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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?

Question Number 8234 Answers: 0 Comments: 4

Question : figure x for (√(x−4)) > 6−x my answer : (1) x−4 > (6−x)^2 (x−5)(x−8) < 0 5<x<8 (2) x−4 ≥ 0 x ≥ 4 so I have for x ⇒ 5<x<8 what′s wrong with this answer, please help me because if x=9 ⇒ (√(9−4)) > 6−9 , it′s true

$$\mathrm{Question}\::\:\mathrm{figure}\:\mathrm{x}\:\mathrm{for} \\ $$$$\sqrt{\mathrm{x}−\mathrm{4}}\:>\:\mathrm{6}−\mathrm{x} \\ $$$$\mathrm{my}\:\mathrm{answer}\:: \\ $$$$\left(\mathrm{1}\right)\:\:\:\mathrm{x}−\mathrm{4}\:>\:\left(\mathrm{6}−\mathrm{x}\right)^{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\left(\mathrm{x}−\mathrm{5}\right)\left(\mathrm{x}−\mathrm{8}\right)\:<\:\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{5}<\mathrm{x}<\mathrm{8} \\ $$$$ \\ $$$$\left(\mathrm{2}\right)\:\:\:\mathrm{x}−\mathrm{4}\:\geqslant\:\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{x}\:\geqslant\:\mathrm{4} \\ $$$$\mathrm{so}\:\mathrm{I}\:\mathrm{have}\:\mathrm{for}\:\mathrm{x}\:\Rightarrow\:\mathrm{5}<\mathrm{x}<\mathrm{8} \\ $$$$\mathrm{what}'\mathrm{s}\:\mathrm{wrong}\:\mathrm{with}\:\mathrm{this}\:\mathrm{answer},\:\mathrm{please}\:\mathrm{help}\:\mathrm{me} \\ $$$$\mathrm{because}\:\mathrm{if}\:\mathrm{x}=\mathrm{9}\:\Rightarrow\:\sqrt{\mathrm{9}−\mathrm{4}}\:>\:\mathrm{6}−\mathrm{9}\:,\:\mathrm{it}'\mathrm{s}\:\mathrm{true} \\ $$

Question Number 8232 Answers: 1 Comments: 0

Every day, for n days, you put either $1, $2, or $3 into a saving account. It is random as to how much you save each day. What is the average amount you will have saved in n days?

$$\mathrm{Every}\:\mathrm{day},\:\mathrm{for}\:{n}\:\mathrm{days},\:\mathrm{you}\:\mathrm{put}\:\mathrm{either} \\ $$$$\$\mathrm{1},\:\$\mathrm{2},\:\mathrm{or}\:\$\mathrm{3}\:\mathrm{into}\:\mathrm{a}\:\mathrm{saving}\:\mathrm{account}. \\ $$$$\mathrm{It}\:\mathrm{is}\:\mathrm{random}\:\mathrm{as}\:\mathrm{to}\:\mathrm{how}\:\mathrm{much}\:\mathrm{you}\:\mathrm{save} \\ $$$$\mathrm{each}\:\mathrm{day}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:{average}\:\mathrm{amount} \\ $$$$\mathrm{you}\:\mathrm{will}\:\mathrm{have}\:\mathrm{saved}\:\mathrm{in}\:{n}\:\mathrm{days}? \\ $$

Question Number 8230 Answers: 1 Comments: 0

If ((x+y)/(x+y+z)) = ((y+z)/(x+y+z)) = ((x+z)/(x+y+z)) =p, then which of the following can be the value of p?

$$\mathrm{If}\:\frac{{x}+{y}}{{x}+{y}+{z}}\:=\:\frac{{y}+{z}}{{x}+{y}+{z}}\:=\:\frac{{x}+{z}}{{x}+{y}+{z}}\:={p},\:\mathrm{then}\:\mathrm{which}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{following}\:\mathrm{can}\:\mathrm{be}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{p}? \\ $$

Question Number 8224 Answers: 1 Comments: 3

if ∅ lies between −(π/4) and (π/4) then prove that ∅^2 =tan^2 ∅ −(1+(1/3))((tan^4 ∅)/2) +(1+(1/3)+(1/5))((tan^6 ∅)/3) +−−−−− −−−to ∞ terms

$${if}\:\:\varnothing\:{lies}\:{between}\:\:\:−\frac{\pi}{\mathrm{4}}\:{and}\:\:\frac{\pi}{\mathrm{4}}\:\:\:{then}\:\:{prove}\:{that} \\ $$$$\varnothing^{\mathrm{2}} =\mathrm{tan}\:^{\mathrm{2}} \varnothing\:−\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{3}}\right)\frac{\mathrm{tan}\:^{\mathrm{4}} \varnothing}{\mathrm{2}}\:+\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{5}}\right)\frac{\mathrm{tan}\:^{\mathrm{6}} \varnothing}{\mathrm{3}}\:+−−−−− \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:−−−{to}\:\infty\:{terms} \\ $$$$ \\ $$

Question Number 8217 Answers: 1 Comments: 5

what is the coefficient of x^3 in the expansion of (1 + x + x^2 + x^3 + x^4 + x^5 )^6

