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AllQuestion and Answers: Page 317

Question Number 187373    Answers: 0   Comments: 0

Solve for natural numbers: 31 + Σ_(k=1) ^n ( ((n),(k) ) ∙ Σ_(m=1) ^n m^(n−k) ) = 31^(30)

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{natural}\:\mathrm{numbers}: \\ $$$$\mathrm{31}\:+\:\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\:\left(\begin{pmatrix}{\mathrm{n}}\\{\mathrm{k}}\end{pmatrix}\:\:\centerdot\:\underset{\boldsymbol{\mathrm{m}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\:\mathrm{m}^{\boldsymbol{\mathrm{n}}−\boldsymbol{\mathrm{k}}} \right)\:=\:\mathrm{31}^{\mathrm{30}} \\ $$

Question Number 187364    Answers: 2   Comments: 3

Question Number 187362    Answers: 2   Comments: 1

Question Number 187359    Answers: 1   Comments: 0

what are the two complex solution to X^(−x) +(−X)^x =0 in addition to ±1 ?

$$ \\ $$$${what}\:{are}\:{the}\:{two}\:{complex}\:{solution}\:{to} \\ $$$${X}^{−{x}} +\left(−{X}\right)^{{x}} =\mathrm{0}\:{in}\:{addition}\:{to}\:\pm\mathrm{1}\:? \\ $$

Question Number 187357    Answers: 1   Comments: 0

Question Number 187355    Answers: 1   Comments: 2

Question Number 187340    Answers: 0   Comments: 0

Question Number 187336    Answers: 1   Comments: 0

solve for x if X^x •5^x −5^(2+x) =0

$${solve}\:{for}\:{x}\:{if} \\ $$$${X}^{{x}} \bullet\mathrm{5}^{{x}} −\mathrm{5}^{\mathrm{2}+{x}} =\mathrm{0} \\ $$$$ \\ $$

Question Number 187335    Answers: 1   Comments: 0

Apply the rotation of coordinates given by the following matrix to the equation xy=1; what is the equation in th uv coordinate system? [(u),(v) ]= [((cos45 sin45)),((−sin45 cos45)) ] [(x),(y) ]

$${Apply}\:{the}\:{rotation}\:{of}\:{coordinates}\:{given} \\ $$$${by}\:{the}\:{following}\:{matrix}\:{to}\:{the}\:{equation}\: \\ $$$${xy}=\mathrm{1};\:{what}\:{is}\:{the}\:{equation}\:{in}\:{th}\:{uv}\:{coordinate}\: \\ $$$${system}? \\ $$$$\begin{bmatrix}{{u}}\\{{v}}\end{bmatrix}=\begin{bmatrix}{{cos}\mathrm{45}\:\:\:\:\:\:\:\:\:\:\:\:\:\:{sin}\mathrm{45}}\\{−{sin}\mathrm{45}\:\:\:\:\:\:\:\:\:\:{cos}\mathrm{45}}\end{bmatrix}\begin{bmatrix}{{x}}\\{{y}}\end{bmatrix} \\ $$

Question Number 187330    Answers: 1   Comments: 0

show that ((1−cosθ)/(1+cosθ))=tan^2 ((1/2)θ)

$$ \\ $$$${show}\:{that}\:\frac{\mathrm{1}−{cos}\theta}{\mathrm{1}+{cos}\theta}={tan}^{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{2}}\theta\right)\:\:\: \\ $$

Question Number 187324    Answers: 1   Comments: 0

calculate the unit of the chord which is 6cm from the center of the circle of radius 10cm

$${calculate}\:\:{the}\:{unit}\:{of}\:{the}\:\:{chord}\:{which}\:{is} \\ $$$$\mathrm{6}{cm}\:{from}\:{the}\:{center}\:{of}\:\:{the}\:\:{circle}\:{of}\:{radius}\:\mathrm{10}{cm} \\ $$$$ \\ $$$$ \\ $$

Question Number 187317    Answers: 2   Comments: 0

∫(dx/(x(√(1−2x))))

$$\int\frac{{dx}}{{x}\sqrt{\mathrm{1}−\mathrm{2}{x}}} \\ $$

Question Number 187312    Answers: 0   Comments: 1

Question Number 187311    Answers: 0   Comments: 2

If f_k (x)=(1/k)(sin^k x+cos^k x) find f_4 (x)−f_6 (x) f_4 (x)−f_6 (x)=(1/4)(sin^4 x +cos^4 x)−(1/6)(sin^6 x+cos^6 x) =(1/2)sin^4 x((1/2)−(1/3)sin^2 x)+(1/2)cos^4 x((1/2)−(1/3)cos^2 x)

$$\mathrm{If}\:{f}_{{k}} \left({x}\right)=\frac{\mathrm{1}}{{k}}\left(\mathrm{sin}\:^{{k}} {x}+\mathrm{cos}\:^{{k}} {x}\right)\:\mathrm{find}\:{f}_{\mathrm{4}} \left({x}\right)−{f}_{\mathrm{6}} \left({x}\right) \\ $$$${f}_{\mathrm{4}} \left({x}\right)−{f}_{\mathrm{6}} \left({x}\right)=\frac{\mathrm{1}}{\mathrm{4}}\left(\mathrm{sin}\:^{\mathrm{4}} {x}\:+\mathrm{cos}\:^{\mathrm{4}} {x}\right)−\frac{\mathrm{1}}{\mathrm{6}}\left(\mathrm{sin}\:^{\mathrm{6}} {x}+\mathrm{cos}\:^{\mathrm{6}} {x}\right) \\ $$$$\:\:\:=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{sin}\:^{\mathrm{4}} {x}\left(\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{3}}\mathrm{sin}\:^{\mathrm{2}} {x}\right)+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cos}\:^{\mathrm{4}} {x}\left(\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{3}}\mathrm{cos}\:^{\mathrm{2}} {x}\right) \\ $$

