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Question Number 115170 Answers: 3 Comments: 0

(1)Given ((P _(n−1)^(2n+1) )/(P _n^(2n−1) )) = (3/5) , find n = ? (2) in how many ways can 6 persons stand in a queue? (3) how many different 4 letter words can be formed by using letters of EDUCATION using each letter at most once ?

$$\left(\mathrm{1}\right){Given}\:\frac{{P}\:_{{n}−\mathrm{1}} ^{\mathrm{2}{n}+\mathrm{1}} }{{P}\:_{{n}} ^{\mathrm{2}{n}−\mathrm{1}} }\:=\:\frac{\mathrm{3}}{\mathrm{5}}\:,\:{find}\:{n}\:=\:? \\ $$$$\left(\mathrm{2}\right)\:{in}\:{how}\:{many}\:{ways}\:{can}\:\mathrm{6}\:{persons} \\ $$$${stand}\:{in}\:{a}\:{queue}? \\ $$$$\left(\mathrm{3}\right)\:{how}\:{many}\:{different}\:\mathrm{4}\:{letter}\:{words} \\ $$$${can}\:{be}\:{formed}\:{by}\:{using}\:{letters}\:{of}\: \\ $$$${EDUCATION}\:{using}\:{each}\:{letter}\:{at}\: \\ $$$${most}\:{once}\:? \\ $$$$ \\ $$

Question Number 115169 Answers: 3 Comments: 0

∫ sin^2 (ln x) dx

$$\int\:\mathrm{sin}\:^{\mathrm{2}} \left(\mathrm{ln}\:{x}\right)\:{dx}\: \\ $$

Question Number 115167 Answers: 1 Comments: 0

lim_(x→0) ((sec (sin^(−1) (1−x)))/(3(√x))) = ?

$$\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sec}\:\:\left(\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{1}−{x}\right)\right)}{\mathrm{3}\sqrt{{x}}}\:=\:? \\ $$

Question Number 115166 Answers: 2 Comments: 0

Question Number 115162 Answers: 4 Comments: 0

(1) lim_(x→1) arc sin (((1−(√x))/(1−x))) =? (2) lim_(x→∞) e^(x^3 +(√(cos ((1/x^2 ))))) =? (3) lim_(x→0) csc x .sin (sin x) =?

$$\:\left(\mathrm{1}\right)\:\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\mathrm{arc}\:\mathrm{sin}\:\left(\frac{\mathrm{1}−\sqrt{{x}}}{\mathrm{1}−{x}}\right)\:=? \\ $$$$\left(\mathrm{2}\right)\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{e}^{{x}^{\mathrm{3}} +\sqrt{\mathrm{cos}\:\left(\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right)}} \:=? \\ $$$$\left(\mathrm{3}\right)\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\mathrm{csc}\:{x}\:.\mathrm{sin}\:\left(\mathrm{sin}\:{x}\right)\:=? \\ $$

Question Number 115151 Answers: 1 Comments: 0

Question Number 115143 Answers: 0 Comments: 0

Question Number 115136 Answers: 1 Comments: 0

Question Number 115135 Answers: 1 Comments: 1

Question Number 115133 Answers: 3 Comments: 0

Given that the sequence {a_n } is defined as a_1 =2, and a_(n+1) =a_n +(2n−1) for all n≥1. Find the last two digits of a_(100) .

$$\mathrm{Given}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sequence}\:\left\{{a}_{{n}} \right\}\:\mathrm{is}\:\mathrm{defined} \\ $$$$\mathrm{as}\:{a}_{\mathrm{1}} =\mathrm{2},\:\mathrm{and}\:{a}_{{n}+\mathrm{1}} ={a}_{{n}} +\left(\mathrm{2}{n}−\mathrm{1}\right)\:\mathrm{for}\:\mathrm{all}\:{n}\geqslant\mathrm{1}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{last}\:\mathrm{two}\:\mathrm{digits}\:\mathrm{of}\:{a}_{\mathrm{100}} . \\ $$

Question Number 115122 Answers: 4 Comments: 1

lim_(x→∞) (√((x^2 +2x)(x^2 +1))) −(√((x^2 +2x)(x^2 +4))) ?

