OCR C3 (Core Mathematics 3) Specimen

1 Solve the inequality \(| 2 x + 1 | > | x - 1 |\).

2

1. Prove the identity $$\sin \left( x + 30 ^ { \circ } \right) + ( \sqrt { } 3 ) \cos \left( x + 30 ^ { \circ } \right) \equiv 2 \cos x$$ where \(x\) is measured in degrees.
2. Hence express \(\cos 15 ^ { \circ }\) in surd form.

3 The sequence defined by the iterative formula $$x _ { n + 1 } = \sqrt [ 3 ] { } \left( 17 - 5 x _ { n } \right)$$ with \(x _ { 1 } = 2\), converges to \(\alpha\).

1. Use the iterative formula to find \(\alpha\) correct to 2 decimal places. You should show the result of each iteration.
2. Find a cubic equation of the form $$x ^ { 3 } + c x + d = 0$$ which has \(\alpha\) as a root.
3. Does this cubic equation have any other real roots? Justify your answer.

4
\includegraphics[max width=\textwidth, alt={}, center]{b6b6e55a-a5ba-466c-ac9f-b5ef5bca7a3c-2_419_707_1576_660} The diagram shows the curve $$y = \frac { 1 } { \sqrt { } ( 4 x + 1 ) }$$ The region \(R\) (shaded in the diagram) is enclosed by the curve, the axes and the line \(x = 2\).

1. Show that the exact area of \(R\) is 1 .
2. The region \(R\) is rotated completely about the \(x\)-axis. Find the exact volume of the solid formed.

5 At time \(t\) minutes after an oven is switched on, its temperature \(\theta ^ { \circ } \mathrm { C }\) is given by $$\theta = 200 - 180 \mathrm { e } ^ { - 0.1 t }$$

1. State the value which the oven's temperature approaches after a long time.
2. Find the time taken for the oven's temperature to reach \(150 ^ { \circ } \mathrm { C }\).
3. Find the rate at which the temperature is increasing at the instant when the temperature reaches \(150 ^ { \circ } \mathrm { C }\).

6 The function f is defined by $$\mathrm { f } : x \mapsto 1 + \sqrt { } x \quad \text { for } x \geqslant 0$$

1. State the domain and range of the inverse function \(\mathrm { f } ^ { - 1 }\).
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
3. By considering the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), show that the solution to the equation $$\mathrm { f } ( x ) = \mathrm { f } ^ { - 1 } ( x )$$ is \(x = \frac { 1 } { 2 } ( 3 + \sqrt { } 5 )\).

7

1. Write down the formula for \(\tan 2 x\) in terms of \(\tan x\).
2. By letting \(\tan x = t\), show that the equation $$4 \tan 2 x + 3 \cot x \sec ^ { 2 } x = 0$$ becomes $$3 t ^ { 4 } - 8 t ^ { 2 } - 3 = 0$$
3. Hence find all the solutions of the equation $$4 \tan 2 x + 3 \cot x \sec ^ { 2 } x = 0$$ which lie in the interval \(0 \leqslant x \leqslant 2 \pi\).

8
\includegraphics[max width=\textwidth, alt={}, center]{b6b6e55a-a5ba-466c-ac9f-b5ef5bca7a3c-4_476_608_287_756} The diagram shows the curve \(y = ( \ln x ) ^ { 2 }\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. The point \(P\) on the curve is the point at which the gradient takes its maximum value. Show that the tangent at \(P\) passes through the point \(( 0 , - 1 )\).

9
\includegraphics[max width=\textwidth, alt={}, center]{b6b6e55a-a5ba-466c-ac9f-b5ef5bca7a3c-4_424_707_1260_724} The diagram shows the curve \(y = \tan ^ { - 1 } x\) and its asymptotes \(y = \pm a\).

1. State the exact value of \(a\).
2. Find the value of \(x\) for which \(\tan ^ { - 1 } x = \frac { 1 } { 2 } a\). The equation of another curve is \(y = 2 \tan ^ { - 1 } ( x - 1 )\).
3. Sketch this curve on a copy of the diagram, and state the equations of its asymptotes in terms of \(a\).
4. Verify by calculation that the value of \(x\) at the point of intersection of the two curves is 1.54 , correct to 2 decimal places. Another curve (which you are not asked to sketch) has equation \(y = \left( \tan ^ { - 1 } x \right) ^ { 2 }\).
5. Use Simpson's rule, with 4 strips, to find an approximate value for \(\int _ { 0 } ^ { 1 } \left( \tan ^ { - 1 } x \right) ^ { 2 } \mathrm {~d} x\).