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AllQuestion and Answers: Page 278

Question Number 193314 Answers: 1 Comments: 0

a ,b, c > 0 & a^2 +b^2 +c^2 =3 prove that (((1+(3/(ab+bc+ca)) )^((a+b+c)^2 ) ))^(1/3) ≤(1+(a/b))(1+(b/c))(1+(c/a))

$$ \\ $$$$\boldsymbol{{a}}\:,\boldsymbol{{b}},\:\boldsymbol{{c}}\:\:>\:\mathrm{0}\:\&\:\:\boldsymbol{{a}}^{\mathrm{2}} +\boldsymbol{{b}}^{\mathrm{2}} +\boldsymbol{{c}}^{\mathrm{2}} =\mathrm{3}\:\boldsymbol{{prove}}\:\boldsymbol{{that}}\: \\ $$$$\sqrt[{\mathrm{3}}]{\left(\mathrm{1}+\frac{\mathrm{3}}{\boldsymbol{{ab}}+\boldsymbol{{bc}}+\boldsymbol{{ca}}}\:\right)^{\left(\boldsymbol{{a}}+\boldsymbol{{b}}+\boldsymbol{{c}}\right)^{\mathrm{2}} } \:}\leqslant\left(\mathrm{1}+\frac{\boldsymbol{{a}}}{\boldsymbol{{b}}}\right)\left(\mathrm{1}+\frac{\boldsymbol{{b}}}{\boldsymbol{{c}}}\right)\left(\mathrm{1}+\frac{\boldsymbol{{c}}}{\boldsymbol{{a}}}\right) \\ $$

Question Number 193309 Answers: 0 Comments: 1

Question Number 193307 Answers: 0 Comments: 0

Question Number 193296 Answers: 2 Comments: 1

If a^2 + b^2 + c^2 = 16, x^2 + y^2 + z^2 = 25 and ax + by + cz = 20 then what is the value of ((a + b + c)/(x + y + z)) ?

$$\mathrm{If}\:{a}^{\mathrm{2}} \:+\:{b}^{\mathrm{2}} \:+\:{c}^{\mathrm{2}} \:=\:\mathrm{16},\:{x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:+\:{z}^{\mathrm{2}} \:=\:\mathrm{25} \\ $$$$\mathrm{and}\:{ax}\:+\:{by}\:+\:{cz}\:=\:\mathrm{20}\:\mathrm{then}\:\mathrm{what}\:\mathrm{is}\:\mathrm{the} \\ $$$$\mathrm{value}\:\mathrm{of}\:\:\frac{{a}\:+\:{b}\:+\:{c}}{{x}\:+\:{y}\:+\:{z}}\:? \\ $$

Question Number 193295 Answers: 2 Comments: 0

Find the value of x from the following equations: 4^((x/y) + (y/x)) = 32 log_3 (x − y) + log_3 (x + y) = 1

$$\boldsymbol{\mathrm{Find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{value}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{{x}}\:\boldsymbol{\mathrm{from}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{following}} \\ $$$$\boldsymbol{\mathrm{equations}}: \\ $$$$\mathrm{4}^{\frac{{x}}{{y}}\:+\:\frac{{y}}{{x}}} \:=\:\mathrm{32} \\ $$$$\mathrm{log}_{\mathrm{3}} \left({x}\:−\:{y}\right)\:+\:\mathrm{log}_{\mathrm{3}} \left({x}\:+\:{y}\right)\:=\:\mathrm{1} \\ $$

Question Number 193294 Answers: 0 Comments: 0

Question Number 193293 Answers: 1 Comments: 0

Question Number 193292 Answers: 1 Comments: 0

(a) A man P has 5 red, 3 blue and 2 white buses. Another man Q has 3 red, 2 blue and 4 white buses. A bus owned by P is involved in an accident with a bus belonging to Q. Calculate the probability that the two buses are not of the same color. (b) A man travels from Nigeria to Ghana by air and from Ghana to Liberia by ship. He returns by the same means. He has 6 airlines and 4 shipping lines to choose from. In how many ways can he make his journey without using the same airline or shipping line twice?

$$ \\ $$(a) A man P has 5 red, 3 blue and 2 white buses. Another man Q has 3 red, 2 blue and 4 white buses. A bus owned by P is involved in an accident with a bus belonging to Q. Calculate the probability that the two buses are not of the same color. (b) A man travels from Nigeria to Ghana by air and from Ghana to Liberia by ship. He returns by the same means. He has 6 airlines and 4 shipping lines to choose from. In how many ways can he make his journey without using the same airline or shipping line twice?

