Question and Answers Forum

How many ways can the letters in the word MATHEMATICS be rearranged such that the word formed neither starts nor ends with a vowel, and any four consecutive letters must contain at least a vowel?

$$\mathrm{How}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{the}\:\mathrm{letters}\:\mathrm{in} \\ $$$$\mathrm{the}\:\mathrm{word}\:\boldsymbol{\mathrm{MATHEMATICS}} \\ $$$$\mathrm{be}\:\mathrm{rearranged}\:\mathrm{such}\:\mathrm{that}\:\mathrm{the}\:\mathrm{word} \\ $$$$\mathrm{formed}\:\mathrm{neither}\:\mathrm{starts}\:\mathrm{nor}\:\mathrm{ends}\:\mathrm{with}\:\mathrm{a}\: \\ $$$$\mathrm{vowel},\:\mathrm{and}\:\mathrm{any}\:\mathrm{four}\:\mathrm{consecutive}\: \\ $$$$\mathrm{letters}\:\mathrm{must}\:\mathrm{contain}\:\mathrm{at}\:\mathrm{least}\:\mathrm{a}\:\mathrm{vowel}? \\ $$

Question Number 110728 Answers: 1 Comments: 0

a+b=9 , ab=20 a−b=?

$${a}+{b}=\mathrm{9}\:\:,\:\:{ab}=\mathrm{20}\:\:\:\:\:{a}−{b}=? \\ $$

Question Number 110727 Answers: 1 Comments: 0

a^2 +b^2 =10 , ab=13 , a^3 +b^3 =?

$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} =\mathrm{10}\:\:\:,\:\:{ab}=\mathrm{13}\:\:\:,\:{a}^{\mathrm{3}} +{b}^{\mathrm{3}} =? \\ $$

Question Number 110706 Answers: 0 Comments: 0

Question Number 110722 Answers: 2 Comments: 0

Question Number 110675 Answers: 2 Comments: 0

Two polynomials P and Q satisfy P(−2x+Q(x))=Q(−2x+P(x)). Given that Q(x)=x^2 −4 and P(x)=ax+b. Find 2a+b.

$$\mathrm{Two}\:\mathrm{polynomials}\:\mathrm{P}\:\mathrm{and}\:\mathrm{Q}\:\mathrm{satisfy} \\ $$$$\mathrm{P}\left(−\mathrm{2x}+\mathrm{Q}\left(\mathrm{x}\right)\right)=\mathrm{Q}\left(−\mathrm{2x}+\mathrm{P}\left(\mathrm{x}\right)\right). \\ $$$$\mathrm{Given}\:\mathrm{that}\:\mathrm{Q}\left(\mathrm{x}\right)=\mathrm{x}^{\mathrm{2}} −\mathrm{4}\:\mathrm{and} \\ $$$$\mathrm{P}\left(\mathrm{x}\right)=\mathrm{ax}+\mathrm{b}.\:\mathrm{Find}\:\mathrm{2a}+\mathrm{b}. \\ $$

Question Number 118674 Answers: 1 Comments: 0

Please integrate ∫_0 ^1 (1/(1+x^c ))dx where c is a constant.

$${Please}\:{integrate} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{\mathrm{1}+{x}^{{c}} }{dx}\:{where}\:{c}\:{is}\:{a}\:{constant}. \\ $$

Question Number 110669 Answers: 2 Comments: 2

Given that f(x)=(3+2x)^3 (4−x)^4 on the interval −(3/2)<x<4. Find the (a) Maximum value of f(x) (b) The value of x that gives the maximum in (a)

$$\mathrm{Given}\:\mathrm{that}\:\mathrm{f}\left(\mathrm{x}\right)=\left(\mathrm{3}+\mathrm{2x}\right)^{\mathrm{3}} \left(\mathrm{4}−\mathrm{x}\right)^{\mathrm{4}} \:\mathrm{on} \\ $$$$\mathrm{the}\:\mathrm{interval}\:−\frac{\mathrm{3}}{\mathrm{2}}<\mathrm{x}<\mathrm{4}.\:\mathrm{Find}\:\mathrm{the} \\ $$$$\left(\mathrm{a}\right)\:\mathrm{Maximum}\:\mathrm{value}\:\mathrm{of}\:\mathrm{f}\left(\mathrm{x}\right) \\ $$$$\left(\mathrm{b}\right)\:\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x}\:\mathrm{that}\:\mathrm{gives}\:\mathrm{the} \\ $$$$\mathrm{maximum}\:\mathrm{in}\:\left(\mathrm{a}\right) \\ $$

