Question and Answers Forum

AlgebraQuestion and Answers: Page 46

Question Number 205431 Answers: 0 Comments: 0

Prove that in any △ABC ((cotA cotB cotC)/(sinA sinB sinC)) ≤ (8/(27))

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{in}\:\mathrm{any}\:\:\bigtriangleup\mathrm{ABC} \\ $$$$\frac{\mathrm{cotA}\:\mathrm{cotB}\:\mathrm{cotC}}{\mathrm{sinA}\:\mathrm{sinB}\:\mathrm{sinC}}\:\leqslant\:\frac{\mathrm{8}}{\mathrm{27}} \\ $$

Question Number 205430 Answers: 0 Comments: 0

Prove that in any △ABC (1/(sinA)) + (1/(sinB)) + (1/(sinC)) ≤ (2/3) (cot(A/2) + cot(B/2) + cot(C/2))

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{in}\:\mathrm{any}\:\:\bigtriangleup\mathrm{ABC} \\ $$$$\frac{\mathrm{1}}{\mathrm{sinA}}\:+\:\frac{\mathrm{1}}{\mathrm{sinB}}\:+\:\frac{\mathrm{1}}{\mathrm{sinC}}\:\leqslant\:\frac{\mathrm{2}}{\mathrm{3}}\:\left(\mathrm{cot}\frac{\mathrm{A}}{\mathrm{2}}\:+\:\mathrm{cot}\frac{\mathrm{B}}{\mathrm{2}}\:+\:\mathrm{cot}\frac{\mathrm{C}}{\mathrm{2}}\right) \\ $$

Question Number 205423 Answers: 1 Comments: 0

If a , b ∈ R Then: a^2 + b^2 ≥ ab + (√((a^4 + b^4 )/2))

$$\mathrm{If} \\ $$$$\mathrm{a}\:,\:\mathrm{b}\:\in\:\mathbb{R} \\ $$$$\mathrm{Then}: \\ $$$$\mathrm{a}^{\mathrm{2}} \:+\:\mathrm{b}^{\mathrm{2}} \:\geqslant\:\mathrm{ab}\:+\:\sqrt{\frac{\mathrm{a}^{\mathrm{4}} \:+\:\mathrm{b}^{\mathrm{4}} }{\mathrm{2}}} \\ $$

Question Number 205422 Answers: 1 Comments: 0

If (a + 1)(b + 1)(c + 1) = 8 Then: a^2 + b^2 + c^2 ≥ 3

$$\mathrm{If} \\ $$$$\left(\mathrm{a}\:+\:\mathrm{1}\right)\left(\mathrm{b}\:+\:\mathrm{1}\right)\left(\mathrm{c}\:+\:\mathrm{1}\right)\:=\:\mathrm{8} \\ $$$$\mathrm{Then}: \\ $$$$\mathrm{a}^{\mathrm{2}} \:+\:\mathrm{b}^{\mathrm{2}} \:+\:\mathrm{c}^{\mathrm{2}} \:\geqslant\:\mathrm{3} \\ $$

Question Number 205420 Answers: 1 Comments: 0

If a^3 + b^3 + a^2 + b^2 = 4 Then: a^4 + b^4 ≥ 2

$$\mathrm{If} \\ $$$$\mathrm{a}^{\mathrm{3}} \:+\:\mathrm{b}^{\mathrm{3}} \:+\:\mathrm{a}^{\mathrm{2}} \:+\:\mathrm{b}^{\mathrm{2}} \:=\:\mathrm{4} \\ $$$$\mathrm{Then}: \\ $$$$\mathrm{a}^{\mathrm{4}} \:+\:\mathrm{b}^{\mathrm{4}} \:\geqslant\:\mathrm{2} \\ $$

Question Number 205421 Answers: 0 Comments: 0

If a,b,c>0 and abc=1 Then: (a/b^(2024) ) + (b/c^(2024) ) + (c/a^(2024) ) ≥ a + b + c

$$\mathrm{If} \\ $$$$\mathrm{a},\mathrm{b},\mathrm{c}>\mathrm{0}\:\:\:\mathrm{and}\:\:\:\mathrm{abc}=\mathrm{1} \\ $$$$\mathrm{Then}: \\ $$$$\frac{\mathrm{a}}{\mathrm{b}^{\mathrm{2024}} }\:+\:\frac{\mathrm{b}}{\mathrm{c}^{\mathrm{2024}} }\:+\:\frac{\mathrm{c}}{\mathrm{a}^{\mathrm{2024}} }\:\geqslant\:\mathrm{a}\:+\:\mathrm{b}\:+\:\mathrm{c} \\ $$

