SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

Functions, 2ADV EQ-Bank 10 MC

The graph of  `y = kx-3`  intersects the graph of  `y = x^2 + 8x`  at two distinct points for

  1. `k = 11`
  2. `k > 8 + 2 sqrt 3 or k < 8-2 sqrt 3`
  3. `5 <= k <= 6`
  4. `8-2 sqrt 3 <= k <= 8 + 2 sqrt 3`
Show Answers Only

`B`

Show Worked Solution

`text(Intersection occurs when:)`

`kx-3` `= x^2 + 8x`
`x^2 + (8-k)x + 3` `= 0`

 
`text(For 2 points of intersection:)`

`Delta` `> 0`
`(8-k)^2-4 (3)` `> 0`
`(8-k)^2` `>12`

 
`:. k < 8-2 sqrt 3\  uu\  k > 8 + 2 sqrt 3`

`=>   B`

Functions, 2ADV EQ-Bank 5 MC

The set of values of `k` for which  `x^2 + 2x-k = 0`  has two real solutions is

  1. `[-1, 1]`
  2. `(-1, oo)`
  3. `(-oo, -1)`
  4. `[-1]`
Show Answers Only

`B`

Show Worked Solution

`text(Two real solutions):`

`b^2-4ac` `> 0`
`4-4 ⋅ 1 ⋅ (-k)` `> 0`
`4k` `> -4`
`k` `> -1`

 
`k in (-1, oo)`

`=>   B`

Functions, 2ADV EQ-Bank 11

Express  \(\dfrac{m^4\, n^{-3}}{m^{-2}\, n^2}\)  with positive indices only.   (1 mark)

Show Answers Only

\(\dfrac{m^6}{n^5}\)

Show Worked Solution

\(\dfrac{m^4\, n^{-3}}{m^{-2}\, n^2}=m^{(4+2)}\, n^{(-3-2)}=m^6\, n^{-5}=\dfrac{m^6}{n^5}\)

Functions, 2ADV EQ-Bank 10

Simplify  \(\left(16p^8\right)^{\frac{3}{4}} \ ÷\ (2p)^3\).   (2 marks)

Show Answers Only

\(p^3\)

Show Worked Solution
\(\left(16^{\frac{1}{4}} \times p^{8 \times \frac{1}{4}}\right)^3 \ ÷\ 8 p^3\) \(=\dfrac{\left(2 p^2\right)^3}{8 p^3}\)
  \(=\dfrac{8 p^6}{8 p^3}\)
  \(=p^3\)

Functions, 2ADV EQ-Bank 9

Simplify  \(\left(27x^6\right)^{\frac{1}{3}} \times(2x)^{-2}\).   (2 marks)

Show Answers Only

\(\dfrac{3}{4}\)

Show Worked Solution
\(\left(27 x^6\right)^{\frac{1}{3}} \times(2 x)^{-2}\) \(=27^{\frac{1}{3}} \times x^{6 \times \frac{1}{3}} \times \dfrac{1}{4 x^2}\)
  \(=3 x^2 \times \dfrac{1}{4 x^2}\)
  \(=\dfrac{3}{4}\)

Functions, 2ADV EQ-Bank 6

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

Show Answers Only

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

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 5

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

Show Answers Only

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

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 4

Simplify  \(\dfrac{a^3 b-a b^3}{a^2+2 a b+b^2}\).   (2 marks)

Show Answers Only

\(\dfrac{a b(a-b)}{a+b}\)

Show Worked Solution
\(\dfrac{a^3 b-a b^3}{a^2+2 a b+b^2}\) \(=\dfrac{a b\left(a^2-b^2\right)}{(a+b)^2}\)
  \(=\dfrac{a b(a+b)(a-b)}{(a+b)^2}\)
  \(=\dfrac{a b(a-b)}{a+b}\)

Functions, 2ADV EQ-Bank 3

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)

Show Answers Only

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

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 2

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

Show Answers Only

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

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 1

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)

Show Answers Only

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

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 20

Find the value of \(k\)  if  \(4kx^2-(3-4k) x+k=0\)  has one root.   (2 marks)

Show Answers Only

\(k=\dfrac{3}{8}\)

Show Worked Solution

\(4kx^2-(3-4k) x+k=0\)

