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AllQuestion and Answers: Page 1533

Question Number 57011 Answers: 1 Comments: 0

f(((x+y)/2))f(((x−y)/2))=g(x) g(x+y)g(x−y)=[f(x)]^2 −[f(y)]^2 f(x),g(x)=?

$${f}\left(\frac{{x}+{y}}{\mathrm{2}}\right){f}\left(\frac{{x}−{y}}{\mathrm{2}}\right)={g}\left({x}\right) \\ $$$${g}\left({x}+{y}\right){g}\left({x}−{y}\right)=\left[{f}\left({x}\right)\right]^{\mathrm{2}} −\left[{f}\left({y}\right)\right]^{\mathrm{2}} \\ $$$${f}\left({x}\right),{g}\left({x}\right)=? \\ $$

Question Number 57010 Answers: 1 Comments: 1

f(((x+y)/2))=((f(x)f(y))/(f(2))) f(x)=?

$${f}\left(\frac{{x}+{y}}{\mathrm{2}}\right)=\frac{{f}\left({x}\right){f}\left({y}\right)}{{f}\left(\mathrm{2}\right)} \\ $$$${f}\left({x}\right)=? \\ $$

Question Number 57007 Answers: 1 Comments: 3

cosec ((π/(14))) − 4 cos (((2π)/7)) = ?

$$\mathrm{cosec}\:\left(\frac{\pi}{\mathrm{14}}\right)\:−\:\mathrm{4}\:\mathrm{cos}\:\left(\frac{\mathrm{2}\pi}{\mathrm{7}}\right)\:\:=\:\:? \\ $$

Question Number 57001 Answers: 0 Comments: 1

construct an analytic function f(z) whose real part is e^x cos y

$${construct}\:{an}\:{analytic}\:{function}\:{f}\left({z}\right)\:{whose}\:{real}\:{part}\:{is}\:{e}^{{x}} \mathrm{cos}\:{y} \\ $$

Question Number 57000 Answers: 3 Comments: 1

If (x+2)^2 is a factor of the polynomial f(x)=mx^3 +x^2 +x+n, find; the values of m and n.

$$\mathrm{If}\:\left(\mathrm{x}+\mathrm{2}\right)^{\mathrm{2}} \:\mathrm{is}\:\mathrm{a}\:\mathrm{factor}\:\mathrm{of}\:\mathrm{the}\:\mathrm{polynomial} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{mx}^{\mathrm{3}} +\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{n},\:\mathrm{find}; \\ $$$$\:\mathrm{the}\:\mathrm{values}\:\mathrm{of}\:\mathrm{m}\:\mathrm{and}\:\mathrm{n}. \\ $$

Question Number 56991 Answers: 0 Comments: 4

x! − x^2 = 8 , Find x

$$\:\:\mathrm{x}!\:−\:\mathrm{x}^{\mathrm{2}} \:\:=\:\:\mathrm{8}\:,\:\:\:\:\mathrm{Find}\:\:\mathrm{x} \\ $$$$ \\ $$

Question Number 56986 Answers: 2 Comments: 1

Question Number 56971 Answers: 1 Comments: 1

Question Number 56962 Answers: 1 Comments: 1

find S_n =Σ_(k=0) ^n k^2 C_n ^k cos(2kx) interms of n.

$${find}\:{S}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:{k}^{\mathrm{2}} \:{C}_{{n}} ^{{k}} \:{cos}\left(\mathrm{2}{kx}\right) \\ $$$${interms}\:{of}\:{n}. \\ $$

Question Number 56961 Answers: 1 Comments: 0

Out of 6 mathematicians and 7 physicists a committee consisting of 3 mathematicians and 3 physicists is to be formed. In how many ways can this be done if two particular mathematicians cannot be on the commitee?

$$\mathrm{Out}\:\mathrm{of}\:\mathrm{6}\:\mathrm{mathematicians}\:\mathrm{and}\:\mathrm{7}\:\mathrm{physicists} \\ $$$$\mathrm{a}\:\mathrm{committee}\:\mathrm{consisting}\:\mathrm{of}\:\mathrm{3}\:\mathrm{mathematicians} \\ $$$$\mathrm{and}\:\mathrm{3}\:\mathrm{physicists}\:\mathrm{is}\:\mathrm{to}\:\mathrm{be}\:\mathrm{formed}.\:\mathrm{In}\:\mathrm{how} \\ $$$$\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{this}\:\mathrm{be}\:\mathrm{done}\:\mathrm{if}\:\mathrm{two}\: \\ $$$$\mathrm{particular}\:\mathrm{mathematicians}\:\mathrm{cannot}\:\mathrm{be} \\ $$$$\mathrm{on}\:\mathrm{the}\:\mathrm{commitee}? \\ $$

Question Number 56954 Answers: 2 Comments: 1

Find minimum value of : cos(ω−φ)+cos(φ−ϕ)+cos (ϕ−ω).

$${Find}\:{minimum}\:{value}\:{of}\:: \\ $$$${cos}\left(\omega−\phi\right)+\mathrm{cos}\left(\phi−\varphi\right)+\mathrm{cos}\:\left(\varphi−\omega\right). \\ $$

