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Question Number 106644 Answers: 0 Comments: 0

Let P(x) be a polynomial of degree n with real coefficients. Prove that Σ_(k=0) ^n ((P^((k)) (0))/((k+1)!))=Σ_(k=0) ^n (((−1)^k P^((k)) (1))/((k+1)!))

$$\mathrm{Let}\:\mathrm{P}\left(\mathrm{x}\right)\:\mathrm{be}\:\mathrm{a}\:\mathrm{polynomial}\:\mathrm{of}\:\mathrm{degree}\:\mathrm{n} \\ $$$$\mathrm{with}\:\mathrm{real}\:\mathrm{coefficients}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\frac{\mathrm{P}^{\left(\mathrm{k}\right)} \left(\mathrm{0}\right)}{\left(\mathrm{k}+\mathrm{1}\right)!}=\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\frac{\left(−\mathrm{1}\right)^{\mathrm{k}} \mathrm{P}^{\left(\mathrm{k}\right)} \left(\mathrm{1}\right)}{\left(\mathrm{k}+\mathrm{1}\right)!} \\ $$$$ \\ $$

Question Number 106637 Answers: 1 Comments: 0

@JS@ The quartic equation x^4 +2x^3 +14x+15=0 has one root equal to 1+2i . Find the other three roots.

$$\:\:\:\:\:\:@\mathrm{JS}@ \\ $$$$\mathrm{The}\:\mathrm{quartic}\:\mathrm{equation}\:\mathrm{x}^{\mathrm{4}} +\mathrm{2x}^{\mathrm{3}} +\mathrm{14x}+\mathrm{15}=\mathrm{0} \\ $$$$\mathrm{has}\:\mathrm{one}\:\mathrm{root}\:\mathrm{equal}\:\mathrm{to}\:\mathrm{1}+\mathrm{2i}\:.\:\mathrm{Find} \\ $$$$\mathrm{the}\:\mathrm{other}\:\mathrm{three}\:\mathrm{roots}.\: \\ $$

Question Number 106633 Answers: 3 Comments: 0

^(@bemath@) lim_(x→0) (((√(1+2sin x)) −(√(1−4sin 4x)))/(4x))

$$\:\:\:\:\:\:\:\overset{@\mathrm{bemath}@} {\:} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{1}+\mathrm{2sin}\:\mathrm{x}}\:−\sqrt{\mathrm{1}−\mathrm{4sin}\:\mathrm{4x}}}{\mathrm{4x}} \\ $$

Question Number 106630 Answers: 2 Comments: 0

Find n in this equation: (−2)^n = 4096

$${Find}\:{n}\:{in}\:{this}\:{equation}: \\ $$$$\left(−\mathrm{2}\right)^{{n}} \:=\:\mathrm{4096} \\ $$

Question Number 106628 Answers: 0 Comments: 0

(((p−1))/p) where p=prime no. Remainder will always be (p−1) or −1 Que. find Remainder ((1!+2!+3!+........................1000!)/(10)) Que. ((1!+2!+3!+........................1000!)/(12)) Que. ((1!+2!+3!+........................1000!)/9) Que. What id the unit digit of below expression 1!+2!+3!+4!+......................1000! ANS. If we divide some number by 100,then remainder is last 2digit similary 1000----Last 3digit 10000 last 4 digit 100000 last 5 digits [(((1+2+3+4+0+0+0+..........+0)/(10))),() ] R=3 unit digit =3

Question Number 106614 Answers: 3 Comments: 0

_(@bemath@) lim_(x→0) ((((1+3sin x ))^(1/3) −((1+sin 3x))^(1/3) )/(2x)) =?

$$\:\:\:\:\:\underset{@\mathrm{bemath}@} {\:} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\sqrt[{\mathrm{3}}]{\mathrm{1}+\mathrm{3sin}\:\mathrm{x}\:}\:−\sqrt[{\mathrm{3}}]{\mathrm{1}+\mathrm{sin}\:\mathrm{3x}}}{\mathrm{2x}}\:=?\: \\ $$

Question Number 106655 Answers: 0 Comments: 1

Question Number 106609 Answers: 1 Comments: 1

Prove that (1−sin^2 θ)sec^2 θ=1

$${Prove}\:{that} \\ $$$$\left(\mathrm{1}−{sin}^{\mathrm{2}} \theta\right){sec}^{\mathrm{2}} \theta=\mathrm{1} \\ $$