\(\text{1 root}\ \Rightarrow \Delta=0\)

  \(\Delta\) \(=b^2-4 a c\)
  \(0\) \(=\left[ -\left( 3-4k \right)\right]^2-4\times 4k \times k\)
  \(0\) \(=9-24k + 16k^2-16k^2\)
  \(24k\) \(=9\)
  \(k\) \(=\dfrac{9}{24}=\dfrac{3}{8}\)

Functions, 2ADV F1 EQ-Bank 10

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

Show Answers Only

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

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

Functions, 2ADV F1 EQ-Bank 13

Show that the parabola  \(2x^2-kx+k-2\)  has at least one real root.   (3 marks)

Show Answers Only

\(2x^2-kx+k-2=0\)

\(\Delta=b^2-4ac=(-k)^2-4 \times 2(k-2) = k^{2}-8k+16\)

\(\text{Real roots:}\ \ \Delta \geqslant 0\)

\(k^2-8k+16\) \(\geqslant 0\)  
\((x-4)^2\) \(\geqslant 0\)  

 
\(\therefore\ \text{At least one root exists for all}\ k\)

Show Worked Solution

\(2x^2-kx+k-2=0\)

\(\Delta=b^2-4ac=(-k)^2-4 \times 2(k-2) = k^{2}-8k+16\)

\(\text{Real roots:}\ \ \Delta \geqslant 0\)

\(k^2-8k+16\) \(\geqslant 0\)  
\((x-4)^2\) \(\geqslant 0\)  

 
\(\therefore\ \text{At least one root exists for all}\ k\)

Functions, 2ADV F1 EQ-Bank 16

  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)

Show Answers Only

a.    `-1/3`

b.    `0.462`

Show Worked Solution

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

b.     `1/(root3(7+pi))` `=0.4619…`
    `=0.462\ \ text{(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)

Show Answers Only

`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)

Show Answers Only

 `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)

Show Answers Only

`:. 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)

Show Answers Only
  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 2015 HSC 12d

For what values of `k` does the quadratic equation  `x^2-8x + k = 0`  have real roots?   (2 marks)

Show Answers Only

`k <= 16`

Show Worked Solution

`x^2-8x + k = 0`

`text(Real roots when)\ Delta >= 0:`

`b^2-4ac` `>= 0`
`(-8)^2-4 xx 1 xx k` `>= 0`
`64-4k` `>= 0`
`4k` `<= 64`
`k` `<= 16`

 
`:.\ text(Real roots exists when)\ k <= 16`

Functions, 2ADV 2004 HSC 2c

For what values of `k` does  `x^2-kx + 4 = 0`  have no real roots?   (2 marks)

Show Answers Only

`-4 < k < 4`

Show Worked Solution

`x^2-kx + 4 = 0`

`text(No real roots when)\ Delta < 0`

`b^2-4ac` `< 0`
`(text(–) k^2)-4 xx 1 xx 4` `< 0`
`k^2-16` `< 0`
`k^2` `< 16`
`:.\ -4 < k` `< 4`

 

`:.\ text(There are no real roots when)\ \ \ -4 < k < 4.`

Functions, 2ADV F1 2005 HSC 1d

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

Show Answers Only

`(8x-13)/10`

Show Worked Solution

`((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)

Show Answers Only

`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)

Show Answers Only

`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)

Show Answers Only

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

Show Worked Solution

`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 2009 HSC 4b

Find the values of `k` for which the quadratic equation 

`x^2-(k + 4)x + (k + 7) = 0`

has equal roots.    (3 marks)

Show Answers Only

 `k = -6 \ \ text(or)\ \  2`

Show Worked Solution

`x^2-(k + 4)x + (k + 7) = 0`

`text(Equal roots when)\ Delta = 0:`

`[-(k + 4)]^2-4(1)(k + 7)` `= 0`
`k^2 + 8k + 16-4k\-28` `= 0`
`k^2 + 4k-12` `= 0`
`(k + 6)(k-2)` `= 0`
`k` `= -6 \ \ text(or)\ \  2`

 
`:.\ text(Equal roots when)\ \ k = -6 or 2`

Functions, 2ADV F1 2011 HSC 1a

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

Show Answers Only

 `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` 
Show Answers Only

`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`