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Question Number 8310    Answers: 1   Comments: 0

Given that cosecA+cotA=3 evaluate cosecA−cotA and cosA.

$${Given}\:{that}\:{cosecA}+{cotA}=\mathrm{3}\:{evaluate} \\ $$$${cosecA}−{cotA}\:{and}\:{cosA}. \\ $$$$ \\ $$

Question Number 8314    Answers: 2   Comments: 9

Question Number 8325    Answers: 0   Comments: 0

In Δ ABC,∠A=90° , AD⊥BC, DE⊥AC, AF⊥FG,GH⊥FC. (a)How many triangles are there? (b)If AB=((16)/9) , ∠B=60°,find the length of GH.

$${In}\:\Delta\:{ABC},\angle{A}=\mathrm{90}°\:,\:{AD}\bot{BC},\:{DE}\bot{AC}, \\ $$$${AF}\bot{FG},{GH}\bot{FC}. \\ $$$$\left({a}\right){How}\:{many}\:{triangles}\:{are}\:{there}? \\ $$$$\left({b}\right){If}\:{AB}=\frac{\mathrm{16}}{\mathrm{9}}\:,\:\angle{B}=\mathrm{60}°,{find}\:{the}\:{length} \\ $$$$\:\:\:\:\:\:{of}\:{GH}. \\ $$

Question Number 8305    Answers: 0   Comments: 3

Question Number 8296    Answers: 0   Comments: 2

Give the integral representation of 2333!

$$\mathrm{Give}\:\mathrm{the}\:\mathrm{integral}\:\mathrm{representation}\:\mathrm{of}\:\:\mathrm{2333}! \\ $$

Question Number 8289    Answers: 1   Comments: 2

what is value sin 36° plese give me answer

$${what}\:{is}\:{value} \\ $$$${sin}\:\mathrm{36}° \\ $$$${plese}\:{give}\:{me}\:{answer} \\ $$

Question Number 8287    Answers: 1   Comments: 0

Show that tan(α+β)=((tanα+tanβ)/(1−tanαtanβ)).

$${Show}\:{that}\:{tan}\left(\alpha+\beta\right)=\frac{{tan}\alpha+{tan}\beta}{\mathrm{1}−{tan}\alpha{tan}\beta}. \\ $$

Question Number 8297    Answers: 1   Comments: 0

B_ y expessing each side of the equation in terms of tanA ,or otherwise show that ((sin2A+cos2A+1)/(sin2A+cos2A−1))=((tan(45°+A))/(tanA))

$$\underset{} {{B}y}\:{expessing}\:{each}\:{side}\:{of}\:{the} \\ $$$${equation}\:{in}\:{terms}\:{of}\:{tanA}\:,{or}\: \\ $$$${otherwise}\:{show}\:{that} \\ $$$$\frac{{sin}\mathrm{2}{A}+{cos}\mathrm{2}{A}+\mathrm{1}}{{sin}\mathrm{2}{A}+{cos}\mathrm{2}{A}−\mathrm{1}}=\frac{{tan}\left(\mathrm{45}°+{A}\right)}{{tanA}} \\ $$

Question Number 8306    Answers: 0   Comments: 1

If 270°<x<360°, simplify (√(2+(√(2+2cosx)))).

$${If}\:\mathrm{270}°<{x}<\mathrm{360}°,\:{simplify} \\ $$$$\sqrt{\mathrm{2}+\sqrt{\mathrm{2}+\mathrm{2}{cosx}}}. \\ $$

Question Number 8302    Answers: 0   Comments: 0

find all possible values of x and y satisfying 1! + 2! + 3! + ... + x! = y^2

$$\mathrm{find}\:\mathrm{all}\:\mathrm{possible}\:\mathrm{values}\:\mathrm{of}\:\mathrm{x}\:\mathrm{and}\:\mathrm{y}\:\mathrm{satisfying}\: \\ $$$$\mathrm{1}!\:+\:\mathrm{2}!\:+\:\mathrm{3}!\:+\:...\:+\:\mathrm{x}!\:=\:\mathrm{y}^{\mathrm{2}} \\ $$

