What about these

Q1) If cos(xy)+cos(yz)+cos(zx)=32cos(x-y)+cos(y-z)+cos(z-x)=\frac{-3}{2} and cosx+cosy+cosz=acosx+cosy+cosz=a and sinx+siny+sinz=bsinx+siny+sinz=b. Find aa and bb

Q2) Two non-conducting infinite wires have shapes as shown above. The linear charge densities of wire (1) and wire (2) are

xifxa and xa0ifa<x<a)x\quad if\quad x\leq -a~ and~ x\geq a\\ \\ 0\quad if\quad -a<x<a)

and

yifya and ya0ifa<y<a)y\quad if\quad y\leq -a~ and~ y\geq a\\ \\ 0\quad if\quad -a<y<a)

respectively where (x,y)(x,y) are coordinates of points on the wire. An electron is released from origin, the unit vector along the direction of velocity of the electron just after its release will be in terms of i^\hat { i } and j^\hat { j }

Note by Tanishq Varshney
4 years, 7 months ago

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1 vote

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cosx+cosy+cosz=a cosx + cosy + cosz = a

cos2x+cos2y+cos2z+2cosxcosy+2cosycosz+2cosxcosy=a2 \cos^2x + \cos^2y + \cos^2z + 2\cos x\cos y + 2\cos y\cos z + 2 \cos x\cos y = a^2

Likewise, sin2x+sin2y+sin2z+2sinxsiny+2sinysinz+2sinxsiny=b2 \sin^2x + \sin^2y + \sin^2z + 2\sin x\sin y + 2\sin y\sin z + 2 \sin x\sin y = b^2

Adding, cos2x+cos2y+cos2z+2cosxcosy+2cosycosz+2cosxcosy+ \cos^2x + \cos^2y + \cos^2z + 2\cos x\cos y + 2\cos y\cos z + 2 \cos x\cos y +

sin2x+sin2y+sin2z+2sinxsiny+2sinysinz+2sinxsiny=a2+b2 \sin^2x + \sin^2y + \sin^2z + 2\sin x\sin y + 2\sin y\sin z + 2 \sin x\sin y = a^2 + b^2

    1+1+1+2(cosxcosy+cosycosz+cosxcosy+sinxsiny+sinysinz+sinxsiny)=a2+b2 \implies 1 + 1 + 1 + 2(\cos x\cos y + \cos y\cos z + \cos x\cos y + \sin x\sin y + \sin y\sin z + \sin x\sin y) = a^2 + b^2

    3+2(cos(xy)+cos(yz)+cos(zx))=a2+b2 \implies 3 + 2(\cos(x-y) + \cos(y-z) + \cos(z-x)) = a^2 + b^2

    3+232=a2+b2 \implies 3 + 2* \frac{-3}{2} = a^2 + b^2

0=a2+b2    a=0&b=0 0 = a^2 + b^2 \implies a = 0\quad \& \quad b = 0

Siddhartha Srivastava - 4 years, 7 months ago

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Thanks for this man.

Raghav Vaidyanathan - 4 years, 7 months ago

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The answer to q 1 is very easy: a=b=0a=b=0.

Raghav Vaidyanathan - 4 years, 7 months ago

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Ans 2 q2 seems to be zero too.

Raghav Vaidyanathan - 4 years, 7 months ago

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can u explain in brief the first part

Tanishq Varshney - 4 years, 7 months ago

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@Tanishq Varshney for first one The simplest way is to assume all the three angle differences equal, which ofcourse then must be 120 degree, so just consider 1 , w ,w^2 , add em up

Mvs Saketh - 4 years, 7 months ago

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