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Question Number 173408    Answers: 1   Comments: 1

Question Number 173406    Answers: 0   Comments: 0

Find: Ω =lim_(n→∞) ((√((1 + n!)^(n!) ))/(n ∙ (n!)!))

$$\mathrm{Find}:\:\:\:\Omega\:=\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\:\frac{\sqrt{\left(\mathrm{1}\:+\:\mathrm{n}!\right)^{\boldsymbol{\mathrm{n}}!} }}{\mathrm{n}\:\centerdot\:\left(\mathrm{n}!\right)!} \\ $$

Question Number 173403    Answers: 0   Comments: 4

Question Number 173401    Answers: 0   Comments: 1

1.show that lim_(n→+∝) ((2^(2022) n^2 )/3^n )=0 ?

$$\mathrm{1}.\mathrm{show}\:\mathrm{that}\:\underset{\mathrm{n}\rightarrow+\propto} {\mathrm{lim}}\frac{\mathrm{2}^{\mathrm{2022}} \mathrm{n}^{\mathrm{2}} }{\mathrm{3}^{\mathrm{n}} }=\mathrm{0}\:? \\ $$

Question Number 173396    Answers: 1   Comments: 0

Consider the statements: x: Birds fly y: The sky is blue Which of the following statements can be represented aa x⇔y? (i) When the sky is blue, the birds fly. (ii) Either the bird is flying or the sky is blue. (iii) Birds fly if and only if the sky is blue. (iv) When birds fly, the sky is blue.

$$\mathrm{Consider}\:\mathrm{the}\:\mathrm{statements}: \\ $$$${x}:\:\mathrm{Birds}\:\mathrm{fly} \\ $$$${y}:\:\mathrm{The}\:\mathrm{sky}\:\mathrm{is}\:\mathrm{blue} \\ $$$$\mathrm{Which}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{statements}\:\mathrm{can} \\ $$$$\mathrm{be}\:\mathrm{represented}\:\mathrm{aa}\:{x}\Leftrightarrow{y}? \\ $$$$\left({i}\right)\:\mathrm{When}\:\mathrm{the}\:\mathrm{sky}\:\mathrm{is}\:\mathrm{blue},\:\mathrm{the}\:\mathrm{birds}\:\mathrm{fly}. \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{Either}\:\mathrm{the}\:\mathrm{bird}\:\mathrm{is}\:\mathrm{flying}\:\mathrm{or}\:\mathrm{the}\:\mathrm{sky}\:\mathrm{is}\:\mathrm{blue}. \\ $$$$\left({iii}\right)\:\mathrm{Birds}\:\mathrm{fly}\:\mathrm{if}\:\mathrm{and}\:\mathrm{only}\:\mathrm{if}\:\mathrm{the}\:\mathrm{sky}\:\mathrm{is}\:\mathrm{blue}. \\ $$$$\left({iv}\right)\:\mathrm{When}\:\mathrm{birds}\:\mathrm{fly},\:\mathrm{the}\:\mathrm{sky}\:\mathrm{is}\:\mathrm{blue}. \\ $$

Question Number 173393    Answers: 0   Comments: 0

If a,b≥0 then prove that: ∫_a ^( b) ∫_a ^( b) ∣ay−ab + bx∣ dydx ≤ a^2 b^2

$$\mathrm{If}\:\:\mathrm{a},\mathrm{b}\geqslant\mathrm{0}\:\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\mathrm{b}}} \int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\mathrm{b}}} \:\mid\mathrm{ay}−\mathrm{ab}\:+\:\mathrm{bx}\mid\:\mathrm{dydx}\:\leqslant\:\mathrm{a}^{\mathrm{2}} \mathrm{b}^{\mathrm{2}} \\ $$

Question Number 173383    Answers: 1   Comments: 0

Question Number 173380    Answers: 1   Comments: 1

Question Number 173373    Answers: 1   Comments: 0

Question Number 173372    Answers: 1   Comments: 0

Question Number 173377    Answers: 1   Comments: 0

If a,b,c,d>0 And (1/(a+1)) + (2/(b+1)) + (3/(c+1)) + (4/(d+1)) = 1 Then find (a/(a+1)) + ((2b)/(b+1)) + ((3c)/(c+1)) + ((4d)/(d+1))

$$\mathrm{If}\:\:\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}>\mathrm{0} \\ $$$$\mathrm{And}\:\:\frac{\mathrm{1}}{\mathrm{a}+\mathrm{1}}\:+\:\frac{\mathrm{2}}{\mathrm{b}+\mathrm{1}}\:+\:\frac{\mathrm{3}}{\mathrm{c}+\mathrm{1}}\:+\:\frac{\mathrm{4}}{\mathrm{d}+\mathrm{1}}\:=\:\mathrm{1} \\ $$$$\mathrm{Then}\:\mathrm{find}\:\:\frac{\mathrm{a}}{\mathrm{a}+\mathrm{1}}\:+\:\frac{\mathrm{2b}}{\mathrm{b}+\mathrm{1}}\:+\:\frac{\mathrm{3c}}{\mathrm{c}+\mathrm{1}}\:+\:\frac{\mathrm{4d}}{\mathrm{d}+\mathrm{1}} \\ $$

