Question and Answers Forum

AllQuestion and Answers: Page 1003

Question Number 117944 Answers: 3 Comments: 0

Find the value of k satisfies the equation ∫ _0^(π/3) (((tan x (√(cos x)))/( (√(2k)))) ) dx = 1−(1/( (√2)))

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{k}\:\mathrm{satisfies}\: \\ $$$$\mathrm{the}\:\mathrm{equation}\:\int\:_{\mathrm{0}} ^{\frac{\pi}{\mathrm{3}}} \:\left(\frac{\mathrm{tan}\:\mathrm{x}\:\sqrt{\mathrm{cos}\:\mathrm{x}}}{\:\sqrt{\mathrm{2k}}}\:\right)\:\mathrm{dx}\:=\:\mathrm{1}−\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}} \\ $$$$ \\ $$

Question Number 117943 Answers: 1 Comments: 0

solve (1/2)+((sin 112°)/(16sin 7°)) −cos 7°.cos 14°.cos 28°.cos 56° =?

$$\mathrm{solve}\:\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{sin}\:\mathrm{112}°}{\mathrm{16sin}\:\mathrm{7}°}\:−\mathrm{cos}\:\mathrm{7}°.\mathrm{cos}\:\mathrm{14}°.\mathrm{cos}\:\mathrm{28}°.\mathrm{cos}\:\mathrm{56}°\:=? \\ $$

Question Number 117953 Answers: 2 Comments: 0

... ◂nice integral▶... please prove : Ω=∫_0 ^( (π/2)) ln^2 (cot(x))dx =(π^3 /8) ...♠m.n.1070♠...

$$\:\:\:\:\:\:\:\:\:...\:\:\blacktriangleleft{nice}\:\:{integral}\blacktriangleright... \\ $$$$\:\:\:\:{please}\:\:{prove}\:: \\ $$$$ \\ $$$$\:\:\Omega=\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} {ln}^{\mathrm{2}} \left({cot}\left({x}\right)\right){dx}\:=\frac{\pi^{\mathrm{3}} }{\mathrm{8}}\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:...\spadesuit{m}.{n}.\mathrm{1070}\spadesuit... \\ $$

Question Number 117938 Answers: 5 Comments: 0

(1) ((∣a^2 −16∣)/(4−a)) − ((∣a^2 −9∣)/(3+a)) − ((∣4−a^2 ∣)/(2−a)) =? (2)(((∣x∣ +x )/(x−1)))^2 −((14x)/(x−1)) + 12 = 0 (3) log _4 (2^(2x) −(√3) cos x−6sin^2 x ) = x where ((5π)/2) ≤x≤4π (4) ((2cos^2 x−(√3) cos x)/(log _4 (sin x))) = 0 , where −3π ≤x≤−((3π)/2)

$$\left(\mathrm{1}\right)\:\frac{\mid{a}^{\mathrm{2}} −\mathrm{16}\mid}{\mathrm{4}−{a}}\:−\:\frac{\mid{a}^{\mathrm{2}} −\mathrm{9}\mid}{\mathrm{3}+{a}}\:−\:\frac{\mid\mathrm{4}−{a}^{\mathrm{2}} \mid}{\mathrm{2}−{a}}\:=? \\ $$$$\left(\mathrm{2}\right)\left(\frac{\mid\mathrm{x}\mid\:+\mathrm{x}\:}{\mathrm{x}−\mathrm{1}}\right)^{\mathrm{2}} −\frac{\mathrm{14x}}{\mathrm{x}−\mathrm{1}}\:+\:\mathrm{12}\:=\:\mathrm{0} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{log}\:_{\mathrm{4}} \left(\mathrm{2}^{\mathrm{2x}} −\sqrt{\mathrm{3}}\:\mathrm{cos}\:\mathrm{x}−\mathrm{6sin}\:^{\mathrm{2}} \mathrm{x}\:\right)\:=\:\mathrm{x} \\ $$$$\mathrm{where}\:\frac{\mathrm{5}\pi}{\mathrm{2}}\:\leqslant\mathrm{x}\leqslant\mathrm{4}\pi \\ $$$$\left(\mathrm{4}\right)\:\frac{\mathrm{2cos}\:^{\mathrm{2}} \mathrm{x}−\sqrt{\mathrm{3}}\:\mathrm{cos}\:\mathrm{x}}{\mathrm{log}\:_{\mathrm{4}} \left(\mathrm{sin}\:\mathrm{x}\right)}\:=\:\mathrm{0}\:,\:\mathrm{where}\: \\ $$$$−\mathrm{3}\pi\:\leqslant\mathrm{x}\leqslant−\frac{\mathrm{3}\pi}{\mathrm{2}} \\ $$

