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AllQuestion and Answers: Page 181

Question Number 204019    Answers: 1   Comments: 2

G is a group : prove that : (G/(Z (G ))) ≅ Inn(G ) Where , Inn(G)= {f ∣ f: G →^(f is an Automorphism) G}

$$ \\ $$$$\:\:\:\:\:{G}\:{is}\:{a}\:{group}\:: \\ $$$$\:\:\:\:\:{prove}\:{that}\::\:\:\frac{{G}}{{Z}\:\left({G}\:\right)}\:\cong\:{Inn}\left({G}\:\right) \\ $$$$\:\:\:\:{Where}\:,\:{Inn}\left({G}\right)=\:\left\{{f}\:\mid\:{f}:\:{G}\:\overset{{f}\:{is}\:{an}\:{Automorphism}} {\rightarrow}\:{G}\right\} \\ $$$$ \\ $$

Question Number 204018    Answers: 1   Comments: 0

Aut (Z )= ? where , Aut (G )= { f ∣ f :G →_(G is a group) ^(f is a isomorphism) G}

$$ \\ $$$$\:\:\:\:\:\:\:\:\:{Aut}\:\left(\mathbb{Z}\:\right)=\:? \\ $$$$\:\:\:\:\:\:\:{where}\:,\:{Aut}\:\left({G}\:\right)=\:\left\{\:{f}\:\mid\:{f}\::{G}\:\underset{{G}\:{is}\:{a}\:{group}} {\overset{{f}\:{is}\:{a}\:{isomorphism}} {\rightarrow}}\:{G}\right\} \\ $$

Question Number 204005    Answers: 2   Comments: 1

Find: cos44° − cos84° + ctg45° = ?

$$\mathrm{Find}: \\ $$$$\mathrm{cos44}°\:−\:\mathrm{cos84}°\:+\:\mathrm{ctg45}°\:=\:? \\ $$

Question Number 203995    Answers: 3   Comments: 0

3x+4x=5

$$\mathrm{3}{x}+\mathrm{4}{x}=\mathrm{5} \\ $$

Question Number 203994    Answers: 2   Comments: 0

d(((x!)/dx))=?

$${d}\left(\frac{{x}!}{{dx}}\right)=? \\ $$

Question Number 203988    Answers: 0   Comments: 0

Question Number 203985    Answers: 0   Comments: 8

Question Number 203980    Answers: 3   Comments: 0

lim_(x→2) ((ax^2 +bx+6)/(x^2 −x−2))=10 ; find a=? ∧ b=?

$$\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\frac{{ax}^{\mathrm{2}} +{bx}+\mathrm{6}}{{x}^{\mathrm{2}} −{x}−\mathrm{2}}=\mathrm{10}\:\:;\:\:\:{find}\:\:{a}=?\:\wedge\:{b}=? \\ $$

Question Number 203975    Answers: 0   Comments: 0

Let P∈C[X] p(x^2 +1)=p^2 (x)+1 p(0)=0 determin all polynom

$$\mathrm{Let}\:\mathrm{P}\in\mathbb{C}\left[\mathrm{X}\right] \\ $$$$\mathrm{p}\left(\mathrm{x}^{\mathrm{2}} +\mathrm{1}\right)=\mathrm{p}^{\mathrm{2}} \left(\mathrm{x}\right)+\mathrm{1} \\ $$$$\mathrm{p}\left(\mathrm{0}\right)=\mathrm{0} \\ $$$$\mathrm{determin}\:\mathrm{all}\:\mathrm{polynom}\: \\ $$

Question Number 203970    Answers: 0   Comments: 0

In a convex quadrilateral ABCD , if sinA = cosB, then find the range of S = 3sinA + 4sinB + 5(√2)sinC + 5(√2)sinD.

$$\mathrm{In}\:\mathrm{a}\:\mathrm{convex}\:\mathrm{quadrilateral}\:{ABCD}\:,\: \\ $$$$\mathrm{if}\:\mathrm{sin}{A}\:=\:\mathrm{cos}{B},\:\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of} \\ $$$${S}\:=\:\mathrm{3sin}{A}\:+\:\mathrm{4sin}{B}\:+\:\mathrm{5}\sqrt{\mathrm{2}}\mathrm{sin}{C}\:+\:\mathrm{5}\sqrt{\mathrm{2}}\mathrm{sin}{D}. \\ $$

Question Number 203969    Answers: 1   Comments: 0

Question Number 203964    Answers: 2   Comments: 0

Advanced calculus ... Q: If , ∫_0 ^1 ∫_0 ^( 1) (( xln(x)ln^2 (y ))/(1−xy)) dxdy = λ Σ_(n=1) ^∞ (( 1)/(n^( 3) ( n+1 )^2 )) ⇒ Find , λ =?

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{Advanced}\:\:{calculus}\:... \\ $$$$\:\:{Q}:\:\:\:{If}\:,\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{x}\mathrm{ln}\left({x}\right)\mathrm{ln}^{\mathrm{2}} \left({y}\:\right)}{\mathrm{1}−{xy}}\:{dxdy}\:=\:\lambda\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:\mathrm{1}}{{n}^{\:\mathrm{3}} \left(\:{n}+\mathrm{1}\:\right)^{\mathrm{2}} }\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Rightarrow\:\:{Find}\:,\:\:\:\:\:\lambda\:=? \\ $$$$ \\ $$

Question Number 203955    Answers: 0   Comments: 10

50%+50%=75%?

$$\mathrm{50\%}+\mathrm{50\%}=\mathrm{75\%}? \\ $$

Question Number 203941    Answers: 5   Comments: 0

If a^(2 ) −a+2=0 and x^2 =a^6 +2a^4 +a^2 then find x? IIT−JEE based question. Find sol^n .

