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Question Number 43023    Answers: 1   Comments: 1

Question Number 43011    Answers: 0   Comments: 2

Question Number 43008    Answers: 1   Comments: 0

(y′)^2 =−1+sin x y=?

$$\left({y}'\right)^{\mathrm{2}} =−\mathrm{1}+\mathrm{sin}\:{x} \\ $$$${y}=? \\ $$

Question Number 43007    Answers: 1   Comments: 1

calculate A_n =Σ_(k=0) ^n k C_n ^k B_n = Σ_(k=0) ^n k^2 C_n ^k C_n =Σ_(k=0) ^n k^3 C_n ^k

$${calculate}\: \\ $$$${A}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:{k}\:{C}_{{n}} ^{{k}} \\ $$$${B}_{{n}} =\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{k}^{\mathrm{2}} \:{C}_{{n}} ^{{k}} \\ $$$${C}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:{k}^{\mathrm{3}} \:{C}_{{n}} ^{{k}} \\ $$

Question Number 43005    Answers: 0   Comments: 19

The base of triangle passes through a fixed point p(a,b) and its sides are respectively bisected at right angles by the line x+y=0 and x=9y if the locus of the third vartex is a circle. then find its equation.

$$\mathrm{The}\:\mathrm{base}\:\mathrm{of}\:\mathrm{triangle}\:\mathrm{passes}\:\mathrm{through} \\ $$$$\mathrm{a}\:\mathrm{fixed}\:\mathrm{point}\:\mathrm{p}\left(\mathrm{a},\mathrm{b}\right)\:\mathrm{and}\:\mathrm{its}\:\mathrm{sides} \\ $$$$\mathrm{are}\:\mathrm{respectively}\:\mathrm{bisected}\:\mathrm{at}\:\mathrm{right}\:\mathrm{angles}\: \\ $$$$\mathrm{by}\:\mathrm{the}\:\mathrm{line}\:\mathrm{x}+\mathrm{y}=\mathrm{0}\:\mathrm{and}\:\mathrm{x}=\mathrm{9y} \\ $$$$\mathrm{if}\:\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:\mathrm{the}\:\mathrm{third}\:\mathrm{vartex}\:\mathrm{is}\:\mathrm{a}\:\mathrm{circle}. \\ $$$$\mathrm{then}\:\mathrm{find}\:\mathrm{its}\:\mathrm{equation}. \\ $$

Question Number 43003    Answers: 1   Comments: 0

let u_n = Σ_(1≤i<j≤n) (1/(√(ij))) 1) find a equivalent of u_n 2)calculate lim_(n→+∞) u_n

$${let}\:{u}_{{n}} =\:\sum_{\mathrm{1}\leqslant{i}<{j}\leqslant{n}} \:\:\frac{\mathrm{1}}{\sqrt{{ij}}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{equivalent}\:{of}\:{u}_{{n}} \\ $$$$\left.\mathrm{2}\right){calculate}\:{lim}_{{n}\rightarrow+\infty} \:{u}_{{n}} \\ $$

Question Number 43001    Answers: 1   Comments: 0

prove that Π_(k=1) ^n (1+(1/k))>1+Σ_(k=1) ^n (1/k)

$${prove}\:{that}\:\prod_{{k}=\mathrm{1}} ^{{n}} \left(\mathrm{1}+\frac{\mathrm{1}}{{k}}\right)>\mathrm{1}+\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}} \\ $$

Question Number 43000    Answers: 2   Comments: 0

prove that (√(2+(√(2+....+(√2))))) =2cos((π/2^n ))

$${prove}\:{that}\:\sqrt{\mathrm{2}+\sqrt{\mathrm{2}+....+\sqrt{\mathrm{2}}}}\:\:=\mathrm{2}{cos}\left(\frac{\pi}{\mathrm{2}^{{n}} }\right) \\ $$

