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Question Number 202302 Answers: 2 Comments: 0

If ((3cosx + 2sinx)/(cosx − sinx)) = 4 find ctgx = ?

$$\mathrm{If}\:\:\:\frac{\mathrm{3cosx}\:+\:\mathrm{2sinx}}{\mathrm{cosx}\:−\:\mathrm{sinx}}\:=\:\mathrm{4}\:\:\:\mathrm{find}\:\:\:\mathrm{ctgx}\:=\:? \\ $$

Question Number 202300 Answers: 1 Comments: 0

Parvin leaves home to go to school. After walking 3/8 of the way, he remembers that he forgot his book. He goes back home, picks up his book, and arrives at school at the time he was originally supposed to. How many times did Parvin increase his initial speed to reach school on time ?

$$ \\ $$Parvin leaves home to go to school. After walking 3/8 of the way, he remembers that he forgot his book. He goes back home, picks up his book, and arrives at school at the time he was originally supposed to. How many times did Parvin increase his initial speed to reach school on time ?

Question Number 202298 Answers: 1 Comments: 0

x ∈ R Find: max((5/(x^2 − 6x + 11))) = ?

$$\mathrm{x}\:\in\:\mathbb{R} \\ $$$$\mathrm{Find}:\:\:\:\boldsymbol{\mathrm{max}}\left(\frac{\mathrm{5}}{\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{6x}\:+\:\mathrm{11}}\right)\:=\:? \\ $$

Question Number 202297 Answers: 2 Comments: 0

If a^2 + a = 5 Find ((a^(−b) + a^(1−b) + a^(2−b) )/a^(−b) ) = ?

$$\mathrm{If}\:\:\:\mathrm{a}^{\mathrm{2}} \:+\:\mathrm{a}\:=\:\mathrm{5} \\ $$$$\mathrm{Find}\:\:\:\frac{\mathrm{a}^{−\boldsymbol{\mathrm{b}}} \:+\:\mathrm{a}^{\mathrm{1}−\boldsymbol{\mathrm{b}}} \:+\:\mathrm{a}^{\mathrm{2}−\boldsymbol{\mathrm{b}}} }{\mathrm{a}^{−\boldsymbol{\mathrm{b}}} }\:=\:? \\ $$

Question Number 202296 Answers: 1 Comments: 0

Find: log_2 (log_3 (log_4 64)) = ?

$$\mathrm{Find}: \\ $$$$\mathrm{log}_{\mathrm{2}} \:\left(\mathrm{log}_{\mathrm{3}} \:\left(\mathrm{log}_{\mathrm{4}} \:\mathrm{64}\right)\right)\:=\:? \\ $$

Question Number 202295 Answers: 1 Comments: 0

psinθcon^2 θ=a pcosθsin^2 θ=b p≠0 θ∈(0 ,(π/2)) prove p=(((a^2 +b^2 )^(3/2) )/(ab))

$${psin}\theta{con}^{\mathrm{2}} \theta={a} \\ $$$${pcos}\theta{sin}^{\mathrm{2}} \theta={b} \\ $$$${p}\neq\mathrm{0}\:\:\:\:\theta\in\left(\mathrm{0}\:,\frac{\pi}{\mathrm{2}}\right) \\ $$$${prove}\:{p}=\frac{\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} \right)^{\frac{\mathrm{3}}{\mathrm{2}}} }{{ab}} \\ $$

Question Number 202290 Answers: 2 Comments: 0

If ((3a − b)/(x + y)) = ((3b − c)/(y + z)) = ((3c − a)/(z + x)) then show that ((a + b + c)/(x + y + z)) = ((a^(2 ) + b^2 + c^2 )/(ax + by + cz)) .