Question Number 117934 Answers: 1 Comments: 0

Let f : [1,∞)→[2,∞) be the function defined by f(x)=x+(1/x) If g : [2,∞)→[1,∞), is a function such that (g○f)(x)=x for all x≥1. Show that g(t)=((t+(√(t^2 −4)))/2)

$$\mathrm{Let}\:{f}\::\:\left[\mathrm{1},\infty\right)\rightarrow\left[\mathrm{2},\infty\right)\:\mathrm{be}\:\mathrm{the}\:\mathrm{function}\:\mathrm{defined}\:\mathrm{by} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{f}\left({x}\right)={x}+\frac{\mathrm{1}}{{x}} \\ $$$$\mathrm{If}\:\mathrm{g}\::\:\left[\mathrm{2},\infty\right)\rightarrow\left[\mathrm{1},\infty\right),\:\mathrm{is}\:\mathrm{a}\:\mathrm{function}\:\mathrm{such}\:\mathrm{that}\:\left(\mathrm{g}\circ{f}\right)\left({x}\right)={x} \\ $$$$\mathrm{for}\:\mathrm{all}\:{x}\geqslant\mathrm{1}.\:\mathrm{Show}\:\mathrm{that}\:\mathrm{g}\left({t}\right)=\frac{{t}+\sqrt{{t}^{\mathrm{2}} −\mathrm{4}}}{\mathrm{2}} \\ $$

Question Number 117926 Answers: 3 Comments: 0

tan^(−1) (((2x)/(x^2 −1))) + cot^(−1) (((x^2 −1)/(2x))) =((2π)/3) (tan^(−1) (x))^2 +(cot^(−1) (x))^2 =((5π)/8)

$$\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{2x}}{\mathrm{x}^{\mathrm{2}} −\mathrm{1}}\right)\:+\:\mathrm{cot}^{−\mathrm{1}} \left(\frac{\mathrm{x}^{\mathrm{2}} −\mathrm{1}}{\mathrm{2x}}\right)\:=\frac{\mathrm{2}\pi}{\mathrm{3}} \\ $$$$\left(\mathrm{tan}^{−\mathrm{1}} \left(\mathrm{x}\right)\right)^{\mathrm{2}} +\left(\mathrm{cot}^{−\mathrm{1}} \left(\mathrm{x}\right)\right)^{\mathrm{2}} =\frac{\mathrm{5}\pi}{\mathrm{8}} \\ $$$$ \\ $$

Question Number 117912 Answers: 1 Comments: 1

Question Number 117910 Answers: 3 Comments: 1

∫ sin^(−1) ((√x)) dx =?

$$\int\:\mathrm{sin}^{−\mathrm{1}} \left(\sqrt{\mathrm{x}}\right)\:\mathrm{dx}\:=? \\ $$

Question Number 117903 Answers: 2 Comments: 0

prove by mathematical induction that n(n+1)(n+2) is an integer multiple of 6

$${prove}\:{by}\:{mathematical}\:{induction} \\ $$$${that}\:{n}\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)\:{is}\:{an}\:{integer}\: \\ $$$${multiple}\:{of}\:\mathrm{6} \\ $$

Question Number 117901 Answers: 0 Comments: 1

Question Number 117895 Answers: 1 Comments: 1

prove by mathematical induction that (1/(n+1))+(1/(n+2))+...+(1/(2n))>(1/2)

$${prove}\:{by}\:{mathematical}\:{induction}\:{that} \\ $$$$\frac{\mathrm{1}}{{n}+\mathrm{1}}+\frac{\mathrm{1}}{{n}+\mathrm{2}}+...+\frac{\mathrm{1}}{\mathrm{2}{n}}>\frac{\mathrm{1}}{\mathrm{2}} \\ $$

Question Number 117866 Answers: 2 Comments: 3

how many four digit numbers can be formed with the digits 0 1 2 3 4 5 6 7 8 9?

$${how}\:{many}\:{four}\:{digit}\:{numbers}\:{can}\:{be} \\ $$$${formed}\:{with}\:{the}\:{digits}\:\mathrm{0}\:\mathrm{1}\:\mathrm{2}\:\mathrm{3}\:\mathrm{4}\:\mathrm{5}\:\mathrm{6}\:\mathrm{7}\:\mathrm{8} \\ $$$$\mathrm{9}? \\ $$