Question Number 42999    Answers: 0   Comments: 1

let f(x)=(√x)+(1/(x−1)) 1) calculate f^((n)) (2) 2) if f(x) =Σ_(n=0) ^∞ a_n (x−2)^n find the sequence a_n

$${let}\:{f}\left({x}\right)=\sqrt{{x}}+\frac{\mathrm{1}}{{x}−\mathrm{1}} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{\left({n}\right)} \left(\mathrm{2}\right) \\ $$$$\left.\mathrm{2}\right)\:{if}\:{f}\left({x}\right)\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:{a}_{{n}} \left({x}−\mathrm{2}\right)^{{n}} \:{find}\:{the} \\ $$$${sequence}\:{a}_{{n}} \\ $$

Question Number 42995    Answers: 0   Comments: 0

Dear Jr inter students use this firmulas Derivates d/dx constant(k)=0 d/dx x^n =n.x^(n−1) d/dx x^2 =2x d/dx (√x) =1/2(√x) d/dx e^x =e^x d/dx a^x =a^x loxa d/dx logx =1/x _ d dx 1/x =−1/x^2

$${Dear}\:\:{Jr}\:{inter}\:{students}\:{use}\:{this}\:{firmulas} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{Derivates}\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:{d}/{dx}\:{constant}\left({k}\right)=\mathrm{0} \\ $$$$\:\:{d}/{dx}\:\:{x}^{{n}} \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:={n}.{x}^{{n}−\mathrm{1}} \\ $$$$\:\:{d}/{dx}\:\:{x}^{\mathrm{2}} \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{2}{x} \\ $$$$\:\:{d}/{dx}\:\:\sqrt{{x}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{1}/\mathrm{2}\sqrt{{x}} \\ $$$$\:\:{d}/{dx}\:\:{e}^{{x}} \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:={e}^{{x}} \:\:\: \\ $$$$\:\:{d}/{dx}\:\:\:{a}^{{x}} \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:={a}^{{x}} \:{loxa} \\ $$$$\:\:{d}/{dx}\:\:{logx}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{1}/{x} \\ $$$$\underset{} {\:}\:{d}\:{dx}\:\:\:\:\mathrm{1}/{x}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=−\mathrm{1}/{x}^{\mathrm{2}} \\ $$$$ \\ $$$$ \\ $$

Question Number 42994    Answers: 0   Comments: 5

∫(√(1+((cos x)/(4tan x))))dx=?

$$\int\sqrt{\mathrm{1}+\frac{\mathrm{cos}\:{x}}{\mathrm{4tan}\:{x}}}{dx}=? \\ $$

Question Number 42988    Answers: 0   Comments: 0

reduce this matrix [(2,3,4,1),(1,7,2,3),((−1),4,2,0),(0,1,1,0) ]

$${reduce}\:{this}\:{matrix}\begin{bmatrix}{\mathrm{2}}&{\mathrm{3}}&{\mathrm{4}}&{\mathrm{1}}\\{\mathrm{1}}&{\mathrm{7}}&{\mathrm{2}}&{\mathrm{3}}\\{−\mathrm{1}}&{\mathrm{4}}&{\mathrm{2}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{1}}&{\mathrm{1}}&{\mathrm{0}}\end{bmatrix} \\ $$

Question Number 42948    Answers: 1   Comments: 5

Question Number 42945    Answers: 2   Comments: 12

∫_0 ^( π/2) (dx/(√(sin x))) = ?

$$\int_{\mathrm{0}} ^{\:\:\pi/\mathrm{2}} \frac{{dx}}{\sqrt{\mathrm{sin}\:{x}}}\:=\:? \\ $$

