SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

Proof, EXT2 P1 2022 HSC 15d

The complex number `z` satisfies `|z-(4)/(z)|=2`.  

Using the triangle inequality, or otherwise, show that `|z| <= sqrt5+1`.  (3 marks)

Show Answers Only

`text{Proof (See Worked Solutions)}`

Show Worked Solution


♦♦♦ Mean mark 27%.

`text{Triangle Inequality:}\ \ absx+absy>=abs(x+y)`

`absz` `<=abs(z-z/4)+abs(4/z)`  
`absz` `<=2+4/absz\ \ \ (text{using}\ |z-(4)/(z)|=2)`  
`absz^2` `<=2absz+4`  
`absz^2-2absz-4` `<=0`  
`absz` `<=(2+sqrt(2^2+4xx4))/2\ \ \ (absz>=0)`  
`absz` `<=(2+sqrt20)/2`  
`absz` `<=1+sqrt5\ \ text{… as required}`  

Vectors, EXT2 V1 2021 HSC 16a

  1. The point  `P(x, y, z)`  lies on the sphere of radius 1 centred at the origin `O`.
  2. Using the position vector of `P, overset->{OP} = x underset~i + y underset~j + z underset~k` , and the triangle inequality, or otherwise, show that `| x | + | y | + |z | ≥ 1`.  (2 marks)

  3. Given the vectors  `underset~a = ((a_1),(a_2),(a_3))`  and  `underset~b = ((b_1),(b_2),(b_3))` , show that 
  4.    `|a_1 b_1 + a_2 b_2 + a_3 b_3 | ≤ sqrt{a_1^2 + a_2^2 + a_3^2 }\ sqrt{b_1^2 + b_2^2 + b_3^2}`.  (3 marks)

  5. As in part (i), the point  `P (x, y, z)`  lies on the sphere of radius 1 centred at the origin `O`.
  6. Using part (ii), or otherwise, show that  `| x | + | y | + | z | ≤ sqrt3`.  (2 marks)

Show Answers Only
  1. `text{See Worked Solution}`
  2. `text{See Worked Solution}`
  3. `text{See Worked Solution}`
Show Worked Solution
i.      `text{Triangle inequality:} \ |x| + |y| ≥ |x + y|`
♦ Mean mark (i) 50%.
`|x| + |y| + |z|` ` = |x_underset~i| + |y_underset~j| + |z_underset~k|`
  `≥ |x_underset~i + y_underset~j| + |z_underset~k|`
  `≥ |x_underset~i + y_underset~j + z_underset~k|`
  `≥ 1\ \ (|overset->{OP}| = | x_underset~i + y_underset~j + z_underset~k | = 1)`

 

ii.   `text{Using the dot product:}`

♦ Mean mark (ii) 42%.

`underset~a * underset~b = a_1 b_1 + a_2 b_2 + a_3 b_3`

`underset~a * underset~b = |underset~a| |underset~b| cos theta`

 

`a_1 b_1 + a_2 b_2 + a_3 b_3` `= sqrt{a_1^2 + a_2^2 + a_3^2} * sqrt{b_1^2 + b_2^2 + b_3^2} * cos theta`
`|a_1 b_1 + a_2 b_2 + a_3 b_3|` `= sqrt{a_1^2 + a_2^2 + a_3^2} * sqrt{b_1^2 + b_2^2 + b_3^2} * |cos theta|`

 

`text{S} text{ince} \ -1 ≤ cos theta ≤ 1 \ => \ |cos theta| ≤ 1`

`:. \ | a_1 b_1 + a_2 b_2 + a_3 b_3 | ≤ sqrt{a_1^2 + a_2^3 + a_3^2} * sqrt{b_1^2 + b_2^2 + b_3^2}`

 

iii.  `text{Using part (ii) with vectors:}`

♦♦♦ Mean mark (iii) 14%.

`underset~a = ((1),(1),(1)) \ , \ underset~b = (( | x| ),( |y| ),( |z| ))`

`|\ |x| + |y| + |z|\ |` `≤ sqrt{1^2 + 1^2 + 1^2} * sqrt{ x^2 + y^2 + z^2}`
`|x| + |y| + |z|` `≤ sqrt3`