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AlgebraQuestion and Answers: Page 235

Question Number 113510 Answers: 1 Comments: 0

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Question Number 113451 Answers: 3 Comments: 0

If 2x=a^n +a^(−n) and 2y=a^n −a^(−n) calculate the value of x^2 −y^(2 ) in its simplest form

$$\mathrm{If}\:\mathrm{2x}=\mathrm{a}^{\mathrm{n}} +\mathrm{a}^{−\mathrm{n}} \:\mathrm{and}\:\mathrm{2y}=\mathrm{a}^{\mathrm{n}} −\mathrm{a}^{−\mathrm{n}} \:\mathrm{calculate} \\ $$$$\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x}^{\mathrm{2}} −\mathrm{y}^{\mathrm{2}\:} \:\mathrm{in}\:\mathrm{its}\:\mathrm{simplest}\:\mathrm{form} \\ $$

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Question Number 113372 Answers: 1 Comments: 2

2^a +2^b +2^c +2^d =57, find a+b+c+d. a≠b≠c≠d and a,b,c,d are positive integers.

$$\mathrm{2}^{\mathrm{a}} +\mathrm{2}^{\mathrm{b}} +\mathrm{2}^{\mathrm{c}} +\mathrm{2}^{\mathrm{d}} =\mathrm{57},\:\mathrm{find}\:\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}. \\ $$$$\mathrm{a}\neq\mathrm{b}\neq\mathrm{c}\neq\mathrm{d}\:\mathrm{and}\:\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\:\mathrm{are}\:\mathrm{positive} \\ $$$$\mathrm{integers}. \\ $$

Question Number 113272 Answers: 2 Comments: 0

Solve the following equations: a)(x^2 −a)^2 −6x^2 +4x+2a=0 b)x^4 −4x^3 −10x^3 +37x−14=0,if it known that the left−hand side of the equation can be decomposed into factors with integral coefficients.

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{following}\:\mathrm{equations}: \\ $$$$\left.\mathrm{a}\right)\left(\mathrm{x}^{\mathrm{2}} −\mathrm{a}\right)^{\mathrm{2}} −\mathrm{6x}^{\mathrm{2}} +\mathrm{4x}+\mathrm{2a}=\mathrm{0} \\ $$$$\left.\mathrm{b}\right)\mathrm{x}^{\mathrm{4}} −\mathrm{4x}^{\mathrm{3}} −\mathrm{10x}^{\mathrm{3}} +\mathrm{37x}−\mathrm{14}=\mathrm{0},\mathrm{if}\:\mathrm{it} \\ $$$$\mathrm{known}\:\mathrm{that}\:\mathrm{the}\:\mathrm{left}−\mathrm{hand}\:\mathrm{side}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equation}\:\mathrm{can}\:\mathrm{be}\:\mathrm{decomposed}\:\mathrm{into} \\ $$$$\mathrm{factors}\:\mathrm{with}\:\mathrm{integral}\:\mathrm{coefficients}. \\ $$

Question Number 113241 Answers: 1 Comments: 0

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Question Number 113188 Answers: 1 Comments: 0

proporsed by m.n july 1790 ∫_0 ^π ln(1−(1/2)sinx)dx

$${proporsed}\:{by}\:{m}.{n}\:{july}\:\mathrm{1790} \\ $$$$\int_{\mathrm{0}} ^{\pi} \mathrm{ln}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{sin}{x}\right){dx} \\ $$

Question Number 113154 Answers: 1 Comments: 0

If ⌊x + (√5)⌋ = ⌊x⌋ + ⌊5⌋ then ⌊x⌋ − x would be greater than (a) (√5) − 2 (b) (√5) − 3 (c) (√5) (d) (√5) + 1 (e) (√5) − 1

$$\mathrm{If}\:\:\:\:\:\:\lfloor\mathrm{x}\:\:+\:\:\sqrt{\mathrm{5}}\rfloor\:\:\:=\:\:\:\lfloor\mathrm{x}\rfloor\:\:+\:\:\lfloor\mathrm{5}\rfloor \\ $$$$\mathrm{then}\:\:\:\:\:\lfloor\mathrm{x}\rfloor\:\:−\:\:\mathrm{x}\:\:\:\:\mathrm{would}\:\mathrm{be}\:\mathrm{greater}\:\mathrm{than} \\ $$$$\left(\mathrm{a}\right)\:\:\:\:\sqrt{\mathrm{5}}\:\:−\:\:\mathrm{2}\:\:\:\:\:\:\:\left(\mathrm{b}\right)\:\:\:\:\sqrt{\mathrm{5}}\:\:−\:\:\mathrm{3}\:\:\:\:\:\:\left(\mathrm{c}\right)\:\:\:\:\sqrt{\mathrm{5}}\:\:\:\:\:\:\:\:\left(\mathrm{d}\right)\:\:\:\:\sqrt{\mathrm{5}}\:\:+\:\:\mathrm{1}\:\:\:\:\:\:\:\left(\mathrm{e}\right)\:\:\:\sqrt{\mathrm{5}}\:\:−\:\:\mathrm{1} \\ $$

