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Question Number 144053 Answers: 1 Comments: 0

........ Calculus........ Ω:=lim(1/π)∫_0 ^( 2π) (Σ_(k=1) ^n ((sin(kx))/( (√2^k ))))^2 dx=?

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:........\:{Calculus}........ \\ $$$$\:\:\:\:\:\:\:\Omega:={lim}\frac{\mathrm{1}}{\pi}\int_{\mathrm{0}} ^{\:\mathrm{2}\pi} \left(\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{{sin}\left({kx}\right)}{\:\sqrt{\mathrm{2}^{{k}} }}\right)^{\mathrm{2}} {dx}=? \\ $$$$ \\ $$

Question Number 144052 Answers: 1 Comments: 0

Evaluate ∫ ((√x)/(sinh x)) dx

$$\mathrm{Evaluate}\: \\ $$$$\:\int\:\frac{\sqrt{{x}}}{\mathrm{sinh}\:{x}}\:{dx} \\ $$

Question Number 144050 Answers: 0 Comments: 1

in a triangle ABC we have { ((2sinA^ +4cosB^ =6)),((4sinB^ +3cosA^ =1)) :} determine C^

$${in}\:{a}\:{triangle}\:{ABC}\:{we}\:{have} \\ $$$$\begin{cases}{\mathrm{2}{sin}\hat {{A}}+\mathrm{4}{cos}\hat {{B}}=\mathrm{6}}\\{\mathrm{4}{sin}\hat {{B}}+\mathrm{3}{cos}\hat {{A}}=\mathrm{1}}\end{cases} \\ $$$${determine}\:\hat {{C}} \\ $$

Question Number 144049 Answers: 1 Comments: 0

Given the equation 1000 = 2000(((1−(1+t)^(−n) )/t)) find the value of t.

$$\:\mathrm{Given}\:\mathrm{the}\:\mathrm{equation}\:\:\mathrm{1000}\:=\:\mathrm{2000}\left(\frac{\mathrm{1}−\left(\mathrm{1}+{t}\right)^{−{n}} }{{t}}\right) \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{t}. \\ $$

Question Number 144199 Answers: 1 Comments: 0

Prove that 1+(1^2 /(6+(3^2 /(6+(5^2 /(6+(7^2 /(6+(9^2 /(6+...))))))))))=(2/(1+((1×2)/(1+((2×3)/(1+((3×4)/(1+((4×5)/(1+((5×6)/(1+...))))))))))))

$$\mathrm{Prove}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}+\frac{\mathrm{1}^{\mathrm{2}} }{\mathrm{6}+\frac{\mathrm{3}^{\mathrm{2}} }{\mathrm{6}+\frac{\mathrm{5}^{\mathrm{2}} }{\mathrm{6}+\frac{\mathrm{7}^{\mathrm{2}} }{\mathrm{6}+\frac{\mathrm{9}^{\mathrm{2}} }{\mathrm{6}+...}}}}}=\frac{\mathrm{2}}{\mathrm{1}+\frac{\mathrm{1}×\mathrm{2}}{\mathrm{1}+\frac{\mathrm{2}×\mathrm{3}}{\mathrm{1}+\frac{\mathrm{3}×\mathrm{4}}{\mathrm{1}+\frac{\mathrm{4}×\mathrm{5}}{\mathrm{1}+\frac{\mathrm{5}×\mathrm{6}}{\mathrm{1}+...}}}}}} \\ $$

Question Number 144042 Answers: 3 Comments: 0

Question Number 144038 Answers: 0 Comments: 0

Question Number 144031 Answers: 1 Comments: 0

Question Number 144026 Answers: 1 Comments: 1

Question Number 144025 Answers: 1 Comments: 0

Between 12 p.m. today and 12 p.m. tomorrow, how many times do the hour hand and the minute hand on a clock form an angle of 120°?

