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AlgebraQuestion and Answers: Page 78

Question Number 188776 Answers: 3 Comments: 1

Prove that n^2 +3n+2 is divisible by 2 for any n∈Z

$$\mathrm{Prove}\:\mathrm{that}\:{n}^{\mathrm{2}} +\mathrm{3}{n}+\mathrm{2}\:\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{2} \\ $$$$\mathrm{for}\:\mathrm{any}\:{n}\in\mathbb{Z} \\ $$

Question Number 188774 Answers: 1 Comments: 0

Question Number 188753 Answers: 1 Comments: 0

x^2 − y^2 = 2023 x, y ∈ N How many pair of (x, y)

$$\:{x}^{\mathrm{2}} \:−\:{y}^{\mathrm{2}} \:=\:\mathrm{2023}\:\:\:\:\:\:\:\:\:{x},\:{y}\:\in\:\mathrm{N} \\ $$$$\:\mathrm{H}{ow}\:{many}\:{pair}\:{of}\:\left({x},\:{y}\right) \\ $$

Question Number 188704 Answers: 1 Comments: 0

Question Number 188660 Answers: 2 Comments: 0

solve x^4 +4x=1

$${solve}\:{x}^{\mathrm{4}} +\mathrm{4}{x}=\mathrm{1} \\ $$

Question Number 188653 Answers: 0 Comments: 1

In how many different ways can the letters of the word ABRAKADABRA be arranged?

$$\mathrm{In}\:\mathrm{how}\:\mathrm{many}\:\mathrm{different}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{the} \\ $$$$\mathrm{letters}\:\mathrm{of}\:\mathrm{the}\:\mathrm{word}\:\mathrm{ABRAKADABRA} \\ $$$$\mathrm{be}\:\mathrm{arranged}? \\ $$

Question Number 188648 Answers: 1 Comments: 0

Question Number 188647 Answers: 0 Comments: 1

Question Number 188646 Answers: 2 Comments: 0

Question Number 188619 Answers: 0 Comments: 1

Question Number 188587 Answers: 0 Comments: 0

Question Number 188572 Answers: 0 Comments: 3

Question Number 188515 Answers: 1 Comments: 0

in AB^Δ C : a=3 , b=6 , c=7 find the value of : E = (a+b )cos(C) + (b+c)cos(A)+ (a+c )cos(B)=?

$$ \\ $$$$\:\:\:\:\:{in}\:\:{A}\overset{\Delta} {{B}C}\:\::\:\:\:{a}=\mathrm{3}\:\:,\:\:{b}=\mathrm{6}\:\:,\:\:{c}=\mathrm{7} \\ $$$$\:\:\: \\ $$$$\: \\ $$$$\:\:\:\:{find}\:\:{the}\:{value}\:\:{of}\:: \\ $$$$\:\:\: \\ $$$$\:\:\:\:\:\:\:{E}\:=\:\left({a}+{b}\:\right){cos}\left({C}\right)\:+\:\left({b}+{c}\right){cos}\left({A}\right)+\:\left({a}+{c}\:\right){cos}\left({B}\right)=?\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\: \\ $$

Question Number 188508 Answers: 2 Comments: 0

Question Number 188482 Answers: 1 Comments: 0

512x^(1−x^(−3) ) =−1 find volue of Σ_(n=1) ^∞ (x^2 )^n =?

$$\mathrm{512}{x}^{\mathrm{1}−{x}^{−\mathrm{3}} } =−\mathrm{1} \\ $$$${find}\:\:{volue}\:\:{of}\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left({x}^{\mathrm{2}} \right)^{{n}} =? \\ $$

Question Number 188430 Answers: 0 Comments: 0

2⌊ x ⌋ − ⌊ −x ⌋ =4 −−−− if x∈Z ⇒ 2x +x = 4 ⇒ x=(4/3) ,impossible if x∉ Z ⇒^(⌊−x⌋=−⌊x⌋−1) 2⌊x⌋+⌊x⌋=3 ⇒ ⌊ x ⌋= 1 ⇒ 1≤ x < 2 ⇒^(x≠1) x∈ (1 , 2) ✓