Question Number 193286 Answers: 1 Comments: 2

Let ′P′ is a prime number (P > 1000). If ′P′ devided by 1000, then remainder is ′r′. How many value of ′r′ ?

$$\mathrm{Let}\:'\mathrm{P}'\:\mathrm{is}\:\mathrm{a}\:\mathrm{prime}\:\mathrm{number}\:\left(\mathrm{P}\:>\:\mathrm{1000}\right). \\ $$$$\mathrm{If}\:\:'\mathrm{P}'\:\mathrm{devided}\:\mathrm{by}\:\mathrm{1000},\:\mathrm{then}\:\mathrm{remainder}\:\mathrm{is}\:'\mathrm{r}'. \\ $$$$\mathrm{How}\:\mathrm{many}\:\mathrm{value}\:\mathrm{of}\:\:'\mathrm{r}'\:? \\ $$

Question Number 193284 Answers: 1 Comments: 0

6y −2xy = 4 8z − yz = 9 10x − 4xz = 8 find x+y +z = ?

$$\:\:\:\mathrm{6}{y}\:−\mathrm{2}{xy}\:=\:\mathrm{4} \\ $$$$\:\:\:\:\mathrm{8}{z}\:−\:{yz}\:=\:\mathrm{9} \\ $$$$\:\:\:\mathrm{10}{x}\:−\:\mathrm{4}{xz}\:=\:\mathrm{8}\: \\ $$$${find}\:{x}+{y}\:+{z}\:=\:? \\ $$

Question Number 193280 Answers: 1 Comments: 0

Question Number 193278 Answers: 1 Comments: 0

Please Help...!! ∫^( ∞) _( 0) x.e^(−x) .sinx.dx

$${Please}\:{Help}...!! \\ $$$$\:\:\:\:\underset{\:\:\:\:\mathrm{0}} {\int}^{\:\:\infty} {x}.{e}^{−{x}} .{sinx}.{dx}\: \\ $$$$ \\ $$

Question Number 193272 Answers: 2 Comments: 0

Question Number 193268 Answers: 1 Comments: 0

Question Number 193267 Answers: 1 Comments: 0

Question Number 193266 Answers: 0 Comments: 0

Question Number 193262 Answers: 1 Comments: 0

lim_(x→+∞) (ln((x+(√(x^2 +1)))/(x+(√(x^2 −1)))).ln^2 ((x+1)/(x−1)))

$$\underset{{x}\rightarrow+\infty} {\mathrm{lim}}\left({ln}\frac{{x}+\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}}{{x}+\sqrt{{x}^{\mathrm{2}} −\mathrm{1}}}.{ln}^{\mathrm{2}} \:\frac{{x}+\mathrm{1}}{{x}−\mathrm{1}}\right) \\ $$$$ \\ $$

Question Number 193256 Answers: 1 Comments: 0

Question Number 193253 Answers: 1 Comments: 0

Select the correct option with explaination: If (1/3)log_3 M + 3log_3 N = 1 + log_(0.008) 5 then a. M^9 = (9/N) b. N^9 = (9/M) c. M^3 = (3/N) d. N^3 = (3/M)

$$\boldsymbol{\mathrm{Select}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{correct}}\:\boldsymbol{\mathrm{option}}\:\boldsymbol{\mathrm{with}}\: \\ $$$$\boldsymbol{\mathrm{explaination}}: \\ $$$$\mathrm{If}\:\frac{\mathrm{1}}{\mathrm{3}}\mathrm{log}_{\mathrm{3}} {M}\:+\:\mathrm{3log}_{\mathrm{3}} {N}\:=\:\mathrm{1}\:+\:\mathrm{log}_{\mathrm{0}.\mathrm{008}} \mathrm{5}\:\mathrm{then} \\ $$$$\mathrm{a}.\:{M}^{\mathrm{9}} \:=\:\frac{\mathrm{9}}{{N}} \\ $$$$\mathrm{b}.\:{N}^{\mathrm{9}} \:=\:\frac{\mathrm{9}}{{M}} \\ $$$$\mathrm{c}.\:{M}^{\mathrm{3}} \:=\:\frac{\mathrm{3}}{{N}} \\ $$$$\mathrm{d}.\:{N}^{\mathrm{3}} \:=\:\frac{\mathrm{3}}{{M}}\: \\ $$

