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Question Number 56169 Answers: 1 Comments: 0

∫^1 _(−∞) (a+bi)^x dx=?

$$\underset{−\infty} {\int}^{\mathrm{1}} \left({a}+{b}\mathrm{i}\right)^{{x}} {dx}=? \\ $$

Question Number 56166 Answers: 0 Comments: 1

Question Number 56165 Answers: 1 Comments: 0

Question Number 56147 Answers: 1 Comments: 6

if ∫_( 1) ^( 2) f(x) dx = (√( 2 )), then ∫_( 1) ^( 4) (1/((√( x )) )) f(x) dx is ?? please help me Sir. I′ve been trying this for 2 days and getting stuck.

$$\mathrm{if}\:\underset{\:\:\mathrm{1}} {\overset{\:\:\mathrm{2}} {\int}}\:{f}\left({x}\right)\:{dx}\:=\:\sqrt{\:\mathrm{2}\:},\:\mathrm{then}\:\underset{\:\:\mathrm{1}} {\overset{\:\:\mathrm{4}} {\int}}\:\frac{\mathrm{1}}{\sqrt{\:{x}\:}\:}\:{f}\left({x}\right)\:{dx} \\ $$$$\:\mathrm{is}\:?? \\ $$$$\:\:\mathrm{please}\:\mathrm{help}\:\mathrm{me}\:\mathrm{Sir}.\:\mathrm{I}'\mathrm{ve}\:\mathrm{been}\:\mathrm{trying} \\ $$$$\:\:\mathrm{this}\:\mathrm{for}\:\mathrm{2}\:\mathrm{days}\:\mathrm{and}\:\mathrm{getting}\:\mathrm{stuck}. \\ $$$$ \\ $$$$ \\ $$

Question Number 56146 Answers: 1 Comments: 1

Given complex number z_1 , z_2 , and z_3 satiesfied z_1 +z_2 +z_3 =0 and ∣z_1 ∣=∣z_2 ∣=∣z_3 ∣=1. Prove that z_1 ^2 +z_2 ^2 +z_3 ^2 =0

$$\mathrm{Given}\:\mathrm{complex}\:\mathrm{number} \\ $$$${z}_{\mathrm{1}} ,\:{z}_{\mathrm{2}} ,\:\mathrm{and}\:{z}_{\mathrm{3}} \:\mathrm{satiesfied}\:{z}_{\mathrm{1}} +{z}_{\mathrm{2}} +{z}_{\mathrm{3}} =\mathrm{0} \\ $$$$\mathrm{and}\:\mid{z}_{\mathrm{1}} \mid=\mid{z}_{\mathrm{2}} \mid=\mid{z}_{\mathrm{3}} \mid=\mathrm{1}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$${z}_{\mathrm{1}} ^{\mathrm{2}} +{z}_{\mathrm{2}} ^{\mathrm{2}} +{z}_{\mathrm{3}} ^{\mathrm{2}} =\mathrm{0} \\ $$

Question Number 56145 Answers: 1 Comments: 0

find residu of function f(z)=(e^(1/z) /(z^2 +1)) in z=0

$$\mathrm{find}\:\mathrm{residu}\:\mathrm{of}\:\mathrm{function} \\ $$$${f}\left({z}\right)=\frac{{e}^{\frac{\mathrm{1}}{{z}}} }{{z}^{\mathrm{2}} +\mathrm{1}}\:\mathrm{in}\:{z}=\mathrm{0} \\ $$

Question Number 56144 Answers: 1 Comments: 0

calculate (i−1)^(49) (cos (π/(40))+i sin (π/(40)))^(10)

$$\mathrm{calculate}\:\left({i}−\mathrm{1}\right)^{\mathrm{49}} \left(\mathrm{cos}\:\frac{\pi}{\mathrm{40}}+{i}\:\mathrm{sin}\:\frac{\pi}{\mathrm{40}}\right)^{\mathrm{10}} \\ $$

Question Number 72830 Answers: 2 Comments: 0

The value of x satisfying the inequalities (((2x−1)(x−1)^4 (x−2)^4 )/((x−2)(x−4)^4 ))≤ 0

$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:{x}\:\mathrm{satisfying}\:\mathrm{the}\: \\ $$$$\mathrm{inequalities}\:\frac{\left(\mathrm{2}{x}−\mathrm{1}\right)\left({x}−\mathrm{1}\right)^{\mathrm{4}} \left({x}−\mathrm{2}\right)^{\mathrm{4}} }{\left({x}−\mathrm{2}\right)\left({x}−\mathrm{4}\right)^{\mathrm{4}} }\leqslant\:\mathrm{0} \\ $$

