H2 Maths: Some short questions I thought of

• eagle

  6 Jul 09, 00:36

  Some of the questions I thought of to train my students at times... feel free to use it or add on :D

  Shall post whenever remember some or got inspiration to create.

   

  Complex Numbers:

  Evaluate square root of i in the form a + bi in exact form, showing your workings clearly.

   

  Integration:

  Integrate x / (x⁴+4) dx

   

  Enjoy :D

• Forbiddensinner

  6 Jul 09, 12:12

  Originally posted by eagle:

  Some of the questions I thought of to train my students at times... feel free to use it or add on :D

  Shall post whenever remember some or got inspiration to create.

   

  Complex Numbers:

  Evaluate square root of i in exact form, showing your workings clearly.

   

  Integration:

  Integrate x / (x⁴+4) dx

   

  Enjoy :D

  I am here.... to spoil market.

   

  square root i = (i)^1/2 = [ ( i X i )^ 1/2 ] ^1/2 = [ (-1)^1/2 ] ^1/2 = (-1)^1/4

   

  x / ( x^4 + 4 )

  = (x/4) X ( 4 / ( 4 + x^4 )

  = (x/4) X ( 1 / [ ( 4 + x^4 ) / 4 ] )

  = (x/4) X ( 1 / [ 1 + (1/4)(x^4)] )

  = (x/4) X ( 1 / [ 1 + [(1/2)(x^2)]^2 ] )

  Here is when students will say "Ya Hor! Why I never see this?"

  After integrating the above, it will become,

  (1/4)(tan^-1 [(1/2)(x^2)]) + constant

   

   

   

   

• deepak.c

  6 Jul 09, 12:32

   

  Eh......

   

  arctan in A level syllabus meh?

   

   

• eagle

  6 Jul 09, 13:05

  Originally posted by Forbiddensinner:

  I am here.... to spoil market.

   

  square root i = (i)^1/2 = [ ( i X i )^ 1/2 ] ^1/2 = [ (-1)^1/2 ] ^1/2 = (-1)^1/4

   

  x / ( x^4 + 4 )

  = (x/4) X ( 4 / ( 4 + x^4 )

  = (x/4) X ( 1 / [ ( 4 + x^4 ) / 4 ] )

  = (x/4) X ( 1 / [ 1 + (1/4)(x^4)] )

  = (x/4) X ( 1 / [ 1 + [(1/2)(x^2)]^2 ] )

  Here is when students will say "Ya Hor! Why I never see this?"

  After integrating the above, it will become,

  (1/4)(tan^-1 [(1/2)(x^2)]) + constant

   

   

   

   

  first one wrong.. but I think is because I never phrase the question properly... Changed it

  2nd one correct

• eagle

  6 Jul 09, 13:06

  Originally posted by deepak.c:

   

  Eh......

   

  arctan in A level syllabus meh?

   

  yup, in calculus, both differentiation and integration

  They can use their MF15 sheet to do as well

• Forbiddensinner

  6 Jul 09, 13:43

  Originally posted by eagle:

  first one wrong.. but I think is because I never phrase the question properly... Changed it

  2nd one correct

  sq rt i

  = i^(1/2)

  = (2i)^(1/2) / 2^(1/2)

  = (1+2i-1)^(1/2) / 2^(1/2)

  = [(1+i)^2]^(1/2) / 2^(1/2)

  = (1+i) / 2^(1/2)

  = 1 / 2^(1/2) + [1 / 2^(1/2)] i

   

• eagle

  6 Jul 09, 14:24

  yup :D

  a shorter method would be to convert

  i = e^(i pi/2)
  sqrt(i) = e^(i pi/4)

  expand Euler's formula to cos + i sin
  sqrt (i) = 1 / 2^(1/2) + [1 / 2^(1/2)] i

• deepak.c

  6 Jul 09, 19:26

  Originally posted by eagle:

  yup, in calculus, both differentiation and integration

  They can use their MF15 sheet to do as well

   

  Ooops.....looks like I was daydreaming during lectures.

   

• wee_ws

  8 Jul 09, 19:53

  Hi,

    Let me add a question that connects three topics (i.e. graphing techniques, applications of differentiation and applications of integration).

    The curve C has equation given by x^2 + 3xy = 4.

    (i) Sketch C, indicating its asymptotes, stationary points and axial intercepts, where appropriate.

    (ii) Prove, algebraically, that there are no points on C at which the tangent is parallel to the x-axis.

    (iii) Find the exact area of the region bounded by C, the line x = 6 and the x-axis.

    (iv) The region bounded by C, the line y = 1 and the axes is rotated through 4 right angles about the y-axis. Find the volume of revolution.

    Thanks!

  Cheers,
  Wen Shih

• Ah Chia

  8 Jul 09, 19:58

  Okay off topic here, but calculus is useful in which area of study?

• eagle

  8 Jul 09, 20:03

  Originally posted by Ah Chia:

  Okay off topic here, but calculus is useful in which area of study?

  
  Engineering?

  Physics?

   

• Ah Chia

  8 Jul 09, 20:06

  Any others?

   

• wee_ws

  8 Jul 09, 21:25

  Hi,

    Business, economics, biology and medicine too. Thanks!

  Cheers,
  Wen Shih

• Lee012lee

  8 Jul 09, 22:47

  Hi Mr Wee,

  Mr Chia is only interested in politics.

  He will not be interested on the applications of calculus to business, economics, biology, medicine, engineering or physics.

  He is only interested if calculus can be applied to politics.

  The url link for the application of differentiation and integration to politics is

  http://www.jstor.org/pss/2778814

  But it might be too mathematical for Mr Chia.

   

• wee_ws

  13 Jul 09, 22:59

  Originally posted by wee_ws:

    The curve C has equation given by x^2 + 3xy = 4.

    (iv) The region bounded by C, the line y = 1 and the axes is rotated through 4 right angles about the y-axis. Find the volume of revolution.

  Hi,

    I have written an article on my website:

    http://www.freewebs.com/weews/solidsofrevolution.htm

    Enjoy!

  Cheers,
  Wen Shih

• absol

  20 Jul 09, 17:57

  Originally posted by eagle:

   

  Complex Numbers:

  Evaluate square root of i in the form a + bi in exact form, showing your workings clearly.

  root of i, wouldnt there be 2 answers?

  let i^1/2 be a+bi

  i = (a+bi)^2

  i = a^2 - b^2 + 2abi

  comparing coeff of real part,

  a^2-b^2 = 0

  => a = b   -(1)

  comparing coeff of imaginary,

  1 = 2ab

  since a=b by (1), a^2 = 0.5 and b^2 = 0.5

  => a = plus minus 1/sqrt2 and b = plus minus 1/sqrt 2.

   

  therefore i^1/2 = 1/sqrt2 + 1/sqrt2 i and -1/sqrt2 - 1/sqrt2 i...?