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Question Number 134049    Answers: 0   Comments: 0

Question Number 134040    Answers: 3   Comments: 1

Find the shortest distance between the curve y^2 = x−1and x^2 = y−1

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{shortest}\:\mathrm{distance}\: \\ $$$$\mathrm{between}\:\mathrm{the}\:\mathrm{curve}\: \\ $$$$\mathrm{y}^{\mathrm{2}} \:=\:\mathrm{x}−\mathrm{1and}\:\mathrm{x}^{\mathrm{2}} \:=\:\mathrm{y}−\mathrm{1} \\ $$

Question Number 134039    Answers: 0   Comments: 0

Question Number 134038    Answers: 1   Comments: 0

Question Number 134037    Answers: 1   Comments: 0

Question Number 134036    Answers: 0   Comments: 2

Is this proposition true?: ∀ x ∈ Z, x^2 +x+3≡0[5] if and only if x≡1[5]

$${Is}\:{this}\:{proposition}\:{true}?: \\ $$$$ \\ $$$$\forall\:{x}\:\in\:\mathbb{Z},\:{x}^{\mathrm{2}} +{x}+\mathrm{3}\equiv\mathrm{0}\left[\mathrm{5}\right]\:{if}\:\:{and}\:{only}\:{if} \\ $$$$\:{x}\equiv\mathrm{1}\left[\mathrm{5}\right] \\ $$

Question Number 134034    Answers: 1   Comments: 0

how many zeros has the number 1000! at the end? and what is the last digit before these zeros?

$${how}\:{many}\:{zeros}\:{has}\:{the}\:{number} \\ $$$$\mathrm{1000}!\:{at}\:{the}\:{end}?\:{and}\:{what}\:{is}\:{the} \\ $$$${last}\:{digit}\:{before}\:{these}\:{zeros}? \\ $$

Question Number 134031    Answers: 0   Comments: 1

Question Number 134020    Answers: 0   Comments: 3

how to write pancham in bengali. please help

$${how}\:{to}\:{write}\:\boldsymbol{{pancham}}\:{in}\:{bengali}.\:{please}\:{help} \\ $$

Question Number 134016    Answers: 1   Comments: 0

?prove :Σ_(n=1) ^∞ (((−1)^(n−1) H_(2n) )/(2n+1))=(π/8)ln(2)..

$$ \\ $$$$\:\:\:\:\:\:?{prove}\::\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} {H}_{\mathrm{2}{n}} }{\mathrm{2}{n}+\mathrm{1}}=\frac{\pi}{\mathrm{8}}{ln}\left(\mathrm{2}\right).. \\ $$

Question Number 134064    Answers: 1   Comments: 1

Question Number 134009    Answers: 0   Comments: 0

E is a vec torial space which has as base B=(i^→ ,j^→ ,k^→ ). f: E→E is a linear application such that f(i^→ )=−i^→ +2k^→ ; f(j^→ )=j^→ +2k^→ and j(k^→ )=2i^→ +2j^→ . 1. Write the matrix of f in base B. 2. Show that the kernel (ker f) of f is a straigh line; give one base of its. 3.Determinate Im f.

$${E}\:{is}\:{a}\:{vec}\:{torial}\:{space}\:{which}\:{has}\:{as} \\ $$$${base}\:\mathscr{B}=\left(\overset{\rightarrow} {{i}},\overset{\rightarrow} {{j}},\overset{\rightarrow} {{k}}\right).\:{f}:\:{E}\rightarrow{E}\:{is}\:{a}\:{linear} \\ $$$${application}\:{such}\:{that} \\ $$$${f}\left(\overset{\rightarrow} {{i}}\right)=−\overset{\rightarrow} {{i}}+\mathrm{2}\overset{\rightarrow} {{k}};\:{f}\left(\overset{\rightarrow} {{j}}\right)=\overset{\rightarrow} {{j}}+\mathrm{2}\overset{\rightarrow} {{k}}\:{and} \\ $$$${j}\left(\overset{\rightarrow} {{k}}\right)=\mathrm{2}\overset{\rightarrow} {{i}}+\mathrm{2}\overset{\rightarrow} {{j}}. \\ $$$$\mathrm{1}.\:\boldsymbol{{W}}{rite}\:{the}\:{matrix}\:{of}\:{f}\:{in}\:{base}\:\mathscr{B}. \\ $$$$\mathrm{2}.\:{Show}\:{that}\:{the}\:{kernel}\:\left({ker}\:{f}\right)\:{of}\:{f} \\ $$$${is}\:{a}\:{straigh}\:{line};\:{give}\:{one}\:{base}\:{of}\:{its}. \\ $$$$\mathrm{3}.{Determinate}\:{Im}\:{f}. \\ $$