$$\mathrm{Between}\:\mathrm{12}\:\mathrm{p}.\mathrm{m}.\:\mathrm{today}\:\mathrm{and}\:\mathrm{12}\:\mathrm{p}.\mathrm{m}. \\ $$$$\mathrm{tomorrow},\:\mathrm{how}\:\mathrm{many}\:\mathrm{times}\:\mathrm{do}\:\mathrm{the} \\ $$$$\mathrm{hour}\:\mathrm{hand}\:\mathrm{and}\:\mathrm{the}\:\mathrm{minute}\:\mathrm{hand}\:\mathrm{on}\:\mathrm{a} \\ $$$$\mathrm{clock}\:\mathrm{form}\:\mathrm{an}\:\mathrm{angle}\:\mathrm{of}\:\mathrm{120}°? \\ $$

Question Number 144021 Answers: 0 Comments: 0

Question Number 144016 Answers: 0 Comments: 0

Question Number 144015 Answers: 0 Comments: 0

Question Number 144007 Answers: 0 Comments: 0

find maximum of x cosec x if 0<x<(Π/6)

$$\mathrm{find}\:\mathrm{maximum}\:\mathrm{of}\:\mathrm{x}\:\mathrm{cosec}\:\mathrm{x} \\ $$$$\mathrm{if}\:\mathrm{0}<\mathrm{x}<\frac{\Pi}{\mathrm{6}} \\ $$

Question Number 144006 Answers: 0 Comments: 0

find minimum value of sec 2A+sec 2B where A+B is constant and A,B∈(o (Π/4))

$$\mathrm{find}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:\mathrm{sec}\:\mathrm{2A}+\mathrm{sec}\:\mathrm{2B} \\ $$$$\mathrm{where}\:\mathrm{A}+\mathrm{B}\:\mathrm{is}\:\mathrm{constant}\:\mathrm{and} \\ $$$$\mathrm{A},\mathrm{B}\in\left(\mathrm{o}\:\frac{\Pi}{\mathrm{4}}\right) \\ $$

Question Number 144001 Answers: 2 Comments: 0

Question Number 144000 Answers: 1 Comments: 0

prove that ∀_m ∈N , a_k ,b_k ∈R cos^(2m) x =Σ_(k=1) ^m a_k cos 2kx cos^(2m−1) x=Σ_(k=1) ^m b_k cos (2k−1)x and find expr of a_k ,b_k in terms of k.

$$\mathrm{prove}\:\mathrm{that}\: \\ $$$$ \\ $$$$\forall_{{m}} \in\mathbb{N}\:,\:{a}_{{k}} ,{b}_{{k}} \in\mathbb{R} \\ $$$$\mathrm{cos}\:^{\mathrm{2}{m}} {x}\:=\underset{{k}=\mathrm{1}} {\overset{{m}} {\sum}}{a}_{{k}} \mathrm{cos}\:\mathrm{2}{kx} \\ $$$$\mathrm{cos}\:^{\mathrm{2}{m}−\mathrm{1}} {x}=\underset{{k}=\mathrm{1}} {\overset{{m}} {\sum}}{b}_{{k}} \mathrm{cos}\:\left(\mathrm{2}{k}−\mathrm{1}\right){x} \\ $$$$ \\ $$$$\mathrm{and}\:\:\mathrm{find}\:\mathrm{expr}\:\:\mathrm{of}\:\:{a}_{{k}} \:,{b}_{{k}} \:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{k}. \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 143997 Answers: 2 Comments: 0

