158  Mathematics Questions Answers
If sin^{1}a + sin^{1}b+ sin^{1}c = π , then the value of { a√(1a^{2}) + b√(1b^{2}) + c√(1c^{2})}
let sin^{1}a = A , sin^{1}b = B and sin^{1}c= C^{ }
so ^{ }{ a√(1a^{2}) + b√(1b^{2}) + c√(1c^{2})} = sinAcosA + sinBcosB + sinCcosC = 1/2 (sin2A+sin2B+sin2C)
= 1/2 (4sinAsinBsinC)
= 2abc
If [cot‾^{1}x] + [cos‾^{1}x] = 0 , then complete set of value of x is ( [ * ] is GIF)?
These are the graphs for cot^{1}x and cos^{1}x
violet for cos and green for cot
from graph it is clear that cos part with integer function will be 0 for cos1<x≤1
and cot part with integer function will be 0 for cot1<x<∞
so answer will be cot1<x≤1
y=px+a/p
this is a special type of differential equation in which p = dy/dx
its solution will be y = cx+(a/c) here c is a constant
yxp=x+yp
on applying p = dy/dx
(ydxxdy)/dx = (xdx+ydy)/dx
or ydxxdy = xdx+ydy
or dy/dx = (yx)/(y+x)
now it is homogeneous
1) Find the invariants of the matrix
0 1 0
0 0 0
0 0 0 of a linear transformation T in A(v) ?
2) Find the companion matrix of the polynomial ( x+1)^{2 } ?
3) show that two real symmetric matrices are congruent if and only if they have the same rank and signature?
rank(A) = 1
det(A) = 0
trace(A) = 0
The matrix is not symmetric.
characteristic polynomial of the given matrix is x^{3}
companion matrix of the polynomial (x+1)^{2}
0 1
1 2
A real symmetric matrix of rank r is congruent over the field of real numbers to a canonical matrix
The integer p is called the index of the matrix and s = p  (r  p) is called the signature.
The index of a symmetric or Hermitian matrix is the number of positive elements when it is transformed to a diagonal matrix. The signature is the number of positive terms diminished by the number of negative terms and the total number of nonzero terms is the rank.
now solve it
A regular hexagon and a regular dodecagon are inscribed in the same circle. if the side of the dodecagon is (√31), then the side of hexagon is
for dodecagon sin(2π/40) = (a/2)/r so r = (a/2)/ sin(π/20)
now for hexagon sin(2π/12) = (x/2)/r so x/2 = r sin(π/6) = (a/2) sin(π/6)/ sin(π/20)
so x = (√31) (1/2) / ((√51)/4) here sin(π/20) = (√51)/4
or x = 4(√31) / 2(√51)
now solve it for further simplification
What are trigonometrical identities?
sin^{2}A + cos^{2}A = 1 and two similar formulae in terms of tanA, cotA, secA and cosecA
these are identities because these are true for all values of A
if secAtanA=3
then find the value of (5cosA+4cotB)??
jus a hint
First replace cotB by cotA.
because it is wrongly printed by you.
answer by NIKHIL is correct
The sum of three positive real numbers x, y and z is 9. If the largest positive difference between any 2 of the 3 numbers is 5 and these 3 numbers are in arithmetic progression then the product of these 3 numbers will be what .please tell me.
let the numbers are ad, a, a+d
then according to the given condition ad+a+a+d = 9 so 3a = 9 so a = 3
and [a+d][ad] = 5 so 2d = 5 so d =5/2
now solve
Importance High!!!!!!!!!!!!!!!!!
Dear Sir / Madam
Myself Rajeev Shrivastava, I am putting a signs series in front of you which based on numbers from 01 to 100. Actually I want to know that what reason of behind Approx equability of signs is
(++), (+), () & (+) at end of the month or year between the below mentioned.
Let's define a kind of mapping:
0↔1
2↔3
4↔5
6↔7
8↔9
(+) means: an Even number
() means: an Odd number
I simply say: for any EO (+) it exists an OE (+) which is paired to.
And that's true since each E is mapped to a O and vice versa (1 to 1 relation where "f(f(x))=x" ) !
E.g.: 29 ↔ 83
There is 25 numbers in the EO (+) group, so 25 in OE (+).
There is 25 numbers in the EE (++) group, so 25 in OO ().
Picking a number at random between 01 and 100 inclusive is choosing equitably in EO, OE, EE or OO group. (25% for each)
So, I really need of your help to solve a puzzle please. I just am trying to make you understand what type of solution I need:
There are some results for your review:
MONTH 
DATE 
DAY 
FIRST_SERIES 
SECOND_SERIES 
FIRST_SERIES 
SECOND_SERIES 

