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Functions, 2ADV EQ-Bank 06

Simplify  \(\dfrac{x^2}{x^2-2 x-15}-\dfrac{x}{x+3}\).   (2 marks)

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\(\dfrac{5 x}{(x+3)(x-5)}\)

Show Worked Solution
\(\dfrac{x^2}{x^2-2 x-15}-\dfrac{x}{x+3}\) \(=\dfrac{x^2}{(x+3)(x-5)}-\dfrac{x}{x+3}\)
  \(=\dfrac{x^2-x(x-5)}{(x+3)(x-5)}\)
  \(=\dfrac{x^2-x^2+5 x}{(x+3)(x-5)}\)
  \(=\dfrac{5 x}{(x+3)(x-5)}\)

Functions, 2ADV EQ-Bank 05

Simplify  \(\dfrac{p+1}{q-q^3} \ ÷ \ \dfrac{p^3+p^2}{q^2-q}\).   (2 marks)

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\(\dfrac{1}{p^2(1+q)}\)

Show Worked Solution
\(\dfrac{p+1}{q-q^3} \ ÷ \ \dfrac{p^3+p^2}{q^2-q}\) \(=\dfrac{p+1}{q\left(1-q^2\right)} \times \dfrac{q\left(1-q\right)}{p^2(p+1)}\)
  \(=\dfrac{(1-q)}{(1+q)(1-q) p^2}\)
  \(=\dfrac{1}{p^2(1+q)}\)

Functions, 2ADV EQ-Bank 03

Find \(a\) and \(b\) such that \(a\) and \(b\) are real and   \(\dfrac{2\sqrt{3}+2}{\sqrt{6}-\sqrt{2}} = a\,\sqrt{2} + b\,\sqrt{6}\).   (2 marks)

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\(a=2\ \ \text{and}\ \ b=1 \)

Show Worked Solution

\(\dfrac{2\sqrt{3}+2}{\sqrt{6}-\sqrt{2}} \times \dfrac{\sqrt{6}+\sqrt{2}}{\sqrt{6}+\sqrt{2}} \)

\(=\dfrac{(2\sqrt{3}+2)(\sqrt{6}+\sqrt{2})}{6-2}\)

\(=\dfrac{2\sqrt{18}+2\sqrt{6}+2\sqrt{6}+2\sqrt{2}}{4}\)

\(=\dfrac{6\sqrt{2}+4\sqrt{6}+2\sqrt{2}}{4}\)

\(=\dfrac{8\sqrt{2}+4\sqrt{6}}{4}\)

\(=2\sqrt{2}+\sqrt{6}\)
 

\(\therefore a=2\ \ \text{and}\ \ b=1 \)

Functions, 2ADV EQ-Bank 02

Rationalise the denominator in \(\dfrac{\sqrt{3}-\sqrt{2}}{\sqrt{5}+\sqrt{2}}\), and express in the simplest form.   (2 marks)

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\(\dfrac{\sqrt{15}-\sqrt{6}-\sqrt{10}+2}{3} \)

Show Worked Solution

\(\dfrac{\sqrt{3}-\sqrt{2}}{\sqrt{5}+\sqrt{2}} \times \dfrac{\sqrt{5}-\sqrt{2}}{\sqrt{5}-\sqrt{2}}\)

\(=\dfrac{(\sqrt{3}-\sqrt{2})(\sqrt{5}-\sqrt{2})}{5-2}\)

\(=\dfrac{\sqrt{15}-\sqrt{6}-\sqrt{10}+2}{3}\)

Functions, 2ADV EQ-Bank 01

Find \(x\) and \(y\) such that \(x\) and \(y\) are real and   \(\dfrac{\sqrt{2}+1}{\sqrt{6}-\sqrt{3}} = x\,\sqrt{3} + y\,\sqrt{6}\).   (2 marks)

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\(x=1\ \ \text{and}\ \ y=\dfrac{2}{3} \)

