# 158 - Mathematics Questions Answers

If  sin-1a + sin-1b+ sin-1c = π , then the value of { a√(1-a2) + b√(1-b2) + c√(1-c2)}

Asked By: AMIT DAS
Joshi sir comment

let sin-1a = A ,    sin-1b = B    and    sin-1c= C

so  { a√(1-a2) + b√(1-b2) + c√(1-c2)}   = sinAcosA + sinBcosB + sinCcosC   = 1/2 (sin2A+sin2B+sin2C)

= 1/2 (4sinAsinBsinC)

= 2abc

If    [cot‾1x] + [cos‾1x] = 0 , then complete set of value of x is ( [ * ] is GIF)?

Asked By: AMIT DAS
Joshi sir comment These are the graphs for cot-1x and cos-1x

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

Joshi sir comment

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

y-xp=x+yp

Joshi sir comment

on applying p = dy/dx

(ydx-xdy)/dx = (xdx+ydy)/dx

or ydx-xdy = xdx+ydy

or dy/dx = (y-x)/(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?

Joshi sir comment

rank(A) = 1
det(A) = 0
trace(A) = 0
The matrix is not symmetric.

characteristic polynomial of the given matrix is x3

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 (√3-1), then the side of hexagon is

Asked By: HEMA RANI
Joshi sir comment

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  = (√3-1) (1/2) / ((√5-1)/4)                      here sin(π/20) = (√5-1)/4

or x = 4(√3-1) / 2(√5-1)

now solve it for further simplification

What are trigonometrical identities?

Asked By: SIDDHANT KAPOOR
Joshi sir comment

sin2A + cos2A = 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 secA-tanA=3

then find the value of (5cosA+4cotB)??

jus a hint

Asked By: AMBUJ SINGLA Solved By: NIKHIL VARSHNEY
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Joshi sir comment

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.

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Joshi sir comment

let the numbers are a-d, a, a+d

then according to the given condition    a-d+a+a+d = 9   so  3a = 9   so a = 3

and [a+d]-[a-d] = 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 22nd and 23rd 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 22nd and 23rd 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

Asked By: RAJEEV SHRIVASTAVA
is this question helpfull: 8 1 read solutions ( 1 ) | submit your answer
Joshi sir comment

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