\sqrt{5}\). [4]

$$\text{f}(x) = e^{3x + 1} - 2, \quad x \in \mathbb{R}.$$

1. State the range of f. [1]

The curve \(y = \text{f}(x)\) meets the \(y\)-axis at the point \(P\) and the \(x\)-axis at the point \(Q\).

1. Find the exact coordinates of \(P\) and \(Q\). [3]
2. Show that the tangent to the curve at \(P\) has the equation $$y = 3ex + e - 2.$$ [4]
3. Find to 3 significant figures the \(x\)-coordinate of the point where the tangent to the curve at \(P\) meets the tangent to the curve at \(Q\). [4]