$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:\mathrm{x}^{\mathrm{3}} \:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion} \\ $$$$\mathrm{of}\:\left(\mathrm{1}\:+\:\mathrm{x}\:+\:\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{x}^{\mathrm{3}} \:+\:\mathrm{x}^{\mathrm{4}} \:+\:\mathrm{x}^{\mathrm{5}} \right)^{\mathrm{6}} \\ $$

Question Number 8176 Answers: 1 Comments: 0

Prove that cos2θ=((1−tan^2 θ)/(1+tan^2 θ)).Hence deduce that tan22(1/2)=(√2)−1.

$${Prove}\:{that}\:{cos}\mathrm{2}\theta=\frac{\mathrm{1}−{tan}^{\mathrm{2}} \theta}{\mathrm{1}+{tan}^{\mathrm{2}} \theta}.{Hence}\:{deduce}\:{that}\: \\ $$$${tan}\mathrm{22}\frac{\mathrm{1}}{\mathrm{2}}=\sqrt{\mathrm{2}}−\mathrm{1}. \\ $$$$ \\ $$$$ \\ $$

Question Number 8175 Answers: 0 Comments: 0

Prove (4^n /n)<^(2n) C_n for all n≥4 and n∈Z^+

$$\mathrm{Prove} \\ $$$$\frac{\mathrm{4}^{{n}} }{{n}}<\:^{\mathrm{2}{n}} {C}_{{n}} \:\mathrm{for}\:\mathrm{all}\:{n}\geqslant\mathrm{4}\:\mathrm{and}\:{n}\in\mathbb{Z}^{+} \\ $$

Question Number 8174 Answers: 0 Comments: 0

Prove that there are infinite prime numbers of the form 10^n +1

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{there}\:\mathrm{are}\:\mathrm{infinite}\:\mathrm{prime} \\ $$$$\mathrm{numbers}\:\mathrm{of}\:\mathrm{the}\:\mathrm{form}\:\mathrm{10}^{{n}} +\mathrm{1} \\ $$

Question Number 8168 Answers: 0 Comments: 3

Find the coefficient of in the expansion of (1+x)(1+x^2 )(1+x^3 )...(1+x^n ).

$${Find}\:{the}\:{coefficient}\:{of}\:{in}\:{the}\:{expansion}\:{of} \\ $$$$\left(\mathrm{1}+{x}\right)\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{x}^{\mathrm{3}} \right)...\left(\mathrm{1}+{x}^{{n}} \right). \\ $$

Question Number 8164 Answers: 1 Comments: 2

if p is prima show that (√p) irasional

$${if}\:\boldsymbol{{p}}\:{is}\:{prima}\:{show}\:{that}\:\sqrt{\boldsymbol{{p}}}\:{irasional} \\ $$

Question Number 8156 Answers: 1 Comments: 1

The value of the sum of the series 3^n C_0 − 8^n C_1 +13^n C_2 −18^n C_3 +...+(−1)^n (3+5n)^n C_n ]

$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{series} \\ $$$$\left.\mathrm{3}\:^{{n}} {C}_{\mathrm{0}} −\:\mathrm{8}\:^{{n}} {C}_{\mathrm{1}} +\mathrm{13}\:^{{n}} {C}_{\mathrm{2}} −\mathrm{18}\:^{{n}} {C}_{\mathrm{3}} +...+\left(−\mathrm{1}\right)^{{n}} \:\left(\mathrm{3}+\mathrm{5}{n}\right)\:^{{n}} {C}_{{n}} \right] \\ $$

Question Number 8155 Answers: 1 Comments: 2

If f(x)=x^n , then the value of f(1) + ((f^1 (1))/1) + ((f^2 (1))/(2!)) + ... + ((f^( n) (1))/(n!)) , where f^( r) (x) denotes the rth order derivative of f(x) with respect to x , is

$$\mathrm{If}\:{f}\left({x}\right)={x}^{{n}} ,\:\mathrm{then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$${f}\left(\mathrm{1}\right)\:+\:\frac{{f}^{\mathrm{1}} \left(\mathrm{1}\right)}{\mathrm{1}}\:+\:\frac{{f}^{\mathrm{2}} \left(\mathrm{1}\right)}{\mathrm{2}!}\:+\:...\:+\:\frac{{f}^{\:{n}} \left(\mathrm{1}\right)}{{n}!}\:,\:\mathrm{where} \\ $$$${f}^{\:{r}} \left({x}\right)\:\mathrm{denotes}\:\mathrm{the}\:{r}\mathrm{th}\:\mathrm{order}\:\mathrm{derivative}\:\mathrm{of} \\ $$$${f}\left({x}\right)\:\mathrm{with}\:\mathrm{respect}\:\mathrm{to}\:{x}\:,\:\mathrm{is} \\ $$

Question Number 8148 Answers: 2 Comments: 2

Determine x^4 − (1/x^4 ) , if x^2 + (1/x^2 )=34 .

$$\mathrm{Determine}\:\mathrm{x}^{\mathrm{4}} −\:\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{4}} }\:,\:\mathrm{if}\:\mathrm{x}^{\mathrm{2}} +\:\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} }=\mathrm{34}\:. \\ $$

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