Question Number 117898 Answers: 1 Comments: 0

lim_(x→0) (( 1)/(sin^4 x)) (sin ((x/(x+1)))−((sin x)/(1+sin x)) ) =?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\:\mathrm{1}}{\mathrm{sin}\:^{\mathrm{4}} \mathrm{x}}\:\left(\mathrm{sin}\:\left(\frac{\mathrm{x}}{\mathrm{x}+\mathrm{1}}\right)−\frac{\mathrm{sin}\:\mathrm{x}}{\mathrm{1}+\mathrm{sin}\:\mathrm{x}}\:\right)\:=? \\ $$

Question Number 117865 Answers: 3 Comments: 0

discuss the lim_(n→∞) (1+(1/n))^n

$${discuss}\:{the}\:{lim}_{{n}\rightarrow\infty} \left(\mathrm{1}+\frac{\mathrm{1}}{{n}}\right)^{{n}} \\ $$

Question Number 117862 Answers: 0 Comments: 1

Question Number 117857 Answers: 1 Comments: 0

Let n∈N . Find the number of polynomials p(x) with coefficients in { 0,1,2,3 } such that p(2)= n

$$\mathrm{Let}\:\mathrm{n}\in\mathbb{N}\:.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\: \\ $$$$\mathrm{polynomials}\:\mathrm{p}\left(\mathrm{x}\right)\:\mathrm{with}\:\mathrm{coefficients} \\ $$$$\mathrm{in}\:\left\{\:\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3}\:\right\}\:\mathrm{such}\:\mathrm{that}\:\mathrm{p}\left(\mathrm{2}\right)=\:\mathrm{n}\: \\ $$

Question Number 117863 Answers: 1 Comments: 0

if y=((x+2)/( (√(x+1)))) find (dy/dx) from first principle.

$${if}\:{y}=\frac{{x}+\mathrm{2}}{\:\sqrt{{x}+\mathrm{1}}}\:{find}\:\frac{{dy}}{{dx}}\:{from}\:{first} \\ $$$${principle}. \\ $$

Question Number 117852 Answers: 3 Comments: 0

Determine the value of (1)(tan ((7π)/(24))+tan ((5π)/(24))).cos (π/(12)) + 2 . (2) (((9−4(√5))/(5x)))^(1/(4 )) .(5(√x) +(√(20x)) )^(0.5) . 2^(−1) = ?

$$\mathrm{Determine}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\: \\ $$$$\left(\mathrm{1}\right)\left(\mathrm{tan}\:\frac{\mathrm{7}\pi}{\mathrm{24}}+\mathrm{tan}\:\frac{\mathrm{5}\pi}{\mathrm{24}}\right).\mathrm{cos}\:\frac{\pi}{\mathrm{12}}\:+\:\mathrm{2}\:. \\ $$$$\left(\mathrm{2}\right)\:\sqrt[{\mathrm{4}\:}]{\frac{\mathrm{9}−\mathrm{4}\sqrt{\mathrm{5}}}{\mathrm{5x}}}\:.\left(\mathrm{5}\sqrt{\mathrm{x}}\:+\sqrt{\mathrm{20x}}\:\right)^{\mathrm{0}.\mathrm{5}} .\:\mathrm{2}^{−\mathrm{1}} \:=\:? \\ $$

Question Number 117848 Answers: 0 Comments: 0

Determine all functions f:R→R such that the equality f([x] y)= f(x) [f(y) ] holds for all x,y ∈R . Here by [x] we denote the greatest integer not exceeding x.

$${Determine}\:{all}\:{functions}\:{f}:\mathbb{R}\rightarrow\mathbb{R} \\ $$$${such}\:{that}\:{the}\:{equality}\:{f}\left(\left[{x}\right]\:{y}\right)=\:{f}\left({x}\right)\:\left[{f}\left({y}\right)\:\right] \\ $$$${holds}\:{for}\:{all}\:{x},{y}\:\in\mathbb{R}\:.\:{Here}\:\:{by}\:\left[{x}\right]\:{we}\: \\ $$$${denote}\:{the}\:{greatest}\:{integer}\:{not}\:{exceeding}\:{x}. \\ $$$$ \\ $$

Question Number 117844 Answers: 0 Comments: 1