Mathematics Paper 2 November 2011

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1. A$\dfrac{-2}{x(x-1)(x+1)}$, from subtracting the numerators 3 and 5 directly
2. B$\dfrac{3 - 2x}{x(x-1)(x+1)}$, after taking $x(x-1)(x+1)$ as the common denominator
3. C$\dfrac{8x - 3}{x(x-1)(x+1)}$, after adding rather than subtracting the second numerator
4. D$\dfrac{3x + 3 - 5x}{(x^2-x)(x^2-1)}$, using the product of the two denominators

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1. A$(2y - 1)(x + z)$, by grouping the x terms and the z terms separately
2. B$(2y + 1)(x - z)$, taking $z$ out of the last two terms with the wrong sign
3. C$x(2y - 1) + z(2y + 1)$, which is as far as the grouping can be taken
4. D$(2xy - x)(2yz - z)$, treating the two grouped pairs as factors

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1. A$x = \pm\sqrt{4b^2 - a^2}$, with the two terms taken the other way round
2. B$x = \pm\sqrt{a^2 - 4b^2}$, after doubling and then squaring both sides
3. C$x = a - 2b$, from taking the square root of each term separately
4. D$x = \pm\sqrt{a^2 - \frac{1}{4}b^2}$, from squaring the $\frac{1}{2}$ instead of clearing it

1. A$\dfrac{2\pi d^3}{3}$ for the hemisphere plus $\pi d^2 h$ for the cylinder
2. B$\dfrac{\pi d^3}{12}$ for the hemisphere plus $\pi d^2 h$ for the cylinder
3. C$\dfrac{\pi d^3}{6}$ for the hemisphere plus $\dfrac{\pi d^2 h}{4}$ for the cylinder, halving the radius but not the sphere
4. D$\dfrac{\pi d^3}{12}$ for the hemisphere plus $\dfrac{\pi d^2 h}{4}$ for the cylinder

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1. A$x = 3,50$ or $x = -0,50$
2. B$x = 2,27$ or $x = -1,77$
3. C$x = 2,77$ or $x = -1,27$
4. D$x = 1,27$ or $x = -2,77$

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1. A$\begin{pmatrix} 3 & 1 \\ 2 & 4 \end{pmatrix}$
2. B$\begin{pmatrix} 3 & 7 \\ 2 & 5 \end{pmatrix}$
3. C$\begin{pmatrix} 3 & 7 \\ 2 & 4 \end{pmatrix}$
4. D$\begin{pmatrix} 3 & 2 \\ 7 & 4 \end{pmatrix}$

1. A$\begin{pmatrix} 0 & -\frac{1}{2} \\ -1 & \frac{3}{2} \end{pmatrix}$
2. B$\begin{pmatrix} 0 & \frac{1}{2} \\ 1 & -\frac{3}{2} \end{pmatrix}$
3. C$\begin{pmatrix} 0 & \frac{1}{3} \\ \frac{1}{2} & 0 \end{pmatrix}$
4. D$\begin{pmatrix} 3 & -1 \\ -2 & 0 \end{pmatrix}$

1. AThe bisector of the angle that AB makes at A, drawn from A
2. BA circle of radius 9 cm drawn with its centre at the midpoint of AB
3. CThe perpendicular bisector of AB, crossing AB at its midpoint
4. DThe pair of lines drawn parallel to AB, one on each side of it

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1. AA circle of radius 5.7 cm drawn with its centre at B
2. BA pair of arcs of radius 5.7 cm struck on each side of B
3. CThe perpendicular bisector of a line of length 5.7 cm from B
4. DA straight line drawn 5.7 cm away from B and parallel to it

1. AAbout $118^\circ$, so the angle at D is only just obtuse and close to a right angle
2. BAbout $141^\circ$, so the angle at D is reflex-free but obtuse
3. CAbout $205^\circ$, so the angle at D is reflex and turns past a straight line
4. DAbout $96^\circ$, so the angle at D is close to a right angle

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1. A496 cm$^2$, since only the two whole cuboids present any surface to the outside at all
2. B546 cm$^2$, since the two hidden circles come off and the curved surface goes on
3. C521 cm$^2$, since both hidden circles come off and no curved surface goes on
4. D571 cm$^2$, since the curved surface is added to both whole cuboid surfaces

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1. A$x = 0$, $x = 1,73$ or $x = -1,73$
2. B$x = 0$ or $x = 3$ only, from taking a square root of each side
3. C$x = 0$, $x = 1,73$ or $x = 3$
4. D$x = 1,73$ or $x = -1,73$ only, from dividing both sides by $x$

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1. AAbout 0 square units, since the curve and the line meet inside the strip and cancel
2. BAbout 3 square units, since the strip is 1,5 wide and about 2 units deep
3. CAbout 1 square unit, since the curve and the line meet inside the strip at $x = 1,41$
4. DAbout 5 square units, since $y$ on the curve reaches 3,375 at the right hand edge

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1. A$23,3^\circ$
2. B$136,7^\circ$
3. C$316,7^\circ$
4. D$363,3^\circ$

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1. A$140 \le h < 150$, the group holding 6 pupils
2. B$180 \le h < 190$, the group holding 1 pupil, the tallest of them all
3. C$160 \le h < 170$, the group holding 7 pupils
4. D$150 \le h < 160$, the group holding 10 pupils

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1. A$\mathbf{b} - \mathbf{a}$
2. B$\frac{1}{2}\mathbf{a} - \frac{1}{2}\mathbf{b}$
3. C$\frac{1}{2}\mathbf{a} + \frac{1}{2}\mathbf{b}$
4. D$\frac{1}{2}\mathbf{b} - \frac{1}{2}\mathbf{a}$

1. A$\frac{5}{4}\mathbf{a} + \frac{5}{6}\mathbf{b}$
2. B$\frac{7}{4}\mathbf{a} + \frac{5}{6}\mathbf{b}$
3. C$\frac{3}{4}\mathbf{a} + \frac{5}{6}\mathbf{b}$
4. D$\frac{5}{4}\mathbf{a} + \frac{5}{3}\mathbf{b}$

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1. AA rotation of $90^\circ$ anticlockwise about the point $(-1; 0)$
2. BA reflection in the line drawn through $(-1; 0)$ and $(0; -1)$
3. CA rotation of $180^\circ$ about the point $(-1,5; -0,5)$ midway between the two
4. DA rotation of $90^\circ$ clockwise about the point $(-1; 0)$

1. AA translation of $\binom{-2}{-4}$ on its own, with no reflection at all
2. BA rotation of $180^\circ$ about the point $(-0,5; 0,5)$ between the shapes
3. CA reflection in the line $y = 1$ on its own, with no further movement along it
4. DA reflection in the line $y = 1$ together with a translation of $\binom{-2}{0}$

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