S4 Math questions

1. Let one of the 2 roots be a, then the other root is 1/a (meaning of reciprocal), so product of roots = a x 1/a = 1. So knowing the coefficients is not necessary.
2.
Length of one wire is x, so side of square formed = x/4, so area of square = x^2/16.
Length of the other wire = (32 - x), side of square formed = (32 - x)/4, so area of square = (32 - x)^2/16
Sum of area of the 2 squares = x^2/16 + (32 - x)^2/16
= 1/16[x^2 + 1024 +x^2 - 64x) = 1/16(2x^2 - 64x + 1024) = 1/8(x^2 - 32x + 512)
By completing square method, this can be expressed as 1/8[(x - 16)^2 - 256 + 512) = 1/8[(x - 16)^2 + 256] = 1/8(x - 16)^2 + 32.
So area is a minimum when (x - 16) = 0, that is x = 16.
3.
Let original intensity = A, so 50dB = 10 log A/B where B is the reference intensity.
Now intensity is 1000A, so loudness = 10 log 1000A/B = 10 [ log 1000 + log A/B] = 10 log 1000 + 10 log A/B = 30 + log A/B = 30 + 50 = 80dB.
So increase in loudness = 30 dB.
4.
Let point E be vertically above point A, so angle BEA is a rt. angle.
So angle EAB = 180 - 33 - 90 = 57 degree. ( angle sum of triangle.)
Angle CAD = 180 - 55 - 57 = 68 degree ( ad. angles on a st. line.)
By sine rule, AC/sin 95 = 50/sin 68, so AC = 53.7215.
Angle ABC = 90 - 33 = 57 degree.
Again by sine rule, BC/sin 55 = AC/sin 57
so height of tall building = BC = AC sin 55/sin 57 = 52.47m.
5.
Since complex roots are in conjugate, if one root is 3 + 4i, the other root will be 3 - 4i.
- b/5 = sum of roots = 3 + 4i + 3 - 4i = 6, so b = - 30.
c/5 = product of roots = (3 + 4i)(3 - 4i) = 3^2 + 4^2 = 9 + 16 = 25
so c = 125.
6.
Let AC and BD intersects at point M. AM = MC = 6
So DM/AM = 9/6 = 3/2.
Correspondingly, MC/MB = 6/4 = 3/2.
Also, angle AMD = angle BMC (vert. opp. angles)
So triangle AMD similar to triangle BMC ( incl. angle equal and side in same ratio).
So angle DAM = angle CBM,
so ABCD is a cyclic quad. ( angle in the same segment equal).