Question Number 56951 Answers: 0 Comments: 2

The deviations from the mean of a set of numbers are (x+2), (2x−11), −9, (x+1)^2 , (x−4)^2 , (1−3x). find the value of x where x>0.

$$\mathrm{The}\:\mathrm{deviations}\:\mathrm{from}\:\mathrm{the}\:\mathrm{mean}\:\mathrm{of}\:\mathrm{a} \\ $$$$\mathrm{set}\:\mathrm{of}\:\mathrm{numbers}\:\mathrm{are}\:\left(\mathrm{x}+\mathrm{2}\right),\:\left(\mathrm{2x}−\mathrm{11}\right), \\ $$$$−\mathrm{9},\:\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{2}} ,\:\left(\mathrm{x}−\mathrm{4}\right)^{\mathrm{2}} ,\:\left(\mathrm{1}−\mathrm{3x}\right).\:\mathrm{find}\:\mathrm{the} \\ $$$$\mathrm{value}\:\mathrm{of}\:\mathrm{x}\:\mathrm{where}\:\mathrm{x}>\mathrm{0}. \\ $$

Question Number 56949 Answers: 0 Comments: 1

Question Number 56939 Answers: 1 Comments: 2

calculate ∫ (dx/((x+1)^3 (x^2 −3x +2))) 2) find the value of ∫_2 ^(+∞) (dx/((x+1)^3 (x^2 −3x+2)))

$${calculate}\:\int\:\:\:\:\frac{{dx}}{\left({x}+\mathrm{1}\right)^{\mathrm{3}} \left({x}^{\mathrm{2}} −\mathrm{3}{x}\:+\mathrm{2}\right)} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{2}} ^{+\infty} \:\:\frac{{dx}}{\left({x}+\mathrm{1}\right)^{\mathrm{3}} \left({x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{2}\right)} \\ $$

Question Number 56938 Answers: 0 Comments: 0

let A_n =∫∫_W_n e^(−xy) (√(x^2 +y^2 ))dxdy with W_n =[(1/n),n[×[(1/n),n[ 1) find A_n interms of n 2) determine lim_(n→+∞) A_n

$${let}\:{A}_{{n}} =\int\int_{{W}_{{n}} } {e}^{−{xy}} \sqrt{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }{dxdy}\:\:\:{with}\:{W}_{{n}} =\left[\frac{\mathrm{1}}{{n}},{n}\left[×\left[\frac{\mathrm{1}}{{n}},{n}\left[\right.\right.\right.\right. \\ $$$$\left.\mathrm{1}\right)\:{find}\:{A}_{{n}} {interms}\:{of}\:{n} \\ $$$$\left.\mathrm{2}\right)\:{determine}\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} \\ $$$$ \\ $$

Question Number 56937 Answers: 0 Comments: 0

1. calculate f(x) =∫_0 ^(π/4) ln(1+xtanθ)dθ 2. calculate ∫_0 ^1 f(x)dx

$$\mathrm{1}.\:{calculate}\:\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:{ln}\left(\mathrm{1}+{xtan}\theta\right){d}\theta \\ $$$$\mathrm{2}.\:\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx} \\ $$

Question Number 56942 Answers: 0 Comments: 0

find U_n =∫_1 ^n (([(√(x+1))]−[(√x)])/x^3 ) dx 2) find nature of the serie Σ U_n

$${find}\:\:{U}_{{n}} =\int_{\mathrm{1}} ^{{n}} \:\frac{\left[\sqrt{{x}+\mathrm{1}}\right]−\left[\sqrt{{x}}\right]}{{x}^{\mathrm{3}} }\:{dx} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{nature}\:{of}\:{the}\:{serie}\:\Sigma\:{U}_{{n}} \\ $$

Question Number 56935 Answers: 0 Comments: 1

1. calculate U_n =∫_0 ^∞ (x^3 −2x+1)e^(−n[x]) dx with n integr natural and n≥1 2. find nature of Σ U_n

$$\mathrm{1}.\:{calculate}\:\:{U}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\left({x}^{\mathrm{3}} −\mathrm{2}{x}+\mathrm{1}\right){e}^{−{n}\left[{x}\right]} {dx}\:\:{with}\:{n}\:{integr}\:{natural}\:{and}\:{n}\geqslant\mathrm{1} \\ $$$$\mathrm{2}.\:{find}\:{nature}\:{of}\:\Sigma\:{U}_{{n}} \\ $$