Question Number 187302    Answers: 0   Comments: 0

f(4)=44, f(m)=52,f(l)=−33 l,m are positive integers such that 4<m<l and f(x) is a polynomial with integer coefficients. Find l+m.

$${f}\left(\mathrm{4}\right)=\mathrm{44},\:{f}\left({m}\right)=\mathrm{52},{f}\left({l}\right)=−\mathrm{33} \\ $$$${l},{m}\:\mathrm{are}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{such}\:\mathrm{that}\:\mathrm{4}<{m}<{l} \\ $$$$\mathrm{and}\:{f}\left({x}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{polynomial}\:\mathrm{with}\:\mathrm{integer}\:\mathrm{coefficients}. \\ $$$$\mathrm{Find}\:{l}+{m}. \\ $$

Question Number 187301    Answers: 2   Comments: 0

Question Number 187300    Answers: 0   Comments: 0

Find the number of integral solutions of (p+q)(q+r)(r+p)=8pqr+2

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{integral}\:\mathrm{solutions}\:\mathrm{of} \\ $$$$\left({p}+{q}\right)\left({q}+{r}\right)\left({r}+{p}\right)=\mathrm{8}{pqr}+\mathrm{2} \\ $$

Question Number 187298    Answers: 0   Comments: 3

if 2^y =25 and 5^x =16, find x+y

$${if}\: \\ $$$$\mathrm{2}^{{y}} =\mathrm{25}\:{and}\:\mathrm{5}^{{x}} =\mathrm{16},\:{find}\:{x}+{y} \\ $$

Question Number 187296    Answers: 0   Comments: 1

Prove that for n≥4, S_n = Σ_(k=1) ^n k^(k!) is never a perfect cube.

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{for}\:\mathrm{n}\geqslant\mathrm{4},\:\mathrm{S}_{{n}} =\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{k}^{{k}!} \:\mathrm{is}\:\mathrm{never}\:\mathrm{a}\:\mathrm{perfect}\:\mathrm{cube}. \\ $$

Question Number 187295    Answers: 0   Comments: 2

solve: ((√(4+(√4))^x )) +[(√(4−(√(4]^x )))) =4^x

$${solve}: \\ $$$$\left(\sqrt{\left.\mathrm{4}+\sqrt{\mathrm{4}}\right)\:^{{x}} }\:+\left[\sqrt{\mathrm{4}−\sqrt{\left.\mathrm{4}\right]^{{x}} }}\:=\mathrm{4}^{{x}} \right.\right. \\ $$

Question Number 187294    Answers: 0   Comments: 1

solve: −1−[x^x −5x+6]^x =1

$${solve}: \\ $$$$−\mathrm{1}−\left[{x}^{{x}} −\mathrm{5}{x}+\mathrm{6}\right]^{{x}} =\mathrm{1} \\ $$

Question Number 187288    Answers: 1   Comments: 0

logx+x!=2 x=?

$${logx}+{x}!=\mathrm{2} \\ $$$${x}=? \\ $$

Question Number 187284    Answers: 1   Comments: 2

Question Number 187270    Answers: 3   Comments: 0

lim_(x→π) ((cos x + sin(2x) +1)/(x^2 −π^2 )) = A. (1/(2π)) C. 1 B. (1/π) D. Does Not exist

$$\underset{{x}\rightarrow\pi} {\mathrm{lim}}\frac{\mathrm{cos}\:{x}\:+\:\mathrm{sin}\left(\mathrm{2}{x}\right)\:+\mathrm{1}}{{x}^{\mathrm{2}} −\pi^{\mathrm{2}} }\:= \\ $$$$\mathrm{A}.\:\frac{\mathrm{1}}{\mathrm{2}\pi}\:\:\:\:\:\:\:\:\:{C}.\:\mathrm{1} \\ $$$${B}.\:\frac{\mathrm{1}}{\pi}\:\:\:\:\:\:\:\:\:{D}.\:\mathrm{Does}\:\mathrm{Not}\:\mathrm{exist} \\ $$

Question Number 187257    Answers: 0   Comments: 2

Question Number 187255    Answers: 1   Comments: 0

Find the directional derivatives of the function f(x,y,z)=2x^2 +3y^2 +z^2 at the point p(2,1,3)

$${Find}\:{the}\:{directional}\:{derivatives}\:{of}\:{the} \\ $$$${function}\: \\ $$$${f}\left({x},\mathrm{y},\mathrm{z}\right)=\mathrm{2}{x}^{\mathrm{2}} +\mathrm{3}{y}^{\mathrm{2}} +{z}^{\mathrm{2}} \:{at}\:{the}\:{point}\:{p}\left(\mathrm{2},\mathrm{1},\mathrm{3}\right) \\ $$

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