Advanced level 2025 North West Regional Mock mathematics statistics 2

Advanced level 2025 North West Regional Mock mathematics statistics 2

Advanced level 2025 North West Regional Mock mathematics statistics 2

1. When f(x)=2x3−3x2−11x+c is divided by $x+2$, the remainder is 0. a) Show that $c=6$. b) Factorize $f(x)$ completely. c) Find the range of values of $x$ for which (x+2)(2x−1)2x3−3x2−11x+6​<7,x=−2,21​.

2. If $p$ and $q$ are the roots of the quadratic equation 9x2−4kx−(1+6k)=0 where $k$ is a constant. Show that an equation whose roots are p1​ and q1​ is −(1+6k)x2+4kx+9=0.

3. In a box of 12 fuses, 4 fuses are chosen at random and inspected. The box is rejected if more than one fuse is found to be faulty and there are 3 faulty fuses in the box. Find the number of ways the box can be accepted.

4. a) Construct a truth table for $(p\vee \sim q)\wedge q$. b) (i) Show that log16​xy=41​log2​x+41​log2​y. (ii) Hence or otherwise, solve the equations: log16​xy=21​ and logx​2−logy​2=0.

5. a) Show that 1+cos2x+cosxsin2x+sinx​=tanx. Hence, find the general solution of 1+cos2x+cosxsin2x+sinx​=1. b) Given that f(x)=cos3x−3​sin3x. (i) Express $f(x)$ in the form $R\cos (3x+\alpha )$, where $R>0$ and $\alpha$ is an acute angle. (ii) Find the maximum value of the function.

6. A mathematical model assumes that $x$ and $y$ are related by the equation y=log10​(bx+d) for some constants $a$ and $b$. Approximate values of $y$ corresponding to the given values of $x$ are tabulated below: | $x$ | 3 | 4 | 5 | 6 | |—–|——–|——–|——–|——–| | $y$ | 0.778 | 0.857 | 0.942 | 1.079 | a) By drawing a suitable linear graph, estimate the value of the constants $a$ and $b$. b) Use the trapezoidal rule to obtain an estimate for ∫36​ydx.

7. a) (i) Given that $z=\cos \theta +i\sin \theta$, find z2+z21​. (ii) Hence, show that cos4θ=cos4θ−6cos2θsin2θ+sin4θ. b) (i) Describe the locus of the point $z$ such that $|z-(3+2i)|=2$. (ii) Find the complex number $w$ such that $|w-3|=|w-2i|$ and arg(w−2iw−3​)=2π​.