Advanced level 2025 South West Regional Mock mathematics statistics 2

Advanced level 2025 South West Regional Mock mathematics statistics 2

Advanced level 2025 South West Regional Mock mathematics statistics 2

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1. (i) Given that $\alpha$ and β2 are the roots of the quadratic equation x2−2x+27=0. Find the quadratic equation whose roots are $\alpha$ and $\beta$. (ii) Given the polynomial f(x)=2x3−x2+ax+b, $a,b\in \mathbb{R}$. The remainder when $f(x)$ is divided by $(x+1)$ is three times the remainder when $f(x)$ is divided by $(x-1)$ and that $f(0)=1$. Find the values of $a$ and $b$. (iii) Given that log3​6=x and log5​6=y, show that log10​=x(y+1)−1.

2. (i) The function $f$ is defined by $f:\mathbb{R}\setminus \{-1\}\to \mathbb{R}\setminus \{2\}$, x↦x+12x−1​. (a) Show that $f$ is bijective. (b) Find the function $g$ such that f∘g(x)=x21​, x=0. (ii) Find the constant term in the expansion of (5x−x22​)6.

3. (i) Given the vectors a” />=(2i^−3j^​+5k^), b” />=(−i^+2k^) and c” />=(2i^+j^​+k^). Find: (a) a” />⋅(a” />×b” />), hence, state a vector perpendicular to a” />. (b) the Cartesian equation of the plane $\pi$ containing a” /> and b” />. (c) the perpendicular distance from c” /> to the plane $\pi$. (ii) Write down the power set of the set $A=\{1,5,7\}$.

4. (i) The angle $\theta$ is such that sin(2θ+4π​)=3cos(2θ+4π​). (a) Show that tan2θ=21​. (b) Hence find, in surd form, the exact value of $\tan \theta$, given that $\theta$ is obtuse. (ii) Express $f(x)=7\cos x-24\sin x$ in the form $R\cos (x+\alpha )$, where $R>0$ and $\alpha$ is an acute angle. Hence, state the range of the function $g(x)=12+7\cos x-24\sin x$, $0\le x\le 2\pi$.

5. (i) Find dxdy​ given that: (a) y=ln(x−1x+1​)3 (b) y=1+x2e2xsinx​