Question Number 193250 Answers: 0 Comments: 0

Solve: y′(x)=y^2 (t)+t^2

$${Solve}: \\ $$$${y}'\left({x}\right)={y}^{\mathrm{2}} \left({t}\right)+{t}^{\mathrm{2}} \: \\ $$

Question Number 193248 Answers: 1 Comments: 0

L= lim_( x→0) (( sin(x )−arcsin(x))/(tan(x)− arctan(x)))=?

$$ \\ $$$$\:\mathrm{L}=\:\mathrm{lim}_{\:{x}\rightarrow\mathrm{0}} \:\frac{\:\mathrm{sin}\left({x}\:\right)−\mathrm{arcsin}\left({x}\right)}{\mathrm{tan}\left({x}\right)−\:\mathrm{arctan}\left({x}\right)}=?\:\:\:\:\:\:\: \\ $$$$ \\ $$$$\:\:\:\: \\ $$

Question Number 193247 Answers: 2 Comments: 0

if f(x) = ((ax+1)/(x+b)) find f^( 100) (x) and f^(101) (x) ?

$${if}\:{f}\left({x}\right)\:=\:\frac{{ax}+\mathrm{1}}{{x}+{b}}\:{find}\:{f}^{\:\mathrm{100}} \left({x}\right)\:{and}\:{f}^{\mathrm{101}} \left({x}\right)\:? \\ $$

Question Number 193245 Answers: 0 Comments: 0

Question Number 193239 Answers: 2 Comments: 0

Question Number 193238 Answers: 1 Comments: 0

s=a+b+c+d+..... number terms :n {a;b;c;d.....}>0 then E=s/(s−a)+s/(s−b)+s/s−c)+.... a) E>=n^2 b)E>=n^2 /(n−1) c) E>=n/(n+1) d) E>=n^2 /(n+1) e) E>=n^2 −1

$${s}={a}+{b}+{c}+{d}+..... \\ $$$${number}\:{terms}\::{n} \\ $$$$\left\{{a};{b};{c};{d}.....\right\}>\mathrm{0} \\ $$$$\left.{then}\:{E}={s}/\left({s}−{a}\right)+{s}/\left({s}−{b}\right)+{s}/{s}−{c}\right)+.... \\ $$$$\left.{a}\left.\right)\:{E}>={n}^{\mathrm{2}} \:\:\:\:\:\:\:\:\:{b}\right){E}>={n}^{\mathrm{2}} /\left({n}−\mathrm{1}\right) \\ $$$$\left.{c}\left.\right)\:{E}>={n}/\left({n}+\mathrm{1}\right)\:\:\:\:\:\:{d}\right)\:{E}>={n}^{\mathrm{2}} /\left({n}+\mathrm{1}\right) \\ $$$$\left.{e}\right)\:{E}>={n}^{\mathrm{2}} −\mathrm{1} \\ $$$$ \\ $$$$ \\ $$

Question Number 193237 Answers: 1 Comments: 0

Prove that: In any acute △ABC, cot^2 A+cot^2 B+cot^2 C≥1. Equality is possible if and only if A=B=C=(π/3).

$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\mathrm{In}\:\mathrm{any}\:\mathrm{acute}\:\bigtriangleup{ABC},\:\mathrm{cot}^{\mathrm{2}} {A}+\mathrm{cot}^{\mathrm{2}} {B}+\mathrm{cot}^{\mathrm{2}} {C}\geqslant\mathrm{1}. \\ $$$$\mathrm{Equality}\:\mathrm{is}\:\mathrm{possible}\:\mathrm{if}\:\mathrm{and}\:\mathrm{only}\:\mathrm{if}\:{A}={B}={C}=\frac{\pi}{\mathrm{3}}. \\ $$

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