Question Number 72829 Answers: 0 Comments: 0

Σ_(k=m) ^n ^k C_r equals

$$\underset{{k}={m}} {\overset{{n}} {\sum}}\:^{{k}} {C}_{{r}} \:\mathrm{equals} \\ $$

Question Number 56142 Answers: 0 Comments: 0

If n is even and rth term has the greatest coefficient in the binomial expansion of (1+x)^n , then

$$\mathrm{If}\:{n}\:\mathrm{is}\:\mathrm{even}\:\mathrm{and}\:{r}\mathrm{th}\:\mathrm{term}\:\mathrm{has}\:\mathrm{the}\:\mathrm{greatest} \\ $$$$\mathrm{coefficient}\:\mathrm{in}\:\mathrm{the}\:\mathrm{binomial}\:\mathrm{expansion}\:\mathrm{of} \\ $$$$\left(\mathrm{1}+{x}\right)^{{n}} ,\:\mathrm{then} \\ $$

Question Number 56141 Answers: 1 Comments: 0

If (1+x)^n =C_0 +C_1 x+C_2 x^2 +...+C_n x^n , then for n odd, C_0 ^2 −C_1 ^2 +C_2 ^2 −C_3 ^2 +...+(−1)^n C_n ^2 is equal to

$$\mathrm{If}\:\left(\mathrm{1}+{x}\right)^{{n}} ={C}_{\mathrm{0}} +{C}_{\mathrm{1}} {x}+{C}_{\mathrm{2}} {x}^{\mathrm{2}} +...+{C}_{{n}} {x}^{{n}} ,\:\mathrm{then} \\ $$$$\mathrm{for}\:{n}\:\mathrm{odd},\:{C}_{\mathrm{0}} \:^{\mathrm{2}} −{C}_{\mathrm{1}} \:^{\mathrm{2}} +{C}_{\mathrm{2}} \:^{\mathrm{2}} −{C}_{\mathrm{3}} \:^{\mathrm{2}} +...+\left(−\mathrm{1}\right)^{{n}} {C}_{{n}} \:^{\mathrm{2}} \\ $$$$\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$

Question Number 56140 Answers: 1 Comments: 0

The value of 2 C_0 +(2^2 /2)C_1 +(2^3 /3)C_2 +(2^4 /4)C_3 +...+(2^(11) /(11))C_(10) is

$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\:\: \\ $$$$\mathrm{2}\:{C}_{\mathrm{0}} +\frac{\mathrm{2}^{\mathrm{2}} }{\mathrm{2}}{C}_{\mathrm{1}} +\frac{\mathrm{2}^{\mathrm{3}} }{\mathrm{3}}{C}_{\mathrm{2}} +\frac{\mathrm{2}^{\mathrm{4}} }{\mathrm{4}}{C}_{\mathrm{3}} +...+\frac{\mathrm{2}^{\mathrm{11}} }{\mathrm{11}}{C}_{\mathrm{10}} \:\:\mathrm{is} \\ $$

Question Number 56139 Answers: 1 Comments: 0

If x+y=1, then Σ_(r=0) ^n r^2 ^n C_r x^r y^(n−r) equals

$$\mathrm{If}\:{x}+{y}=\mathrm{1},\:\mathrm{then}\:\:\underset{{r}=\mathrm{0}} {\overset{{n}} {\sum}}\:{r}^{\mathrm{2}} \:\:^{{n}} {C}_{{r}} \:{x}^{{r}} \:{y}^{{n}−{r}} \:\mathrm{equals} \\ $$

Question Number 56138 Answers: 1 Comments: 2

The pisitive value of a so that the coefficient of x^5 and x^(15) are equal in the expansion of (x^2 + (a/x^3 ))^(10)

$$\mathrm{The}\:\mathrm{pisitive}\:\mathrm{value}\:\mathrm{of}\:\:{a}\:\mathrm{so}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{coefficient}\:\mathrm{of}\:{x}^{\mathrm{5}} \:\mathrm{and}\:{x}^{\mathrm{15}} \:\mathrm{are}\:\mathrm{equal}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{expansion}\:\mathrm{of}\:\left({x}^{\mathrm{2}} +\:\frac{{a}}{{x}^{\mathrm{3}} }\right)^{\mathrm{10}} \\ $$

Question Number 56137 Answers: 1 Comments: 1

If (1+2x+x^2 )^n = Σ_(r=0) ^(2n) a_r x^r , then a_r =

$$\mathrm{If}\:\left(\mathrm{1}+\mathrm{2}{x}+{x}^{\mathrm{2}} \right)^{{n}} \:=\:\underset{{r}=\mathrm{0}} {\overset{\mathrm{2}{n}} {\sum}}\:{a}_{{r}} \:{x}^{{r}} ,\:\mathrm{then}\:{a}_{{r}} = \\ $$

