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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 .

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

Question Number 143963 Answers: 1 Comments: 0

A=x^(ln(y/x)) .y^(ln(z/x)) .z^(ln(x/y)) =??

$$\mathrm{A}=\mathrm{x}^{\mathrm{ln}\frac{\mathrm{y}}{\mathrm{x}}} .\mathrm{y}^{\mathrm{ln}\frac{\mathrm{z}}{\mathrm{x}}} .\mathrm{z}^{\mathrm{ln}\frac{\mathrm{x}}{\mathrm{y}}} =?? \\ $$

Question Number 143962 Answers: 2 Comments: 0

∫_0 ^(π/2) ((2xsin^2 t)/(cos^2 t+x^2 sin^2 t))dt=.....???

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{2}{xsin}^{\mathrm{2}} {t}}{{cos}^{\mathrm{2}} {t}+{x}^{\mathrm{2}} {sin}^{\mathrm{2}} {t}}{dt}=.....??? \\ $$

Question Number 143961 Answers: 1 Comments: 0

I=∫_(π/6) ^(π/3) ((cos^6 x)/(1−3sin^2 xcos^2 x))dx=?

$$\mathrm{I}=\int_{\frac{\pi}{\mathrm{6}}} ^{\frac{\pi}{\mathrm{3}}} \frac{\mathrm{cos}^{\mathrm{6}} \mathrm{x}}{\mathrm{1}−\mathrm{3sin}^{\mathrm{2}} \mathrm{xcos}^{\mathrm{2}} \mathrm{x}}\mathrm{dx}=? \\ $$

Question Number 143960 Answers: 2 Comments: 0

The value of lim_(x→0) ((√(1−cos x^2 ))/(1−cos x)) =?

$$\:\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{1}−\mathrm{cos}\:\mathrm{x}^{\mathrm{2}} }}{\mathrm{1}−\mathrm{cos}\:\mathrm{x}}\:=? \\ $$

Question Number 143959 Answers: 1 Comments: 0

If g(x)=(4cos^4 x−2cos 2x−(1/2)cos 4x−x^7 )^(1/7) then tbe value of g(g(100)) is equal to ...

$$\:\:\mathrm{If}\:\mathrm{g}\left(\mathrm{x}\right)=\left(\mathrm{4cos}\:^{\mathrm{4}} \mathrm{x}−\mathrm{2cos}\:\mathrm{2x}−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cos}\:\mathrm{4x}−\mathrm{x}^{\mathrm{7}} \right)^{\frac{\mathrm{1}}{\mathrm{7}}} \\ $$$$\mathrm{then}\:\mathrm{tbe}\:\mathrm{value}\:\mathrm{of}\:\mathrm{g}\left(\mathrm{g}\left(\mathrm{100}\right)\right)\:\mathrm{is} \\ $$$$\mathrm{equal}\:\mathrm{to}\:... \\ $$

Question Number 143958 Answers: 1 Comments: 0

Question Number 143951 Answers: 0 Comments: 0

∫_0 ^(π/2) ((2xsin^2 t)/(cos^2 t+x^2 sin^2 t))dt=.....????

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{2}{xsin}^{\mathrm{2}} {t}}{{cos}^{\mathrm{2}} {t}+{x}^{\mathrm{2}} {sin}^{\mathrm{2}} {t}}{dt}=.....???? \\ $$

Question Number 143952 Answers: 3 Comments: 0

Question Number 143920 Answers: 2 Comments: 0

Question Number 143913 Answers: 1 Comments: 0

If ((cos α)/(cos β)) = m and ((cos α)/(sin β)) = n then prove that (m^2 +n^2 )cos^2 β = n^2

$$\mathrm{If}\:\:\frac{\mathrm{cos}\:\alpha}{{cos}\:\beta}\:=\:\mathrm{m}\:\mathrm{and}\:\frac{\mathrm{cos}\:\alpha}{\mathrm{sin}\:\beta}\:=\:\mathrm{n}\:\mathrm{then} \\ $$$$\mathrm{prove}\:\mathrm{that}\:\left(\mathrm{m}^{\mathrm{2}} +\mathrm{n}^{\mathrm{2}} \right)\mathrm{cos}^{\mathrm{2}} \beta\:=\:{n}^{\mathrm{2}} \\ $$

Question Number 143932 Answers: 4 Comments: 0

Question Number 143923 Answers: 0 Comments: 1

What is the value of [sin^(−1) (sin(((2π)/3)))]?

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\left[\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{sin}\left(\frac{\mathrm{2}\pi}{\mathrm{3}}\right)\right)\right]? \\ $$

Question Number 143904 Answers: 1 Comments: 0

Question Number 143906 Answers: 1 Comments: 0

Question Number 143909 Answers: 2 Comments: 3

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