OCR C3 (Core Mathematics 3)

1. (i) Differentiate \(x ^ { 3 } \ln x\) with respect to \(x\).
   (ii) Given that

$$x = \frac { y + 1 } { 3 - 2 y }$$ find and simplify an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(y\).

2.
\includegraphics[max width=\textwidth, alt={}, center]{687756c0-2038-4077-8c5c-fe0ca0f6ce65-1_638_677_749_443} The diagram shows the curves \(y = 3 + 2 \mathrm { e } ^ { x }\) and \(y = \mathrm { e } ^ { x + 2 }\) which cross the \(y\)-axis at the points \(A\) and \(B\) respectively.

1. Write down the coordinates of \(A\) and \(B\). The two curves intersect at the point \(C\).
2. Find an expression for the \(x\)-coordinate of \(C\) and show that the \(y\)-coordinate of \(C\) is \(\frac { 3 \mathrm { e } ^ { 2 } } { \mathrm { e } ^ { 2 } - 2 }\).

3. The functions f and g are defined by $$\begin{aligned} & \mathrm { f } ( x ) \equiv 6 x - 1 , \quad x \in \mathbb { R } ,
& \mathrm {~g} ( x ) \equiv \log _ { 2 } ( 3 x + 1 ) , \quad x \in \mathbb { R } , \quad x > - \frac { 1 } { 3 } . \end{aligned}$$

1. Evaluate \(\mathrm { gf } ( 1 )\).
2. Find an expression for \(\mathrm { g } ^ { - 1 } ( x )\).
3. Find, in terms of natural logarithms, the solution of the equation $$\mathrm { fg } ^ { - 1 } ( x ) = 2$$

1. (i) Use the identity for \(\cos ( A + B )\) to prove that

$$\cos 2 x \equiv 2 \cos ^ { 2 } x - 1$$ (ii) Prove that, for \(\cos x \neq 0\), $$2 \cos x - \sec x \equiv \sec x \cos 2 x$$ (iii) Hence, or otherwise, find the values of \(x\) in the interval \(0 \leq x \leq 180 ^ { \circ }\) for which $$2 \cos x - \sec x \equiv 2 \cos 2 x$$

1. (i) Show that the equation

$$2 \sin x + \sec \left( x + \frac { \pi } { 6 } \right) = 0$$ can be written as $$\sqrt { 3 } \sin x \cos x + \cos ^ { 2 } x = 0$$ (ii) Hence, or otherwise, find in terms of \(\pi\) the solutions of the equation $$2 \sin x + \sec \left( x + \frac { \pi } { 6 } \right) = 0$$ for \(x\) in the interval \(0 \leq x \leq \pi\).

6.
\includegraphics[max width=\textwidth, alt={}, center]{687756c0-2038-4077-8c5c-fe0ca0f6ce65-2_444_825_1571_516} The diagram shows the curve with equation \(y = \sqrt { \frac { x } { x + 1 } }\).
The shaded region is bounded by the curve, the \(x\)-axis and the line \(x = 3\).

1. Use Simpson's rule with six strips to estimate the area of the shaded region. The shaded region is rotated through four right angles about the \(x\)-axis.
2. Show that the volume of the solid formed is \(\pi ( 3 - \ln 4 )\).

7. (i) Sketch on the same diagram the graphs of \(y = 4 a ^ { 2 } - x ^ { 2 }\) and \(y = | 2 x - a |\), where \(a\) is a positive constant. Show, in terms of \(a\), the coordinates of any points where each graph meets the coordinate axes.
(ii) Find the exact solutions of the equation $$4 - x ^ { 2 } = | 2 x - 1 |$$

1. A curve has the equation \(y = \frac { \mathrm { e } ^ { 2 } } { x } + \mathrm { e } ^ { x } , x \neq 0\).
   1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
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   2. Show that the curve has a stationary point in the interval [1.3,1.4].
   The point \(A\) on the curve has \(x\)-coordinate 2 .
2. Show that the tangent to the curve at \(A\) passes through the origin. The tangent to the curve at \(A\) intersects the curve again at the point \(B\).
   The \(x\)-coordinate of \(B\) is to be estimated using the iterative formula $$x _ { n + 1 } = - \frac { 2 } { 3 } \sqrt { 3 + 3 x _ { n } \mathrm { e } ^ { x _ { n } - 2 } }$$ with \(x _ { 0 } = - 1\).
3. Find \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\) to 7 significant figures and hence state the \(x\)-coordinate of \(B\) to 5 significant figures.