Question Number 56124 Answers: 2 Comments: 0

Question Number 56120 Answers: 1 Comments: 0

Question Number 56107 Answers: 2 Comments: 1

∫_(−1) ^0 ∣x sin (πx)∣ dx

$$\int_{−\mathrm{1}} ^{\mathrm{0}} \:\mid{x}\:\mathrm{sin}\:\left(\pi{x}\right)\mid\:{dx} \\ $$

Question Number 56104 Answers: 2 Comments: 0

((1/(27)))^z +9^z .9=(1/3^(−4) ) solve for z

$$\left(\frac{\mathrm{1}}{\mathrm{27}}\right)^{\mathrm{z}} +\mathrm{9}^{\mathrm{z}} .\mathrm{9}=\frac{\mathrm{1}}{\mathrm{3}^{−\mathrm{4}} } \\ $$$$\mathrm{solve}\:\mathrm{for}\:\mathrm{z} \\ $$

Question Number 56095 Answers: 2 Comments: 0

The coefficient of x^5 in the expansion of (1+x)^(21) +(1+x)^(22) +...+(1+x)^(30) is

$$\mathrm{The}\:\mathrm{coefficient}\:\mathrm{of}\:{x}^{\mathrm{5}} \:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of} \\ $$$$\left(\mathrm{1}+{x}\right)^{\mathrm{21}} \:+\left(\mathrm{1}+{x}\right)^{\mathrm{22}} +...+\left(\mathrm{1}+{x}\right)^{\mathrm{30}} \:\:\mathrm{is} \\ $$

Question Number 56092 Answers: 2 Comments: 0

How to rationalize a denominator in a fraction? Like this. (((x)^(1/3) +2)/((x)^(1/3) −2))

$${How}\:{to}\:{rationalize}\:{a}\:{denominator}\:{in} \\ $$$${a}\:{fraction}?\:{Like}\:{this}. \\ $$$$\frac{\sqrt[{\mathrm{3}}]{{x}}+\mathrm{2}}{\sqrt[{\mathrm{3}}]{{x}}−\mathrm{2}}\: \\ $$

Question Number 56090 Answers: 0 Comments: 5

Question Number 56075 Answers: 1 Comments: 0

Prove that If a set consist of n number of terms then its Power Set would contain 2^n number of terms. [Use formulas of sequence and series]

$${Prove}\:{that}\:{If}\:{a}\:{set}\:{consist}\:{of}\:{n}\:{number}\:{of} \\ $$$${terms}\:{then}\:{its}\:{Power}\:{Set}\:{would}\:{contain} \\ $$$$\mathrm{2}^{{n}} \:{number}\:{of}\:{terms}. \\ $$$$\left[{Use}\:{formulas}\:{of}\:{sequence}\:{and}\:{series}\right] \\ $$

Question Number 56074 Answers: 2 Comments: 2

In a sequence if r^(th) term is given by T_r =2×T_(r−1) +1 then give it′s n^(th) term in terms of it′s 1^(st) term [Given : T_1 =2]

$${In}\:{a}\:{sequence}\:{if}\:{r}^{{th}} \:{term}\:{is}\:{given}\:{by} \\ $$$${T}_{{r}} =\mathrm{2}×{T}_{{r}−\mathrm{1}} +\mathrm{1} \\ $$$${then}\:{give}\:{it}'{s}\:{n}^{{th}} \:{term}\:{in}\:{terms}\:{of}\:\:{it}'{s}\:\mathrm{1}^{{st}} \:{term} \\ $$$$\left[{Given}\::\:\:\:\:\:{T}_{\mathrm{1}} =\mathrm{2}\right] \\ $$

Question Number 56069 Answers: 0 Comments: 5

Find the expansion and the convergemce of the following in the power of x: (i)sinx (ii)ln (1+x) (iii)tan^(−1) x

$${Find}\:{the}\:{expansion}\:{and}\:{the}\:{convergemce} \\ $$$${of}\:{the}\:{following}\:{in}\:{the}\:{power}\:{of}\:{x}: \\ $$$$\left({i}\right){sinx} \\ $$$$\left({ii}\right)\mathrm{ln}\:\left(\mathrm{1}+{x}\right) \\ $$$$\left({iii}\right)\mathrm{tan}^{−\mathrm{1}} {x} \\ $$

Question Number 56061 Answers: 1 Comments: 0

∫_0 ^1 e^(−x^2 ) dx correct to 3 decimal place.

$$\int_{\mathrm{0}} ^{\mathrm{1}} {e}^{−{x}^{\mathrm{2}} } {dx}\:{correct}\:{to}\:\mathrm{3}\:{decimal}\:{place}. \\ $$

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