Question Number 134008    Answers: 0   Comments: 0

we consider that application n≥1 det : M_n (R)→R A det(A) 1−verify that ∀H∈M_n (R) and t∈R if A=I_n ⇒det(A+tH)=1+t.Tr(H)+○(t) 2−suppose that: A∈GL_n (R) prouve that the differntial of det in A is given by: H Tr[(com(A))^T H] 3−determinate the differential of determinant of a matrix in general case. (Use the density of GL_n (R) in M_n (R) Tr: trace of matrix (com(A))^T : transpose of the comatrix

$$\:\:\:\:{we}\:{consider}\:{that}\:{application}\:{n}\geqslant\mathrm{1} \\ $$$$\:\:{det}\::\:{M}_{{n}} \left(\mathbb{R}\right)\rightarrow\mathbb{R} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{A} {det}\left({A}\right) \\ $$$$\mathrm{1}−{verify}\:{that}\:\forall{H}\in{M}_{{n}} \left(\mathbb{R}\right)\:{and}\:{t}\in\mathbb{R} \\ $$$$\:{if}\:{A}={I}_{{n}} \Rightarrow{det}\left({A}+{tH}\right)=\mathrm{1}+{t}.{Tr}\left({H}\right)+\circ\left({t}\right) \\ $$$$\mathrm{2}−{suppose}\:{that}:\:{A}\in{GL}_{{n}} \left(\mathbb{R}\right) \\ $$$$\:{prouve}\:{that}\:{the}\:{differntial}\:{of}\:{det}\:{in}\:{A}\:{is}\:{given}\:{by}: \\ $$$$\:\:\:{H} {Tr}\left[\left({com}\left({A}\right)\right)^{{T}} {H}\right] \\ $$$$\mathrm{3}−{determinate}\:{the}\:{differential}\:{of}\:{determinant}\:{of}\:{a}\:{matrix}\:{in}\:{general}\:{case}. \\ $$$$\left({Use}\:{the}\:{density}\:{of}\:{GL}_{{n}} \left(\mathbb{R}\right)\:{in}\:{M}_{{n}} \left(\mathbb{R}\right)\right. \\ $$$$\:{Tr}:\:{trace}\:{of}\:{matrix} \\ $$$$\left({com}\left({A}\right)\right)^{{T}} :\:{transpose}\:{of}\:{the}\:{comatrix} \\ $$

Question Number 134006    Answers: 1   Comments: 0

Question Number 134005    Answers: 1   Comments: 0

solve x^3 −2⌊x⌋=5

$${solve}\:{x}^{\mathrm{3}} −\mathrm{2}\lfloor{x}\rfloor=\mathrm{5} \\ $$

Question Number 134002    Answers: 1   Comments: 1

What will be the minimum area of a heptagon inscribed in an unit square?

$${What}\:{will}\:{be}\:{the}\:{minimum}\:{area}\:{of}\:{a}\:{heptagon}\:{inscribed}\:{in} \\ $$$${an}\:{unit}\:{square}? \\ $$

Question Number 133997    Answers: 1   Comments: 0

lim_(n→∞) (((1+(1/2)+(1/3)+...+(1/n))/n^2 ))^n

$${lim}_{{n}\rightarrow\infty} \left(\frac{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}+...+\frac{\mathrm{1}}{{n}}}{{n}^{\mathrm{2}} }\right)^{{n}} \\ $$

Question Number 133996    Answers: 2   Comments: 0

show that (√2)<log_2 3<(√3)

$${show}\:{that}\:\sqrt{\mathrm{2}}<\mathrm{log}_{\mathrm{2}} \:\mathrm{3}<\sqrt{\mathrm{3}} \\ $$

Question Number 133995    Answers: 0   Comments: 2

in how many ways can n men and n women be arranged in a row such that men and women alternate?

$${in}\:{how}\:{many}\:{ways}\:{can}\:{n}\:{men}\:{and} \\ $$$${n}\:{women}\:{be}\:{arranged}\:{in}\:{a}\:{row}\:{such} \\ $$$${that}\:{men}\:{and}\:{women}\:{alternate}? \\ $$

Question Number 133994    Answers: 1   Comments: 0

Given a function f satisfies f(x)= { ((x+a(√2) sin x ; 0≤x<(π/4))),((2x cot x +b ; (π/4)≤x≤(π/2))),((a cos 2x−bsin x ; (π/2)<x≤π)) :} continuous in [ 0,π ], then find the value of a and b.