Question Number 110647 Answers: 1 Comments: 0

1)lim_(x→0) sin((1/x))=? 2) lim_(x→0) cos ((1/x))=?

$$\left.\:\mathrm{1}\right)\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}sin}\left(\frac{\mathrm{1}}{\mathrm{x}}\right)=? \\ $$$$\left.\mathrm{2}\right)\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}cos}\:\left(\frac{\mathrm{1}}{\mathrm{x}}\right)=? \\ $$$$ \\ $$

Question Number 110642 Answers: 0 Comments: 0

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Question Number 110619 Answers: 0 Comments: 2

someone posted this problem and my solution followed solve ∫_0 ^1 ((ln(x+(√(1+x^2 ))))/(√(1+x^2 )))dx solution y=ln(x+(√(1+x^2 ))) and dy=(1/(√(1+x^2 )))dx at x=0,y=0 and at x=1,y=ln(1+(√2)) hence I=∫_0 ^(ln(1+(√2))) (y/(√(1+x^2 )))×(√(1+x^2 ))dy=∫_0 ^(ln(1+(√2))) ydy I=[(y^2 /2)]_0 ^(ln(1+(√2))) =(1/2)ln^2 (1+(√2)) ∵∫_0 ^1 ((ln(x+(√(1+x^2 ))))/(√(1+x^2 )))dx=(1/2)ln^2 (1+(√2))

Question Number 110616 Answers: 1 Comments: 0

lim_(n→∞) ((e^(1/n) +e^(2/n) +......+e^(n/n) )/n) ho is can you help me step by step

$${lim}_{{n}\rightarrow\infty} \frac{{e}^{\frac{\mathrm{1}}{{n}}} +{e}^{\frac{\mathrm{2}}{{n}}} +......+{e}^{\frac{{n}}{{n}}} }{{n}}\: \\ $$$${ho}\:{is}\:{can}\:{you}\:{help}\:{me}\:{step}\:{by}\:{step} \\ $$

Question Number 110614 Answers: 1 Comments: 0

Question Number 110608 Answers: 0 Comments: 1

(√x) +1=0 Solution (√(x )) = −1 recalled that ϱ^(iΠ) = −1 (√x) =ϱ^(iΠ) x=(ϱ^(iΠ) )^2 ⇒ x=ϱ^(2iΠ) or x= 2cosΠ+isinΠ x has no real value!

$$\sqrt{{x}}\:+\mathrm{1}=\mathrm{0} \\ $$$${Solution} \\ $$$$\sqrt{{x}\:}\:=\:−\mathrm{1} \\ $$$${recalled}\:{that}\:\varrho^{{i}\Pi} =\:−\mathrm{1} \\ $$$$\sqrt{{x}}\:=\varrho^{{i}\Pi} \\ $$$${x}=\left(\varrho^{{i}\Pi} \right)^{\mathrm{2}} \\ $$$$\Rightarrow\:{x}=\varrho^{\mathrm{2}{i}\Pi} \\ $$$${or}\:{x}=\:\mathrm{2}{cos}\Pi+{isin}\Pi \\ $$$${x}\:{has}\:{no}\:{real}\:{value}! \\ $$

Question Number 110704 Answers: 3 Comments: 0

Find the minimum value of ((n^2 +1)/n)+(n/(n^2 +1)),n>0

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of} \\ $$$$\frac{\mathrm{n}^{\mathrm{2}} +\mathrm{1}}{\mathrm{n}}+\frac{\mathrm{n}}{\mathrm{n}^{\mathrm{2}} +\mathrm{1}},\mathrm{n}>\mathrm{0} \\ $$

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