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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

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

Question Number 115035 Answers: 0 Comments: 2

solve ∣ 4−(3/x) ∣ < 8

$${solve}\:\mid\:\mathrm{4}−\frac{\mathrm{3}}{{x}}\:\mid\:<\:\mathrm{8} \\ $$

Question Number 115034 Answers: 0 Comments: 0

what the equation of the hyperbola with the given asymtotes y=43x+13 and y=−43x+13 , a vertex at (−1,7)

$${what}\:{the}\:{equation}\:{of}\:{the}\:{hyperbola}\: \\ $$$${with}\:{the}\:{given}\:{asymtotes}\:{y}=\mathrm{43}{x}+\mathrm{13} \\ $$$${and}\:{y}=−\mathrm{43}{x}+\mathrm{13}\:,\:{a}\:{vertex}\:{at}\:\left(−\mathrm{1},\mathrm{7}\right) \\ $$

Question Number 115033 Answers: 2 Comments: 0

If a and b positive real number where a^(505) + b^(505) = 1, then minimum value a^(2020) + b^(2020) is __

$${If}\:{a}\:{and}\:{b}\:{positive}\:{real}\:{number}\:{where} \\ $$$${a}^{\mathrm{505}} \:+\:{b}^{\mathrm{505}} \:=\:\mathrm{1},\:{then}\:{minimum}\:{value} \\ $$$${a}^{\mathrm{2020}} \:+\:{b}^{\mathrm{2020}} \:{is}\:\_\_ \\ $$

Question Number 115031 Answers: 1 Comments: 0

circle of centre P touches externally both the circle x^2 +y^2 −4x+3=0 and x^2 +y^2 −6y+5=0 . The locus of P is (3/4)x^2 −3xy+2y^2 = λ(y−x) where λ is __

$$\:{circle}\:{of}\:{centre}\:{P}\:\:{touches}\:{externally}\:{both} \\ $$$${the}\:{circle}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} −\mathrm{4}{x}+\mathrm{3}=\mathrm{0}\:{and}\: \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} −\mathrm{6}{y}+\mathrm{5}=\mathrm{0}\:.\:{The}\:{locus}\:{of}\:{P}\:{is} \\ $$$$\frac{\mathrm{3}}{\mathrm{4}}{x}^{\mathrm{2}} −\mathrm{3}{xy}+\mathrm{2}{y}^{\mathrm{2}} \:=\:\lambda\left({y}−{x}\right)\:{where}\:\lambda\:{is}\:\_\_ \\ $$

Question Number 115030 Answers: 2 Comments: 0

∫_(−(π/2)) ^(π/2) (√(sec x−cos x)) dx =?

$$\underset{−\frac{\pi}{\mathrm{2}}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\sqrt{\mathrm{sec}\:{x}−\mathrm{cos}\:{x}}\:{dx}\:=? \\ $$

Question Number 115027 Answers: 1 Comments: 0

If lim_(x→3) ((17 ((ax+3))^(1/(3 )) +b)/(x−3)) = ((136)/(27)) then 8a+b = ?

$$\mathrm{If}\:\underset{{x}\rightarrow\mathrm{3}} {\mathrm{lim}}\:\frac{\mathrm{17}\:\sqrt[{\mathrm{3}\:}]{\mathrm{ax}+\mathrm{3}}\:+\mathrm{b}}{\mathrm{x}−\mathrm{3}}\:=\:\frac{\mathrm{136}}{\mathrm{27}} \\ $$$$\mathrm{then}\:\mathrm{8a}+\mathrm{b}\:=\:? \\ $$

Question Number 115026 Answers: 2 Comments: 0

Solve: ∫_(1/π) ^(1/2) ln ⌊(1/x)⌋dx

$$\mathrm{Solve}:\:\:\int_{\mathrm{1}/\pi} ^{\mathrm{1}/\mathrm{2}} \mathrm{ln}\:\lfloor\frac{\mathrm{1}}{{x}}\rfloor{dx} \\ $$

Question Number 115023 Answers: 2 Comments: 0

solve { ((7x−5y+3z=6)),((2x+4y−5z=−5)),((9x−8y+2z=−1)) :}. Find x+y+z

$${solve}\:\begin{cases}{\mathrm{7}{x}−\mathrm{5}{y}+\mathrm{3}{z}=\mathrm{6}}\\{\mathrm{2}{x}+\mathrm{4}{y}−\mathrm{5}{z}=−\mathrm{5}}\\{\mathrm{9}{x}−\mathrm{8}{y}+\mathrm{2}{z}=−\mathrm{1}}\end{cases}.\:{Find}\:{x}+{y}+{z}\: \\ $$

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