Question Number 106604 Answers: 7 Comments: 1

Question Number 106596 Answers: 4 Comments: 0

Question Number 106594 Answers: 0 Comments: 4

Question Number 106591 Answers: 1 Comments: 7

Question Number 106588 Answers: 0 Comments: 0

@bemath@ x (d^2 y/dx^2 ) + 3x−(dy/dx) = e^(3x)

$$\:\:\:\:\:\:\:@\mathrm{bemath}@ \\ $$$$\mathrm{x}\:\frac{\mathrm{d}^{\mathrm{2}} \mathrm{y}}{\mathrm{dx}^{\mathrm{2}} }\:+\:\mathrm{3x}−\frac{\mathrm{dy}}{\mathrm{dx}}\:=\:\mathrm{e}^{\mathrm{3x}} \: \\ $$

Question Number 106583 Answers: 1 Comments: 0

@bemath@ ∫_0 ^(π/2) ((sin x dx)/(sin x+cos x)) =?

$$\:\:\:\:\:\:\:\:\:\:\:@\mathrm{bemath}@ \\ $$$$\underset{\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}\:\frac{\mathrm{sin}\:\mathrm{x}\:\mathrm{dx}}{\mathrm{sin}\:\mathrm{x}+\mathrm{cos}\:\mathrm{x}}\:=? \\ $$

Question Number 106579 Answers: 1 Comments: 0

Given cos x+cos y=(3/2)+cos (x+y) where x,y ∈ [0,2π ]. find x & y

$$\mathcal{G}\mathrm{iven}\:\mathrm{cos}\:\mathrm{x}+\mathrm{cos}\:\mathrm{y}=\frac{\mathrm{3}}{\mathrm{2}}+\mathrm{cos}\:\left(\mathrm{x}+\mathrm{y}\right) \\ $$$$\mathrm{where}\:\mathrm{x},\mathrm{y}\:\in\:\left[\mathrm{0},\mathrm{2}\pi\:\right].\:\mathrm{find}\:\mathrm{x}\:\&\:\mathrm{y}\: \\ $$

Question Number 106572 Answers: 2 Comments: 2

Question Number 106570 Answers: 3 Comments: 0

(1)∫_0 ^a ((√(a−x))/((√(a−x))+(√x))) dx =? (a) 0 (b) (a/2) (c) a (d) 2a (e) (5/2)a (2) ∫_0 ^(π/4) ((1−tan x)/(1+tan x)) dx =? (a) 0 (b) ln 2 (c) −ln 2 (d) πln 2 (e)(1/2)ln 2 (3) ((√3)+2)^x > 7−4(√3) , find the solution set

$$\left(\mathrm{1}\right)\underset{\mathrm{0}} {\overset{\mathrm{a}} {\int}}\:\frac{\sqrt{\mathrm{a}−\mathrm{x}}}{\sqrt{\mathrm{a}−\mathrm{x}}+\sqrt{\mathrm{x}}}\:\mathrm{dx}\:=? \\ $$$$\left(\mathrm{a}\right)\:\mathrm{0}\:\:\:\:\:\:\left(\mathrm{b}\right)\:\frac{\mathrm{a}}{\mathrm{2}}\:\:\:\:\:\:\left(\mathrm{c}\right)\:\mathrm{a}\:\:\:\:\:\:\:\:\left(\mathrm{d}\right)\:\mathrm{2a}\:\:\:\:\:\:\left(\mathrm{e}\right)\:\frac{\mathrm{5}}{\mathrm{2}}\mathrm{a} \\ $$$$\left(\mathrm{2}\right)\:\underset{\mathrm{0}} {\overset{\pi/\mathrm{4}} {\int}}\frac{\mathrm{1}−\mathrm{tan}\:\mathrm{x}}{\mathrm{1}+\mathrm{tan}\:\mathrm{x}}\:\mathrm{dx}\:=? \\ $$$$\left(\mathrm{a}\right)\:\mathrm{0}\:\:\:\:\left(\mathrm{b}\right)\:\mathrm{ln}\:\mathrm{2}\:\:\:\:\:\left(\mathrm{c}\right)\:−\mathrm{ln}\:\mathrm{2}\:\:\:\:\:\left(\mathrm{d}\right)\:\pi\mathrm{ln}\:\mathrm{2}\:\:\:\left(\mathrm{e}\right)\frac{\mathrm{1}}{\mathrm{2}}\mathrm{ln}\:\mathrm{2} \\ $$$$\left(\mathrm{3}\right)\:\left(\sqrt{\mathrm{3}}+\mathrm{2}\right)^{\mathrm{x}} \:>\:\mathrm{7}−\mathrm{4}\sqrt{\mathrm{3}}\:,\:\mathrm{find}\:\mathrm{the}\:\mathrm{solution}\:\mathrm{set} \\ $$

Question Number 107247 Answers: 1 Comments: 0