Advanced level 2025 South West regional mock Maths statistics 2

Here are the questions on pages 2 and 3 of the document:

Page 2

(i) Given that $α^{2} β$ and $β^{2} α$ are the roots of the quadratic equation $x^{2} - 2 x + 27 = 0$ Find the quadratic equation whose roots are a and B. (4 marks)
(ii) Given the polynomial $f (x) = 2 x^{3} - x^{2} + a x + b$ , a, $b \in R$ The remainder when $f (x)$ is divided by $(x + 1)$ is three times the remainder when divided by $(x - 1)$ and that $f (0) = 1$ .
Find the values of a and b. (5 marks)
(iii) Given that $l o g_{3} 6 = x$ and $l o g_{6} 5 = y$ show that $l o g_{3} 10 = x (y + 1) - 1$
(i) The function f is defined by $f : R - {- 1} \to R - {2},$ $x \mapsto x + 1 2 ( x - 1 )$ (a) Show that is bijective. (4 marks)
(b) Find the function g such that $f \circ g (x) = x 1, x \neq = 0.$ (3 marks)
(ii) Find the constant term in the expansion of $(5 x - 2 x 1)^{8}$ (4 marks)
(1) Given the vectors $a = (2 i - 3 j + 5 k),$ $b = (- i + 2 k)$ and $c = (21 + j + k)$ Find
(a) $a \cdot (a \times b)$ and hence, state a vector perpendicular to a (3 marks)
(b) the Cartesian equation of the plane r containing a and (2 marks)
(c) the perpendicular distance from c to the plane π (2 marks)
(ii) Write down the power set of the set $A = {1, 5, 7} .$ (2 marks)
(1) The angle is such that (a) Show that $t an 2 θ = 2 1$ (2 marks) (b) Hence find, in surd form, the exact value of $t an θ$ , given that is obtuse. (2 marks) (ii) Express $f (x) = 7 cos x - 24 s in x$ in the form

Page 3

$R cos (x + α)$ where R>0 and $0 < α < 90$
Hence solve the equation $7 cos x - 24 s in x = 5$ for $0 < x < 360$
(5 marks)
5. (i) Differentiate with respect to x,
(a) $y = x^{2} e^{- x}$
(b) $y = l n 1 - x 1 + x, - 1 < x < 1$
(ii) The parametric equation of a curve are
$x = t - s in t,$ $y = 1 - cos t$
Find the value of $d x d y$ at the point where $t = 3 π$
6. (i) Evaluate
$\int_{0 2 π} s i n^{3} x co s^{2} x d x$
(ii) Find the area of the region enclosed between the curves with equations $y = x^{2} - 2 x$ and $y = 4 x - x^{2}$
7. (i) Solve the differential equation
$d x d y = x (1 + y^{2})$ given that $y = 1$ when $x = 1$
Give your answer in the form $y = f (x)$
(ii) Given that $f (x) = x + 1 x$
(a) Find the values of a and b such that $f^{'} (x) = ( x + 1 ) ^{b} a$
Henee,
(b) Show that the graph of $y = f (x)$ has no turning points.
(c) Study the monotonicity of f in its domain.
(d) Sketch the graph of $y = f (x)$ showing clearly its intercepts with the coordinate axes and behavior near its asymptotes.

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