212)2f(x) + f(-x) = 1+x
2f(-x) + f(x) = 1-x
now solve for f(x)
1. If f(x) = \lim_{x\rightarrow \infty } \frac{1}{1+nsin^{2}\pi x} , find f(x) for all real values of x.
2. 2f(x) + f(-x) = 1+x, find f'(10).
3. If a,b,c are in GP such that a+b+c = bx. Find all x satisfying the above relation.
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13 Answers
1708$(3) Let $b=ar$ and $c=ar^2.$ Where $-1<r<1$.Then\\\\ $a+ar+ar^2=ar.x\Leftrightarrow 1+r+r^2=rx$\\\\ $r^2+r(1-x)+1=0.$\\\\ Now solve The Quadratic equation Where $-1<r<1$
71Q:=> Find all integers n1,n2 satisfying n1n2 = 2n1-n2
Q:=> Find the locus of (3secθ,4cotθ) where θ varies.
23n1n2 = 2n1-n2
implies 2= (n1+1) (2-n2)
now use the arguement that both are integers
1In the first sum are you sure it is x→∞ and not n→∞..??
1if its n→∞.
Then the limit wud give 1 for all integral values of x as nSin2\PiX wud be =0..
T all other values of x limit wud be =0 as denominator is ∞
11n1n2=2n1-n2.
thus,n2(n1+1)2 =n1.
thus,n2 =2n1n1+1.
n2= 2 - 2n1+1.
n1 can take values 1, 0, -2 and -3 otherwise n2 becomes a fraction.
the answer is 4??