CAIE A-Level Mathematics 3.1 Algebra Question Bank

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 partial quotient of the form 3 x^3 +/- 3 x^2
or 3 x^4-2 x^2 (x+1)(A x^3+B x^2+C x+D)+E x+F, and reach A=3 and B= +/- 3

M1

May be seen in synthetic division.

Obtain quotient 3 x^3-3 x^2+x-1 Do not ISW

A1

Don't need to state which is the quotient and which
is remainder. However, if clearly muddled, then
M1A1A0 for both expressions correct.

Obtain remainder of 1

A1

Do not ISW.

Marking guidance:

Allow e.g. 3 x^3-3 x^2+x-1+1/x+1 but NOT
remainder =1/x+1.

Alternative Method for Question 2

f(-1)=3-2=1= remainder

B1

Do not ISW.

Use division or inspection or compare coefficients

M1

3 x^4-2 x^2-1 (x+1)(3 x^3-3 x^2+x-1)

Obtain quotient 3 x^3-3 x^2+x-1

A1

Do not ISW.
Allow e.g. 3 x^3-3 x^2+x-1+1/x+1 but NOT
remainder =1/x+1.

Use f(-2 a)=-22 a^3

M1

Or use long division and equate a constant
remainder to -22 a^3.

Obtain -24 a^3+4 p a^3-14 a^3+q a^3=-22 a^3

A1

Must evaluate the terms.

OE, e.g. 4 p+q=16.

Use f(a/3)=-a^3

M1

Or use long division and equate a constant
remainder to -a^3.

Obtain 1/9 a^3+1/9 p a^3+7/3 a^3+q a^3=-a^3

A1

Must evaluate the terms.

OE, e.g. p+9 q=-31

Obtain p=5, q=-4

A1