161 - Questions Answers
for n = 4, we can form a square with integral coordinates of all the vertices.
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
If cos-1x - cos-1(y/2) = α , then 4x2-4xycosα + y2 is equal to ?
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
If sin-1a + sin-1b+ sin-1c = π , then the value of { a√(1-a2) + b√(1-b2) + c√(1-c2)}
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)?
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
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
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?
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
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?
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