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Question Number 11035 Answers: 1 Comments: 0

A= [(1,(−1)),((−4),(−2)) ] A^(2016) =.....?

$${A}=\begin{bmatrix}{\mathrm{1}}&{−\mathrm{1}}\\{−\mathrm{4}}&{−\mathrm{2}}\end{bmatrix} \\ $$$${A}^{\mathrm{2016}} =.....? \\ $$

Question Number 11034 Answers: 0 Comments: 1

u= [(1,(−1),2) ] A= [(3,2),(1,3),(0,1) ] v= [(2,(−1)) ] uAv^t =....?

$${u}=\begin{bmatrix}{\mathrm{1}}&{−\mathrm{1}}&{\mathrm{2}}\end{bmatrix} \\ $$$${A}=\begin{bmatrix}{\mathrm{3}}&{\mathrm{2}}\\{\mathrm{1}}&{\mathrm{3}}\\{\mathrm{0}}&{\mathrm{1}}\end{bmatrix} \\ $$$${v}=\begin{bmatrix}{\mathrm{2}}&{−\mathrm{1}}\end{bmatrix} \\ $$$${uAv}^{{t}} =....? \\ $$

Question Number 11214 Answers: 1 Comments: 0

Question Number 11212 Answers: 1 Comments: 0

Question Number 11217 Answers: 0 Comments: 0

∃e_i :i∈N e_i is a basis vector A∈C^n A=Σ_(i∈N) e_i A_i = ((A_1 ),(A_2 ),(⋮),(A_n ) ) = ∣A⟩ ⟨B∣=(∣B^∗ ⟩)^T ∣A⟩⟨B∣=???

$$\exists{e}_{{i}} :{i}\in\mathbb{N}\:\:\:\:\:\:\:\:{e}_{{i}} \:\mathrm{is}\:\mathrm{a}\:\mathrm{basis}\:\mathrm{vector} \\ $$$$\boldsymbol{{A}}\in\mathbb{C}^{{n}} \\ $$$$\boldsymbol{{A}}=\underset{{i}\in\mathbb{N}} {\sum}{e}_{{i}} {A}_{{i}} =\begin{pmatrix}{{A}_{\mathrm{1}} }\\{{A}_{\mathrm{2}} }\\{\vdots}\\{{A}_{{n}} }\end{pmatrix}\:\:\:=\:\mid{A}\rangle \\ $$$$\langle{B}\mid=\left(\mid{B}^{\ast} \rangle\right)^{\mathrm{T}} \\ $$$$\mid{A}\rangle\langle{B}\mid=??? \\ $$

Question Number 11215 Answers: 0 Comments: 0

Question Number 11024 Answers: 1 Comments: 0

P(x+1)×P(x−1)=4x^2 +8x+a−5 a=?

$${P}\left({x}+\mathrm{1}\right)×{P}\left({x}−\mathrm{1}\right)=\mathrm{4}{x}^{\mathrm{2}} +\mathrm{8}{x}+{a}−\mathrm{5} \\ $$$${a}=? \\ $$

Question Number 11023 Answers: 1 Comments: 0

(x^3 +6)×P(x)+6x=ax^3 +2ax+b+3 ⇒b=?

$$\left({x}^{\mathrm{3}} +\mathrm{6}\right)×{P}\left({x}\right)+\mathrm{6}{x}={ax}^{\mathrm{3}} +\mathrm{2}{ax}+{b}+\mathrm{3} \\ $$$$\Rightarrow{b}=? \\ $$

Question Number 11019 Answers: 2 Comments: 0

Question Number 11020 Answers: 2 Comments: 0

Question Number 11013 Answers: 0 Comments: 0

Euler vs. Newton

$$\mathrm{Euler}\:\mathrm{vs}.\:\mathrm{Newton} \\ $$

Question Number 11011 Answers: 2 Comments: 0

if tan(xy)=x then (dy/dx)=

$${if}\:{tan}\left({xy}\right)={x}\:{then}\:\frac{{dy}}{{dx}}= \\ $$

Question Number 11009 Answers: 1 Comments: 0

If x^3 = y^3 Is it always true that x = y ?

$$\mathrm{If}\:\:{x}^{\mathrm{3}} \:=\:{y}^{\mathrm{3}} \\ $$$$\mathrm{Is}\:\mathrm{it}\:\mathrm{always}\:\mathrm{true}\:\mathrm{that}\:{x}\:=\:{y}\:? \\ $$

Question Number 11005 Answers: 1 Comments: 0

prove that (9/4) < (log _2 3)^2 < ((25)/9) .

$${prove}\:{that}\:\frac{\mathrm{9}}{\mathrm{4}}\:<\:\left(\mathrm{log}\:_{\mathrm{2}} \mathrm{3}\right)^{\mathrm{2}} \:<\:\frac{\mathrm{25}}{\mathrm{9}}\:\:. \\ $$

Question Number 10995 Answers: 1 Comments: 1

Prove that 3>(log_2 3)^2 >2.

