# 48 - Calculus Questions Answers

**Asked By: LUFFY**

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**Joshi sir comment**

length for one turn = 3

Draw diagonal of a rectangle then turn it as a cylinder, you will get the construction given in the question.

$Intrianglel=\sqrt{{3}^{2}+{n}^{2}}\phantom{\rule{0ex}{0ex}}Sototallengthofwirehavingnturn=n\sqrt{{3}^{2}+{n}^{2}}\phantom{\rule{0ex}{0ex}}Comparewith20andsolve$

**Asked By: KANDUKURI ASHISH**

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**Joshi sir comment**

$Thegeneralsolutionofthiskindof\phantom{\rule{0ex}{0ex}}differentialeq.isy={e}^{ax}\phantom{\rule{0ex}{0ex}}Letafunctiong\left(x\right)={e}^{ax}f\left(x\right)\left(1\right)\phantom{\rule{0ex}{0ex}}differentiateeq.\left(1\right)thriceforgetting{g}^{,,,}\left(x\right)\phantom{\rule{0ex}{0ex}}nowforgettingthesamefunctionasgiven\phantom{\rule{0ex}{0ex}}finda.yougetthesamefunctionfor\phantom{\rule{0ex}{0ex}}a=1/2.clearly,formin5zerosofg\left(x\right),\phantom{\rule{0ex}{0ex}}therearemin2zerosfor{g}^{,,,}\left(x\right).\phantom{\rule{0ex}{0ex}}solveandinformiffaceanyproblem.$

Find

lim n→∞ ⁿ√(n!) / n

**Asked By: LUFFY**

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**Joshi sir comment**

1 . Iₙ = ∫ ( 1/ ( x² + a²)ⁿ )dx

How to do this by substituting x = a tan Α

2. Is this true

∫ f(x) d(kg(x)) = k ∫ f(x) d(g(x))

where k is a constant

**Asked By: LUFFY**

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**Solution by Joshi sir**

d/dx((1+x^2+x^4)/(1+x+x^2))=ax+b

then a=?, b=?

**Asked By: RAJIV**

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**Joshi sir comment**

1+x^{2}+x^{4} = (1+x+x^{2})(1-x+x^{2})

now solve

Find a formula for a function g(x) satisfying the following conditions

a) domain of g is (-∞ , ∞ ) b) range of g is [-2 , 8] c) g has a period π d) g(2) = 3

**Asked By: VAIBHAV GUPTA**

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**Joshi sir comment**

g(x) = 3-5sin(2x-4)

Let f(x) = x^{135} + x^{125} - x^{115 }+ x^{5} +1. If f(x) is divided by x^{3}-x then the remainder is some function of x say g(x). Find the value of g(10)

**Asked By: VAIBHAV GUPTA**

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**Joshi sir comment**

for getting reminder put x^{3}= x so

x^{135} + x^{125} - x^{115 }+ x^{5} +1 will give

x^{45} + x^{41}*x^{2} - x^{38}*x +x*x^{2} +1

x^{15} + x^{13}*x^{2}*x^{2} - x^{12}*x^{2}*x +x*x^{2} +1

x^{17 }+ x^{3} +1

x^{5}*x^{2} + x + 1

x^{7 }+ x + 1

x^{2}*x + x +1

x^{3} + x +1

x+x+1

2x+1

now put x = 10