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Question Number 75930 Answers: 0 Comments: 0

consider the function f(x) = { ((1+2x^2 , if x is rational)),((1 + x^4 , if x is irrational)) :} Use the sandwich(pinchin) theorem to prove that lim_(x→0) f(x) = 1.

$${consider}\:{the}\:{function} \\ $$$$\:{f}\left({x}\right)\:=\:\begin{cases}{\mathrm{1}+\mathrm{2}{x}^{\mathrm{2}} ,\:{if}\:{x}\:{is}\:{rational}}\\{\mathrm{1}\:+\:{x}^{\mathrm{4}} ,\:{if}\:{x}\:{is}\:{irrational}}\end{cases} \\ $$$${Use}\:{the}\:{sandwich}\left({pinchin}\right)\:{theorem}\:{to} \\ $$$${prove}\:{that}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{f}\left({x}\right)\:=\:\mathrm{1}. \\ $$

Question Number 75929 Answers: 0 Comments: 2

Evaluate lim_(x→−∞) [(√(1−xe^x )) ]

$${Evaluate} \\ $$$$\underset{{x}\rightarrow−\infty} {\:\mathrm{lim}}\:\left[\sqrt{\mathrm{1}−{xe}^{{x}} }\:\right] \\ $$

Question Number 75925 Answers: 0 Comments: 0

Let OA^(→) = i+3j−2k and OB^(→) =3i+j−2k. The vector OC^(→) bisecting the ∠AOB and C being a point on the line AB is

$$\mathrm{Let}\:\overset{\rightarrow} {{OA}}=\:\boldsymbol{\mathrm{i}}+\mathrm{3}\boldsymbol{\mathrm{j}}−\mathrm{2}\boldsymbol{\mathrm{k}}\:\mathrm{and}\:\overset{\rightarrow} {{OB}}=\mathrm{3}\boldsymbol{\mathrm{i}}+\boldsymbol{\mathrm{j}}−\mathrm{2}\boldsymbol{\mathrm{k}}. \\ $$$$\mathrm{The}\:\mathrm{vector}\:\overset{\rightarrow} {{OC}}\:\mathrm{bisecting}\:\mathrm{the}\:\angle{AOB}\:\mathrm{and} \\ $$$${C}\:\:\mathrm{being}\:\mathrm{a}\:\mathrm{point}\:\mathrm{on}\:\mathrm{the}\:\mathrm{line}\:{AB}\:\mathrm{is} \\ $$

Question Number 75924 Answers: 0 Comments: 3

Thr value of ∫_(0 ) ^(π/2) cosec (x−(π/3))cosec (x−(π/6))dx is

$$\mathrm{Thr}\:\mathrm{value}\:\mathrm{of} \\ $$$$\:\underset{\mathrm{0}\:} {\overset{\pi/\mathrm{2}} {\int}}\mathrm{cosec}\:\left({x}−\frac{\pi}{\mathrm{3}}\right)\mathrm{cosec}\:\left({x}−\frac{\pi}{\mathrm{6}}\right){dx}\:\:\mathrm{is} \\ $$

Question Number 75921 Answers: 0 Comments: 1

Question Number 75916 Answers: 0 Comments: 1

x^3 +x^2 −24x+36=0 prove that x=2,3,−6.

$${x}^{\mathrm{3}} +{x}^{\mathrm{2}} −\mathrm{24}{x}+\mathrm{36}=\mathrm{0} \\ $$$${prove}\:{that}\:{x}=\mathrm{2},\mathrm{3},−\mathrm{6}. \\ $$

Question Number 75913 Answers: 0 Comments: 1

x^3 −7x+6=0 prove that x=2,−3,1 .

$${x}^{\mathrm{3}} −\mathrm{7}{x}+\mathrm{6}=\mathrm{0} \\ $$$${prove}\:{that}\:{x}=\mathrm{2},−\mathrm{3},\mathrm{1}\:. \\ $$

Question Number 75918 Answers: 1 Comments: 0

Question Number 75917 Answers: 0 Comments: 0

Question Number 75905 Answers: 0 Comments: 1

Question Number 75902 Answers: 1 Comments: 1

Question Number 75901 Answers: 1 Comments: 0

Question Number 75899 Answers: 0 Comments: 0

A group of 30 men participate in a survey of telecom companies. The number of men who use both Idea and Airtel internet was equal to the number of men who use neither of these inernets. The number of men who use Idea is 4 more than those who use Airtel. How many use Airtel internet.

