9.


Rationalise the denominator for all the terms, multiplying each term by its conjugate.

The denominator is in the form (a + b)(a - b), which equals a² - b². The square roots have been resolved and we get an integer.

Eg.

1/(√2000 + √2004)

= 1/(√2000 + √2004) × (√2000 - √2004)/(√2000 - √2004)

= (√2000 - √2004)/((√2000)² - (√2004)²)

= (√2000 - √2004)/(2000 - 2004)

= (√2000 - √2004)/(-4)


Do this for every term and you will get :


= (√2000 - √2004)/(-4) + (√2004 - √2008)/(-4) + (√2008 - √2012)/(-4) + (√2012 - √2016)/(-4)


= (√2000 - √2004 + √2004 - √2008 + √2008 - √2012 + √2012 - √2016) / (-4)

= (√2000 - √2016) / (-4)

= (√(400×5) - √(144×14)) / (-4)

= (√400√5 - √144√14) / (-4)

= (20√5 - 12√14)/(-4)

= -5√5 + 3√14

= 3√14 - 5√5

(A)

c = sin 102°, d = sin 169°

sin → opposite / hypotenuse

Remember from the graph of sin x , it rises from 0 when x = 0° to 1 when x = 90°. It then falls to 0 again from x = 90° to x = 180° (second quadrant)

Since 169° > 102° , sin 169° < sin 102°

d < c




a = cos 282°, b = cos 349°

cos → adjacent/hypotenuse

Likewise, these two are in thr 4th quadrant where cos x is positive between 270° and 360° (exclusive of the former)

Here, cos x increases from 0 when x = 270° to 1 when x = 360°


Since 349° > 282°, cos 282°< cos 349°

a < b



From these two conclusions, options (C),(D) and (E) are eliminated.


Comparing (A) and (B),

We just need to decide if c > a or a > c

a = cos 282°, c = sin 102°

Now recall sin x = cos (90° - x) and cos x = sin (90° - x)

(Either by looking at the two acute angles of a right - angled triangle, or by translating the graphs of sin x and cos x)


But we also know that the graphs repeat every 360° i.e sin x = sin (x + 360°) and cos x = cos (x + 360°)


So cos 282° = cos ( -78° + 360°)

= cos (-78°)

= cos 78° since cos x = cos (-x)

= cos (90° - 12°)

= sin 12°

= sin (180° - 12°)

(Supplement angle identity, based on symmetry of the triangles drawn)

= sin 168°


And sin 168° < sin 102° since sin x decreases from 0 to -1 from 90° to 180° (2nd quadrant)

So c > a


∴ b > c > a > d

Option (B) is correct.

If we convert all the cos to sin, we get :


c = sin 102°
d = sin 169°

a = cos 282° = sin 168° (shown earlier)


b = cos 349°
= cos (360° - 11°)
= cos (-11°)
= cos 11°
= sin (90° - 11°)
= sin 79°
= sin (180° - 79°)
= sin 101°


sin 101° > sin 102° > sin 168° > sin 169°

b > c > a > d