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Question Number 17080    Answers: 2   Comments: 0

sin^4 θ/2+cos^4 θ/2≥1/2

$$\mathrm{sin}^{\mathrm{4}} \theta/\mathrm{2}+\mathrm{cos}\:^{\mathrm{4}} \theta/\mathrm{2}\geqslant\mathrm{1}/\mathrm{2} \\ $$

Question Number 17093    Answers: 0   Comments: 16

If f(x) is a polynomial function satisfying f(x).f((1/x)) = f(x) + f((1/x)) ; x ∈ R − {0} and f(3) = 28, then f(4) is equal to

$$\mathrm{If}\:{f}\left({x}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{polynomial}\:\mathrm{function} \\ $$$$\mathrm{satisfying}\:{f}\left({x}\right).{f}\left(\frac{\mathrm{1}}{{x}}\right)\:=\:{f}\left({x}\right)\:+\:{f}\left(\frac{\mathrm{1}}{{x}}\right)\:; \\ $$$${x}\:\in\:{R}\:−\:\left\{\mathrm{0}\right\}\:\mathrm{and}\:{f}\left(\mathrm{3}\right)\:=\:\mathrm{28},\:\mathrm{then}\:{f}\left(\mathrm{4}\right)\:\mathrm{is} \\ $$$$\mathrm{equal}\:\mathrm{to} \\ $$

Question Number 17075    Answers: 2   Comments: 1

Given that: log((x/(y − z))) = log((y/(z − x))) = log((z/(x − y))) Show that : x^x × y^y × z^z = 1

$$\mathrm{Given}\:\mathrm{that}:\:\:\mathrm{log}\left(\frac{\mathrm{x}}{\mathrm{y}\:−\:\mathrm{z}}\right)\:=\:\mathrm{log}\left(\frac{\mathrm{y}}{\mathrm{z}\:−\:\mathrm{x}}\right)\:=\:\mathrm{log}\left(\frac{\mathrm{z}}{\mathrm{x}\:−\:\mathrm{y}}\right) \\ $$$$\mathrm{Show}\:\mathrm{that}\::\:\:\:\mathrm{x}^{\mathrm{x}} \:×\:\mathrm{y}^{\mathrm{y}} \:×\:\mathrm{z}^{\mathrm{z}} \:=\:\mathrm{1} \\ $$

Question Number 17073    Answers: 0   Comments: 4

Question Number 17065    Answers: 0   Comments: 4

if y=x^x what is the range of this function?

$$\mathrm{if}\:\mathrm{y}=\mathrm{x}^{\mathrm{x}} \\ $$$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of}\:\mathrm{this}\:\mathrm{function}? \\ $$

Question Number 17095    Answers: 1   Comments: 0

The total number of solutions of the equation tan x + sec x = 2 which lie in the interval [0, 2π] is

$$\mathrm{The}\:\mathrm{total}\:\mathrm{number}\:\mathrm{of}\:\mathrm{solutions}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equation}\:\mathrm{tan}\:{x}\:+\:\mathrm{sec}\:{x}\:=\:\mathrm{2}\:\mathrm{which}\:\mathrm{lie}\:\mathrm{in} \\ $$$$\mathrm{the}\:\mathrm{interval}\:\left[\mathrm{0},\:\mathrm{2}\pi\right]\:\mathrm{is} \\ $$

Question Number 17068    Answers: 2   Comments: 0

prove that 4tan^(−1) ((1/5))−tan^(−1) ((1/(239))) =π/4

$$\mathrm{prove}\:\mathrm{that}\: \\ $$$$ \\ $$$$\mathrm{4tan}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{5}}\right)−\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{239}}\right)\:=\pi/\mathrm{4} \\ $$

Question Number 17869    Answers: 1   Comments: 1

Question Number 17034    Answers: 1   Comments: 0

∫_0 ^( Π) (dx/(3+2sinx+cosx))

$$\int_{\mathrm{0}} ^{\:\Pi} \frac{\mathrm{dx}}{\mathrm{3}+\mathrm{2sinx}+\mathrm{cosx}} \\ $$

Question Number 17033    Answers: 1   Comments: 0

∫_((Π )/2) ^( 0) ((sinx cosx dx)/(2cos^2 x+3sin^2 x))

$$\int_{\frac{\Pi\:}{\mathrm{2}}} ^{\:\mathrm{0}} \:\frac{\mathrm{sinx}\:\mathrm{cosx}\:\mathrm{dx}}{\mathrm{2cos}^{\mathrm{2}} \mathrm{x}+\mathrm{3sin}^{\mathrm{2}} \mathrm{x}} \\ $$

Question Number 17030    Answers: 0   Comments: 1

∫ cot^2 (x^3 ) dx

$$\int\:\mathrm{cot}^{\mathrm{2}} \left(\mathrm{x}^{\mathrm{3}} \right)\:\mathrm{dx} \\ $$

