Q10.

Q:::) (x-1)(x-2)(x-3)(x-4)=15
find no.of +ive solutions..
A:) simplyfy to get x4-10x3+35x2-50x+9=0 ---a
+ - + - + : 4 sign changes imply 4 real roots exist.
differentiate to get 4x3-30x2+70x-50=0 ----b
differentiate to get 12x2-60x+70=0 -----c
both roots of 'c' are positive...(obvious)..and f(0)>0
hence dy/dx =0 for 'b' at points say 'p','q' where both>0
and fb(0)<0> all 3 roots of (B) are positive... since for (b) f(inf)=inf & f(-inf)=-inf
coming bak to (a)...dy/dx=0 at 3 points ....and all are positive....further f(0)>0 and f(inf)=f(-inf)=inf.....hence all roots are positive...

thus answer should be 4 but no such option exixts..

An Ode to Opinion sheet

o thee opinion sheet,
given,when teachers deem fit.

for us,its a holy grail,
for it, we pale nd wail.

as days go by,
we wait to say them 'hi'.

some're waiting with a file,
hoping they would pile.

In delhi,this big city,
for them,there's no pity.

o teachers,i do sympathize,
but no, i can't empathize.

write you will, good-good things,
give,your imagination wings.

your opinion , may not change,
you've got a very small range.

but, just in case it does,
we'll have nothing much to fuss.

we'll not say 'woof--woof'
coz we'll have a good proof-proof....

4th may

life has become monotonous.....its under a spell of non-action and listlessness....there 's a kind of calm that has set into my life--the kind that usually spells out the onset of a tempest- a tempest so immensely powerful that it threatens to engulf your identity. Yet, giving up is the first sign of defeat..and so have i decided to wage on the battle and take on life with a new found curiosity and purpose. It has been tough with one blow after the other...as what i thought were my strongholds...crumpled like ninepins...and the cards that held my hope fell like dominoes..

to lose and not to concede defeat is the way of life. And today i do realise 1 thing for sure--to change the world is slightly less difficult than changing oneself...