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{3}>\left(\mathrm{log}_{\mathrm{2}} \mathrm{3}\right)^{\mathrm{2}} >\mathrm{2}. \\ $$

Question Number 10994 Answers: 0 Comments: 3

Γ(2407) = (2406)! = ∫_0 ^∞ e^(−x) x^(2406) dx How evaluate ∫e^(−x) x^(2406) dx ???

$$\Gamma\left(\mathrm{2407}\right)\:=\:\left(\mathrm{2406}\right)!\:=\:\int_{\mathrm{0}} ^{\infty} {e}^{−{x}} {x}^{\mathrm{2406}} \:\mathrm{dx} \\ $$$$\mathrm{How}\:\mathrm{evaluate}\:\int{e}^{−{x}} {x}^{\mathrm{2406}} \:\mathrm{dx}\:??? \\ $$

Question Number 10989 Answers: 1 Comments: 0

find the image of the point(5,2) under a rotation of 90° clockwise

$${find}\:{the}\:{image}\:{of}\:{the}\:{point}\left(\mathrm{5},\mathrm{2}\right)\: \\ $$$${under}\:{a}\:{rotation}\:{of}\:\mathrm{90}°\:{clockwise} \\ $$

Question Number 10987 Answers: 1 Comments: 0

If • f(2x + 1) + g(3 − x) = x • f(((3x + 5)/( x + 1))) + 2g(((2x + 1)/(x + 1))) = (x/(x + 1)) for every x ∈ R, x ≠ −1 Find f(x) !!

$$\mathrm{If}\: \\ $$$$\bullet\:{f}\left(\mathrm{2}{x}\:+\:\mathrm{1}\right)\:+\:{g}\left(\mathrm{3}\:−\:{x}\right)\:=\:{x} \\ $$$$\bullet\:{f}\left(\frac{\mathrm{3}{x}\:+\:\mathrm{5}}{\:{x}\:+\:\mathrm{1}}\right)\:+\:\mathrm{2}{g}\left(\frac{\mathrm{2}{x}\:+\:\mathrm{1}}{{x}\:+\:\mathrm{1}}\right)\:=\:\frac{{x}}{{x}\:+\:\mathrm{1}} \\ $$$$\mathrm{for}\:\mathrm{every}\:{x}\:\in\:\mathbb{R},\:\:{x}\:\neq\:−\mathrm{1} \\ $$$$\mathrm{Find}\:{f}\left({x}\right)\:!! \\ $$

Question Number 10981 Answers: 1 Comments: 1

Question Number 10970 Answers: 1 Comments: 0

x^x^x^⋰^2 = 2 What is the value of x ?

$${x}^{{x}^{{x}^{\iddots^{\mathrm{2}} } } } \:=\:\mathrm{2} \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{x}\:? \\ $$

Question Number 10973 Answers: 3 Comments: 1

∣∣x∣+2x∣≤3, interval x=...? A.−3≤x≤3 B. x≥0 C. x≤0 D. −1≤x≤1 E. x≤2

$$\mid\mid{x}\mid+\mathrm{2x}\mid\leqslant\mathrm{3},\:\mathrm{interval}\:\mathrm{x}=...? \\ $$$$\mathrm{A}.−\mathrm{3}\leqslant{x}\leqslant\mathrm{3} \\ $$$$\mathrm{B}.\:{x}\geqslant\mathrm{0} \\ $$$$\mathrm{C}.\:{x}\leqslant\mathrm{0} \\ $$$$\mathrm{D}.\:−\mathrm{1}\leqslant{x}\leqslant\mathrm{1} \\ $$$$\mathrm{E}.\:{x}\leqslant\mathrm{2} \\ $$

Question Number 10956 Answers: 0 Comments: 4

y = ((x − 2)/(2(x − 1)^(3/2) )) Let p = x − 1 ⇒ y = ((p − 1)/(2p^(3/2) )) Is it true that (dy/dx) = (dy/dp) ?

