CAIE A-Level Mathematics A2 3.1 Algebra Questions

Square both sides:
$$(2x+3)^2>9(x+2)^2.$$
Then
$$4x^2+12x+9>9x^2+36x+36,$$
so
$$5x^2+24x+27<0.$$
Factorising,
(5x+9)(x+3)<0.
The critical values are x=-3 and $x=-\frac95$. The product is negative between them, hence
$$-3<x<-\frac95.$$

B1 for the non-modular squared inequality or equivalent linear equations. M1 for solving the quadratic/equations. A1 for critical values -3 and $-\frac95$. A1 for $-3<x<-\frac95$, with strict inequalities.

Commence division and reach quotient of the form $2 x \pm 1$
Or by inspection $8 x^{3}+4 x^{2}+2 x+7=\left(4 x^{2}+1\right)(2 x \pm 1) +\mathrm{r}$
Or by inspection $8 x^{3}+4 x^{2}+2 x+7=\left(4 x^{2}+1\right)(2 x \pm 1)$
+r

Obtain (quotient) 2 x+1
Obtain (remainder) 6
3

Substitute x=-3 to obtain value of p(-3)
Obtain p(-3)=0 and hence given result
Alternative method for Question 10(a)

Divide p(x) by ( x+3 ) to obtain quotient $x^{2} \pm 2 x+\ldots$
Obtain quotient $x^{2}+2 x+25$, with zero remainder and hence given result

State or imply the form $A+\frac{B}{2 x+3}+\frac{C}{x-4}$$\frac{D x+E}{2 x+3}+\frac{F}{x-4}$ and $\frac{P}{2 x+3}+\frac{Q x+R}{x-4}$ are also valid.

Use a correct method for finding a constant
SC: If score B0, they can score M1 A1 for one correct constant.
B0 M1 A0 available if they substitute two values to form simultaneous equations but get an incorrect answer, or they substitute one value and make an arithmetic error.

Obtain one of A=3, B=-2 and C=4
SC: If the horizontal equation is correct apart from an incorrect
value for A, the other A marks may be available.

Obtain a second value
SC: If denominator factorised as $\left(x+\frac{3}{2}\right)(x-4)$ can score a

Marking guidance:

maximum of B0 M1 A1 A1 A0 for a split involving 3 terms.

Obtain a third value
ISW
Statement of the final split is not required.

2

Alternative method for Question 2

Divide numerator by denominator
Obtain $3\left(+\frac{P x+Q}{2 x^{2}-5 x-12}\right)$$\left(3+\frac{6 x+20}{(2 x+3)(x-4)}\right)$

State or imply the form $\frac{P x+Q}{2 x^{2}-5 x-12}=\frac{D}{2 x+3}+\frac{E}{x-4}$
Must deal with the 3 separately or include it correctly on both sides in their split.

Obtain one of D=-2 and E=4
SC: If denominator factorised as $\left(x+\frac{3}{2}\right)(x-4)$, then can score a
maximum of B0 M1 A1 A1 A0 for a split involving three terms.

Obtain a second value
ISW
Statement of the final split is not required.

State correct unsimplified first two terms of the expansion of $(1-2 x)^{\frac{1}{2}}$,
e.g. $1+\frac{1}{2}(-2 x)$
Symbolic coefficients are not sufficient.
1-x

State correct unsimplified term in $x^{2}$, e.g. $\frac{\frac{1}{2}\left(\frac{1}{2}-1\right)(-2 x)^{2}}{2!}$
Symbolic coefficients are not sufficient.
$-\frac{1}{2} x^{2}$

Obtain sufficient terms of the product of (3+x) and the expansion up to the term
in $x^{2}$
Obtain final answer $3-2 x-\frac{5}{2} x^{2}$