$$\boldsymbol{{If}}\:\boldsymbol{{a}}^{\mathrm{2}\:} −\boldsymbol{{a}}+\mathrm{2}=\mathrm{0}\:\boldsymbol{{and}}\:\boldsymbol{{x}}^{\mathrm{2}} =\boldsymbol{{a}}^{\mathrm{6}} +\mathrm{2}\boldsymbol{{a}}^{\mathrm{4}} +\boldsymbol{{a}}^{\mathrm{2}} \:\boldsymbol{{then}}\:\boldsymbol{{find}}\:\boldsymbol{{x}}? \\ $$$$\boldsymbol{{IIT}}−\boldsymbol{{JEE}}\:\boldsymbol{{based}}\:\boldsymbol{{question}}.\:\boldsymbol{{Find}}\:\boldsymbol{{sol}}^{\boldsymbol{{n}}} . \\ $$

Question Number 203935    Answers: 2   Comments: 0

((cos(x/3)+sin (x/3))/(cos (x/3)−sin (x/3)))=?

$$\frac{{cos}\frac{{x}}{\mathrm{3}}+{sin}\:\frac{{x}}{\mathrm{3}}}{{cos}\:\frac{{x}}{\mathrm{3}}−{sin}\:\frac{{x}}{\mathrm{3}}}=? \\ $$

Question Number 203933    Answers: 2   Comments: 0

a^2 +b^2 +2c^2 =22 prove a+2b+c≤11

$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} +\mathrm{2}{c}^{\mathrm{2}} =\mathrm{22} \\ $$$${prove}\:{a}+\mathrm{2}{b}+{c}\leqslant\mathrm{11} \\ $$

Question Number 203919    Answers: 1   Comments: 0

. Z^5 = 1 ??

$$ . \\ $$$$\:\:\:\:\:\:\:\mathrm{Z}^{\mathrm{5}} \:=\:\mathrm{1}\:\:\:?? \\ $$

Question Number 203911    Answers: 2   Comments: 0

if 11^(800) divide by 9 what will be the remainder?

$${if}\:\mathrm{11}^{\mathrm{800}} \:{divide}\:{by}\:\mathrm{9}\:{what}\:{will}\:{be}\:{the}\: \\ $$$${remainder}? \\ $$

Question Number 203910    Answers: 2   Comments: 0

if y=(1−x)(2−x)(3−x)∙∙∙(n−x) y^′ =?

$${if}\:\:\:{y}=\left(\mathrm{1}−{x}\right)\left(\mathrm{2}−{x}\right)\left(\mathrm{3}−{x}\right)\centerdot\centerdot\centerdot\left({n}−{x}\right) \\ $$$${y}^{'} =? \\ $$

Question Number 203909    Answers: 2   Comments: 0

prove that 0^n =0 , n>0

$${prove}\:{that}\:\:\:\mathrm{0}^{{n}} =\mathrm{0}\:\:\:,\:\:\:\:{n}>\mathrm{0} \\ $$

Question Number 203905    Answers: 0   Comments: 8

let ABC a given triangle. Can we find three positions I,J,K on the side AB,AC,BC Such that IJK is equilateral?

$${let}\:{ABC}\:{a}\:{given}\:{triangle}.\:{Can}\:{we}\:{find} \\ $$$${three}\:{positions}\:{I},{J},{K}\:{on}\:{the}\:{side}\:{AB},{AC},{BC} \\ $$$${Such}\:{that}\:{IJK}\:{is}\:{equilateral}? \\ $$

Question Number 203921    Answers: 0   Comments: 3

prove that 0^0 =1

$${prove}\:{that}\:\mathrm{0}^{\mathrm{0}} =\mathrm{1} \\ $$

Question Number 203900    Answers: 0   Comments: 0

Let A ∈ R^(N×N) be a symmetric positive definite matrix and b ∈ R^N a vector. If x ∈ R^N , evaluate the integral Z(A,b) = ∫e^(−(1/2)x^T Ax + b^T x) dx as a function of A and b.

$${Let}\:{A}\:\in\:{R}^{{N}×{N}} \:{be}\:{a}\:{symmetric}\:{positive} \\ $$$${definite}\:{matrix}\:{and}\:{b}\:\in\:{R}^{{N}} \:{a}\:{vector}. \\ $$$${If}\:{x}\:\in\:{R}^{{N}} ,\:{evaluate}\:{the}\:{integral} \\ $$$${Z}\left({A},{b}\right)\:=\:\int{e}^{−\frac{\mathrm{1}}{\mathrm{2}}{x}^{{T}} {Ax}\:+\:{b}^{{T}} {x}} {dx}\:{as}\:{a}\:{function} \\ $$$${of}\:{A}\:{and}\:{b}. \\ $$

Question Number 203898    Answers: 0   Comments: 2

Let f(W) be a function of vector W ∈ R^N , i.e. f(W) = (1/(1 + e^(−W^T x) )) Determine the first derivative and matrix of second derivatives of f with respect to W

$${Let}\:{f}\left({W}\right)\:{be}\:{a}\:{function}\:{of}\:{vector}\:{W}\:\in\: {R}^{{N}} , \\ $$$${i}.{e}.\:{f}\left({W}\right)\:=\:\frac{\mathrm{1}}{\mathrm{1}\:+\:{e}^{−{W}^{{T}} {x}} } \\ $$$${Determine}\:{the}\:{first}\:{derivative}\:{and} \\ $$$${matrix}\:{of}\:{second}\:{derivatives}\:{of}\:{f}\:{with} \\ $$$${respect}\:{to}\:{W} \\ $$$$ \\ $$

Question Number 203897    Answers: 3   Comments: 0

Question Number 203891    Answers: 1   Comments: 0

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