Question Number 173371    Answers: 0   Comments: 2

The number of distinct terms in the expassion of (x_1 +x_2 +x_3 +.......+ x_n )^(4 ) is (a)

$${The}\:{number}\:{of}\:{distinct}\:{terms}\:{in}\:{the}\:{expassion}\:{of}\:\left({x}_{\mathrm{1}} +{x}_{\mathrm{2}} \:+{x}_{\mathrm{3}} \:+.......+\:{x}_{{n}} \:\right)^{\mathrm{4}\:} {is} \\ $$$$\left({a}\right)\: \\ $$

Question Number 173369    Answers: 1   Comments: 0

Question Number 173354    Answers: 0   Comments: 2

A three−digit odd number less than 500 is to be formed from 1,2,3,4 and 5. If repetition of digits is allowed, in how many ways can this be done?

$$\mathrm{A}\:\mathrm{three}−\mathrm{digit}\:\mathrm{odd}\:\mathrm{number}\:\mathrm{less}\:\mathrm{than}\:\mathrm{500} \\ $$$$\mathrm{is}\:\mathrm{to}\:\mathrm{be}\:\mathrm{formed}\:\mathrm{from}\:\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4}\:\mathrm{and}\:\mathrm{5}.\:\mathrm{If}\:\mathrm{repetition} \\ $$$$\mathrm{of}\:\mathrm{digits}\:\mathrm{is}\:\mathrm{allowed},\:\mathrm{in}\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways} \\ $$$$\mathrm{can}\:\mathrm{this}\:\mathrm{be}\:\mathrm{done}? \\ $$

Question Number 173352    Answers: 1   Comments: 0

Question Number 173351    Answers: 1   Comments: 0

Question Number 173350    Answers: 1   Comments: 0

Question Number 173339    Answers: 0   Comments: 1

Question Number 173334    Answers: 1   Comments: 0

Let lim_(x→∞) (3^x /2^(f(x)) )=9 what is f(x)?

$$\mathrm{Let}\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{3}^{{x}} }{\mathrm{2}^{{f}\left({x}\right)} }=\mathrm{9} \\ $$$$\mathrm{what}\:\mathrm{is}\:{f}\left({x}\right)? \\ $$

Question Number 173332    Answers: 1   Comments: 0

A shop keeper gave a bill to a customer. The bill was in two digits but was mistakenly interchanged thereby undercharging the customer by $45.00. If the sum of the digits is 9, find the actual value of the bill.

$$\mathrm{A}\:\mathrm{shop}\:\mathrm{keeper}\:\mathrm{gave}\:\mathrm{a}\:\mathrm{bill}\:\mathrm{to}\:\mathrm{a}\:\mathrm{customer}. \\ $$$$\mathrm{The}\:\mathrm{bill}\:\mathrm{was}\:\mathrm{in}\:\mathrm{two}\:\mathrm{digits}\:\mathrm{but}\:\mathrm{was}\:\mathrm{mistakenly} \\ $$$$\mathrm{interchanged}\:\mathrm{thereby}\:\mathrm{undercharging}\:\mathrm{the} \\ $$$$\mathrm{customer}\:\mathrm{by}\:\$\mathrm{45}.\mathrm{00}.\:\mathrm{If}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{digits}\:\mathrm{is}\: \\ $$$$\mathrm{9},\:\mathrm{find}\:\mathrm{the}\:\mathrm{actual}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{bill}. \\ $$

Question Number 173321    Answers: 1   Comments: 0

The diagonal of a rectangualar field is 169m. If the ratio of the lengh to the width is 12:5, find its dimensions.

$$\mathrm{The}\:\mathrm{diagonal}\:\mathrm{of}\:\mathrm{a}\:\mathrm{rectangualar}\:\mathrm{field}\:\mathrm{is}\:\mathrm{169m}. \\ $$$$\mathrm{If}\:\mathrm{the}\:\mathrm{ratio}\:\mathrm{of}\:\mathrm{the}\:\mathrm{lengh}\:\mathrm{to}\:\mathrm{the}\:\mathrm{width}\:\mathrm{is} \\ $$$$\mathrm{12}:\mathrm{5},\:\mathrm{find}\:\mathrm{its}\:\mathrm{dimensions}. \\ $$

Question Number 173386    Answers: 1   Comments: 0

Question Number 173311    Answers: 4   Comments: 2

Question Number 173309    Answers: 2   Comments: 0

lim_(x→∞) ((1+(1/x))^x x−ex)=?

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left(\left(\mathrm{1}+\frac{\mathrm{1}}{{x}}\right)^{{x}} {x}−{ex}\right)=? \\ $$

Question Number 173302    Answers: 2   Comments: 2

Question Number 173303    Answers: 1   Comments: 0

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