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Question Number 26693 Answers: 0 Comments: 3

STATEMENT-1: The angle between one of the lines represented by ax^2 + 2hxy + by^2 = 0 and one of the lines represented by (a + 2008)x^2 + 2hxy + (b + 2008)y^2 = 0 is equal to angle between other two lines of the system. and STATEMENT-2: The pair of lines given by a^2 x^2 + 2008(a + b)xy + b^2 y^2 = 0 is equally inclined to the pair given by ax^2 + 2008xy + by^2 = 0.

$${STATEMENT}-\mathrm{1}:\:{The}\:{angle}\:{between} \\ $$$${one}\:{of}\:{the}\:{lines}\:{represented}\:{by}\:{ax}^{\mathrm{2}} \:+ \\ $$$$\mathrm{2}{hxy}\:+\:{by}^{\mathrm{2}} \:=\:\mathrm{0}\:{and}\:{one}\:{of}\:{the}\:{lines} \\ $$$${represented}\:{by}\:\left({a}\:+\:\mathrm{2008}\right){x}^{\mathrm{2}} \:+\:\mathrm{2}{hxy} \\ $$$$+\:\left({b}\:+\:\mathrm{2008}\right){y}^{\mathrm{2}} \:=\:\mathrm{0}\:{is}\:{equal}\:{to}\:{angle} \\ $$$${between}\:{other}\:{two}\:{lines}\:{of}\:{the} \\ $$$${system}. \\ $$$$\boldsymbol{{and}} \\ $$$${STATEMENT}-\mathrm{2}:\:{The}\:{pair}\:{of}\:{lines} \\ $$$${given}\:{by}\:{a}^{\mathrm{2}} {x}^{\mathrm{2}} \:+\:\mathrm{2008}\left({a}\:+\:{b}\right){xy}\:+\:{b}^{\mathrm{2}} {y}^{\mathrm{2}} \\ $$$$=\:\mathrm{0}\:{is}\:{equally}\:{inclined}\:{to}\:{the}\:{pair} \\ $$$${given}\:{by}\:{ax}^{\mathrm{2}} \:+\:\mathrm{2008}{xy}\:+\:{by}^{\mathrm{2}} \:=\:\mathrm{0}. \\ $$

Question Number 26686 Answers: 1 Comments: 1

Question Number 26681 Answers: 0 Comments: 1

f:R×R→R such that f(x+iy)=(√(x^2 +y^2 .)) Then f is a) many−one and into function b) one−one and onto function c) many−one and onto function d) one−one and into function

$${f}:{R}×{R}\rightarrow{R}\:{such}\:{that}\:{f}\left({x}+{iy}\right)=\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} .} \\ $$$${Then}\:{f}\:{is} \\ $$$$\left.{a}\right)\:{many}−{one}\:{and}\:{into}\:{function} \\ $$$$\left.{b}\right)\:{one}−{one}\:{and}\:{onto}\:{function} \\ $$$$\left.{c}\right)\:{many}−{one}\:{and}\:{onto}\:{function} \\ $$$$\left.{d}\right)\:{one}−{one}\:{and}\:{into}\:{function} \\ $$

Question Number 26680 Answers: 1 Comments: 0

(b−c)x^2 +(c−a)x+(a−b)=0 if the eqation roots are eqal you proved that 2b=a+c.

$$\left({b}−{c}\right){x}^{\mathrm{2}} +\left({c}−{a}\right){x}+\left({a}−{b}\right)=\mathrm{0}\:{if}\:{the}\: \\ $$$${eqation}\:{roots}\:\:{are}\:{eqal}\:{you}\:{proved}\:{that} \\ $$$$\mathrm{2}{b}={a}+{c}. \\ $$

Question Number 26652 Answers: 0 Comments: 2

Question Number 26649 Answers: 1 Comments: 1

Solve the equation:x+y=5−−−i x^x + y^y =31−−−ii No trial and error

$${Solve}\:{the}\:{equation}:{x}+{y}=\mathrm{5}−−−{i} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{x}^{{x}} +\:{y}^{{y}} =\mathrm{31}−−−{ii} \\ $$$${No}\:{trial}\:{and}\:{error} \\ $$$$ \\ $$

Question Number 26644 Answers: 1 Comments: 0

Question Number 26642 Answers: 2 Comments: 0

9x^2 +41x−204=0. solved it.

$$\mathrm{9}{x}^{\mathrm{2}} +\mathrm{41}{x}−\mathrm{204}=\mathrm{0}.\:{solved}\:{it}. \\ $$

