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AllQuestion and Answers: Page 994

Question Number 120444    Answers: 0   Comments: 0

Question Number 120438    Answers: 3   Comments: 0

Question Number 120431    Answers: 1   Comments: 2

Question Number 120427    Answers: 2   Comments: 0

lim_(x→0) (((√(cos x+(√(2sin x+x)))) −1)/(sin x))

$$\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{cos}\:{x}+\sqrt{\mathrm{2sin}\:{x}+{x}}}\:−\mathrm{1}}{\mathrm{sin}\:{x}} \\ $$

Question Number 120425    Answers: 4   Comments: 0

lim_(x→0) ((2^x −cos x)/(sin x)) ?

$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{2}^{{x}} −\mathrm{cos}\:{x}}{\mathrm{sin}\:{x}}\:? \\ $$

Question Number 120422    Answers: 0   Comments: 4

Question Number 120414    Answers: 0   Comments: 0

Question Number 120413    Answers: 1   Comments: 1

Question Number 120409    Answers: 3   Comments: 1

I/.∫((1/(lnx)) − (1/(ln^2 x)))dx (Helpe me please)

$$\mathrm{I}/.\int\left(\frac{\mathrm{1}}{\mathrm{lnx}}\:−\:\frac{\mathrm{1}}{\mathrm{ln}^{\mathrm{2}} \mathrm{x}}\right)\mathrm{dx} \\ $$$$\left(\mathrm{Helpe}\:\mathrm{me}\:\mathrm{please}\right) \\ $$

Question Number 120395    Answers: 3   Comments: 0

Question Number 120393    Answers: 0   Comments: 4

Question Number 120389    Answers: 3   Comments: 2

Question Number 120383    Answers: 3   Comments: 0

I/ .∫_0 ^1 ((lnt )/((1+t)^2 ))dt =? (Helpe me please)

$$\mathrm{I}/\:.\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{lnt}\:}{\left(\mathrm{1}+\mathrm{t}\right)^{\mathrm{2}} }\mathrm{dt}\:\:=? \\ $$$$\:\left(\mathrm{Helpe}\:\mathrm{me}\:\mathrm{please}\right) \\ $$

Question Number 120380    Answers: 2   Comments: 0

Given { ((log _2 (10)=(a/(a−1)))),((log _3 (5)=(1/b))) :} find the value of 1+log _(12) (15) ?

$${Given}\:\begin{cases}{\mathrm{log}\:_{\mathrm{2}} \left(\mathrm{10}\right)=\frac{{a}}{{a}−\mathrm{1}}}\\{\mathrm{log}\:_{\mathrm{3}} \left(\mathrm{5}\right)=\frac{\mathrm{1}}{{b}}}\end{cases} \\ $$$${find}\:{the}\:{value}\:{of}\:\mathrm{1}+\mathrm{log}\:_{\mathrm{12}} \left(\mathrm{15}\right)\:? \\ $$

Question Number 120378    Answers: 1   Comments: 2

find the point on the curve 3x^2 −4y^2 =72 closest to line 3x+2y−1=0

$${find}\:{the}\:{point}\:{on}\:{the}\:{curve}\:\mathrm{3}{x}^{\mathrm{2}} −\mathrm{4}{y}^{\mathrm{2}} =\mathrm{72} \\ $$$${closest}\:{to}\:{line}\:\mathrm{3}{x}+\mathrm{2}{y}−\mathrm{1}=\mathrm{0} \\ $$

Question Number 120374    Answers: 3   Comments: 0

∫ ((√(1+x^2 ))/x) dx

$$\:\:\:\:\int\:\frac{\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}{{x}}\:{dx}\: \\ $$

Question Number 120365    Answers: 0   Comments: 4

A body is released from rest at the top ofa plane inclined at 30 to the horizontaland 4.0m high . If the coefficiet of friction between the body and plane is 0.3. calculate to the time theb ody takes to reach the bottom of. thea

$$\mathrm{A}\:\mathrm{body}\:\mathrm{is}\:\mathrm{released}\:\mathrm{from}\:\mathrm{rest}\:\mathrm{at}\:\mathrm{the}\:\mathrm{top}\: \\ $$$$\mathrm{ofa}\:\mathrm{plane}\:\mathrm{inclined}\:\mathrm{at}\:\mathrm{30}\:\mathrm{to}\:\mathrm{the}\: \\ $$$$\mathrm{horizontaland}\:\mathrm{4}.\mathrm{0m}\:\mathrm{high}\:.\:\mathrm{If}\:\mathrm{the} \\ $$$$\mathrm{coefficiet}\:\mathrm{of}\:\mathrm{friction}\:\mathrm{between}\:\mathrm{the}\:\mathrm{body} \\ $$$$\mathrm{an}{d}\:\mathrm{plane}\:\mathrm{is}\:\mathrm{0}.\mathrm{3}.\:\mathrm{calculate}\:\mathrm{to}\:\mathrm{the}\:\mathrm{time}\:\mathrm{theb} \\ $$$$\mathrm{ody}\:\mathrm{takes}\:\mathrm{to}\:\mathrm{reach}\:\mathrm{the}\:\mathrm{bottom}\:\mathrm{of}.\:\mathrm{thea} \\ $$