Question Number 17027    Answers: 0   Comments: 0

Question Number 17019    Answers: 1   Comments: 0

Σ_(r = 1) ^3 2r − 1 = ?

$$\underset{\mathrm{r}\:=\:\mathrm{1}} {\overset{\mathrm{3}} {\sum}}\:\mathrm{2r}\:−\:\mathrm{1}\:\:=\:\:? \\ $$

Question Number 17018    Answers: 0   Comments: 8

Find the cube root of: 55 + 63 (√2)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{cube}\:\mathrm{root}\:\mathrm{of}:\:\:\:\mathrm{55}\:+\:\mathrm{63}\:\sqrt{\mathrm{2}} \\ $$

Question Number 17017    Answers: 1   Comments: 0

solve the simultaenous equation x+y=3 (2^x /x)=(2^y /y) find xand y.show ur workings....

$${solve}\:{the}\:{simultaenous}\:{equation} \\ $$$$\:{x}+{y}=\mathrm{3} \\ $$$$\:\frac{\mathrm{2}^{{x}} }{{x}}=\frac{\mathrm{2}^{{y}} }{{y}} \\ $$$$\:{find}\:{xand}\:{y}.{show}\:{ur}\:{workings}.... \\ $$

Question Number 17011    Answers: 1   Comments: 0

∫_0 ^( a) x(√((a^2 −x^2 )/(a^2 +x^2 )))dx

$$\int_{\mathrm{0}} ^{\:\mathrm{a}} \mathrm{x}\sqrt{\frac{\mathrm{a}^{\mathrm{2}} −\mathrm{x}^{\mathrm{2}} }{\mathrm{a}^{\mathrm{2}} +\mathrm{x}^{\mathrm{2}} }}\mathrm{dx} \\ $$

Question Number 17010    Answers: 1   Comments: 0

∫_0 ^( 1) x(tan^(−1) x)^2 dx

$$\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\mathrm{x}\left(\mathrm{tan}^{−\mathrm{1}} \mathrm{x}\right)^{\mathrm{2}} \mathrm{dx} \\ $$

Question Number 17009    Answers: 0   Comments: 1

∫_0 ^( a) cos^(−1) ((1−x^2 )/(1+x^2 ))dx

$$\int_{\mathrm{0}} ^{\:\mathrm{a}} \:\:\mathrm{cos}^{−\mathrm{1}} \frac{\mathrm{1}−\mathrm{x}^{\mathrm{2}} }{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\mathrm{dx} \\ $$

Question Number 17008    Answers: 1   Comments: 0

∫_((1 )/Π) ^(2/Π) (1/x^2 )sin(1/x)dx

$$\int_{\frac{\mathrm{1}\:}{\Pi}} ^{\frac{\mathrm{2}}{\Pi}} \:\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} }\mathrm{sin}\frac{\mathrm{1}}{\mathrm{x}}\mathrm{dx} \\ $$

Question Number 17053    Answers: 1   Comments: 0

Question Number 19167    Answers: 1   Comments: 1

Two particles A and B move with constant velocities v_1 and v_2 along two mutually perpendicular straight lines towards the intersection point O. At moment t = 0, the particles were located at distances d_1 and d_2 from O respectively. Find the time, when they are nearest and also this shortest distance.

$$\mathrm{Two}\:\mathrm{particles}\:{A}\:\mathrm{and}\:{B}\:\mathrm{move}\:\mathrm{with} \\ $$$$\mathrm{constant}\:\mathrm{velocities}\:{v}_{\mathrm{1}} \:\mathrm{and}\:{v}_{\mathrm{2}} \:\mathrm{along}\:\mathrm{two} \\ $$$$\mathrm{mutually}\:\mathrm{perpendicular}\:\mathrm{straight}\:\mathrm{lines} \\ $$$$\mathrm{towards}\:\mathrm{the}\:\mathrm{intersection}\:\mathrm{point}\:{O}.\:\mathrm{At} \\ $$$$\mathrm{moment}\:{t}\:=\:\mathrm{0},\:\mathrm{the}\:\mathrm{particles}\:\mathrm{were} \\ $$$$\mathrm{located}\:\mathrm{at}\:\mathrm{distances}\:{d}_{\mathrm{1}} \:\mathrm{and}\:{d}_{\mathrm{2}} \:\mathrm{from}\:{O} \\ $$$$\mathrm{respectively}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{time},\:\mathrm{when}\:\mathrm{they} \\ $$$$\mathrm{are}\:\mathrm{nearest}\:\mathrm{and}\:\mathrm{also}\:\mathrm{this}\:\mathrm{shortest} \\ $$$$\mathrm{distance}. \\ $$

Question Number 17169    Answers: 2   Comments: 0

The base of a pyramid is an equilateral triangle of side length 6 cm. The other edges of the pyramid are each of length (√(15)) cm. Find the volume of the pyramid.