Show Worked Solution

\(\dfrac{\sqrt{2}+1}{\sqrt{6}-\sqrt{3}} \times \dfrac{\sqrt{6}+\sqrt{3}}{\sqrt{6}+\sqrt{3}} \)

\(=\dfrac{(\sqrt{2}+1)(\sqrt{6}+\sqrt{3})}{6-3}\)

\(=\dfrac{\sqrt{12}+2\sqrt{6}+\sqrt{3}}{3}\)

\(=\dfrac{3\sqrt{3}+2\sqrt{6}}{3}\)

\(= \sqrt{3}+\dfrac{2}{3} \sqrt{6}\)
 

\(\therefore x=1\ \ \text{and}\ \ y=\dfrac{2}{3} \)

Functions, 2ADV F1 EQ-Bank 10

Fully simplify the expression  \(\dfrac{4}{x^2-9}-\dfrac{2x+1}{x+3}\)   (3 marks)

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\(\dfrac{-(2x-7)(x+1)}{(x^2-9)}\)

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\(\dfrac{4}{x^2-9}-\dfrac{2x+1}{x+3}\) \(=\dfrac{4-(2x+1)(x-3)}{(x+3)(x-3)}\)  
  \(=\dfrac{4-(2x^2-5x-3)}{(x+3)(x-3)}\)  
  \(=\dfrac{-(2x^2-5x-7)}{(x+3)(x-3)}\)  
  \(=\dfrac{-(2x-7)(x+1)}{(x^2-9)}\)  

Functions, 2ADV F1 SM-Bank 53

  1. If  `1/(root3(7+pi)) = (7+pi)^x`, find  `x`.  (1 mark)

  2. Calculate the value of  `1/(root3(7+pi))`  to 3 significant figures.   (1 mark)

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  1. `-1/3`
  2. `0.462`
Show Worked Solution

i.    `1/(root3(7+pi)) = (7+pi)^(-1/3)`

ii.     `1/(root3(7+pi))` `=0.4619…`
    `=0.462\ \ text{(to 3 sig. fig.)}`

Functions, 2ADV F1 SM-Bank 52

Find `a`  and  `b`  such that  `a,b`  are real numbers and

`(6sqrt3-sqrt5)/(2sqrt5)= a + b sqrt15`.   (2 marks)

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`a= -1/2, \ b=3/5`

Show Worked Solution
`(6sqrt3-sqrt5)/(2sqrt5)` `=(6sqrt3-sqrt5)/(2sqrt5) xx (2sqrt5)/(2sqrt5)`  
  `=(2sqrt5(6sqrt3 – sqrt5))/(4 xx5)`  
  `=(12sqrt15-10)/20`  
  `=- 1/2 + 3/5 sqrt15`  

 
`:. a= -1/2, \ b=3/5`

Functions, 2ADV F1 SM-Bank 51

Find `a`  and  `b`  such that  `a,b`  are real numbers and 

`(sqrt3-2)/(2sqrt3)= a + b sqrt3`.   (2 marks)

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 `a = 1/2, \ b = – 1/3`

Show Worked Solution
`(sqrt3-2)/(2sqrt3)` `= (sqrt3-2)/(2sqrt3) xx (2sqrt3)/(2sqrt3)`
  `= (2sqrt3(sqrt3-2))/(4 xx 3)`
  `= (6-4sqrt3)/12`
  `=1/2 – 1/3 sqrt3`

 
`:.\ a = 1/2, \ b = – 1/3`

Functions, 2ADV F1 SM-Bank 46

Find `a` and `b` such that  `a, b`  are real numbers and

`(8-sqrt27)/(2sqrt3) = a + bsqrt3`.   (2 marks)

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`:. a =-3/2, \ b = 4/3`

Show Worked Solution
`(8-sqrt27)/(2sqrt3) xx (2sqrt3)/(2sqrt3)` `=(2sqrt3(8-3sqrt3))/(2sqrt3)^2`
  `= (16sqrt3-18)/12`
  `= -3/2 + 4/3sqrt3`

 
`:. a = -3/2, \ b = 4/3`

Functions, 2ADV F1 EQ-Bank 22

Worker A picks a bucket of blueberries in `a` hours. Worker B picks a bucket of blueberries in `b` hours.