Question Number 26638 Answers: 1 Comments: 0

x^3 −8x^2 +7 factorise it.

$$\mathrm{x}^{\mathrm{3}} −\mathrm{8x}^{\mathrm{2}} +\mathrm{7}\:\:\mathrm{factorise}\:\mathrm{it}. \\ $$

Question Number 26634 Answers: 1 Comments: 1

Question Number 26633 Answers: 0 Comments: 0

find the consumption function when MPC is c′(y)=0.8+0.1(√(y )) and that C=Y when Y=100

$$\mathrm{find}\:\mathrm{the}\:\mathrm{consumption}\:\mathrm{function}\:\mathrm{when}\:\mathrm{MPC}\:\mathrm{is}\:\mathrm{c}'\left(\mathrm{y}\right)=\mathrm{0}.\mathrm{8}+\mathrm{0}.\mathrm{1}\sqrt{\mathrm{y}\:} \\ $$$$\mathrm{and}\:\mathrm{that}\:\mathrm{C}=\mathrm{Y}\:\mathrm{when}\:\mathrm{Y}=\mathrm{100} \\ $$

Question Number 26631 Answers: 0 Comments: 1

∫_0 ^∞ (1/x^2 )dx

$$\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} }\mathrm{dx} \\ $$$$ \\ $$

Question Number 26626 Answers: 1 Comments: 0

Question Number 26625 Answers: 1 Comments: 0

3y−2x+7=0 x^2 −4y^2 −21=0

$$\mathrm{3y}−\mathrm{2x}+\mathrm{7}=\mathrm{0} \\ $$$$\mathrm{x}^{\mathrm{2}} −\mathrm{4y}^{\mathrm{2}} −\mathrm{21}=\mathrm{0} \\ $$

Question Number 26623 Answers: 2 Comments: 0

distance between 2 places A and B on road is 70 km. a car starts from A and other from B .if they travel in same direction they will meet after 7 hours. if they travel towards each other they will meet after 1 hour then find their speeds

$$\mathrm{distance}\:\mathrm{between}\:\mathrm{2}\:\mathrm{places}\:\mathrm{A}\:\mathrm{and}\:\mathrm{B}\:\mathrm{on} \\ $$$$\mathrm{road}\:\mathrm{is}\:\mathrm{70}\:\mathrm{km}.\:\mathrm{a}\:\mathrm{car}\:\mathrm{starts}\:\mathrm{from}\:\mathrm{A}\:\mathrm{and}\:\mathrm{other}\: \\ $$$$\mathrm{from}\:\mathrm{B}\:.\mathrm{if}\:\mathrm{they}\:\mathrm{travel}\:\mathrm{in}\:\mathrm{same}\:\mathrm{direction} \\ $$$$\mathrm{they}\:\mathrm{will}\:\mathrm{meet}\:\mathrm{after}\:\mathrm{7}\:\mathrm{hours}.\:\mathrm{if}\:\mathrm{they}\:\mathrm{travel} \\ $$$$\mathrm{towards}\:\mathrm{each}\:\mathrm{other}\:\mathrm{they}\:\mathrm{will}\:\mathrm{meet}\:\mathrm{after} \\ $$$$\mathrm{1}\:\mathrm{hour}\:\mathrm{then}\:\mathrm{find}\:\mathrm{their}\:\mathrm{speeds} \\ $$

Question Number 26591 Answers: 1 Comments: 1

Prove lim_(x→−1) (1+x)ln (1+x)=0

$$\mathrm{Prove} \\ $$$$\underset{{x}\rightarrow−\mathrm{1}} {\mathrm{lim}}\left(\mathrm{1}+{x}\right)\mathrm{ln}\:\left(\mathrm{1}+{x}\right)=\mathrm{0} \\ $$

Question Number 26583 Answers: 0 Comments: 2

find the decomposition in C[x] then R[x] for the rationsl fraction F(x)= ((1 )/(x^(2n) −1)) .with n integer not 0

$${find}\:{the}\:{decomposition}\:{in}\:\mathbb{C}\left[{x}\right]\:{then}\:\mathbb{R}\left[{x}\right] \\ $$$${for}\:{the}\:{rationsl}\:{fraction} \\ $$$${F}\left({x}\right)=\:\:\frac{\mathrm{1}\:}{{x}^{\mathrm{2}{n}} −\mathrm{1}}\:\:.{with}\:{n}\:{integer}\:{not}\:\mathrm{0} \\ $$