$$\mathrm{The}\:\mathrm{base}\:\mathrm{of}\:\mathrm{a}\:\mathrm{pyramid}\:\mathrm{is}\:\mathrm{an}\:\mathrm{equilateral} \\ $$$$\mathrm{triangle}\:\mathrm{of}\:\mathrm{side}\:\mathrm{length}\:\mathrm{6}\:\mathrm{cm}.\:\mathrm{The}\:\mathrm{other} \\ $$$$\mathrm{edges}\:\mathrm{of}\:\mathrm{the}\:\mathrm{pyramid}\:\mathrm{are}\:\mathrm{each}\:\mathrm{of}\:\mathrm{length} \\ $$$$\sqrt{\mathrm{15}}\:\mathrm{cm}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{volume}\:\mathrm{of}\:\mathrm{the}\:\mathrm{pyramid}. \\ $$

Question Number 16983    Answers: 1   Comments: 0

If 15 men or 24 women or 36 boys do a piece of work in 12 days, working 8 hours a day, how many men must associated with 12 women and 6 boys to do another piece of work 2(1/4) times as great in 30 days working 6 hrs a day?

$$\mathrm{If}\:\mathrm{15}\:\mathrm{men}\:\mathrm{or}\:\mathrm{24}\:\mathrm{women}\:\mathrm{or}\:\mathrm{36}\:\mathrm{boys}\:\mathrm{do}\:\mathrm{a}\: \\ $$$$\mathrm{piece}\:\mathrm{of}\:\mathrm{work}\:\mathrm{in}\:\mathrm{12}\:\mathrm{days},\:\mathrm{working}\:\mathrm{8}\:\mathrm{hours} \\ $$$$\mathrm{a}\:\mathrm{day},\:\mathrm{how}\:\mathrm{many}\:\mathrm{men}\:\mathrm{must}\:\mathrm{associated} \\ $$$$\mathrm{with}\:\mathrm{12}\:\mathrm{women}\:\mathrm{and}\:\mathrm{6}\:\mathrm{boys}\:\mathrm{to}\:\mathrm{do}\:\mathrm{another} \\ $$$$\mathrm{piece}\:\mathrm{of}\:\mathrm{work}\:\mathrm{2}\frac{\mathrm{1}}{\mathrm{4}}\:\mathrm{times}\:\mathrm{as}\:\mathrm{great}\:\mathrm{in}\:\mathrm{30} \\ $$$$\mathrm{days}\:\mathrm{working}\:\mathrm{6}\:\mathrm{hrs}\:\mathrm{a}\:\mathrm{day}? \\ $$

Question Number 17096    Answers: 1   Comments: 0

The total number of solutions of the equation tan 3x − tan 2x − tan 3x tan 2x = 1 in [0, 2π] is

$$\mathrm{The}\:\mathrm{total}\:\mathrm{number}\:\mathrm{of}\:\mathrm{solutions}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equation} \\ $$$$\mathrm{tan}\:\mathrm{3}{x}\:−\:\mathrm{tan}\:\mathrm{2}{x}\:−\:\mathrm{tan}\:\mathrm{3}{x}\:\mathrm{tan}\:\mathrm{2}{x}\:=\:\mathrm{1}\:\mathrm{in} \\ $$$$\left[\mathrm{0},\:\mathrm{2}\pi\right]\:\mathrm{is} \\ $$

Question Number 16980    Answers: 0   Comments: 1

To Q16066: I have posted my solution there. Those who are intetested in this interesting question please have a critical view at it. Maybe there are alternative solutions which are easier and more direct and straight on.

$$\mathrm{To}\:\mathrm{Q16066}: \\ $$$$\mathrm{I}\:\mathrm{have}\:\mathrm{posted}\:\mathrm{my}\:\mathrm{solution}\:\mathrm{there}. \\ $$$$\mathrm{Those}\:\mathrm{who}\:\mathrm{are}\:\mathrm{intetested}\:\mathrm{in}\:\mathrm{this}\:\mathrm{interesting} \\ $$$$\mathrm{question}\:\mathrm{please}\:\mathrm{have}\:\mathrm{a}\:\mathrm{critical}\:\mathrm{view}\:\mathrm{at} \\ $$$$\mathrm{it}.\:\mathrm{Maybe}\:\mathrm{there}\:\mathrm{are}\:\mathrm{alternative}\:\mathrm{solutions} \\ $$$$\mathrm{which}\:\mathrm{are}\:\mathrm{easier}\:\mathrm{and}\:\mathrm{more}\:\mathrm{direct}\:\mathrm{and} \\ $$$$\mathrm{straight}\:\mathrm{on}. \\ $$

Question Number 16973    Answers: 1   Comments: 0

5^(log(x)) = x^(log(2)) , find x.

$$\mathrm{5}^{\mathrm{log}\left(\mathrm{x}\right)} \:=\:\mathrm{x}^{\mathrm{log}\left(\mathrm{2}\right)} ,\:\:\:\:\:\mathrm{find}\:\:\mathrm{x}. \\ $$

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