OCR C3 (Core Mathematics 3) 2012 January

1 Show that \(\int _ { \sqrt { 2 } } ^ { \sqrt { 6 } } \frac { 2 } { x } \mathrm {~d} x = \ln 3\).

2
\includegraphics[max width=\textwidth, alt={}, center]{89e54367-bb83-483a-add5-0527b71a5cac-2_490_713_447_660} The diagram shows part of the curve \(y = \frac { 6 } { ( 2 x + 1 ) ^ { 2 } }\). The shaded region is bounded by the curve and the lines \(x = 0 , x = 1\) and \(y = 0\). Find the exact volume of the solid produced when this shaded region is rotated completely about the \(x\)-axis.

3 Find the equation of the normal to the curve \(y = \frac { x ^ { 2 } + 4 } { x + 2 }\) at the point \(\left( 1 , \frac { 5 } { 3 } \right)\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

4 The acute angles \(\alpha\) and \(\beta\) are such that $$2 \cot \alpha = 1 \text { and } 24 + \sec ^ { 2 } \beta = 10 \tan \beta \text {. }$$

1. State the value of \(\tan \alpha\) and determine the value of \(\tan \beta\).
2. Hence find the exact value of \(\tan ( \alpha + \beta )\).

5
\includegraphics[max width=\textwidth, alt={}, center]{89e54367-bb83-483a-add5-0527b71a5cac-3_844_837_242_621} It is given that f is a one-one function defined for all real values. The diagram shows the curve with equation \(y = \mathrm { f } ( x )\). The coordinates of certain points on the curve are shown in the following table.

1. State the value of \(\mathrm { ff } ( 6 )\) and the value of \(\mathrm { f } ^ { - 1 } ( 8 )\).
2. On the copy of the diagram, sketch the curve \(y = \mathrm { f } ^ { - 1 } ( x )\), indicating how the curves \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) are related.
3. Use Simpson's rule with 6 strips to find an approximation to \(\int _ { 2 } ^ { 14 } \mathrm { f } ( x ) \mathrm { d } x\).

6
\includegraphics[max width=\textwidth, alt={}, center]{89e54367-bb83-483a-add5-0527b71a5cac-4_476_709_251_683} The diagram shows the curve with equation \(x = \ln \left( y ^ { 3 } + 2 y \right)\). At the point \(P\) on the curve, the gradient is 4 and it is given that \(P\) is close to the point with coordinates (7.5,12).

1. Find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\).
2. Show that the \(y\)-coordinate of \(P\) satisfies the equation $$y = \frac { 12 y ^ { 2 } + 8 } { y ^ { 2 } + 2 }$$
3. By first using an iterative process based on the equation in part (ii), find the coordinates of \(P\), giving each coordinate correct to 3 decimal places.

7

1. Substance \(A\) is decaying exponentially and its mass is recorded at regular intervals. At time \(t\) years, the mass, \(M\) grams, of substance \(A\) is given by $$M = 40 \mathrm { e } ^ { - 0.132 t }$$ (a) Find the time taken for the mass of substance \(A\) to decrease to \(25 \%\) of its value when \(t = 0\).
   (b) Find the rate at which the mass of substance \(A\) is decreasing when \(t = 5\).
2. Substance \(B\) is also decaying exponentially. Initially its mass was 40 grams and, two years later, its mass is 31.4 grams. Find the mass of substance \(B\) after a further year.

8

1. Express \(\cos 4 \theta\) in terms of \(\sin 2 \theta\) and hence show that \(\cos 4 \theta\) can be expressed in the form \(1 - k \sin ^ { 2 } \theta \cos ^ { 2 } \theta\), where \(k\) is a constant to be determined.
2. Hence find the exact value of \(\sin ^ { 2 } \left( \frac { 1 } { 24 } \pi \right) \cos ^ { 2 } \left( \frac { 1 } { 24 } \pi \right)\).
3. By expressing \(2 \cos ^ { 2 } 2 \theta - \frac { 8 } { 3 } \sin ^ { 2 } \theta \cos ^ { 2 } \theta\) in terms of \(\cos 4 \theta\), find the greatest and least possible values of $$2 \cos ^ { 2 } 2 \theta - \frac { 8 } { 3 } \sin ^ { 2 } \theta \cos ^ { 2 } \theta$$ as \(\theta\) varies.
   \includegraphics[max width=\textwidth, alt={}, center]{89e54367-bb83-483a-add5-0527b71a5cac-5_606_926_267_552} The function f is defined for all real values of \(x\) by $$\mathrm { f } ( x ) = k \left( x ^ { 2 } + 4 x \right) ,$$ where \(k\) is a positive constant. The diagram shows the curve with equation \(y = \mathrm { f } ( x )\).
4. The curve \(y = x ^ { 2 }\) can be transformed to the curve \(y = \mathrm { f } ( x )\) by the following sequence of transformations: a translation parallel to the \(x\)-axis,
   a translation parallel to the \(y\)-axis,
   a stretch. a translation parallel to the \(x\)-axis, a translation parallel to the \(y\)-axis, a stretch.
   Give details, in terms of \(k\) where appropriate, of these transformations.
5. Find the range of f in terms of \(k\).
6. It is given that there are three distinct values of \(x\) which satisfy the equation \(| \mathrm { f } ( x ) | = 20\). Find the value of \(k\) and determine exactly the three values of \(x\) which satisfy the equation in this case.