$$\mathrm{Given}\:\mathrm{a}\:\mathrm{function}\:\mathrm{f}\:\mathrm{satisfies} \\ $$$$\:\mathrm{f}\left(\mathrm{x}\right)=\begin{cases}{{x}+{a}\sqrt{\mathrm{2}}\:\mathrm{sin}\:{x}\:;\:\mathrm{0}\leqslant{x}<\frac{\pi}{\mathrm{4}}}\\{\mathrm{2}{x}\:\mathrm{cot}\:{x}\:+{b}\:;\:\frac{\pi}{\mathrm{4}}\leqslant{x}\leqslant\frac{\pi}{\mathrm{2}}}\\{{a}\:\mathrm{cos}\:\mathrm{2}{x}−{b}\mathrm{sin}\:{x}\:;\:\frac{\pi}{\mathrm{2}}<{x}\leqslant\pi}\end{cases} \\ $$$$\:\mathrm{continuous}\:\mathrm{in}\:\left[\:\mathrm{0},\pi\:\right],\:\mathrm{then}\:\mathrm{find} \\ $$$$\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{a}\:\mathrm{and}\:{b}.\: \\ $$

Question Number 133991    Answers: 1   Comments: 0

∣ 3x−∣ 4x+2 ∣∣ ≥ 4 > ∣ 5x+8 ∣

$$\:\mid\:\mathrm{3x}−\mid\:\mathrm{4x}+\mathrm{2}\:\mid\mid\:\geqslant\:\mathrm{4}\:>\:\mid\:\mathrm{5x}+\mathrm{8}\:\mid\: \\ $$

Question Number 133989    Answers: 1   Comments: 0

(x/(x2+1))<arctan(x)<x

$$\frac{{x}}{{x}\mathrm{2}+\mathrm{1}}<{arctan}\left({x}\right)<{x} \\ $$$$ \\ $$

Question Number 133988    Answers: 0   Comments: 2

(2+(π/e))(((17)/(16))+(π/(4e)))(((82)/(81))+(π/(9e)))(((257)/(256))+(π/(16e)))...

$$\left(\mathrm{2}+\frac{\pi}{{e}}\right)\left(\frac{\mathrm{17}}{\mathrm{16}}+\frac{\pi}{\mathrm{4}{e}}\right)\left(\frac{\mathrm{82}}{\mathrm{81}}+\frac{\pi}{\mathrm{9}{e}}\right)\left(\frac{\mathrm{257}}{\mathrm{256}}+\frac{\pi}{\mathrm{16}{e}}\right)... \\ $$

Question Number 133976    Answers: 1   Comments: 0

sin^(−1) ((3/5))+tan^(−1) ((1/7))=?

$$\:\mathrm{sin}^{−\mathrm{1}} \left(\frac{\mathrm{3}}{\mathrm{5}}\right)+\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{7}}\right)=? \\ $$

Question Number 133973    Answers: 1   Comments: 0

Given { ((f(x)=((x+(√(x^2 +(1/(27))))))^(1/3) +((x−(√(x^2 +(1/(27))))))^(1/3) )),((g(x)=x^3 +x+1)) :} Find ∫_0 ^4 (g○f○g)(x) dx .

$$\:\mathrm{Given}\:\begin{cases}{\mathrm{f}\left(\mathrm{x}\right)=\sqrt[{\mathrm{3}}]{\mathrm{x}+\sqrt{\mathrm{x}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{27}}}}+\sqrt[{\mathrm{3}}]{\mathrm{x}−\sqrt{\mathrm{x}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{27}}}}}\\{\mathrm{g}\left(\mathrm{x}\right)=\mathrm{x}^{\mathrm{3}} +\mathrm{x}+\mathrm{1}}\end{cases} \\ $$$$\mathrm{Find}\:\underset{\mathrm{0}} {\overset{\mathrm{4}} {\int}}\:\left(\mathrm{g}\circ\mathrm{f}\circ\mathrm{g}\right)\left(\mathrm{x}\right)\:\mathrm{dx}\:. \\ $$

Question Number 133972    Answers: 1   Comments: 0

H = ∫ (((2x−1)^7 )/((2x+1)^9 )) dx

$$\mathscr{H}\:=\:\int\:\frac{\left(\mathrm{2x}−\mathrm{1}\right)^{\mathrm{7}} }{\left(\mathrm{2x}+\mathrm{1}\right)^{\mathrm{9}} }\:\mathrm{dx}\: \\ $$

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