# Question 1

**(a)** [2 marks]
- M1: Apply Vieta's formulas to $2x^2 + 6x + 7 = 0$
- A1: $a + b = -3$ and $ab = 3.5$ (or $\frac{7}{2}$)

**(b)** [5 marks]
- M1: Find $a^2 - 1 + b^2 - 1 = (a+b)^2 - 2ab - 2$
- M1: Calculate $(a+b)^2 - 2ab - 2 = 9 - 7 - 2 = 0$
- M1: Find $(a^2-1)(b^2-1) = a^2b^2 - a^2 - b^2 + 1$
- M1: Calculate $a^2b^2 - (a^2 + b^2) + 1 = 12.25 - 7 + 1 = 6.25$
- A1: Quadratic equation is $4x^2 - 25 = 0$ (or equivalent with integer coefficients)

**(c)** [2 marks]
- M1: Use roots of quadratic from part (b)
- A1: $a^2 = \frac{5}{2}$ and $b^2 = \frac{5}{2}$ (or $a^2 = 2.5$, $b^2 = 2.5$)

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# Question 2

**(a)** [1 mark]
- B1: The integrand has a singularity (vertical asymptote) at $x = 0$ within the interval of integration

**(b)** [4 marks]
- M1: Set up limit: $\lim_{\epsilon \to 0^+} \int_{\epsilon}^{4} (4x - 4)x^{-1.5} \, dx$
- M1: Integrate to get $\left[-8x^{0.5} - 8x^{-0.5}\right]_{\epsilon}^{4}$
- M1: Evaluate at limits
- DM1: Take limit as $\epsilon \to 0^+$
- A1: The integral does not have a finite value (or explain divergence to $-\infty$)

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# Question 3

**(a)** [3 marks]
- M1: Rationalize $\frac{3}{2+i}$ by multiplying by $\frac{2-i}{2-i}$
- M1: Calculate numerator and denominator
- A1: $(2+i)^3 = 2 + 11i$ (expressed in form $2 + bi$ where $b = 11$)

**(b)(i)** [4 marks]
- M1: Substitute $z = 2 + i$ into $z^3 + pz + q = 0$
- M1: Calculate $(2+i)^3 = 2 + 11i$
- M1: Set up equation $(2 + 11i) + p(2+i) + q = 0$
- M1: Equate real and imaginary parts
- A1: $p = -11$ and $q = -50$

**(b)(ii)** [2 marks]
- M1: Use that $2-i$ is also a root (complex conjugate pair)
- A1: Quadratic factor is $z^2 - 4z + 5$ (or equivalent)

**(b)(iii)** [2 marks]
- M1: Divide $z^3 - 11z - 50$ by $z^2 - 4z + 5$
- A1: Real root is $z = -2$

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# Question 4

**(a)** [5 marks]
- M1: Solve $\sin(3x + 45°) = \frac{1}{2}$
- M1: Identify general angles: $3x + 45° = 30° + 360°n$ or $3x + 45° = 150° + 360°n$
- M1: Rearrange to find $x$
- A1: $x = -5° + 120°n$ or $x = 35° + 120°n$
- A1: Complete general solution stated clearly

**(b)** [1 mark]
- A1: Solution closest to $200°$ is $x = 155°$ (or appropriate value from general solution)

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# Question 5

**(a)** [4 marks]
- M1: Set up matrix equation $\begin{pmatrix} -2 & c \\ d & 3 \end{pmatrix} \begin{pmatrix} 5 \\ 2 \end{pmatrix} = \begin{pmatrix} -2 \\ 1 \end{pmatrix}$
- M1: Form equations $-10 + 2c = -2$ and $5d + 6 = 1$
- A1: $c = 4$
- A1: $d = -1$

**(b)(i)** [2 marks]
- M1: Calculate successive powers of $B$
- A1: $B^4 = 4I$ (so $k = 4$)

**(b)(ii)** [5 marks]
- B1: Identify that $B$ represents a rotation (by $45°$ counterclockwise)
- B1: Identify that $B$ represents a scaling (by factor $\sqrt{2}$)
- A1: Described as rotation by $45°$ combined with enlargement scale factor $\sqrt{2}$
- A1: Clear description of order and combination
- A1: Centre of rotation stated (origin) if required

**(b)(iii)** [2 marks]
- M1: Use $B^{17} = B^{16} \cdot B = (B^4)^4 \cdot B = (4I)^4 \cdot B = 256I \cdot B$
- A1: $B^{17} = 256B = \begin{pmatrix} 128\sqrt{2} & 128\sqrt{2} \\ -128\sqrt{2} & 128\sqrt{2} \end{pmatrix}$ (or equivalent form)

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# Question 6

**(a)** [2 marks]
- B1: Sketch of hyperbola with correct shape and orientation
- A1: State intercepts: $(\pm 3, 0)$; no $y$-intercepts

**(b)** [3 marks]
- M1: Apply translation to get equation of $C_2$
- M1: Use condition that $C_2$ passes through origin
- A1: Asymptotes are $y = \pm \frac{4}{3}(x + 3)$ (or equivalent form)

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# Question 7

**(a)(i)** [2 marks]
- M1: Evaluate $f(39)$ and $f(40)$ for $f(x) = 2x^3 + 5x^2 + 3x - 132000$
- A1: Show sign change confirms $39 < a < 40$

**(a)(ii)** [3 marks]
- M1: Apply Newton-Raphson: $x_2 = x_1 - \frac{f(x_1)}{f'(x_1)}$ with $x_1 = 40$
- M1: Calculate $f(40)$ and $f'(40) = 6x^2 + 10x + 3$
- A1: $x_2 = 39.49$ (to 2 d.p.)

**(b)** [5 marks]
- M1: Expand $2r(3r + 2) = 6r^2 + 4r$
- M1: Write as $\Sigma 6r^2 + 4\Sigma