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AllQuestion and Answers: Page 170

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

Question Number 202184 Answers: 2 Comments: 0

with f(x)=x^2 +12x+30 and x∈R solve f(f(f(f(f(x)))))=0

$${with}\:{f}\left({x}\right)={x}^{\mathrm{2}} +\mathrm{12}{x}+\mathrm{30}\:{and}\:{x}\in{R} \\ $$$${solve}\:{f}\left({f}\left({f}\left({f}\left({f}\left({x}\right)\right)\right)\right)\right)=\mathrm{0} \\ $$

Question Number 202183 Answers: 1 Comments: 1

what is (√2) over 2

$$\boldsymbol{{what}}\:\boldsymbol{{is}}\:\sqrt{\mathrm{2}}\:\boldsymbol{{over}}\:\mathrm{2} \\ $$

Question Number 202562 Answers: 2 Comments: 0

Question Number 202172 Answers: 1 Comments: 0

If log_(12) 18 = A and log_(24) 54 = B then prove that AB + 5(A − B) = 1.

$$\mathrm{If}\:\mathrm{log}_{\mathrm{12}} \mathrm{18}\:=\:\mathrm{A}\:\mathrm{and}\:\mathrm{log}_{\mathrm{24}} \mathrm{54}\:=\:\mathrm{B}\:\mathrm{then}\:\mathrm{prove} \\ $$$$\mathrm{that}\:\mathrm{AB}\:+\:\mathrm{5}\left(\mathrm{A}\:−\:\mathrm{B}\right)\:=\:\mathrm{1}. \\ $$

Question Number 202170 Answers: 1 Comments: 0

Question Number 202167 Answers: 2 Comments: 0

∫^1 _0 ∫^1 _x sin(y^2 )dydx = ¿

$$\underset{\mathrm{0}} {\int}^{\mathrm{1}} \underset{{x}} {\int}^{\mathrm{1}} {sin}\left({y}^{\mathrm{2}} \right){dydx}\:=\:¿ \\ $$

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