  1.  Write an algebraic expression for the fraction of a bucket of blueberries that could be picked in one hour if A and B worked together.  (2 marks)

  2.  What does the reciprocal of this fraction represent?  (1 mark)

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  1. `(a + b)/(ab)`
  2. `text(The reciprocal represents the number of hours it would)`
  3. `text(take to fill one bucket, with A and B working together.)`
Show Worked Solution

i.    `text(In one hour:)`

COMMENT: Note that the question asks for “a fraction”.

`text(Worker A picks)\ 1/a\ text(bucket.)`

`text(Worker B picks)\ 1/b\ text(bucket.)`
 

`:.\ text(Fraction picked in 1 hour working together)`

`= 1/a + 1/b`

`= (a + b)/(ab)`
 

ii.   `text(The reciprocal represents the number of hours it would)`

`text(take to fill one bucket, with A and B working together.)`

Functions, 2ADV F1 2005 HSC 1d

Express  `((2x-3))/2-((x-1))/5`  as a single fraction in its simplest form.  (2 marks)

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`(8x-13)/10`

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`((2x-3))/2-((x-1))/5`

`= (5(2x-3)-2(x-1))/10`

`= (10x-15-2x + 2)/10`

`= (8x-13)/10`

Functions, 2ADV F1 2004 HSC 1d

 Find integers  `a`  and  `b`  by showing working to expand and simplify 

`(3-sqrt2)^2 = a-b sqrt2`.  (2 marks)

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`a = 11,\ b = 6`

Show Worked Solution
`(3-sqrt2)^2` `= 9-6 sqrt2 + (sqrt2)^2`
  `= 9-6 sqrt2 + 2`
  `= 11-6 sqrt2`
   
`:.\ a = 11, \ \ b = 6`

Functions, 2ADV F1 2004 HSC 1c

Solve   `(x-5)/3-(x+1)/4 = 5`.   (2 marks)

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`83`

Show Worked Solution
`(x-5)/3-(x+1)/4` `= 5`
`12((x-5)/3)-12((x+1)/4)` `= 12 xx 5`
`4x-20-3x-3` `= 60`
`x-23` `= 60`
`:. x` `= 83`

Functions, 2ADV F1 2008 HSC 1c

Simplify  `2/n-1/(n+1)`.   (2 marks)

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`(n + 2)/(n(n+1))`

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`2/n-1/(n+1)`

`= (2(n+1)-1(n))/(n(n+1))`

`= (2n + 2-n)/(n(n+1))`

`= (n+2)/(n(n+1))`

Functions, 2ADV F1 2011 HSC 1a

Evaluate  `root(3)(651/(4pi))`  correct to four significant figures.  (2 marks)

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 `3.728\ text{(to 4 sig. figures)}`

Show Worked Solution
 MARKER’S COMMENT: Show answer to 5 or 6 decimals before rounding. Correct rounding of a wrong answer still receives half marks.
`root(3)(651/(4pi))` `=3.72783…`
  `=3.728\ text{(to 4 sig. figures)}`

Functions, 2ADV F1 2012 HSC 2 MC

Which of the following is equal to  `1/(2sqrt5-sqrt3)`? 

  1. `(2sqrt5\-sqrt3)/7`  
  2. `(2sqrt5 + sqrt3)/7`
  3. `(2sqrt5\-sqrt3)/17` 
  4. `(2sqrt5 + sqrt3)/17` 
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`D`

Show Worked Solution

`1/(2sqrt5\-sqrt3) xx (2sqrt5 + sqrt3)/(2sqrt5 + sqrt3)`

`= (2sqrt5 + sqrt3)/( (2sqrt5\-sqrt3)(2sqrt5 + sqrt3) )`

`= (2sqrt5 + sqrt3)/{(2sqrt5)^2-(sqrt3)^2)`

`= (2sqrt5 + sqrt3)/17`

 
`=>  D`