Question Number 56932 Answers: 1 Comments: 0

let f_n (t) =∫_0 ^∞ (dx/((x^2 +t^2 )^n )) with n from N and n≥1 1. find a explicit form of f_n (t) 2. what is the value of g_n (t)=∫_0 ^∞ ((t dx)/((x^2 +t^2 )^(n+1) )) ? 3. calculate ∫_0 ^∞ (dx/((x^2 +3)^4 )) and ∫_0 ^∞ (dx/((x^2 +16)^3 ))

$${let}\:{f}_{{n}} \left({t}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:+{t}^{\mathrm{2}} \right)^{{n}} } \\ $$$${with}\:{n}\:{from}\:{N}\:{and}\:{n}\geqslant\mathrm{1} \\ $$$$\mathrm{1}.\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}_{{n}} \left({t}\right) \\ $$$$\mathrm{2}.\:{what}\:{is}\:{the}\:{value}\:{of} \\ $$$${g}_{{n}} \left({t}\right)=\int_{\mathrm{0}} ^{\infty} \:\frac{{t}\:{dx}}{\left({x}^{\mathrm{2}} +{t}^{\mathrm{2}} \right)^{{n}+\mathrm{1}} }\:? \\ $$$$\mathrm{3}.\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{3}\right)^{\mathrm{4}} } \\ $$$${and}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:+\mathrm{16}\right)^{\mathrm{3}} } \\ $$

Question Number 56931 Answers: 1 Comments: 1

calculate ∫_(−∞) ^(+∞) ((x^2 −1)/((x^2 −x+3)^2 ))dx

$${calculate}\:\int_{−\infty} ^{+\infty} \:\:\frac{{x}^{\mathrm{2}} −\mathrm{1}}{\left({x}^{\mathrm{2}} −{x}+\mathrm{3}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 56921 Answers: 1 Comments: 0

Given : f(xy)=f(x).f(y)∀x,yεR and f(0)≠0 then f(x)=?

$${Given}\:: \\ $$$${f}\left({xy}\right)={f}\left({x}\right).{f}\left({y}\right)\forall{x},{y}\epsilon\mathbb{R}\:{and}\:{f}\left(\mathrm{0}\right)\neq\mathrm{0} \\ $$$${then}\:{f}\left({x}\right)=? \\ $$

Question Number 56914 Answers: 3 Comments: 0

The shortest distance between the point ((3/2),0) and the curve y=(√x) ,(x>0) is ?

$${The}\:{shortest}\:{distance}\:{between}\:{the}\:{point} \\ $$$$\left(\frac{\mathrm{3}}{\mathrm{2}},\mathrm{0}\right)\:{and}\:{the}\:{curve}\:{y}=\sqrt{{x}}\:,\left({x}>\mathrm{0}\right)\:{is}\:? \\ $$

Question Number 56913 Answers: 1 Comments: 2

How many possible solution sets that satisfy x_1 + x_2 + x_3 + x_4 = 5 with 0 ≤ x_1 ≤ 3 0 ≤ x_2 ≤ 3 0 ≤ x_3 ≤ 2 0 ≤ x_4 ≤ 2

$$\mathrm{How}\:\mathrm{many}\:\mathrm{possible}\:\mathrm{solution}\:\mathrm{sets}\:\mathrm{that}\:\mathrm{satisfy}\: \\ $$$${x}_{\mathrm{1}} \:+\:{x}_{\mathrm{2}} \:+\:{x}_{\mathrm{3}} \:+\:{x}_{\mathrm{4}} \:=\:\mathrm{5} \\ $$$$\mathrm{with} \\ $$$$\mathrm{0}\:\leqslant\:{x}_{\mathrm{1}} \:\leqslant\:\mathrm{3}\:\: \\ $$$$\mathrm{0}\:\leqslant\:{x}_{\mathrm{2}} \:\leqslant\:\mathrm{3} \\ $$$$\mathrm{0}\:\leqslant\:{x}_{\mathrm{3}} \:\leqslant\:\mathrm{2} \\ $$$$\mathrm{0}\:\leqslant\:{x}_{\mathrm{4}} \:\leqslant\:\mathrm{2} \\ $$

Question Number 56912 Answers: 1 Comments: 4

If a and b are positive integers such that (1+ab) divides (a^2 +b^2 ) show that the integer ((a^2 +b^2 )/(1+ab)) must be a perfect square.

$$\mathrm{If}\:{a}\:\mathrm{and}\:{b}\:\mathrm{are}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{such}\:\mathrm{that}\: \\ $$$$\left(\mathrm{1}+{ab}\right)\:\mathrm{divides}\:\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} \right)\:\mathrm{show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{integer} \\ $$$$\frac{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }{\mathrm{1}+{ab}}\:\mathrm{must}\:\mathrm{be}\:\mathrm{a}\:\mathrm{perfect}\:\mathrm{square}. \\ $$

Question Number 56904 Answers: 1 Comments: 0

If α and β are the roots of of the equation 3x^2 −x−3=0, find thevalue of (α^2 −β^2 ) if α>β.

$$\mathrm{If}\:\alpha\:\mathrm{and}\:\beta\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\mathrm{3x}^{\mathrm{2}} −\mathrm{x}−\mathrm{3}=\mathrm{0},\:\mathrm{find}\:\mathrm{thevalue}\:\mathrm{of}\:\left(\alpha^{\mathrm{2}} −\beta^{\mathrm{2}} \right) \\ $$$$\mathrm{if}\:\alpha>\beta. \\ $$

Question Number 56900 Answers: 1 Comments: 0

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