CAIE IGCSE Additional Math Algebra and Functions Questions

Use this algebra-and-functions unit hub to connect function structure, quadratics and polynomials with equations, inequalities, simultaneous systems and logarithms.

Exam points

Analyse functions, quadratics and polynomial factors using domains, ranges, extrema, discriminants and theorems.
Solve modulus, quadratic, polynomial and simultaneous equations or inequalities with valid roots and intervals.
Use logarithmic and exponential graphs, laws and equations with domain checks.

Covered syllabus points

1. Functions 2. Quadratic functions 3. Factors of polynomials 4. Equations, inequalities and graphs 5. Simultaneous equations 6. Logarithmic and exponential functions

[Maximum number: 5]

The polynomial p is such that $\mathrm{p}(x)=x^{3}+a x^{2}+b x-2, \quad$ where a and b are constants.
It is given that:
- x+2 is a factor of p(x)
- when p(x) is divided by x-3 the remainder is 40.

Find the values of a and b.

[Maximum number: 5]

the equation $\left|2 x^{2}+x-10\right|=5$ .

$2 x^{2}+x-10=5$

leading to $2 x^{2}+x-15=0$ with a valid attempt to solve to obtain 2 values for x M1
$x=\frac{5}{2}, \quad-3$ A1 For both
Mark final answer for this equation,
A0 if - 3 is rejected
$x^{2}+x-10=-5$

leading to $2 x^{2}+x-5=0$ M1 Must be a correct 3-term quadratic
equated to zero
$x=\frac{-1 \pm \sqrt{41}}{4}$ oe 2 Dep M1 for a valid method of
solution to obtain 2 values for x.
Mark final answer for this equation,
A0 if negative root is rejected
Denominator of final answer must be positive

[Maximum number: 4]

the axes, sketch the graph of $y=5 \ln (4 x+3)$ .
State the intercepts with the axes.
State the equation of any asymptote.

Figure for Question 3 — CAIE IGCSE Additional Math

Figure for Answer to Question 3 — CAIE IGCSE Additional Math

B1 for correct shape in first three quadrants tending towards a
vertical asymptote in the second and third quadrants.
Dep B1 for curve passing through x=-0.5 oe on the x-axis, or stated
and no other points on the x-axis
Dep B1 for curve passing through y $=5 \ln 3$ oe on the y-axis, or stated
and no other points on the y-axis
B1 for $x=-\frac{3}{4}$ oe shown on graph
or stated as the equation of the asymptote. May be implied by a
dotted or straight vertical line
through $x=-\frac{3}{4}$ . This is not a
dependent mark.

[Maximum number: 3]

Figure for Question 4 — CAIE IGCSE Additional Math

The diagram shows the graph of y=|3x+3|.
Use a graphical method to solve the inequality $|3x+3| \geqslant |x-2|$ .

Correct graph of y=|x-2| and $x \leqslant -2.5,\ x \geqslant -0.25$
B1 for correct graph of y=|x-2|.
B1 for $x \leqslant -2.5$ or $x \geqslant -0.25$ .
B1 STRICT FT for their critical values from the two intersections of their straight-line section of graph, providing it has negative gradient.

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Covered syllabus points

Question 1

Question 2

Question 3

Question 4