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AlgebraQuestion and Answers: Page 57

Question Number 200424 Answers: 0 Comments: 4

Question Number 200417 Answers: 1 Comments: 0

find: Ω = ∫_1 ^( ∞) ((√x)/((1 + x^2 ))) dx = ?

$$\mathrm{find}:\:\:\:\Omega\:=\:\int_{\mathrm{1}} ^{\:\infty} \:\frac{\sqrt{\mathrm{x}}}{\left(\mathrm{1}\:+\:\mathrm{x}^{\mathrm{2}} \right)}\:\mathrm{dx}\:=\:? \\ $$

Question Number 200415 Answers: 1 Comments: 0

Question Number 200412 Answers: 1 Comments: 0

If (√((x-6)^2 + (y+1)^2 )) + (√((x-9)^2 + (y+5)^2 )) find: minumum = ?

$$\mathrm{If} \\ $$$$\sqrt{\left(\mathrm{x}-\mathrm{6}\right)^{\mathrm{2}} \:+\:\left(\mathrm{y}+\mathrm{1}\right)^{\mathrm{2}} }\:+\:\sqrt{\left(\mathrm{x}-\mathrm{9}\right)^{\mathrm{2}} \:+\:\left(\mathrm{y}+\mathrm{5}\right)^{\mathrm{2}} } \\ $$$$ \\ $$$$\mathrm{find}:\:\:\:\mathrm{minumum}\:=\:? \\ $$

Question Number 200388 Answers: 1 Comments: 0

find all values for k such that the eq. x^3 −13x+k=0 has three integer roots.

$${find}\:{all}\:{values}\:{for}\:{k}\:{such}\:{that}\:{the}\:{eq}. \\ $$$${x}^{\mathrm{3}} −\mathrm{13}{x}+{k}=\mathrm{0}\:{has}\:{three}\:{integer}\:{roots}. \\ $$

Question Number 200353 Answers: 3 Comments: 0

If the roots of x^3 +3px^2 +3qx+r=0 are in harmonic progression, then prove that 2q^3 =r(3pq−r)

$$\mathrm{If}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{x}^{\mathrm{3}} +\mathrm{3px}^{\mathrm{2}} +\mathrm{3qx}+\mathrm{r}=\mathrm{0}\:\mathrm{are} \\ $$$$\mathrm{in}\:\mathrm{harmonic}\:\mathrm{progression},\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\mathrm{2q}^{\mathrm{3}} =\mathrm{r}\left(\mathrm{3pq}−\mathrm{r}\right) \\ $$

Question Number 200351 Answers: 1 Comments: 0

Solve the equation x^3 −12x^2 −6x−10=0 by cardon′s method

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{x}^{\mathrm{3}} −\mathrm{12x}^{\mathrm{2}} −\mathrm{6x}−\mathrm{10}=\mathrm{0}\: \\ $$$$\mathrm{by}\:\mathrm{cardon}'\mathrm{s}\:\mathrm{method} \\ $$

Question Number 200350 Answers: 0 Comments: 1

Question Number 200336 Answers: 1 Comments: 0

Question Number 200330 Answers: 4 Comments: 0

2^x − 3^x = (√(6^x − 9^x )) find: x = ?

$$\mathrm{2}^{\boldsymbol{\mathrm{x}}} \:−\:\mathrm{3}^{\boldsymbol{\mathrm{x}}} \:=\:\sqrt{\mathrm{6}^{\boldsymbol{\mathrm{x}}} \:−\:\mathrm{9}^{\boldsymbol{\mathrm{x}}} } \\ $$$$\mathrm{find}:\:\:\:\mathrm{x}\:=\:? \\ $$

Question Number 200242 Answers: 3 Comments: 0

Question Number 200240 Answers: 0 Comments: 0

the local minimum value of the function f(x,y) = x^2 +xy+y^2 −3x−3y+11

$$\mathrm{the}\:\mathrm{local}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{function} \\ $$$${f}\left({x},{y}\right)\:=\:{x}^{\mathrm{2}} +{xy}+{y}^{\mathrm{2}} −\mathrm{3}{x}−\mathrm{3}{y}+\mathrm{11} \\ $$

Question Number 200186 Answers: 2 Comments: 0

Question Number 200169 Answers: 1 Comments: 0

Rationalise the deniminator of the following fraction: (1/( (√6) − (√3) + (√2) + 1)) = ?

$$\mathrm{Rationalise}\:\mathrm{the}\:\mathrm{deniminator}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{following}\:\mathrm{fraction}: \\ $$$$\frac{\mathrm{1}}{\:\sqrt{\mathrm{6}}\:−\:\sqrt{\mathrm{3}}\:+\:\sqrt{\mathrm{2}}\:+\:\mathrm{1}}\:=\:? \\ $$

Question Number 200168 Answers: 1 Comments: 2

If f(x) = 2^x + 86 and g(x) = 3x^2 + x − 4 Then find: g[f^(−1) (g(14))] = ?

