smc-1088-10-Coefficients

Combinatorics, EXT1 A1 2022 HSC 11c

Find the coefficients of `x^(2)` and `x^(3)` in the expansion of `(1-(x)/(2))^(8)`. (2 marks)

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`x^2: 7, \ x^3:-7`

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`text{General Term:}\ (1-(x)/(2))^(8) `

`T_k`	`=((8),(k))(1)^(8-k)(- x/2)^k`
	`=((8),(k))(- 1/2)^k x^k`

`text{Coefficient of}\ \ x^2 = ((8),(2))(- 1/2)^2=7`

`text{Coefficient of}\ \ x^3 = ((8),(3))(- 1/2)^3=-7`

Combinatorics, EXT1 A1 2019 13b

In the expansion of `(5x + 2)^20`, the coefficients of `x^r` and `x^(r + 1)` are equal.

What is the value of `r` ? (3 marks)

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`14`

Show Worked Solution

`(5x + 2)^20`

COMMENT: Arithmetic becomes easier by expanding `(2 + 5x)^20`.

`text(General term:)\ (2 + 5x)^20`

`T_r`	`= \ ^20C_r · 2^(20 – r) · (5x)^r`
	`= \ ^20C_r · 2^(20 – r) · 5^r · x^r`
`T_(r + 1)`	`=\ ^20C_(r + 1) · 2^(19 – r) · 5^(r + 1) · x^(r + 1)`

`text(Equating co-efficients:)`

`(20!)/(r!(20 – r)!) · 2^(20 – r) · 5^r`	`= (20!)/((r + 1)!(19 – r)!) · 2^(19 – r) · 5 ^(r + 1)`
`2/(20 – r)`	`= 5/(r + 1)`
`2r + 2`	`= 100 – 5r`
`7r`	`= 98`
`r`	`= 14`

Combinatorics, EXT1 A1 EQ-Bank 3

Find the coefficient of `x^4` in the expansion of `(x^2 - 3/x)^5`. (2 marks)

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`90`

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`text(General term:)`

`T_k`	`= \ ^5C_k(x^2)^(5 – k) · (−3/x)^k`
	`= \ ^5C_k · x^(10 – 2k)(−3)^k · x^(−k)`
	`= \ ^5C_k · x^(10 – 3k) · (−3)^k`

`text(Coefficient of)\ \ x^4\ \ text(occurs when)`

`10 – 3k`	`= 4`
`3k`	`= 6`
`k`	`= 2`

`:.\ text(Coefficient of)\ \ x^4`

`= \ ^5C_2·(−3)^2`

`= 90`

Combinatorics, EXT1 A1 2015 HSC 13b

Consider the binomial expansion

`(2x + 1/(3x))^18 = a_0x^(18) + a_1x^(16) + a_2x^(14) + …`

where `a_0, a_1, a_2`, . . . are constants.

Find an expression for `a_2`. (2 marks)

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Find an expression for the term independent of `x`. (2 marks)

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`(\ ^18C_2 · 2^(16))/(3^2)`
`(\ ^(18)C_9 · 2^9)/(3^9)`

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i. `text(Need co-efficient of)\ x^(14)`

`text(General term of)\ (2x + 1/(3x))^(18)`

`T_k`	`= \ ^(18)C_k(2x)^(18 − k) · (1/(3x))^k`
	`= \ ^(18)C_k · 2^(18 − k) · x^(18 − k) · 3^(−k) · x^(−k)`
	`= \ ^(18)C_k · 2^(18 − k) · 3^(−k) · x^(18 − 2k)`

`a_2\ text(occurs when:)`

`18 − 2k`	`= 14`
`2k`	`= 4`
`k`	`= 2`

`:.a_2`	`= \ ^(18)C_2 · 2^(18 − 2) · 3^(−2)`
	`= (\ ^(18)C_2 · 2^(16))/(3^2)`

ii. `text(Independent term occurs when:)`

`18 − 2k`	`= 0`
`2k`	`= 18`
`k`	`= 9`

`:.\ text(Independent term)`

`= \ ^(18)C_9 · 2^(18− 9) · 3^(−9)`

`= (\ ^(18)C_9 · 2^9)/(3^9)`

Combinatorics, EXT1 A1 2008 HSC 1d

Find an expression for the coefficient of `x^8 y^4` in the expansion of `(2x + 3y)^12`. (2 marks)

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`10\ 264\ 320`

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`text(Find co-efficient of)\ x^8 y^4 :`

MARKER’S COMMENT: More errors were made by students who used `T_(k+1)` as the general term rather than `T_k` (both are possible). The Worked Solution uses the more successful approach.

`T_k =\ text(General term of)\ (2x + 3y)^12 `

`T_k`	`= ((12),(k)) (2x)^(12 – k) * (3y)^k`
	`= ((12),(k)) * 2^(12 – k) * 3^k * x^(12 – k) * y^k`

`x^8 y^4\ text(occurs when)\ k = 4`

`T_4`	`= ((12),(4)) * 2^(12 – 4) * 3^4 * x^8 y^4`

`:.\ text(Co-efficient of)\ x^8y^4`
	`= ((12),(4)) * 2^8 * 3^4`
	`= 10\ 264\ 320`

Combinatorics, EXT1 A1 2011 HSC 2c

Find an expression for the coefficient of `x^2` in the expansion of `(3x - 4/x)^8`. (2 marks)

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`-870\ 912`

Show Worked Solution

`(3x – 4/x)^8 = sum_(k=0)^8\ ^8C_k *(3x)^(8\ – k) *(–4/x)^k`

`text(General term)`	`=\ ^8C_k * 3^(8\ – k) * x^(8\ – k) * (–1)^k4^k x^(-k)`
	`=\ ^8C_k(–1)^k 3^(8\ – k) * 4^k * x^(8\ – 2k)`

`text(Co-efficient of)\ \ x^2=2\ \ text(occurs when,)`

` 8 – 2k`	`= 2`
`2k`	`= 6`
`k`	`= 3`

`:.\ text(Co-efficient of)\ \ x^2`

`=\ ^8C_3 * (–1)^3*3^5 * 4^3`

`= -56 xx 243 xx 64`

`= -870\ 912`