Question Number 42942    Answers: 2   Comments: 0

Question Number 42934    Answers: 0   Comments: 2

Question Number 42933    Answers: 0   Comments: 1

Suppose that f and g are two functions such that lim_(x→a) g(x) = 0 and lim_(x→a) ((f(x))/(g(x))) exist. Prove that lim_(x→a) f(x) = 0

$$\mathrm{Suppose}\:\mathrm{that}\:{f}\:\mathrm{and}\:{g}\:\mathrm{are}\:\mathrm{two}\:\mathrm{functions}\:\mathrm{such}\:\mathrm{that} \\ $$$$\underset{{x}\rightarrow{a}} {\mathrm{lim}}\:{g}\left({x}\right)\:=\:\mathrm{0}\:\:\:\:\mathrm{and}\:\:\:\:\underset{{x}\rightarrow{a}} {\mathrm{lim}}\:\frac{{f}\left({x}\right)}{{g}\left({x}\right)}\:\:\:\mathrm{exist}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\underset{{x}\rightarrow{a}} {\mathrm{lim}}\:{f}\left({x}\right)\:=\:\mathrm{0} \\ $$

Question Number 42910    Answers: 1   Comments: 0

whatshouldbesubtructedfrom(2a+8b+90)toget(−3a+7b+16)?

$${whatshouldbesubtructedfrom}\left(\mathrm{2}{a}+\mathrm{8}{b}+\mathrm{90}\right){toget}\left(−\mathrm{3}{a}+\mathrm{7}{b}+\mathrm{16}\right)? \\ $$

Question Number 42909    Answers: 1   Comments: 0

How many pairs of (a, b, c, d) so that a! + b! + c! = d! which a, b, c, d ∈ positive integers .

$${How}\:\:{many}\:\:{pairs}\:{of}\:\:\left({a},\:{b},\:{c},\:{d}\right)\:\:{so}\:\:{that} \\ $$$$\:\:\:\:\:\:\:\:\:{a}!\:+\:\:{b}!\:\:+\:\:{c}!\:\:=\:\:{d}! \\ $$$${which}\:\:\:{a},\:{b},\:{c},\:{d}\:\:\in\:\:{positive}\:\:{integers}\:. \\ $$

Question Number 42908    Answers: 1   Comments: 0

Solve 21^a + 28^b = 35^c if a, b, and c are positive integers.

$${Solve}\: \\ $$$$\:\:\:\:\:\mathrm{21}^{{a}} \:+\:\mathrm{28}^{{b}} \:\:=\:\:\mathrm{35}^{{c}} \\ $$$${if} \\ $$$${a},\:{b},\:\:{and}\:\:{c}\:\:{are}\:\:{positive}\:\:{integers}. \\ $$

Question Number 42906    Answers: 3   Comments: 0

x^x +y^y =31 x+y = 5 Find x and y

$${x}^{{x}} +{y}^{{y}} =\mathrm{31} \\ $$$${x}+{y}\:=\:\mathrm{5} \\ $$$${Find}\:{x}\:{and}\:{y} \\ $$

Question Number 42897    Answers: 2   Comments: 1

Question Number 42895    Answers: 1   Comments: 1

Question Number 42878    Answers: 1   Comments: 0

Question Number 42870    Answers: 0   Comments: 0

let 0<x<1 and Γ(x) =∫_0 ^∞ t^(x−1) e^(−t) dt 1) prove that Γ(x).Γ(1−x) =(π/(sin(πx))) (compliments formulae) 2) calculate Γ(n) and Γ(n+(1/2)) with n from N.

$${let}\:\mathrm{0}<{x}<\mathrm{1}\:\:{and}\:\Gamma\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:{t}^{{x}−\mathrm{1}} \:{e}^{−{t}} \:{dt}\: \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\Gamma\left({x}\right).\Gamma\left(\mathrm{1}−{x}\right)\:=\frac{\pi}{{sin}\left(\pi{x}\right)}\:\:\:\left({compliments}\:{formulae}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\Gamma\left({n}\right)\:{and}\:\Gamma\left({n}+\frac{\mathrm{1}}{\mathrm{2}}\right)\:{with}\:{n}\:{from}\:{N}. \\ $$

Question Number 42866    Answers: 2   Comments: 0

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