$${y}\:=\:\frac{{x}\:−\:\mathrm{2}}{\mathrm{2}\left({x}\:−\:\mathrm{1}\right)^{\mathrm{3}/\mathrm{2}} } \\ $$$$\mathrm{Let}\:\:{p}\:=\:{x}\:−\:\mathrm{1} \\ $$$$\Rightarrow\:{y}\:=\:\frac{{p}\:−\:\mathrm{1}}{\mathrm{2}{p}^{\mathrm{3}/\mathrm{2}} } \\ $$$$ \\ $$$$\mathrm{Is}\:\mathrm{it}\:\mathrm{true}\:\mathrm{that}\:\:\:\frac{{dy}}{{dx}}\:\:\:=\:\:\frac{{dy}}{{dp}}\:\:? \\ $$

Question Number 10955 Answers: 2 Comments: 0

If ((cos θ)/(1 − sin θ)) = a a ≠ (π/2) + 2kπ So, tan (θ/2) = ... (A) (a/(a + 1)) (D) ((a + 1)/(a − 1)) (B) (1/(a + 1)) (E) (a/(a − 1)) (C) ((a − 1)/(a + 1))

$$\mathrm{If}\:\:\frac{\mathrm{cos}\:\theta}{\mathrm{1}\:−\:\mathrm{sin}\:\theta}\:=\:{a}\:\:\:\:\:\:\:\:\:\:\:{a}\:\neq\:\frac{\pi}{\mathrm{2}}\:+\:\mathrm{2}{k}\pi \\ $$$$\mathrm{So},\:\:\mathrm{tan}\:\frac{\theta}{\mathrm{2}}\:=\:... \\ $$$$\left(\mathrm{A}\right)\:\:\frac{{a}}{{a}\:+\:\mathrm{1}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{D}\right)\:\:\frac{{a}\:+\:\mathrm{1}}{{a}\:−\:\mathrm{1}} \\ $$$$\left(\mathrm{B}\right)\:\:\frac{\mathrm{1}}{{a}\:+\:\mathrm{1}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{E}\right)\:\:\frac{{a}}{{a}\:−\:\mathrm{1}} \\ $$$$\left(\mathrm{C}\right)\:\:\frac{{a}\:−\:\mathrm{1}}{{a}\:+\:\mathrm{1}} \\ $$

Question Number 10948 Answers: 1 Comments: 0

If cos^(−1) (x/a)+cos^(−1) (y/b)=α prove (x^2 /a^2 )−((2xy)/(ab))cos α+(y^2 /b^2 )=sin^2 α

$$\mathrm{If}\:\mathrm{cos}^{−\mathrm{1}} \frac{{x}}{{a}}+\mathrm{cos}^{−\mathrm{1}} \frac{{y}}{{b}}=\alpha \\ $$$${prove}\: \\ $$$$\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }−\frac{\mathrm{2}{xy}}{{ab}}\mathrm{cos}\:\alpha+\frac{{y}^{\mathrm{2}} }{{b}^{\mathrm{2}} }=\mathrm{sin}^{\mathrm{2}} \alpha \\ $$

Question Number 10947 Answers: 0 Comments: 0

In a triangle ABC prove the following (((a+b+c)^2 )/(a^2 +b^2 +c^2 )) = ((cot (A/2)+cot (B/2)+cot (C/2))/(cot A+cot B+cot C))

$$\mathrm{In}\:\mathrm{a}\:\mathrm{triangle}\:\mathrm{ABC}\:\mathrm{prove}\:\mathrm{the}\:\mathrm{following} \\ $$$$\frac{\left({a}+{b}+{c}\right)^{\mathrm{2}} }{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} }\:=\:\frac{\mathrm{cot}\:\frac{{A}}{\mathrm{2}}+\mathrm{cot}\:\frac{{B}}{\mathrm{2}}+\mathrm{cot}\:\frac{{C}}{\mathrm{2}}}{\mathrm{cot}\:{A}+\mathrm{cot}\:{B}+\mathrm{cot}\:{C}} \\ $$

Question Number 10944 Answers: 1 Comments: 2

Find all ordered pairs (a,b) so that ((ab)/(a+b)) is an integer. (a and b are integers).

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{ordered}\:\mathrm{pairs}\:\left(\mathrm{a},\mathrm{b}\right)\:\mathrm{so}\:\mathrm{that}\:\frac{\mathrm{ab}}{\mathrm{a}+\mathrm{b}}\:\mathrm{is}\:\mathrm{an}\:\mathrm{integer}. \\ $$$$\left(\mathrm{a}\:\mathrm{and}\:\mathrm{b}\:\mathrm{are}\:\mathrm{integers}\right). \\ $$

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