Question Number 120363    Answers: 1   Comments: 0

Determine the sum of the 1st nth term of the Sequence 1,4,10,22,46,.... ★Almighty Formula

$$ \\ $$$$\mathrm{Determine}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{1st}\:\mathrm{nth}\:\mathrm{term} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{Sequence} \\ $$$$ \\ $$$$\:\mathrm{1},\mathrm{4},\mathrm{10},\mathrm{22},\mathrm{46},.... \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\bigstar\mathrm{Almighty}\:\mathrm{Formula} \\ $$

Question Number 120355    Answers: 3   Comments: 2

Question Number 120342    Answers: 0   Comments: 0

Question Number 120331    Answers: 0   Comments: 8

Question Number 120325    Answers: 0   Comments: 6

Let f:R→R be a function satisfying the functional relation (f(x))^y +(f(y))^x =2f(xy) for all x, y ∈R and it is given that f(1)=1/2. Answer the following questions. (i) f(x+y)= (A) f(x)+f(y) (B) f(x)f(y) (C) f(x^y y^x ) (D) ((f(x))/(f(y))) (ii) f(xy)= (A) f(x)f(y) (B) f(x)+f(y) (C) (f(x))^y (D) (f(xy))^(xy) (iii) Σ_(k=0) ^∞ f(k)= (A) 5/2 (B) 3/2 (C) 3 (D) 2

$$\mathrm{Let}\:{f}:\mathbb{R}\rightarrow\mathbb{R}\:\mathrm{be}\:\mathrm{a}\:\mathrm{function}\:\mathrm{satisfying}\:\mathrm{the} \\ $$$$\mathrm{functional}\:\mathrm{relation} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left({f}\left(\mathrm{x}\right)\right)^{\mathrm{y}} +\left({f}\left(\mathrm{y}\right)\right)^{\mathrm{x}} =\mathrm{2}{f}\left(\mathrm{xy}\right) \\ $$$$\mathrm{for}\:\mathrm{all}\:\mathrm{x},\:\mathrm{y}\:\in\mathbb{R}\:\mathrm{and}\:\mathrm{it}\:\mathrm{is}\:\mathrm{given}\:\mathrm{that}\:{f}\left(\mathrm{1}\right)=\mathrm{1}/\mathrm{2}.\:\mathrm{Answer} \\ $$$$\mathrm{the}\:\mathrm{following}\:\mathrm{questions}. \\ $$$$\left(\boldsymbol{\mathrm{i}}\right)\:\:\:\:{f}\left(\mathrm{x}+\mathrm{y}\right)= \\ $$$$\:\:\:\:\:\:\:\:\:\left(\mathrm{A}\right)\:{f}\left(\mathrm{x}\right)+{f}\left(\mathrm{y}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{B}\right)\:\:{f}\left(\mathrm{x}\right){f}\left(\mathrm{y}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\left(\mathrm{C}\right)\:{f}\left(\mathrm{x}^{\mathrm{y}} \mathrm{y}^{\mathrm{x}} \right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{D}\right)\:\:\:\frac{{f}\left(\mathrm{x}\right)}{{f}\left(\mathrm{y}\right)} \\ $$$$\left(\boldsymbol{\mathrm{ii}}\right)\:\:\:\:{f}\left(\mathrm{xy}\right)= \\ $$$$\:\:\:\:\:\:\:\:\:\:\left(\mathrm{A}\right)\:{f}\left(\mathrm{x}\right){f}\left(\mathrm{y}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{B}\right)\:{f}\left(\mathrm{x}\right)+{f}\left(\mathrm{y}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\left(\mathrm{C}\right)\:\left({f}\left(\mathrm{x}\right)\right)^{\mathrm{y}} \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{D}\right)\:\:\left({f}\left(\mathrm{xy}\right)\right)^{\mathrm{xy}} \\ $$$$\left(\boldsymbol{\mathrm{iii}}\right)\:\:\underset{\mathrm{k}=\mathrm{0}} {\overset{\infty} {\sum}}{f}\left({k}\right)= \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{A}\right)\:\mathrm{5}/\mathrm{2}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{B}\right)\:\mathrm{3}/\mathrm{2}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{C}\right)\:\mathrm{3}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{D}\right)\:\mathrm{2} \\ $$

Question Number 120324    Answers: 1   Comments: 0

we are in C. (E): z^3 +(4−5i)z^2 +(8−20i)z−40i=0 1) Show that (E) has one imaginary pure root 2) solve (E)

$$\mathrm{we}\:\mathrm{are}\:\mathrm{in}\:\mathbb{C}. \\ $$$$\left(\mathrm{E}\right):\:\mathrm{z}^{\mathrm{3}} +\left(\mathrm{4}−\mathrm{5i}\right)\mathrm{z}^{\mathrm{2}} +\left(\mathrm{8}−\mathrm{20i}\right)\mathrm{z}−\mathrm{40i}=\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:\mathrm{Show}\:\mathrm{that}\:\left(\mathrm{E}\right)\:\mathrm{has}\:\mathrm{one}\:\mathrm{imaginary}\:\mathrm{pure}\:\mathrm{root} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{solve}\:\left(\mathrm{E}\right) \\ $$

Question Number 120322    Answers: 1   Comments: 3

Question Number 120323    Answers: 0   Comments: 0

Question Number 120316    Answers: 1   Comments: 0

calculate ∫_0 ^(π/2) ((xcosx)/(cos(2x)))dx

$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{{xcosx}}{{cos}\left(\mathrm{2}{x}\right)}{dx} \\ $$

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