Question Number 205380 Answers: 3 Comments: 0

Question Number 205367 Answers: 1 Comments: 0

Question Number 205353 Answers: 2 Comments: 0

If ax^2 + bx + c = 0 had two roots p and q and p^2 + q^2 = p^3 + q^3 then show that b^3 − 2a^2 c + ab^2 = 3abc.

$$\mathrm{If}\:{ax}^{\mathrm{2}} \:+\:{bx}\:+\:{c}\:=\:\mathrm{0}\:\mathrm{had}\:\mathrm{two}\:\mathrm{roots}\:{p}\:\mathrm{and}\:{q} \\ $$$$\mathrm{and}\:{p}^{\mathrm{2}} \:+\:{q}^{\mathrm{2}} \:=\:{p}^{\mathrm{3}} \:+\:{q}^{\mathrm{3}} \:\mathrm{then}\:\mathrm{show}\:\mathrm{that} \\ $$$${b}^{\mathrm{3}} \:−\:\mathrm{2}{a}^{\mathrm{2}} {c}\:+\:{ab}^{\mathrm{2}} \:=\:\mathrm{3}{abc}. \\ $$

Question Number 205324 Answers: 3 Comments: 0

Compare: 37^(37) and 36^(38)

$$\mathrm{Compare}: \\ $$$$\mathrm{37}^{\mathrm{37}} \:\:\:\mathrm{and}\:\:\:\mathrm{36}^{\mathrm{38}} \\ $$

Question Number 205319 Answers: 0 Comments: 0

Question Number 205297 Answers: 2 Comments: 0

Find the′′ range ′′ of : i : f (x) =⌊ (( x)/( ⌊ x ⌋)) ⌋ ii: f(x) = (( x)/(⌊ x ⌋ + ⌊ −x ⌋))

$$ \\ $$$$\:\:\:{Find}\:\:{the}''\:{range}\:''\:{of}\:\:: \\ $$$$ \\ $$$$\:\:\:{i}\::\:\:\:{f}\:\left({x}\right)\:=\lfloor\:\frac{\:{x}}{\:\lfloor\:{x}\:\rfloor}\:\rfloor \\ $$$$\:\:\:{ii}:\:{f}\left({x}\right)\:=\:\frac{\:{x}}{\lfloor\:{x}\:\rfloor\:+\:\lfloor\:−{x}\:\rfloor} \\ $$$$\:\: \\ $$

Question Number 205273 Answers: 3 Comments: 0

If f(x−1)=2x+2 Find f(x)=?

$$\mathrm{If}\:\:\:\mathrm{f}\left(\mathrm{x}−\mathrm{1}\right)=\mathrm{2x}+\mathrm{2} \\ $$$$\mathrm{Find}\:\:\:\mathrm{f}\left(\mathrm{x}\right)=? \\ $$

Question Number 205256 Answers: 2 Comments: 2

Find: lim_(x→0) ((1/x)) = ?

$$\mathrm{Find}:\:\:\:\underset{\boldsymbol{\mathrm{x}}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{\mathrm{1}}{\mathrm{x}}\right)\:=\:? \\ $$

Question Number 207920 Answers: 2 Comments: 2

Given p,q ,r and s real positive numbers such that { ((p^2 +q^2 = r^2 +s^2 )),((p^2 +s^2 −ps = q^2 +r^2 +qr.)) :} Find ((pq+rs)/(ps+qr)) .

$$\:\:\:\:\mathrm{Given}\:\mathrm{p},\mathrm{q}\:,\mathrm{r}\:\mathrm{and}\:\mathrm{s}\:\mathrm{real}\:\mathrm{positive}\: \\ $$$$\:\:\mathrm{numbers}\:\mathrm{such}\:\mathrm{that}\: \\ $$$$\:\:\:\:\begin{cases}{\mathrm{p}^{\mathrm{2}} +\mathrm{q}^{\mathrm{2}} =\:\mathrm{r}^{\mathrm{2}} +\mathrm{s}^{\mathrm{2}} }\\{\mathrm{p}^{\mathrm{2}} +\mathrm{s}^{\mathrm{2}} −\mathrm{ps}\:=\:\mathrm{q}^{\mathrm{2}} +\mathrm{r}^{\mathrm{2}} +\mathrm{qr}.}\end{cases} \\ $$$$\:\:\mathrm{Find}\:\:\frac{\mathrm{pq}+\mathrm{rs}}{\mathrm{ps}+\mathrm{qr}}\:. \\ $$

