Sue Pemberton, Julianne Hughes and, Julian Gilbey Solutions for Chapter: Cross-Topic Review Exercise 1, Exercise 1: CROSS-TOPIC REVIEW EXERCISE 1

Solve the equation $\left|3x+4\right|=\left|3x-11\right|.$

Hence, using logarithms, solve the equation $\left|3\times 2^y+4\right|=\left|3\times 2^y-11\right|,$ giving the answer correct to $3$ significant figures.

Solve the equation $2\left|5^x-1\right|=3\left|5^x\right|$, giving your answer correct to $3$ significant figures.

solve the equation $5 \sin 2\theta +2 \cos 2\theta =4,$ giving all solutions in the interval $0°⩽\theta ⩽360°,$

Determine the least value of $\frac{1}{(10 \sin 2\theta +4 \cos 2\theta {)}^2}$ as $\theta$ varies.

The polynomial $\mathrm{p}\left(x\right)$ is defined by $\mathrm{p}\left(x\right)=a{x}^3+3x^2+bx+12,$ where $a$ and $b$ are constants. It is given that $(x+3)$ is a factor of $\mathrm{p}\left(x\right).$ It is also given that the remainder is $18$ when $\mathrm{p}\left(x\right)$ is divided by $(x+2).$

Find the values of $a$ and $b.$

The polynomial $\mathrm{p}\left(x\right)$ is defined by $\mathrm{p}\left(x\right)=a{x}^3+3x^2+bx+12,$ where $a$ and $b$ are constants. It is given that $(x+3)$ is a factor of $\mathrm{p}\left(x\right).$ It is also given that the remainder is $18$ when $\mathrm{p}\left(x\right)$ is divided by $(x+2).$ When $a=2$ and $b=-5$ have these values, show that the equation $\mathrm{p}\left(x\right)=0$ has exactly one real root,

The polynomial $\mathrm{p}\left(x\right)$ is defined by $\mathrm{p}\left(x\right)=a{x}^3+3x^2+bx+12,$ where $a$ and $b$ are constants. It is given that $(x+3)$ is a factor of $\mathrm{p}\left(x\right).$ It is also given that the remainder is $18$ when $\mathrm{p}\left(x\right)$ is divided by $(x+2).$ When $a=2$ and $b=-5$ have these values, solve the equation $\mathrm{p}(\sec y)=0$ for $-180°<y<180°$