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Tuesday, September 29, 2015

2015/089) lf z= \frac{a+3i}{2+ai} , a is real.

Show that there is only one value of a for which arg z = \frac{pi}{4}  and find it
Solution
arg(z) = \frac{pi}{4}

z must be located at first quadrant...



now \frac{a+3i}{2+ai} = \frac{(a+3i)(2-ai)}{(4+a^2)}
= \dfrac{5a + (6-a^2)i}{4+ a^2}
Arg z = 1 when 5a = 6- a^2
or a^2 + 5a – 6 = 0
(a+6)(a-1) = 0
a = -6 or 1
a = -6 means arg(z) = \frac{3pi}{4} and hence this is invalid value and so a = 1 is the correct ans.
Refer to https://in.answers.yahoo.com/question/index?qid=20111204094456AAQhinr

Sunday, September 27, 2015

2015/088)If a,b,c are real and distinct, then show that

a^2(1+b^2)+b^2(1+c^2)+c^2(1+a^2)\gt6abc

taking 6abc to the left and reordering we get
(a^2b^2-2bac + c^2) + (a^2c^2-2abc + b^2) + (b ^2c^2- 2abc + a^2)
= (ab-c)^2 + (ac-b)^2 + (bc-a)^2 \ge 0
the expression can be zero only if ab= c, ac = b , bc = a or abc = 1 or zero
if a,b,c all are different then above cannot be zero

2015/087) If two vertices of an equilateral triangle are (0,0),(3,\sqrt{3}) then find the third vertex

one point is origin (0,0) and another is (3,\sqrt3)

the length of the side = \sqrt{12} = 2\sqrt3


So it is at 30 degrees with x axis.
For it to be equilateral triangle the 2nd side shall be at 60 degrees more or 60 degrees less
that is there are 2 triangle one is at 30 degrees more that is at 90 degrees which gives co-ordinates (0, 2\sqrt3)
another triangle at 30 degrees less that is at -30 degrees which gives co-ordinates (3, -\sqrt{3})

Friday, September 25, 2015

2015/086) Let S=y^2+20y+12 y is a positive integer.

What is the sum of all possible values of y for which S is a perfect square.

Solution
let S = x^2
so x^2 = (y+10)^2 – 88
or 88 = (y+10)^2-x^2 = (y+10 + x) (y+10-x)
both the terms on RHS has to be even as one add and one even shall give fractional x and y
so (y+10+x) = 44, (y + 10 – x) = 2 giving y = 13, x = 21
or (y+10+x) = 22, (y+10-x) = 4 giving y=3 , x = 9
so sum of all possible values of y = 16

Tuesday, September 22, 2015

2015/085) Prove that

(x+y)^3 + (y+z)^3 + (z+x)^3 -3(x+y)(y+z)(z+x)=2(x^3+y^3+z^3 – 3xyz)
Solution 
knowing
a^3+b^3 + c^3 – 3abc = \frac{1}{2}(a+b+c)((a-b)^2 + (b-c)^2 + (c-a)^2)\cdots (1)
putting a = x + y , b= y + z, c= z+x we get


(x+y)^3 + (y+z)^3 + (z+x)^3 -3(x+y)(y+z)(z+x)
= \frac{1}{2}(2x+2y+ 2z)((x-z)^2 + (y-x)^2 + (z-y)^2)
= 2 * \frac{1}{2}(x+y+z)((x-y)^2+(y-z)^2 + (z-x)^2)
= 2(x^3+y^3+z^3 – 3xyz) using (1)

Saturday, September 19, 2015

2015/084) Prove that the locus of the center of the circle

\frac{1}{2}(x^2 + y^2) + x \cos(\theta) + y \sin(\theta) - 4 = 0 is x^2 + y^2 = 1
Solution
we have (x^2 + 2 x \cos \theta) + (y^2 + 2y \sin \theta ) = 8
or (x^2 + 2 x \cos \theta+1) + (y^2 + 2y \sin \theta+ 1 ) = 10

or ( x + \cos \theta)^2 + (y + \sin \theta ) ^2 = 10
So the locus of the centre
x = - \cos \theta
y = - \sin \theta
to eliminate \theta
or x^2 + y ^2 = 1
above is the locus
so above is true