$$ \\ $$$$\:\:\:\:\mathrm{2}\lfloor\:{x}\:\rfloor\:−\:\lfloor\:−{x}\:\rfloor\:=\mathrm{4} \\ $$$$\:\:\:−−−− \\ $$$$\:\:{if}\:\:{x}\in\mathbb{Z}\:\Rightarrow\:\:\mathrm{2}{x}\:+{x}\:=\:\mathrm{4}\:\Rightarrow\:{x}=\frac{\mathrm{4}}{\mathrm{3}}\:\:,{impossible} \\ $$$$\:\:{if}\:{x}\notin\:\mathbb{Z}\:\overset{\lfloor−{x}\rfloor=−\lfloor{x}\rfloor−\mathrm{1}} {\Rightarrow}\mathrm{2}\lfloor{x}\rfloor+\lfloor{x}\rfloor=\mathrm{3} \\ $$$$\:\:\:\:\:\Rightarrow\:\lfloor\:{x}\:\rfloor=\:\mathrm{1}\:\Rightarrow\:\:\mathrm{1}\leqslant\:{x}\:<\:\mathrm{2}\:\:\:\:\overset{{x}\neq\mathrm{1}} {\Rightarrow}\:{x}\in\:\left(\mathrm{1}\:,\:\mathrm{2}\right)\:\:\:\checkmark \\ $$$$ \\ $$

Question Number 188407 Answers: 3 Comments: 0

xf(x) = f(x + 2) f(2) = 2 f(8) = ?

$${xf}\left({x}\right)\:=\:{f}\left({x}\:+\:\mathrm{2}\right) \\ $$$${f}\left(\mathrm{2}\right)\:=\:\mathrm{2} \\ $$$${f}\left(\mathrm{8}\right)\:=\:?\: \\ $$

Question Number 188379 Answers: 0 Comments: 0

Question Number 189462 Answers: 0 Comments: 0

Question Number 188366 Answers: 1 Comments: 0

xlnx=7 x?

$${xlnx}=\mathrm{7}\:\:\:\:\: \\ $$$${x}? \\ $$

Question Number 188536 Answers: 2 Comments: 0

solve the equation; {: ((x + y − z =5)),((z − yx = 7)),((z = 1 + x)) } x;y;z =??

$$ \\ $$$$\:\:\:\:\boldsymbol{{solve}}\:\boldsymbol{{the}}\:\boldsymbol{{equation}}; \\ $$$$\:\:\:\:\:\:\:\left.\begin{matrix}{\boldsymbol{{x}}\:+\:\boldsymbol{{y}}\:−\:\boldsymbol{{z}}\:=\mathrm{5}}\\{\boldsymbol{{z}}\:−\:\boldsymbol{{yx}}\:=\:\mathrm{7}}\\{\boldsymbol{{z}}\:=\:\mathrm{1}\:+\:\boldsymbol{{x}}}\end{matrix}\right\}\:\:\boldsymbol{{x}};\boldsymbol{{y}};\boldsymbol{{z}}\:=??\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 188535 Answers: 1 Comments: 0

Question Number 188343 Answers: 1 Comments: 0

Find: Ω = Σ_(n=1) ^∞ (lim_(x→0) (1 - ((5 - (√(25 - x^2 )))/x) )^((5n)/x) )

$$\mathrm{Find}:\:\Omega\:=\:\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\:\left(\underset{\boldsymbol{\mathrm{x}}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\mathrm{1}\:-\:\frac{\mathrm{5}\:-\:\sqrt{\mathrm{25}\:-\:\mathrm{x}^{\mathrm{2}} }}{\mathrm{x}}\:\right)^{\frac{\mathrm{5}\boldsymbol{\mathrm{n}}}{\boldsymbol{\mathrm{x}}}} \:\right) \\ $$

Question Number 188286 Answers: 1 Comments: 0

when sin(x)+cos(x)=a find sec(x)+csc(x)=?

$${when}\:\:\:\:\:\:{sin}\left({x}\right)+{cos}\left({x}\right)={a} \\ $$$${find}\:\:\:\:\:\:\:\:\:{sec}\left({x}\right)+{csc}\left({x}\right)=? \\ $$

Question Number 188280 Answers: 0 Comments: 2

Prove that 1+2+3+4+..... = −(1/(12))

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{1}+\mathrm{2}+\mathrm{3}+\mathrm{4}+.....\:=\:−\frac{\mathrm{1}}{\mathrm{12}} \\ $$

Question Number 188270 Answers: 1 Comments: 1

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