$$\mathrm{If}\:\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\mathrm{2}^{\boldsymbol{\mathrm{x}}} \:+\:\mathrm{86}\:\:\mathrm{and}\:\:\mathrm{g}\left(\mathrm{x}\right)\:=\:\mathrm{3x}^{\mathrm{2}} \:+\:\mathrm{x}\:−\:\mathrm{4} \\ $$$$\mathrm{Then}\:\mathrm{find}:\:\:\mathrm{g}\left[\mathrm{f}^{−\mathrm{1}} \left(\mathrm{g}\left(\mathrm{14}\right)\right)\right]\:=\:? \\ $$

Question Number 200167 Answers: 3 Comments: 0

Given f:R→R is a quadratic polynomial f(1) = 1 , f(2) = (1/2) and f(3) = (1/3) Find: f(4) = ?

$$\mathrm{Given}\:\:\:\mathrm{f}:\mathbb{R}\rightarrow\mathbb{R}\:\:\mathrm{is}\:\mathrm{a}\:\mathrm{quadratic}\:\mathrm{polynomial} \\ $$$$\mathrm{f}\left(\mathrm{1}\right)\:=\:\mathrm{1}\:,\:\mathrm{f}\left(\mathrm{2}\right)\:=\:\frac{\mathrm{1}}{\mathrm{2}}\:\:\mathrm{and}\:\:\mathrm{f}\left(\mathrm{3}\right)\:=\:\frac{\mathrm{1}}{\mathrm{3}} \\ $$$$\mathrm{Find}:\:\:\mathrm{f}\left(\mathrm{4}\right)\:=\:? \\ $$

Question Number 200139 Answers: 2 Comments: 0

If x : y : z = (1/7) : (1/3) : (1/(21)) 5x − 2y + z = 16 Find: y = ?

$$\mathrm{If} \\ $$$$\mathrm{x}\::\:\mathrm{y}\::\:\mathrm{z}\:=\:\frac{\mathrm{1}}{\mathrm{7}}\::\:\frac{\mathrm{1}}{\mathrm{3}}\::\:\frac{\mathrm{1}}{\mathrm{21}} \\ $$$$\mathrm{5x}\:−\:\mathrm{2y}\:+\:\mathrm{z}\:=\:\mathrm{16} \\ $$$$ \\ $$$$\mathrm{Find}:\:\:\:\mathrm{y}\:=\:? \\ $$

Question Number 200134 Answers: 0 Comments: 0

Question Number 200087 Answers: 1 Comments: 2

if ω ≠ 1 is a root of unity aand z is a complex number such that ∣z∣ = 1 then ∣((2+3ω+4zω^2 )/(4ω+3ω^2 z+2z))∣= ?

$$\:\:\mathrm{if}\:\omega\:\neq\:\mathrm{1}\:\mathrm{is}\:\mathrm{a}\:\mathrm{root}\:\mathrm{of}\:\mathrm{unity}\:\mathrm{aand}\:\mathrm{z}\:\mathrm{is}\:\mathrm{a}\: \\ $$$$\mathrm{complex}\:\mathrm{number}\:\mathrm{such}\:\mathrm{that}\:\mid{z}\mid\:=\:\mathrm{1}\:\mathrm{then} \\ $$$$\:\:\mid\frac{\mathrm{2}+\mathrm{3}\omega+\mathrm{4}{z}\omega^{\mathrm{2}} }{\mathrm{4}\omega+\mathrm{3}\omega^{\mathrm{2}} {z}+\mathrm{2}{z}}\mid=\:? \\ $$

Question Number 200066 Answers: 1 Comments: 0

If ((1 + x)/( (√3))) = 3 find x + (1/x) − 1 = ?

$$\mathrm{If}\:\:\:\frac{\mathrm{1}\:+\:\mathrm{x}}{\:\sqrt{\mathrm{3}}}\:=\:\mathrm{3}\:\:\:\mathrm{find}\:\:\:\mathrm{x}\:+\:\frac{\mathrm{1}}{\mathrm{x}}\:−\:\mathrm{1}\:=\:? \\ $$

Question Number 200035 Answers: 1 Comments: 2

Question Number 200025 Answers: 3 Comments: 0

Question Number 200019 Answers: 1 Comments: 0

Question Number 200013 Answers: 0 Comments: 3

Question Number 200012 Answers: 0 Comments: 0

Question Number 199932 Answers: 2 Comments: 0

1. If 3 ∙ ab^(−) + bc^(−) = 115 Find: max(a+b+c)=? 2. a,b,c∈N If (a/2) + (b/3) = (c/4) Find: min(a+b+c)=?

$$\mathrm{1}. \\ $$$$\mathrm{If}\:\:\:\mathrm{3}\:\centerdot\:\overline {\mathrm{ab}}\:+\:\overline {\mathrm{bc}}\:=\:\mathrm{115} \\ $$$$\mathrm{Find}:\:\:\:\mathrm{max}\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right)=? \\ $$$$ \\ $$$$\mathrm{2}. \\ $$$$\mathrm{a},\mathrm{b},\mathrm{c}\in\mathbb{N} \\ $$$$\mathrm{If}\:\:\:\frac{\mathrm{a}}{\mathrm{2}}\:\:+\:\:\frac{\mathrm{b}}{\mathrm{3}}\:\:=\:\:\frac{\mathrm{c}}{\mathrm{4}} \\ $$$$\mathrm{Find}:\:\:\:\mathrm{min}\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right)=? \\ $$

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