Question Number 205238 Answers: 1 Comments: 1

4^x + x = 260 find the possible values of x

$$\mathrm{4}^{{x}} \:+\:{x}\:=\:\mathrm{260} \\ $$$${find}\:{the}\:{possible}\:{values}\:{of}\:{x} \\ $$$$ \\ $$

Question Number 205220 Answers: 2 Comments: 0

cos^4 x − sin^4 x = cos^2 x ⇒ x = ?

$$\mathrm{cos}^{\mathrm{4}} \:\mathrm{x}\:−\:\mathrm{sin}^{\mathrm{4}} \:\mathrm{x}\:=\:\mathrm{cos}^{\mathrm{2}} \:\mathrm{x} \\ $$$$\Rightarrow\:\mathrm{x}\:=\:? \\ $$

Question Number 205219 Answers: 2 Comments: 0

{ ((sinx + cosx = 1)),((sinx − cosy = 1)) :} ⇒ x = ?

$$\begin{cases}{\mathrm{sinx}\:+\:\mathrm{cosx}\:=\:\mathrm{1}}\\{\mathrm{sinx}\:−\:\mathrm{cosy}\:=\:\mathrm{1}}\end{cases} \\ $$$$\Rightarrow\:\mathrm{x}\:=\:? \\ $$

Question Number 205218 Answers: 1 Comments: 0

∣ sinx∣ + ∣ cosx ∣ = 1 ⇒ x = ?

$$\mid\:\mathrm{sinx}\mid\:+\:\mid\:\mathrm{cosx}\:\mid\:=\:\mathrm{1} \\ $$$$\Rightarrow\:\mathrm{x}\:=\:? \\ $$

Question Number 205217 Answers: 1 Comments: 0

4 sin (x/2) ∙ cos (x/2) = 1 ⇒ x = ?

$$\mathrm{4}\:\mathrm{sin}\:\frac{\mathrm{x}}{\mathrm{2}}\:\centerdot\:\mathrm{cos}\:\frac{\mathrm{x}}{\mathrm{2}}\:=\:\mathrm{1} \\ $$$$\Rightarrow\:\mathrm{x}\:=\:? \\ $$

Question Number 205211 Answers: 1 Comments: 0

Question Number 205203 Answers: 0 Comments: 0

If x,y,z>0 then in △ABC holds: Σ ((yz)/h_a ^2 ) ≤ (R^2 /(4F^2 )) (x + y + z)^2

$$\mathrm{If}\:\:\:\mathrm{x},\mathrm{y},\mathrm{z}>\mathrm{0}\:\:\:\mathrm{then}\:\mathrm{in}\:\:\:\bigtriangleup\mathrm{ABC}\:\:\:\mathrm{holds}: \\ $$$$\Sigma\:\:\frac{\mathrm{yz}}{\mathrm{h}_{\boldsymbol{\mathrm{a}}} ^{\mathrm{2}} }\:\:\leqslant\:\:\frac{\mathrm{R}^{\mathrm{2}} }{\mathrm{4F}^{\mathrm{2}} }\:\:\left(\mathrm{x}\:+\:\mathrm{y}\:+\:\mathrm{z}\right)^{\mathrm{2}} \\ $$

Question Number 205174 Answers: 2 Comments: 0

lim_(x→∞) {x^(1/x) } = ? where {.} is a fractional part of x

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left\{{x}^{\mathrm{1}/{x}} \right\}\:=\:?\:{where}\:\left\{.\right\}\:{is}\:{a}\:{fractional}\:{part}\:{of}\:{x} \\ $$

Question Number 205173 Answers: 1 Comments: 2

find S=Σ_(n=1) ^∞ (1/((a^2 +n^2 )^2 ))

$${find} \\ $$$${S}=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\left({a}^{\mathrm{2}} +{n}^{\mathrm{2}} \right)^{\mathrm{2}} } \\ $$

Question Number 205147 Answers: 2 Comments: 0

solve for z∈C zln z =z−2

$$\mathrm{solve}\:\mathrm{for}\:{z}\in\mathbb{C} \\ $$$${z}\mathrm{ln}\:{z}\:={z}−\mathrm{2} \\ $$

Question Number 205130 Answers: 1 Comments: 1

Pg 41 Pg 42 Pg 43 Pg 44 Pg 45 Pg 46 Pg 47 Pg 48 Pg 49 Pg 50

Contact: info@tinkutara.com