The maximum value of y = (√((x−3)^2 +(x^2 −2)^2 ))−(√(x^2 +(x^2 −1)^2 )) is (A) (√(10)) (C) 4 (B) 2(√5) (D) 10

$$\:{The}\:{maximum}\:{value}\:{of}\: \\ $$$${y}\:=\:\sqrt{\left({x}−\mathrm{3}\right)^{\mathrm{2}} +\left({x}^{\mathrm{2}} −\mathrm{2}\right)^{\mathrm{2}} }−\sqrt{{x}^{\mathrm{2}} +\left({x}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{2}} } \\ $$$${is}\:\left({A}\right)\:\sqrt{\mathrm{10}}\:\:\:\:\:\:\:\left({C}\right)\:\mathrm{4}\: \\ $$$$\:\:\:\:\:\:\left({B}\right)\:\mathrm{2}\sqrt{\mathrm{5}}\:\:\:\:\:\left({D}\right)\:\mathrm{10}\: \\ $$

Question Number 143995 Answers: 1 Comments: 0

lim_(x→π/4) ((π−4x)/( (√(1−(√(sin 2x)))))) =?

$$\:\:\underset{{x}\rightarrow\pi/\mathrm{4}} {\mathrm{lim}}\frac{\pi−\mathrm{4}{x}}{\:\sqrt{\mathrm{1}−\sqrt{\mathrm{sin}\:\mathrm{2}{x}}}}\:=? \\ $$

Question Number 143993 Answers: 1 Comments: 0

Question Number 143987 Answers: 1 Comments: 0

If f(x^2 −6x+6)+f(x^2 −4x+4)=2x ∀x∈R then f(−3)+f(9)−5f(1)=?

$${If}\:{f}\left({x}^{\mathrm{2}} −\mathrm{6}{x}+\mathrm{6}\right)+{f}\left({x}^{\mathrm{2}} −\mathrm{4}{x}+\mathrm{4}\right)=\mathrm{2}{x} \\ $$$$\forall{x}\in{R}\:{th}\mathrm{e}{n}\:{f}\left(−\mathrm{3}\right)+{f}\left(\mathrm{9}\right)−\mathrm{5}{f}\left(\mathrm{1}\right)=? \\ $$

Question Number 143980 Answers: 1 Comments: 0

Question Number 143979 Answers: 1 Comments: 0

Question Number 143970 Answers: 1 Comments: 0

Given that ω is a complex number, ω^7 =1, ω≠1, find the value of ω^1 +ω^2 +ω^3 +ω^4 +ω^5 +ω^6 .

$$\mathrm{Given}\:\mathrm{that}\:\omega\:\mathrm{is}\:\mathrm{a}\:\mathrm{complex}\:\mathrm{number}, \\ $$$$\omega^{\mathrm{7}} =\mathrm{1},\:\omega\neq\mathrm{1},\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\omega^{\mathrm{1}} +\omega^{\mathrm{2}} +\omega^{\mathrm{3}} +\omega^{\mathrm{4}} +\omega^{\mathrm{5}} +\omega^{\mathrm{6}} . \\ $$

Question Number 143965 Answers: 1 Comments: 1

Question Number 143964 Answers: 1 Comments: 2

x_1 and x_2 are solutions of equality : cos (((πx+π)/6)) − sin (((πx−π)/6)) = (1/2) (√3) , 0 ≤ x ≤ 12 Find the value of x_1 + x_2 .

$${x}_{\mathrm{1}} \:{and}\:\:{x}_{\mathrm{2}} \:\:{are}\:\:{solutions}\:\:{of}\:\:{equality}\:: \\ $$$$\:\:\mathrm{cos}\:\left(\frac{\pi{x}+\pi}{\mathrm{6}}\right)\:−\:\mathrm{sin}\:\left(\frac{\pi{x}−\pi}{\mathrm{6}}\right)\:=\:\frac{\mathrm{1}}{\mathrm{2}}\:\sqrt{\mathrm{3}}\:\:\:,\:\:\:\mathrm{0}\:\leqslant\:{x}\:\leqslant\:\mathrm{12} \\ $$$${Find}\:\:{the}\:\:{value}\:\:{of}\:\:{x}_{\mathrm{1}} +\:{x}_{\mathrm{2}} \:. \\ $$

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