160 - Mathematics Questions Answers

in the given diagram red lines are angle bisectors (interior and exterior)and X, Y, Z, W are the projections of A on these red lines

by angle bisector property all these 4 points will lie on the line BC (either externally or internally) so these points will be in a line

according to the given condition 

cos^-1x  =  cos^-1(y/2)  +  a

so x = cos ( cos^-1y/2   +  a)

so x = y/2 cosa  -  sin cos^-1y/2 + sina

so x = y/2 cosa  - [1-(y²/4)]^1/2 + sina

so x - sina - y/2 cosa = -[1-(y²/4)]^1/2

now square both side and solve

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                                                    

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

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

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

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

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

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

First replace cotB by cotA. 

because it is wrongly printed by you.

answer by NIKHIL is correct