EduNinja

[Maximum number: 9]

(a)

Find the first three terms in the expansion of $\left(1+\frac{x}{7}\right)^5$ , in ascending powers of x. Simplify the coefficient of each term.

[ 2 ]

$1+\frac57x+\frac{10}{49}x^2$
B1 for any two correct terms or for the three terms listed but not summed

(b)

The expansion of $7(1+x)^n\left(1+\frac{x}{7}\right)^5$ , where n is a positive integer, is written in ascending powers of x. The first two terms in the expansion are 7+89x. Find the value of n.

[ 2 ]

12
B1 for 7n+5=89 or $7(n+\frac57)=89$ oe or ${}^nC_1=12$

(c)

In the expansion of $(k-2x)^8$ , where k is a constant, the coefficient of $x^4$ divided by the coefficient of $x^2$ is $\frac{5}{8}$ . The coefficient of x is positive. Form an equation and hence find the value of k.

[ 5 ]

${}^8C_4\times k^4\times(-2)^4[x^4]$
oe, or $1120k^4[x^4]$ soi
${}^8C_2\times k^6\times(-2)^2[x^2]$
oe, or $112k^6[x^2]$ soi
$\frac{1120k^4}{112k^6}=\frac58$
oe, soi
M1 FT providing at least B1 awarded and correct terms attempted
$k^2=16$
soi
For coefficient of x to be positive, k<0, therefore k=-4

[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

Question 1

Question (a)

Question (b)

Question (c)

Question 2

Question (a)

Question (b)

Question (c)