Question Number 26582 Answers: 0 Comments: 1

p is a polynomial having the roots x_1 ,x_2 ,...x_n with x_i ≠ x_j fori≠j give the decomposition of the fravtion F(x)= ((p^′ (x))/(p(x)))

$${p}\:{is}\:{a}\:{polynomial}\:{having}\:{the}\:{roots}\:{x}_{\mathrm{1}} ,{x}_{\mathrm{2}} ,...{x}_{{n}} \\ $$$${with}\:{x}_{{i}} \neq\:{x}_{{j}} \:{fori}\neq{j}\:{give}\:{the}\:{decomposition} \\ $$$${of}\:{the}\:{fravtion}\:{F}\left({x}\right)=\:\frac{{p}^{'} \left({x}\right)}{{p}\left({x}\right)} \\ $$

Question Number 26581 Answers: 0 Comments: 0

Question Number 26580 Answers: 0 Comments: 0

Question Number 26573 Answers: 0 Comments: 0

let put H_n = Σ_(k=1) ^(k=n) (1/k) prove that Σ_(n=1) ^∝ (H_n /n^2 ) = 2ξ(3)

$${let}\:{put}\:\:{H}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{k}={n}} \:\:\frac{\mathrm{1}}{{k}}\:\:\:\:\:\:{prove}\:{that}\:\:\:\sum_{{n}=\mathrm{1}} ^{\propto} \:\frac{{H}_{{n}} }{{n}^{\mathrm{2}} }\:\:=\:\:\mathrm{2}\xi\left(\mathrm{3}\right) \\ $$

Question Number 26572 Answers: 0 Comments: 0

find the radius ofconvergence for the serie Σ_(n≥0) ( e^(√(n+1)) −e^(√n) )z^n with z from C

$${find}\:{the}\:{radius}\:{ofconvergence}\:{for}\:{the}\:{serie} \\ $$$$\sum_{{n}\geqslant\mathrm{0}} \left(\:{e}^{\sqrt{{n}+\mathrm{1}}} −{e}^{\sqrt{{n}}} \right){z}^{{n}} \:\:\:\:\:\:{with}\:{z}\:{from}\:\mathbb{C} \\ $$$$ \\ $$

Question Number 26571 Answers: 0 Comments: 1

let give I(x)= ∫_1 ^∝ ((t−E(t))/t^(x+1) )dt prove that ξ(x)= (x/(x−1)) −xI(x) then chow that (x−1)_(x−1^(+ ew) ) ξ(x)−−>1 we remind ξ(x) = Σ_(n≥1) (1/n^x ) and x>1

$${let}\:{give}\:\:{I}\left({x}\right)=\:\:\int_{\mathrm{1}} ^{\propto} \:\frac{{t}−{E}\left({t}\right)}{{t}^{{x}+\mathrm{1}} }{dt}\:\:\:{prove}\:{that} \\ $$$$\xi\left({x}\right)=\:\frac{{x}}{{x}−\mathrm{1}}\:−{xI}\left({x}\right)\:{then}\:{chow}\:{that}\:\left({x}−\mathrm{1}\right)_{{x}−\mathrm{1}^{+\:{ew}} } \xi\left({x}\right)−−>\mathrm{1} \\ $$$${we}\:{remind}\:\:\xi\left({x}\right)\:=\:\sum_{{n}\geqslant\mathrm{1}} \:\frac{\mathrm{1}}{{n}^{{x}} }\:\:\:{and}\:\:{x}>\mathrm{1} \\ $$

Question Number 26570 Answers: 0 Comments: 1

find the value of ∫_0 ^∞ e^(−px) /sinx/dx with p>0

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} {e}^{−{px}} /{sinx}/{dx}\:\:\:{with}\:{p}>\mathrm{0} \\ $$

Question Number 26569 Answers: 0 Comments: 1

find the value of ∫_0 ^(1 ) x E((1/x))dx

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}\:} {x}\:{E}\left(\frac{\mathrm{1}}{{x}}\right){dx}\: \\ $$

Question Number 26568 Answers: 0 Comments: 0

let give ξ(x)= Σ_(n=1) ^∝ (1/n^x ) prove that ξ(x)−_(x−>∝) 1∼2^(−x)

$${let}\:{give}\:\xi\left({x}\right)=\:\sum_{{n}=\mathrm{1}} ^{\propto} \frac{\mathrm{1}}{{n}^{{x}} }\:\:{prove}\:{that}\:\xi\left({x}\right)−_{{x}−>\propto} \mathrm{1}\sim\mathrm{2}^{−{x}} \\ $$

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