CAIE IGCSE Additional Math 12 Series Questions

Covered syllabus points

12.1. Use the binomial theorem 12.2. Use the general binomial term 12.3. Recognise arithmetic and geometric progressions 12.4. Use finite progression formulas 12.5. Use convergence and sum to infinity

[Maximum number: 5]

In this question a, b and n are constants.
When $5(2+a x)^{n}$ is written in ascending powers of x, the first three terms are $640+b^{2} x+30240 x^{2}$ . Find the value of a and the possible values of b.

n=7 soi
$5\times\frac{7\times6}{2}\times2^5\times a^2=30240$ oe
$5\times7\times2^6\times a=b^2$ a=3 only, nfww
$b=\pm\sqrt{6720}$ oe, isw, nfww
If first B1 only scored then SC1 for $a=3\sqrt5$ or $\sqrt{45}$ or 6.71 or 6.708... only, and SC1 for $b=\pm\sqrt{1344\sqrt5}$ or $\pm54.8...$ .

[Maximum number: 6]

Find the exact value of the term independent of x in the expansion of $\left(2+\frac{3}{x^{2}}\right)^{10}\left(1-4 x^{2}\right)^{2}$ .

$2^{10}+10\times2^9\times\frac{3}{x^2}+45\times2^8\times\left(\frac{3}{x^2}\right)^2\ \text{oe, soi}$
and
$1-8x^2+16x^4\ \text{soi}$
B2 for the first expansion;
B1 for two correct terms in the first expansion;
B1 for $1-8x^2+16x^4$ .
$2^{10}+$ their $15360\times$ their (-8)+ their $103680\times$ their 16
M2 FT expansions;
M1 FT for any two correct terms in this sum.
1537024

[Maximum number: 15]

An arithmetic progression, A, has first term a and common difference d. The 2nd, 14th and 17th terms of A form the first three terms of a convergent geometric progression, G, with common ratio r .

(a)

Given that $d\neq0$ , find two expressions for r in terms of a and d and hence show that a=-17d.

[ 6 ]

$a+d,\quad a+13d,\quad a+16d$
soi
$\frac{a+13d}{a+d}=\frac{a+16d}{a+13d}$
oe
$a^2+26ad+169d^2=a^2+17ad+16d^2$ $9ad+153d^2=0$ $9d(a+17d)=0\Rightarrow a=-17d$
oe

(b)

Find the value of r .

[ 2 ]

r=0.25
oe
M1 for $\frac{-17d+13d}{-17d+d}$ or $\frac{-17d+16d}{-17d+13d}$

(c)

The first term of the geometric progression, G , is q and the sum to infinity is $\frac{256}{3}$ . Find the sum of the first 20 terms of the arithmetic progression, A .

[ 7 ]

$\frac{q}{1-0.25}=\frac{256}{3}$
oe
q=64-17d+d=64
oe
$d=-4,\quad a=68$ $S_{20}=600$

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CAIE IGCSE Additional Math 12 Series Topic Practice

Covered syllabus points

Question 1

Question 2

Question 3

Question (a)

Question (b)

Question (c)