Question Number 113114 Answers: 1 Comments: 1

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Question Number 112992 Answers: 1 Comments: 1

Question Number 112805 Answers: 1 Comments: 0

If x=(((√(a+2b))+ (√(a−2b)))/( (√(a+2b))− (√(a−2b)))), then the value of bx^2 −ax+b is

$$\mathrm{If}\:\mathrm{x}=\frac{\sqrt{\mathrm{a}+\mathrm{2b}}+\:\sqrt{\mathrm{a}−\mathrm{2b}}}{\:\sqrt{\mathrm{a}+\mathrm{2b}}−\:\sqrt{\mathrm{a}−\mathrm{2b}}},\:\mathrm{then}\:\mathrm{the}\:\mathrm{value} \\ $$$$\mathrm{of}\:\mathrm{bx}^{\mathrm{2}} −\mathrm{ax}+\mathrm{b}\:\mathrm{is} \\ $$

Question Number 112804 Answers: 1 Comments: 0

If ((97)/(19))= a+(1/(b+(1/c))), where a,b and c are positive integers, then what is the sum of a,b and c?

$$\mathrm{If}\:\frac{\mathrm{97}}{\mathrm{19}}=\:\mathrm{a}+\frac{\mathrm{1}}{\mathrm{b}+\frac{\mathrm{1}}{\mathrm{c}}},\:\mathrm{where}\:\mathrm{a},\mathrm{b}\:\mathrm{and}\:\mathrm{c}\:\mathrm{are} \\ $$$$\mathrm{positive}\:\mathrm{integers},\:\mathrm{then}\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum} \\ $$$$\mathrm{of}\:\mathrm{a},\mathrm{b}\:\mathrm{and}\:\mathrm{c}? \\ $$

Question Number 112803 Answers: 1 Comments: 0

If x+y+z=0, then the value of ((x^2 +y^2 +z^2 )/(x^2 −yz)) is

$$\mathrm{If}\:\mathrm{x}+\mathrm{y}+\mathrm{z}=\mathrm{0},\: \\ $$$$\mathrm{then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\frac{\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{z}^{\mathrm{2}} }{\mathrm{x}^{\mathrm{2}} −\mathrm{yz}}\:\mathrm{is} \\ $$

Question Number 112802 Answers: 3 Comments: 0

If xy+yz+zx=0, then ((1/(x^2 −yz))+(1/(y^2 −zx))+(1/(z^2 −xy)))(x,y,z ≠ 0) is equal to

$$\mathrm{If}\:\mathrm{xy}+\mathrm{yz}+\mathrm{zx}=\mathrm{0},\:\mathrm{then} \\ $$$$ \\ $$$$\left(\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} −\mathrm{yz}}+\frac{\mathrm{1}}{\mathrm{y}^{\mathrm{2}} −\mathrm{zx}}+\frac{\mathrm{1}}{\mathrm{z}^{\mathrm{2}} −\mathrm{xy}}\right)\left(\mathrm{x},\mathrm{y},\mathrm{z}\:\neq\:\mathrm{0}\right)\:\mathrm{is} \\ $$$$\mathrm{equal}\:\mathrm{to} \\ $$$$ \\ $$

Question Number 112796 Answers: 0 Comments: 2

For any real numbers x,y and z if x+y+z=2, then prove that xyz≥8(1−x)(1−y)(1−z).

$${For}\:{any}\:{real}\:{numbers}\:{x},{y}\:{and}\:{z}\:{if} \\ $$$${x}+{y}+{z}=\mathrm{2},\:{then}\:{prove}\:{that} \\ $$$${xyz}\geqslant\mathrm{8}\left(\mathrm{1}−{x}\right)\left(\mathrm{1}−{y}\right)\left(\mathrm{1}−{z}\right). \\ $$

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If a+b=5, a^2 +b^2 =13, the value of a−b (where a>b) is

$$\mathrm{If}\:\mathrm{a}+\mathrm{b}=\mathrm{5},\:\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} =\mathrm{13},\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mathrm{a}−\mathrm{b}\:\left(\mathrm{where}\:\mathrm{a}>\mathrm{b}\right)\:\mathrm{is} \\ $$

Question Number 112624 Answers: 3 Comments: 0

Minimum value of x^2 +(1/(x^2 +1))−3 is

$$\mathrm{Minimum}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} +\mathrm{1}}−\mathrm{3}\:\mathrm{is} \\ $$

Question Number 112528 Answers: 0 Comments: 2

If ((n−7)/(n−17)) and ((n−14)/(n−24)) are integers numbers n=?

$${If}\:\:\:\:\:\frac{{n}−\mathrm{7}}{{n}−\mathrm{17}}\:\:\:\:\:\:\:{and}\:\:\:\:\frac{{n}−\mathrm{14}}{{n}−\mathrm{24}}\:\:\:{are}\:{integers}\:{numbers} \\ $$$${n}=? \\ $$

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