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\sqrt{\left({x}^{\mathrm{2}} +\mathrm{2}{x}\right)\left({x}^{\mathrm{2}} +\mathrm{1}\right)}\:−\sqrt{\left({x}^{\mathrm{2}} +\mathrm{2}{x}\right)\left({x}^{\mathrm{2}} +\mathrm{4}\right)}\:? \\ $$

Question Number 115121 Answers: 1 Comments: 2

solve { ((tan x + cot x = p)),((sec x − cos x = q)) :}

$${solve}\:\begin{cases}{\mathrm{tan}\:{x}\:+\:\mathrm{cot}\:{x}\:=\:{p}}\\{\mathrm{sec}\:{x}\:−\:\mathrm{cos}\:{x}\:=\:{q}}\end{cases} \\ $$

Question Number 115117 Answers: 1 Comments: 0

What is the value of a and b when 3x^4 +6x^3 −ax^2 −bx−12 is completely divisible by x^2 −3 ?

$${What}\:{is}\:{the}\:{value}\:{of}\:{a}\:{and}\:{b}\:{when}\: \\ $$$$\mathrm{3}{x}^{\mathrm{4}} +\mathrm{6}{x}^{\mathrm{3}} −{ax}^{\mathrm{2}} −{bx}−\mathrm{12}\:{is}\:{completely} \\ $$$${divisible}\:{by}\:{x}^{\mathrm{2}} −\mathrm{3}\:? \\ $$

Question Number 115112 Answers: 1 Comments: 0

What is minimum distance between xy = 4 and x^2 +y^2 = 4 ?

$${What}\:{is}\:{minimum}\:{distance}\:{between}\: \\ $$$${xy}\:=\:\mathrm{4}\:{and}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \:=\:\mathrm{4}\:?\: \\ $$$$ \\ $$

Question Number 115111 Answers: 2 Comments: 0

...mathematical analysis... prove that: Ω=∫_0 ^( 1) (((x^8 +1)ln(x))/(x^(10) −1)) dx=((π^2 ϕ^2 )/(25)) ✓ m.n.july 1970

$$\:\:\:\:\:\:\:\:\:\:\:\:\:...{mathematical}\:\:{analysis}...\:\:\:\:\: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{prove}\:\:{that}: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\left({x}^{\mathrm{8}} +\mathrm{1}\right){ln}\left({x}\right)}{{x}^{\mathrm{10}} −\mathrm{1}}\:{dx}=\frac{\pi^{\mathrm{2}} \varphi^{\mathrm{2}} }{\mathrm{25}}\:\:\checkmark \\ $$$$\:\:\:\:\:\:\:\:\:\:{m}.{n}.{july}\:\mathrm{1970} \\ $$$$ \\ $$

Question Number 115105 Answers: 1 Comments: 0

Question Number 115103 Answers: 3 Comments: 1

lim_(x→∞) (√((x−a)(x+2))) −(√(x(x+1))) = 2 then a = ?

$$\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\sqrt{\left({x}−{a}\right)\left({x}+\mathrm{2}\right)}\:−\sqrt{{x}\left({x}+\mathrm{1}\right)}\:=\:\mathrm{2} \\ $$$${then}\:{a}\:=\:? \\ $$

Question Number 115085 Answers: 0 Comments: 1

Question Number 115074 Answers: 0 Comments: 4

Question Number 115072 Answers: 2 Comments: 1

Question Number 115071 Answers: 1 Comments: 0

∫_0 ^(π/2) ((cos x)/( (√(1−sin x)))) dx ?

$$\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\:\frac{\mathrm{cos}\:{x}}{\:\sqrt{\mathrm{1}−\mathrm{sin}\:{x}}}\:{dx}\:? \\ $$

Question Number 115062 Answers: 2 Comments: 0

If 2cosθ−sinθ=(1/( (√2))) (0°<θ<90°) then 2sinθ+cosθ= ¿

$$\mathrm{If}\:\:\mathrm{2cos}\theta−\mathrm{sin}\theta=\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\:\:\left(\mathrm{0}°<\theta<\mathrm{90}°\right) \\ $$$$\mathrm{then}\:\:\mathrm{2sin}\theta+\mathrm{cos}\theta=\:¿ \\ $$

Question Number 115058 Answers: 1 Comments: 0

.... nice calculus ... a , b , c , d ∈N and (1/a)+(1/b)+(1/c)+(1/d)=(1/2) find max(a+b+c+d) =??? ...m.n.july.1970...

Question Number 115056 Answers: 1 Comments: 0

Question Number 115055 Answers: 2 Comments: 0

... nice calculus... evaluation : χ=∫_0 ^( 1) log(1−x).log(1+x) =??? ...m.n.july.197o...

$$\:\:\:\:\:\:\:\:\:\:\:...\:\:{nice}\:\:{calculus}... \\ $$$$\:\:\:\:\:\:\:\:{evaluation}\:: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\chi=\int_{\mathrm{0}} ^{\:\mathrm{1}} {log}\left(\mathrm{1}−{x}\right).{log}\left(\mathrm{1}+{x}\right)\:=??? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...{m}.{n}.{july}.\mathrm{197}{o}... \\ $$$$\: \\ $$

Question Number 115051 Answers: 2 Comments: 6

∫x^2 (√(x^2 −2))dx

$$\int{x}^{\mathrm{2}} \sqrt{{x}^{\mathrm{2}} −\mathrm{2}}{dx} \\ $$

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