Question Number 8300    Answers: 0   Comments: 2

Question Number 8301    Answers: 1   Comments: 1

Is { (ω+i)^0 , (ω+i)^1 , (ω+i)^2 , ...., (ω+i)^n } cyclic for any value of n? Determine the smallest such n if it exists. ω is a complex cuberoot of unity and i=(√(−1))

$$\mathrm{Is}\:\:\left\{\:\left(\omega+\mathrm{i}\right)^{\mathrm{0}} ,\:\left(\omega+\mathrm{i}\right)^{\mathrm{1}} ,\:\left(\omega+\mathrm{i}\right)^{\mathrm{2}} ,\:....,\:\left(\omega+\mathrm{i}\right)^{\mathrm{n}} \:\right\} \\ $$$$\mathrm{cyclic}\:\mathrm{for}\:\mathrm{any}\:\mathrm{value}\:\mathrm{of}\:\mathrm{n}? \\ $$$$\mathrm{Determine}\:\mathrm{the}\:\mathrm{smallest}\:\mathrm{such}\:\mathrm{n}\:\mathrm{if}\:\mathrm{it}\:\mathrm{exists}. \\ $$$$\omega\:\mathrm{is}\:\mathrm{a}\:\mathrm{complex}\:\mathrm{cuberoot}\:\mathrm{of}\:\mathrm{unity}\:\mathrm{and} \\ $$$$\mathrm{i}=\sqrt{−\mathrm{1}} \\ $$

Question Number 8282    Answers: 1   Comments: 3

Find x, y in R { ((x^2 + y^2 = 1)),((x^8 + y^8 = x^(10) + y^(10) )) :}

$$\mathrm{Find}\:\mathrm{x},\:\mathrm{y}\:\mathrm{in}\:\mathbb{R} \\ $$$$\begin{cases}{\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{2}} \:=\:\mathrm{1}}\\{\mathrm{x}^{\mathrm{8}} \:+\:\mathrm{y}^{\mathrm{8}} \:=\:\mathrm{x}^{\mathrm{10}} \:+\:\mathrm{y}^{\mathrm{10}} }\end{cases} \\ $$

Question Number 8281    Answers: 1   Comments: 0

∫((6 sinx cosx)/(sinx + cosx)) dx

$$\int\frac{\mathrm{6}\:\mathrm{sinx}\:\mathrm{cosx}}{\mathrm{sinx}\:+\:\mathrm{cosx}}\:\mathrm{dx} \\ $$

Question Number 8277    Answers: 0   Comments: 0

Show that one representation for π≈3.14... is π=12cos^(−1) [((3/4))^(1/4) (1+Σ_(r=1) ^∞ ((Π_(k=1) ^(2r) ((3/2)−k))/((2r)!))(((−1)/3))^r )].

$$\mathrm{Show}\:\mathrm{that}\:\mathrm{one}\:\mathrm{representation}\:\mathrm{for}\:\pi\approx\mathrm{3}.\mathrm{14}... \\ $$$$\mathrm{is}\:\pi=\mathrm{12cos}^{−\mathrm{1}} \left[\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{1}/\mathrm{4}} \left(\mathrm{1}+\underset{\mathrm{r}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{2r}} {\prod}}\left(\frac{\mathrm{3}}{\mathrm{2}}−\mathrm{k}\right)}{\left(\mathrm{2r}\right)!}\left(\frac{−\mathrm{1}}{\mathrm{3}}\right)^{\mathrm{r}} \right)\right]. \\ $$$$ \\ $$

Question Number 8275    Answers: 0   Comments: 2

Show that the followings (i)sin(a+b)=sina cosb +cosa sinb (ii)cos(a−b)=cosa cosb +sina sinb

$${Show}\:{that}\:{the}\:{followings} \\ $$$$\left({i}\right){sin}\left({a}+{b}\right)={sina}\:{cosb}\:+{cosa}\:{sinb} \\ $$$$\left({ii}\right){cos}\left({a}−{b}\right)={cosa}\:{cosb}\:+{sina}\:{sinb} \\ $$$$ \\ $$

Question Number 8273    Answers: 1   Comments: 0

Express sinα+(√3)cosα in the form Rsin(α+β) where R>0 and 0°<β<90°. Hence solve the equation sinα+(√3)cosα=2 for 0°<α<270°.

$${Express}\:{sin}\alpha+\sqrt{\mathrm{3}}{cos}\alpha\:{in}\:{the}\:{form}\: \\ $$$${Rsin}\left(\alpha+\beta\right)\:{where}\:{R}>\mathrm{0}\:{and}\:\mathrm{0}°<\beta<\mathrm{90}°. \\ $$$${Hence}\:{solve}\:{the}\:{equation}\:{sin}\alpha+\sqrt{\mathrm{3}}{cos}\alpha=\mathrm{2} \\ $$$${for}\:\mathrm{0}°<\alpha<\mathrm{270}°. \\ $$

Question Number 8269    Answers: 1   Comments: 1

Question Number 8267    Answers: 1   Comments: 0

Show that sinA+sinB=2sin((A+B)/2) cos((A−B)/2).