$$\mathrm{A}\:\mathrm{group}\:\mathrm{of}\:\mathrm{30}\:\mathrm{men}\:\mathrm{participate}\:\mathrm{in}\:\mathrm{a}\: \\ $$$$\mathrm{survey}\:\mathrm{of}\:\mathrm{telecom}\:\mathrm{companies}.\:\mathrm{The} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{men}\:\mathrm{who}\:\mathrm{use}\:\mathrm{both}\:\mathrm{Idea}\: \\ $$$$\mathrm{and}\:\mathrm{Airtel}\:\mathrm{internet}\:\mathrm{was}\:\mathrm{equal}\:\mathrm{to} \\ $$$$\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{men}\:\mathrm{who}\:\mathrm{use}\:\mathrm{neither} \\ $$$$\mathrm{of}\:\mathrm{these}\:\mathrm{inernets}.\:\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{men} \\ $$$$\mathrm{who}\:\mathrm{use}\:\mathrm{Idea}\:\mathrm{is}\:\mathrm{4}\:\mathrm{more}\:\mathrm{than}\:\mathrm{those} \\ $$$$\mathrm{who}\:\mathrm{use}\:\mathrm{Airtel}.\:\mathrm{How}\:\mathrm{many}\:\mathrm{use}\:\mathrm{Airtel} \\ $$$$\mathrm{internet}. \\ $$

Question Number 75896 Answers: 1 Comments: 0

One pipe can fill a tank in 40 minutes and an outlet pipe can empty the full tank in 24 minutes. If both the pipes are opened simultaneously, what time will it take for the full tank to be emptied?

$$\mathrm{One}\:\mathrm{pipe}\:\mathrm{can}\:\mathrm{fill}\:\mathrm{a}\:\mathrm{tank}\:\mathrm{in}\:\mathrm{40}\:\mathrm{minutes} \\ $$$$\mathrm{and}\:\mathrm{an}\:\mathrm{outlet}\:\mathrm{pipe}\:\mathrm{can}\:\mathrm{empty}\:\mathrm{the}\:\mathrm{full} \\ $$$$\mathrm{tank}\:\mathrm{in}\:\mathrm{24}\:\mathrm{minutes}.\:\mathrm{If}\:\mathrm{both}\:\mathrm{the}\:\mathrm{pipes} \\ $$$$\mathrm{are}\:\mathrm{opened}\:\mathrm{simultaneously},\:\mathrm{what}\:\mathrm{time} \\ $$$$\mathrm{will}\:\mathrm{it}\:\mathrm{take}\:\mathrm{for}\:\mathrm{the}\:\mathrm{full}\:\mathrm{tank}\:\mathrm{to}\:\mathrm{be} \\ $$$$\mathrm{emptied}? \\ $$

Question Number 75895 Answers: 1 Comments: 0

A garrison had provisions for 1500 men for 30 days. After some days, 300 more men joined the garrison. The provisions lasted for a total of 26 days from the beginning. After how many days did the new men join?

$$\mathrm{A}\:\mathrm{garrison}\:\mathrm{had}\:\mathrm{provisions}\:\mathrm{for}\:\mathrm{1500}\:\mathrm{men} \\ $$$$\mathrm{for}\:\mathrm{30}\:\mathrm{days}.\:\mathrm{After}\:\mathrm{some}\:\mathrm{days},\:\mathrm{300}\:\mathrm{more} \\ $$$$\mathrm{men}\:\mathrm{joined}\:\mathrm{the}\:\mathrm{garrison}.\:\mathrm{The}\:\mathrm{provisions} \\ $$$$\mathrm{lasted}\:\mathrm{for}\:\mathrm{a}\:\mathrm{total}\:\mathrm{of}\:\mathrm{26}\:\mathrm{days}\:\mathrm{from}\:\mathrm{the} \\ $$$$\mathrm{beginning}.\:\mathrm{After}\:\mathrm{how}\:\mathrm{many}\:\mathrm{days}\:\mathrm{did} \\ $$$$\mathrm{the}\:\mathrm{new}\:\mathrm{men}\:\mathrm{join}? \\ $$

Question Number 75894 Answers: 0 Comments: 0

When 616 is divided by a certain positive number, which is 66(2/3)% of the quotient, it leaves 16 as the remainder. Find the divisor.

$$\mathrm{When}\:\mathrm{616}\:\mathrm{is}\:\mathrm{divided}\:\mathrm{by}\:\mathrm{a}\:\mathrm{certain}\: \\ $$$$\mathrm{positive}\:\mathrm{number},\:\mathrm{which}\:\mathrm{is}\:\mathrm{66}\frac{\mathrm{2}}{\mathrm{3}}\%\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{quotient},\:\mathrm{it}\:\mathrm{leaves}\:\mathrm{16}\:\mathrm{as}\:\mathrm{the}\:\mathrm{remainder}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{divisor}. \\ $$

Question Number 75890 Answers: 0 Comments: 0

calculate ∫_0 ^∞ ((arctan(2x+3))/(x^2 +4))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{arctan}\left(\mathrm{2}{x}+\mathrm{3}\right)}{{x}^{\mathrm{2}} +\mathrm{4}}{dx} \\ $$

Question Number 75889 Answers: 0 Comments: 0

find ∫ (√((x+1)(x+2)(2x−1)))dx

$${find}\:\int\:\sqrt{\left({x}+\mathrm{1}\right)\left({x}+\mathrm{2}\right)\left(\mathrm{2}{x}−\mathrm{1}\right)}{dx} \\ $$

Question Number 75888 Answers: 0 Comments: 1

find ∫_0 ^1 (√(1+x^4 ))dx