$$\mathrm{If}\:\frac{\mathrm{3}{a}\:−\:{b}}{{x}\:+\:{y}}\:=\:\frac{\mathrm{3}{b}\:−\:{c}}{{y}\:+\:{z}}\:=\:\frac{\mathrm{3}{c}\:−\:{a}}{{z}\:+\:{x}}\:\mathrm{then}\:\mathrm{show} \\ $$$$\mathrm{that}\:\frac{{a}\:+\:{b}\:+\:{c}}{{x}\:+\:{y}\:+\:{z}}\:=\:\frac{{a}^{\mathrm{2}\:} \:+\:{b}^{\mathrm{2}} \:+\:{c}^{\mathrm{2}} }{{ax}\:+\:{by}\:+\:{cz}}\:. \\ $$

Question Number 202287 Answers: 2 Comments: 1

Question Number 202276 Answers: 2 Comments: 0

Question Number 203668 Answers: 1 Comments: 0

Question Number 203665 Answers: 0 Comments: 0

Question Number 202258 Answers: 3 Comments: 0

If (x/a) = (y/b) then show that ((x^3 + 3xy^2 )/(a^3 + 3ab^2 )) = (( y^3 + 3x^2 y)/(b^3 + 3a^2 b)) .

$$\mathrm{If}\:\frac{{x}}{{a}}\:=\:\frac{{y}}{{b}}\:\mathrm{then}\:\mathrm{show}\:\mathrm{that}\: \\ $$$$\frac{{x}^{\mathrm{3}} \:+\:\mathrm{3}{xy}^{\mathrm{2}} }{{a}^{\mathrm{3}} \:+\:\mathrm{3}{ab}^{\mathrm{2}} }\:=\:\frac{\:{y}^{\mathrm{3}} \:+\:\mathrm{3}{x}^{\mathrm{2}} {y}}{{b}^{\mathrm{3}} \:+\:\mathrm{3}{a}^{\mathrm{2}} {b}}\:. \\ $$

Question Number 202257 Answers: 0 Comments: 2

Question Number 202255 Answers: 2 Comments: 0

calcul f_n ′(x) f_n (x)=n^α x(1−x)^(n )

$${calcul}\:{f}_{{n}} '\left({x}\right) \\ $$$${f}_{{n}} \left({x}\right)={n}^{\alpha} {x}\left(\mathrm{1}−{x}\right)^{{n}\:} \: \\ $$

Question Number 202251 Answers: 1 Comments: 4

(1/(1×2×3)) + (1/(2×3×4)) + (1/(3×4×5)) + .............. + (1/(n(n+1)(n+2))) = ?

$$\frac{\mathrm{1}}{\mathrm{1}×\mathrm{2}×\mathrm{3}}\:+\:\frac{\mathrm{1}}{\mathrm{2}×\mathrm{3}×\mathrm{4}}\:+\:\frac{\mathrm{1}}{\mathrm{3}×\mathrm{4}×\mathrm{5}}\:+\:..............\:+\:\frac{\mathrm{1}}{\mathrm{n}\left(\mathrm{n}+\mathrm{1}\right)\left(\mathrm{n}+\mathrm{2}\right)}\:=\:?\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$

Question Number 202249 Answers: 0 Comments: 0

C = lim_(x→0) ((x ((cos x))^(1/3) − sin x)/x^5 ) =?

$$\:\:\:\:\mathrm{C}\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{x}\:\sqrt[{\mathrm{3}}]{\mathrm{cos}\:\mathrm{x}}\:−\:\mathrm{sin}\:\mathrm{x}}{\mathrm{x}^{\mathrm{5}} }\:=? \\ $$

Question Number 202250 Answers: 2 Comments: 0

(a^2 − bc)x^2 + 2(b^2 − ca)x + (c^2 − ab) = 0 has two equal roots. Show that either b = 0 or (a^2 /(bc)) + (b^2 /(ca)) + (c^2 /(ab)) = 3.

