So....here goes some of the SHORTCUTS
in complex number
|z - z1| + |z - z2| = K---------(K = const)....
(condition....co eff of z mst be same , z1 & z2 can be real )
then this is the e=n of an ELLIPSE
IF coeff of z are diff....then it will be a PARABOLA
2) wen given.....
coeff of x2 & y2 be same , the e=n will be of a circle
if co eff of x2 & y2 both + ive then ELLIPSE AND if coeff of 1 is -ve then HYPERBOLA
3) In AP etc....
we all knw about arthmetic mean , GM and HM
d in case of
AM = (b-a)/(n+1)
GM = (b/a)1/(n+1)
BUT...wats n--?
n is not the no. of total terms
n is the number of INSERTED TERMS
4) in an AGP , 1st term will always be 1
and last term will be product of "n" terms of AP & GP
5) sum to infinity of an AGP
= a / 1-r + dr / (1-r)2
but r is always < 1
6) If a,b,c are in AP as well as GP then
a = b = c (necesserily)
7)co-eff of x99 in exp of
(x-1)(x-2)(x-3)...........(x-100) =
1 + 2 + 3 + ................... + 100
and of x98 = 1 + 2 + 3 + 4 + 5 + .......... + 99
i hope u got wat i meant to say (1 extra)
8) I FORGOT 1 IN COMPLEX
(z1/z2) + (z2/z1) = 1 and z3 = 0 , its an equilateral triangle wid 1 of vertices at origin
9) have problems in qs solving like
1.2 + 1.2.3 + 1.2.3.4 + ........................
= ????
if given the choices (only for multiple choice qs)
like if choice given n2(n+1)/2
so jst take n = 2 ,
add 1st 2 terms
2 + 6 = 8.....
put n = 2 in the given
it doesnt comes out to be 8 so it cant be the answer
like this qs....
sum of series
3.8 + 6.11 + 9.14 + ................
= ?
options
a) n
b) 3n(n+1)
c)3n(n+1)(n+3)
d) N.O.T
so do that thing
3.8 + 6.11 = 90
put n = 2 in option 1,2,3 resp....
u get option 3 as the answer
P.S -> WEN U GET 2 OPTIONS AS THE ANSWER.....TRY PUTTING n=3 and so on....
LOG WALE HAR QS KA ANS 0 hoga
10) wen given series
8-6i , 7-4i , 6-2i........
last term (an) = 0
and Sn = 0 too
11)in tangents and NORMALS........
length of a subnormal = (y) (dy/dx)
12) In increasing and decreasing func.....
f(x) = kx3 - 9x2 + 9x + 3 is increasing on R
take F'(x)
its a quadratic , and increasing on R
so vaise toh v take D >= 0
but we take
D > 0
HOPE ALL THESE HELP U :) :)
MORE TO FOLLOW
cheers!!!!!!!!!!!
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