$${Show}\:{that}\:{sinA}+{sinB}=\mathrm{2}{sin}\frac{{A}+{B}}{\mathrm{2}}\:{cos}\frac{{A}−{B}}{\mathrm{2}}. \\ $$$$ \\ $$

Question Number 8262    Answers: 0   Comments: 1

∣x−1∣ < 2 ⇒ ∣x−3∣

$$\mid{x}−\mathrm{1}\mid\:<\:\mathrm{2}\:\Rightarrow\:\mid{x}−\mathrm{3}\mid \\ $$

Question Number 8259    Answers: 1   Comments: 0

Question Number 8257    Answers: 1   Comments: 0

If A+B+C=90° ,show that tanA tanB+tanB tanC+tanC tanA=1.

$${If}\:{A}+{B}+{C}=\mathrm{90}°\:,{show}\:{that}\: \\ $$$${tanA}\:{tanB}+{tanB}\:{tanC}+{tanC}\:{tanA}=\mathrm{1}. \\ $$

Question Number 8252    Answers: 1   Comments: 0

Question Number 8244    Answers: 1   Comments: 0

Show that the curve y=ln(((5−7x)/(8+x))) has no stationary point for all real values of x.

$${Show}\:{that}\:{the}\:{curve}\:{y}={ln}\left(\frac{\mathrm{5}−\mathrm{7}{x}}{\mathrm{8}+{x}}\right)\:{has} \\ $$$${no}\:{stationary}\:{point}\:{for}\:{all}\:{real}\:{values} \\ $$$${of}\:{x}. \\ $$

Question Number 8243    Answers: 1   Comments: 0

Find the equation of the perpendicular bisector of the line joining the points (−5,4) to the point (9,−3)

$${Find}\:{the}\:{equation}\:{of}\:{the}\:{perpendicular}\:{bisector}\:{of}\:{the}\:{line}\:{joining}\:{the}\:{points}\:\left(−\mathrm{5},\mathrm{4}\right)\:{to}\:{the}\:{point}\:\left(\mathrm{9},−\mathrm{3}\right) \\ $$$$ \\ $$

Question Number 8236    Answers: 1   Comments: 2

Define a 3×3 matrix whose entries are the first 9 positive integers. Let s_k be the sum of the elements across the kth row. Is there such a matrix where s_1 : s_2 : s_3 = 1 : 2 : 3 ? −−−−−−−−−−−−−−−−−−−− What about n×n matrices whose elements are the first n^2 positive integers? Is there a matrix such that s_1 : s_2 : s_3 : s_4 :.....: s_n = 1 : 2 : 3 :...: n?

$$\mathrm{Define}\:\mathrm{a}\:\mathrm{3}×\mathrm{3}\:\mathrm{matrix}\:\mathrm{whose}\:\mathrm{entries} \\ $$$$\mathrm{are}\:\mathrm{the}\:\mathrm{first}\:\mathrm{9}\:\mathrm{positive}\:\mathrm{integers}. \\ $$$$\mathrm{Let}\:\mathrm{s}_{\mathrm{k}} \:\mathrm{be}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{elements} \\ $$$$\mathrm{across}\:\mathrm{the}\:\mathrm{kth}\:\mathrm{row}.\:\mathrm{Is}\:\mathrm{there}\:\mathrm{such}\:\mathrm{a}\: \\ $$$$\mathrm{matrix}\:\mathrm{where}\:\mathrm{s}_{\mathrm{1}} \::\:\mathrm{s}_{\mathrm{2}} \::\:\mathrm{s}_{\mathrm{3}} \:=\:\mathrm{1}\::\:\mathrm{2}\::\:\mathrm{3}\:? \\ $$$$−−−−−−−−−−−−−−−−−−−− \\ $$$$\mathrm{What}\:\mathrm{about}\:\mathrm{n}×\mathrm{n}\:\mathrm{matrices}\:\mathrm{whose} \\ $$$$\mathrm{elements}\:\mathrm{are}\:\mathrm{the}\:\mathrm{first}\:\mathrm{n}^{\mathrm{2}} \:\mathrm{positive} \\ $$$$\mathrm{integers}?\:\mathrm{Is}\:\mathrm{there}\:\mathrm{a}\:\mathrm{matrix}\:\mathrm{such} \\ $$$$\mathrm{that}\:\mathrm{s}_{\mathrm{1}} \::\:\mathrm{s}_{\mathrm{2}} \::\:\mathrm{s}_{\mathrm{3}} \::\:\mathrm{s}_{\mathrm{4}} \::.....:\:\mathrm{s}_{\mathrm{n}} =\:\mathrm{1}\::\:\mathrm{2}\::\:\mathrm{3}\::...:\:\mathrm{n}? \\ $$$$ \\ $$

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