# 1.The solution of x+ 1/x-2 = 2+ 1/x-1 2. The total area enclosed by lines y=|x| , |x|=1 and y=0 is 3. The minimum value of 3tan^2θ + 12cot^2θ is 4. If midpoint of join (x,y+1) and (x+1,y+2) is {3/2,5/2} then the midpoint of join of (x-1,y+1) and (x+1, y-1) is 5. If pand q are roots of equation x^2+px+q=0, then two values for p are 6.if the roots of x^2-ax+b=0 differ by unity then: a. B^2=1+4a b. a^2 =1+4b c. if x belongs to real no.s, a^2>4b d. a^2 +4b =1 7. Total no. of solutions of 2^x+3^x+4^x-5^x=0 is If a,b, c are three positive real numbers, then mnimum value of expression b+c/a + a+c/b + a+b/c is

1
by Swrp

2015-01-13T15:24:18+05:30

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The above equation seems to have one real irrational root, and two imaginary roots.
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There are two triangles for the area given. O(0,0), (1,0) and (1,1) form one triangle.  O(0,0), (-1,0), and (-1,1) form another triangle. Are a of both is same = 1/2.  Total area = 1.
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(x+x+1)/2=3/2  => x = 1    and  (y+1+y+2)/2 = 5/2      =>  y = 1
mid point of (2,2) and (2, 0) is    (2, 1)
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as p is a root,  substitute in place of x .
p² +p*p + q = 0    => 2p² = - q
q² + p q + q = 0    =>  4p⁴ - 2 p³ -2p²  = 0 ,
one possibility ,  p = 0, and q = 0.  cancel 2 p² in above equation.
2 p² - p -1 = 0
p =  [ 1 + - sqroot(1 +8) ] / 4  =  1  or  -1/2
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let roots be = s+1/2 and  s - 1/2
sum of roots =  a = 2s    =>  s = a/2
product of roots  =  b = s² - 1/4  = (a²-1)/4  =>  a² = 1+4b
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if x is real, then discriminant >= 0,,  so a² - 4b >= 0
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Z = (2^x + 3^x + 4^x) / 5^x;; Z < (4^x + 4^x + 4^x) / 5^x ;; Z < 3 *4^x /5^x ;; Z < 3 * (0.8)^x ;; Log (Z/3) < x Log 0.8 ;; - Log (Z/3) > - x Log0.8 ;; x Log 1.25 < Log (3/Z) ;; x < Log (3/Z) / Log 1.25 ;; if Z = 1, x < Log 3 / Log 1.25 ;; x < 4.9;;
if Z> 1, x < 4.9. Z = 1, x = 4.9 ; Z < 1, x > 4.9;;
as X = (2^x+3^x+4^x)/5^x < (3*4^x/5^x), we see that X is = 1 only for x < 4.9.
let x is negative, x = -a and a>0.;; X = 1/2^a +1/3^a+1/4^a - 1/5^a ;; X > 0 as 1/4^a > 1/5^a. So X will not be zero.
check for values between x = 0 and 5. u find only x = 2.373279... for which X = 0