$$\left({a}^{\mathrm{2}} \:−\:{bc}\right){x}^{\mathrm{2}} \:+\:\mathrm{2}\left({b}^{\mathrm{2}} \:−\:{ca}\right){x}\:+\:\left({c}^{\mathrm{2}} \:−\:{ab}\right)\:=\:\mathrm{0} \\ $$$$\mathrm{has}\:\mathrm{two}\:\mathrm{equal}\:\mathrm{roots}.\:\mathrm{Show}\:\mathrm{that}\:\mathrm{either}\: \\ $$$${b}\:=\:\mathrm{0}\:\mathrm{or}\:\frac{{a}^{\mathrm{2}} }{{bc}}\:+\:\frac{{b}^{\mathrm{2}} }{{ca}}\:+\:\frac{{c}^{\mathrm{2}} }{{ab}}\:=\:\mathrm{3}. \\ $$

Question Number 202247 Answers: 0 Comments: 0

α>1 calcul f_n ^′ (x) f_n (x)=n^α x(1−x)^n

$$\alpha>\mathrm{1} \\ $$$${calcul}\:\:\:{f}_{{n}} ^{'} \left({x}\right) \\ $$$${f}_{{n}} \left({x}\right)={n}^{\alpha} {x}\left(\mathrm{1}−{x}\right)^{{n}} \\ $$

Question Number 202224 Answers: 2 Comments: 0

Question Number 202212 Answers: 1 Comments: 1

Question Number 202208 Answers: 0 Comments: 15

Question Number 202205 Answers: 0 Comments: 4

Question Number 202203 Answers: 1 Comments: 0

2^(2023) = abc.............^(______________) a+b+c = ?

$$\:\: \\ $$$$ \\ $$$$\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}^{\mathrm{2023}} =\:\overset{\_\_\_\_\_\_\_\_\_\_\_\_\_\_} {\boldsymbol{\mathrm{abc}}.............} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{b}}+\boldsymbol{\mathrm{c}}\:=\:? \\ $$$$ \\ $$$$ \\ $$$$\: \\ $$$$ \\ $$

Question Number 202198 Answers: 2 Comments: 0

If α, β are the roots of x^2 + ax − b = 0 and γ, δ are the roots of x^2 + ax + c = 0 then show that ((α − γ)/(β − γ)) = ((β − δ)/(α − δ)) .

$$\mathrm{If}\:\alpha,\:\beta\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:{x}^{\mathrm{2}} \:+\:{ax}\:−\:{b}\:=\:\mathrm{0}\: \\ $$$$\mathrm{and}\:\gamma,\:\delta\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:{x}^{\mathrm{2}} \:+\:{ax}\:+\:{c}\:=\:\mathrm{0}\: \\ $$$$\mathrm{then}\:\mathrm{show}\:\mathrm{that}\:\frac{\alpha\:−\:\gamma}{\beta\:−\:\gamma}\:=\:\frac{\beta\:−\:\delta}{\alpha\:−\:\delta}\:\:. \\ $$

Question Number 202193 Answers: 2 Comments: 0

If α, β are the roots of x^2 + ax + b = 0 and α + δ, β + δ are the roots of x^2 + px + q = 0 then show that a^2 − p^2 = 4(b − q).

$$\mathrm{If}\:\alpha,\:\beta\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:{x}^{\mathrm{2}} \:+\:{ax}\:+\:{b}\:=\:\:\mathrm{0}\: \\ $$$$\mathrm{and}\:\alpha\:+\:\delta,\:\beta\:+\:\delta\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\: \\ $$$${x}^{\mathrm{2}} \:+\:{px}\:+\:{q}\:=\:\mathrm{0}\:\mathrm{then}\:\mathrm{show}\:\mathrm{that}\: \\ $$$${a}^{\mathrm{2}} \:−\:{p}^{\mathrm{2}} \:=\:\mathrm{4}\left({b}\:−\:{q}\right). \\ $$

Question Number 202187 Answers: 2 Comments: 4

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