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Combinatorics, EXT1 A1 EQ-Bank 15

\(\left(\sqrt{5}-2 \right)^5=x+y \sqrt{5}\)

Find the values of \(x\) and \(y\) using binomial expansion.   (2 marks)

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\(x=-682,  y=305\)

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  \( \left(\sqrt{5}-2\right)^5\) \(={ }^5 C_0 \cdot\left(\sqrt{5}\right)^5+{ }^5 C_1\left(\sqrt{5}\right)^4 \cdot(-2)+{ }^5 C_2\left(\sqrt{5}\right)^3 \cdot(-2)^2\)
    \(+{ }^5 C_3\left(\sqrt{5}\right)^2 \cdot(-2)^3 +{ }^5 C_4\left(\sqrt{5}\right) \cdot(-2)^4+{ }^5 C_5(-2)^5 \)
    \( = 25 \sqrt{5}-250+200 \sqrt{5}-400+80 \sqrt{5}-32\)
    \(=  305 \sqrt{5}-682 \)

 
\(\therefore x=-682,  y=305\)

Combinatorics, EXT1 A1 2021 HSC 11b

Expand and simplify  `(2a - b)^4`.  (2 marks)

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`16a^4 – 32a^3b + 24a^2b^2 – 8ab^3 + b^4`

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`(2a – b)^4`

`= (2a)^4 + \ ^4C_1(2a)^3(-b) + \ ^4C_2(2a)^2(-b)^2 + \ ^4C_3(2a)(-b)^3 + (-b)^4`

`= 16a^4 – 4 · 8a^3b + 6 · 4a^2b^2 – 4 · 2ab^3 + b^4`

`= 16a^4 – 32a^3b + 24a^2b^2 – 8ab^3 + b^4`

Combinatorics, EXT1 A1 SM-Bank 9

`(2 - sqrt3)^5 = a + bsqrt3`.

Evaluate `a` and `b` using a binomial expansion.  (2 marks)

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`a = 362, \ b = -209`

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`(2 – sqrt3)^5` `= \ ^5C_0 *2^5 + \ ^5C_1* 2^4(−sqrt3) + \ ^5C_2 *2^3(−sqrt3)^2 + \ ^5C_3 *2^2(−sqrt3)^3`
               `+ \ ^5C_4 *2(−sqrt3)^4 + \ ^5C_5(−sqrt3)^5`
  `= 32 – 80sqrt3 + 240 – 120sqrt3 + 90 – 9sqrt3`
  `= 362 – 209sqrt3`

 
`:. a = 362, \ b = -209`

Combinatorics, EXT1 A1 SM-Bank 1

`(1-2sqrt2)^6 = x + ysqrt2`

Evaluate the value of `x` and `y`.  (2 marks)

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`x = 1593, \ y =-1100`

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`text(Using the binomial expansion:)`

`(1-2sqrt2)^6`

`= \ ^6C_0 + \ ^6C_1(-2sqrt2) + \ ^6C_2(-2sqrt2)^2 + \ ^6C_3(-2sqrt2)^3 + \ ^6C_4(-2sqrt2)^4`

`+ \ ^6C_5(-2sqrt2)^5 + \ ^6C_6(-2sqrt2)^6`
 

`= 1-12sqrt2 + 120 -320sqrt2 + 960-768sqrt2 + 512`

`= 1593-1100sqrt2`
 

`:.x = 1593, \ y =-1100`