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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 = ¿

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

Question Number 202161    Answers: 2   Comments: 0

Question Number 202160    Answers: 1   Comments: 0

Question Number 202154    Answers: 3   Comments: 1

$$\:\:\:\downharpoonleft\underline{\:} \\ $$

Question Number 202153    Answers: 1   Comments: 0

Question Number 202146    Answers: 0   Comments: 1

There are three children in a family. Two of the children have blood group III, and one has blood group I. If the father of the children has blood type III, what is the probability that the mother will also have blood type III ?

$$ \\ $$There are three children in a family. Two of the children have blood group III, and one has blood group I. If the father of the children has blood type III, what is the probability that the mother will also have blood type III ?

Question Number 202143    Answers: 0   Comments: 5

find x in the interval 0°<x<360° if f(x)= 6sinx is −12

$$\:{find}\:{x}\:{in}\:{the}\:{interval}\:\:\mathrm{0}°<{x}<\mathrm{360}°\:{if} \\ $$$$\:\:{f}\left({x}\right)=\:\mathrm{6}{sinx}\:{is}\:−\mathrm{12} \\ $$

Question Number 202129    Answers: 1   Comments: 0

If p = 9999 then ((4p^3 − p)/((2p + 1)(6p − 3))) = ?

$$\mathrm{If}\:{p}\:=\:\mathrm{9999}\:\mathrm{then}\:\frac{\mathrm{4}{p}^{\mathrm{3}} \:−\:{p}}{\left(\mathrm{2}{p}\:+\:\mathrm{1}\right)\left(\mathrm{6}{p}\:−\:\mathrm{3}\right)}\:=\:? \\ $$

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