OCR C3 (Core Mathematics 3) 2016 June

1 Find the equation of the tangent to the curve $$y = 3 x ^ { 2 } ( x + 2 ) ^ { 6 }$$ at the point \(( - 1,3 )\), giving your answer in the form \(y = m x + c\).

2 Find

1. \(\int \left( 2 - \frac { 1 } { x } \right) ^ { 2 } \mathrm {~d} x\),
2. \(\int ( 4 x + 1 ) ^ { \frac { 1 } { 3 } } \mathrm {~d} x\).

3 The mass of a substance is decreasing exponentially. Its mass is \(m\) grams at time \(t\) years. The following table shows certain values of \(t\) and \(m\).

1. Find the values missing from the table.
2. Determine the value of \(t\), correct to the nearest integer, for which the mass is 50 grams.

4 It is given that \(A\) and \(B\) are angles such that $$\sec ^ { 2 } A - \tan A = 13 \quad \text { and } \quad \sin B \sec ^ { 2 } B = 27 \cos B \operatorname { cosec } ^ { 2 } B$$ Find the possible exact values of \(\tan ( A - B )\).

5
\includegraphics[max width=\textwidth, alt={}, center]{6d15cb4d-f540-488b-b94e-7a494f192ba5-2_469_721_1932_662} The diagram shows the curves \(y = \mathrm { e } ^ { 2 x }\) and \(y = 8 \mathrm { e } ^ { - x }\). The shaded region is bounded by the curves and the \(y\)-axis. Without using a calculator,

1. solve an appropriate equation to show that the curves intersect at a point for which \(x = \ln 2\),
2. find the area of the shaded region, giving your answer in simplified form.

6 The curves \(C _ { 1 }\) and \(C _ { 2 }\) have equations $$y = \ln ( 4 x - 7 ) + 18 \quad \text { and } \quad y = a \left( x ^ { 2 } + b \right) ^ { \frac { 1 } { 2 } }$$ respectively, where \(a\) and \(b\) are positive constants. The point \(P\) lies on both curves and has \(x\)-coordinate 2 . It is given that the gradient of \(C _ { 1 }\) at \(P\) is equal to the gradient of \(C _ { 2 }\) at \(P\). Find the values of \(a\) and \(b\).

7

1. By sketching the curves \(y = x ( 2 x + 5 )\) and \(y = \cos ^ { - 1 } x\) (where \(y\) is in radians) in a single diagram, show that the equation \(x ( 2 x + 5 ) = \cos ^ { - 1 } x\) has exactly one real root.
2. Use the iterative formula $$x _ { n + 1 } = \frac { \cos ^ { - 1 } x _ { n } } { 2 x _ { n } + 5 } \text { with } x _ { 1 } = 0.25$$ to find the root correct to 3 significant figures. Show the result of each iteration correct to at least 4 significant figures.
3. Two new curves are obtained by transforming each of the curves \(y = x ( 2 x + 5 )\) and \(y = \cos ^ { - 1 } x\) by the pair of transformations:
   reflection in the \(x\)-axis followed by reflection in the \(y\)-axis.
   State an equation of each of the new curves and determine the coordinates of their point of intersection, giving each coordinate correct to 3 significant figures.

8 The functions f and g are defined for all real values of \(x\) by $$\mathrm { f } ( x ) = | 2 x + a | + 3 a \quad \text { and } \quad \mathrm { g } ( x ) = 5 x - 4 a$$ where \(a\) is a positive constant.

1. State the range of f and the range of g .
2. State why f has no inverse, and find an expression for \(\mathrm { g } ^ { - 1 } ( x )\).
3. Solve for \(x\) the equation \(\operatorname { gf } ( x ) = 31 a\).
4. Show that \(\sin 2 \theta ( \tan \theta + \cot \theta ) \equiv 2\).
5. Hence
   (a) find the exact value of \(\tan \frac { 1 } { 12 } \pi + \tan \frac { 1 } { 8 } \pi + \cot \frac { 1 } { 12 } \pi + \cot \frac { 1 } { 8 } \pi\),
   (b) solve the equation \(\sin 4 \theta ( \tan \theta + \cot \theta ) = 1\) for \(0 < \theta < \frac { 1 } { 2 } \pi\),
   (c) express \(( 1 - \cos 2 \theta ) ^ { 2 } \left( \tan \frac { 1 } { 2 } \theta + \cot \frac { 1 } { 2 } \theta \right) ^ { 3 }\) in terms of \(\sin \theta\).