H2 Maths: Some short questions I thought of

eagle SG

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 SG

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 SG

Eh......

arctan in A level syllabus meh?

eagle SG

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 SG

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 SG

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 SG

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 SG

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 SG

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 SG

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

eagle SG

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

Engineering?

Physics?

Ah Chia SG

Any others?

wee_ws SG

Hi,

Business, economics, biology and medicine too. Thanks!

Cheers,
Wen Shih

Lee012lee SG

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 SG

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 SG

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...?