APR 
1 
SUN 
+ 
 
 
+ 
+ 
+ 
APR 
2 
MON 
+ 
+ 
 
+ 
++ 
+ 
APR 
3 
TUE 
+ 
 
+ 
 
+ 
+ 
APR 
4 
WED 
+ 
+ 
+ 
+ 
++ 
++ 
APR 
5 
THU 
 
+ 
 
+ 
+ 
+ 
APR 
6 
FRI 
+ 
+ 
+ 
 
++ 
+ 
APR 
7 
SAT 
+ 
 
 
+ 
+ 
+ 
APR 
8 
SUN 
 
+ 
+ 
 
+ 
+ 
APR 
9 
MON 
 
+ 
 
+ 
+ 
+ 
APR 
10 
TUE 
+ 
+ 
+ 
+ 
++ 
++ 
APR 
11 
WED 
+ 
+ 
 
 
++ 
 
APR 
12 
THU 
 
+ 
 
 
+ 
 
APR 
13 
FRI 
+ 
+ 
+ 
+ 
++ 
++ 
APR 
14 
SAT 
+ 
 
+ 
 
+ 
+ 
APR 
15 
SUN 
+ 
 
+ 
 
+ 
+ 
APR 
16 
MON 
 
 
+ 
+ 
 
++ 
APR 
17 
TUE 
 
 
+ 
+ 
 
++ 
APR 
18 
WED 






APR 
19 
THU 






APR 
20 
FRI 






APR 
21 
SAT 






APR 
22 







APR 
23 







(Complete sheets for the year 2011 and 2012 are attached with this mail)
You can see that there are two times falls in a day of pair make by even (+) and odd () signs, i.e. (++), (+), () & (+) as above said.
The total of pair combination signs become approx equal to each other at the end of the month or year (every year so, I am sending you attachment to go through the year’s result).
By this analysis:
1 I want to know that what is the relation between current falling signs and past fell signs.
2 How can I come to know, what type of combination of sign would be any particular date or day.
3 As you can seen on 22^{nd} and 23^{rd} April’2012 I don’t know what combination of sign is.
4 There is surety that there is some relation between current First fall of combination of signs and past fall or First fall of combination of signs and Second fall of combination of signs.
Please help me to how come to know what type of fall may be for next day by reviewing past or first combination of signs. So reply or elaborate me with an example.
I would be very grateful to you till entire life.
Regards
RAJEEV SHRIVASTAVA
Importance High!!!!!!!!!!!!!!!!!
Dear Sir / Madam
Myself Rajeev Shrivastava, I am putting a signs series in front of you which based on numbers from 01 to 100. Actually I want to know that what reason of behind Approx equability of signs is
(++), (+), () & (+) at end of the month or year between the below mentioned.
Let's define a kind of mapping:
0↔1
2↔3
4↔5
6↔7
8↔9
(+) means: an Even number
() means: an Odd number
I simply say: for any EO (+) it exists an OE (+) which is paired to.
And that's true since each E is mapped to a O and vice versa (1 to 1 relation where "f(f(x))=x" ) !
E.g.: 29 ↔ 83
There is 25 numbers in the EO (+) group, so 25 in OE (+).
There is 25 numbers in the EE (++) group, so 25 in OO ().
Picking a number at random between 01 and 100 inclusive is choosing equitably in EO, OE, EE or OO group. (25% for each)
So, I really need of your help to solve a puzzle please. I just am trying to make you understand what type of solution I need:
There are some results for your review:
MONTH 
DATE 
DAY 
FIRST_SERIES 
SECOND_SERIES 
FIRST_SERIES 
SECOND_SERIES 

APR 
1 
SUN 
+ 
 
 
+ 
+ 
+ 
APR 
2 
MON 
+ 
+ 
 
+ 
++ 
+ 
APR 
3 
TUE 
+ 
 
+ 
 
+ 
+ 
APR 
4 
WED 
+ 
+ 
+ 
+ 
++ 
++ 
APR 
5 
THU 
 
+ 
 
+ 
+ 
+ 
APR 
6 
FRI 
+ 
+ 
+ 
 
++ 
+ 
APR 
7 
SAT 
+ 
 
 
+ 
+ 
+ 
APR 
8 
SUN 
 
+ 
+ 
 
+ 
+ 
APR 
9 
MON 
 
+ 
 
+ 
+ 
+ 
APR 
10 
TUE 
+ 
+ 
+ 
+ 
++ 
++ 
APR 
11 
WED 
+ 
+ 
 
 
++ 
 
APR 
12 
THU 
 
+ 
 
 
+ 
 
APR 
13 
FRI 
+ 
+ 
+ 
+ 
++ 
++ 
APR 
14 
SAT 
+ 
 
+ 
 
+ 
+ 
APR 
15 
SUN 
+ 
 
+ 
 
+ 
+ 
APR 
16 
MON 
 
 
+ 
+ 
 
++ 
APR 
17 
TUE 
 
 
+ 
+ 
 
++ 
APR 
18 
WED 






APR 
19 
THU 






APR 
20 
FRI 






APR 
21 
SAT 






APR 
22 







APR 
23 







(Complete sheets for the year 2011 and 2012 are attached with this mail)
You can see that there are two times falls in a day of pair make by even (+) and odd () signs, i.e. (++), (+), () & (+) as above said.
The total of pair combination signs become approx equal to each other at the end of the month or year (every year so, I am sending you attachment to go through the year’s result).
By this analysis:
1 I want to know that what is the relation between current falling signs and past fell signs.
2 How can I come to know, what type of combination of sign would be any particular date or day.
3 As you can seen on 22^{nd} and 23^{rd} April’2012 I don’t know what combination of sign is.
4 There is surety that there is some relation between current First fall of combination of signs and past fall or First fall of combination of signs and Second fall of combination of signs.
Please help me to how come to know what type of fall may be for next day by reviewing past or first combination of signs. So reply or elaborate me with an example.
I would be very grateful to you till entire life.
Regards
RAJEEV SHRIVASTAVA
If you are making the sequence on the basis of dates then you should take signs accordingly then you can make this sequence for many years
for example if you take 23 june then first sign series will be +, + for even number and  for odd number , now as conversion rule as you gave in your explanation 23 will be converted to 32 so second series will be +
similarly you can make it for any date
for example 29 december first series will be + and its conversion is 38 so next series will be +
if you are thinking something else then explain your question correctly