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        1.   If f(x),g(x),h(x) are continuous for all x belongs to 'R'. and
h(x)=Lt n→∞ f(x)+x2ng(x)/1+x2n , f(1)/g(1)=K, f(-1)/g(-1)=L then k+L=?
2. A and B are two fixed points on fixed circle and 'p' is a moving    point on the above circle,angleAPB=60 , AB=8, range of PA*PB is ?
3. The population of a country increases by 2% per year.In 100 years population  increases by 'p' times then [p]=?  ([.] denotes greatest integer function)


8 years ago

Jitender Singh
IIT Delhi
158 Points
										Ans:3.Population increases by 2% every year. Let ‘p’ be the population at any time & p0be the initial population.Given:$\frac{dp}{dt} = .02p$$\int \frac{dp}{p} = \int .02dt$$\int_{p_{0}}^{p} \frac{dp}{p} = \int_{0}^{100} .02dt$$(lnp)_{p_{0}}^{p} = (.02t)_{0}^{100}$$ln\frac{p}{p_{0}} = (.02.100) = 2$$p = p_{0}e^{2}$Increase in the population:$p - p_{0} = p_{0} e^{2} -p_{0} = p_{0} (e^{2}-1)$$[e^{2}-1] = [2.718^{2}-1] = [6.3875] = 6$3.What is the range. Its not clear. Please rewrite this part again.1.$h(x) = \lim_{n\rightarrow \infty }(f(x) + \frac{x^{2n}.g(x)}{1+x^{2n}})$Since functions h(x), f(x) & g(x) are continuous in R. There limit should exist at -1 & 1.Limit at x =1LHL = RHL$\lim_{x\rightarrow 1^{-}}h(x) = \lim_{x\rightarrow 1^{+}}h(x)$$LHL = \lim_{n\rightarrow \infty }\lim_{h\rightarrow 0}(f(1-h)+\frac{g(1-h).(1-h)^{2n}}{1+(1-h)^{2n}})$$LHL = f(1)$$RHL = \lim_{n\rightarrow \infty }\lim_{h\rightarrow 0}(f(1+h)+\frac{g(1+h).(1+h)^{2n}}{1+(1+h)^{2n}})$$RHL = f(1) + g(1)$$LHL =RHL$$f(1) = f(1)+g(1)$$\Rightarrow g(1) = 0$Limit at x = -1$LHL = \lim_{n\rightarrow \infty }\lim_{h\rightarrow 0}(f(-1-h)+\frac{g(-1-h).(-1-h)^{2n}}{1+(-1-h)^{2n}})$$LHL = \lim_{n\rightarrow \infty }\lim_{h\rightarrow 0}(f(-1-h)+\frac{g(-1-h).(1+h)^{2n}}{1+(1+h)^{2n}})$$LHL = f(-1) + g(-1)$$RHL = \lim_{n\rightarrow \infty }\lim_{h\rightarrow 0}(f(-1+h)+\frac{g(-1+h).(-1+h)^{2n}}{1+(-1+h)^{2n}})$$RHL = \lim_{n\rightarrow \infty }\lim_{h\rightarrow 0}(f(-1+h)+\frac{g(-1+h).(1-h)^{2n}}{1+(1-h)^{2n}})$$RHL = f(-1)$$LHL=RHL$$\Rightarrow g(-1) = 0$Thanks & RegardsJitender SinghIIT DelhiaskIITians Faculty

3 years ago
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