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AllQuestion and Answers: Page 117

Question Number 211679 Answers: 1 Comments: 1

Inverse root formula: x=((2c)/(−b±(√(b^2 −4ac)))) (1)The“ antiroot formula” is derivedr fom the abovementionedt antiroo formula.

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\mathrm{Inverse}\:\mathrm{root}\:\mathrm{formula}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\boldsymbol{{x}}=\frac{\mathrm{2}\boldsymbol{{c}}}{−\boldsymbol{{b}}\pm\sqrt{\boldsymbol{{b}}^{\mathrm{2}} −\mathrm{4}\boldsymbol{{ac}}}} \\ $$$$\left(\mathrm{1}\right)\mathrm{The}``\:\mathrm{antiroot}\:\mathrm{formula}''\:\mathrm{is}\:\mathrm{derivedr} \\ $$$$\mathrm{fom}\:\mathrm{the}\:\mathrm{abovementionedt} \\ $$$$\mathrm{antiroo}\:\mathrm{formula}. \\ $$

Question Number 213343 Answers: 1 Comments: 0

Question Number 213342 Answers: 0 Comments: 1

Question Number 213341 Answers: 2 Comments: 3

Question Number 213340 Answers: 0 Comments: 3

Question Number 213369 Answers: 2 Comments: 0

Old question 203835 ∫_0 ^(√2) ((√(6−(√(25x^4 −50x^2 +36))))/( (√5)))dx=?

$$\mathrm{Old}\:\mathrm{question}\:\mathrm{203835} \\ $$$$\underset{\mathrm{0}} {\overset{\sqrt{\mathrm{2}}} {\int}}\frac{\sqrt{\mathrm{6}−\sqrt{\mathrm{25}{x}^{\mathrm{4}} −\mathrm{50}{x}^{\mathrm{2}} +\mathrm{36}}}}{\:\sqrt{\mathrm{5}}}{dx}=? \\ $$

Question Number 211675 Answers: 2 Comments: 1

x^2 =2^x Find more than three solutions

$$ \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{x}}^{\mathrm{2}} =\mathrm{2}^{\boldsymbol{{x}}} \\ $$$$\:\mathrm{Find}\:\mathrm{more}\:\mathrm{than}\:\mathrm{three}\:\mathrm{solutions} \\ $$$$ \\ $$

Question Number 211673 Answers: 1 Comments: 0

Resoudre ^x (√(x/(x−1))) =(x−1)^((x−2)) .

$$\:\:\:\boldsymbol{{Resoudre}} \\ $$$$\:\:^{\boldsymbol{\mathrm{x}}} \sqrt{\frac{\boldsymbol{\mathrm{x}}}{\boldsymbol{\mathrm{x}}−\mathrm{1}}}\:\:\:=\left(\boldsymbol{\mathrm{x}}−\mathrm{1}\right)^{\left(\boldsymbol{\mathrm{x}}−\mathrm{2}\right)} . \\ $$

Question Number 211672 Answers: 0 Comments: 2

In a triangle the bisector of the side c is perpendicular to side b. Prove that 2tanC + tanA = 0.

$$\mathrm{In}\:\mathrm{a}\:\mathrm{triangle}\:\mathrm{the}\:\mathrm{bisector}\:\mathrm{of}\:\mathrm{the}\:\mathrm{side}\:{c}\:\mathrm{is} \\ $$$$\mathrm{perpendicular}\:\mathrm{to}\:\mathrm{side}\:{b}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\mathrm{2tanC}\:+\:\mathrm{tanA}\:=\:\mathrm{0}. \\ $$

Question Number 211659 Answers: 2 Comments: 1

Question Number 211658 Answers: 2 Comments: 0

(√(19−8(√3))) x^2 −ax+ b =0 a b a + b .

$$\:\: \sqrt{\mathrm{19}−\mathrm{8}\sqrt{\mathrm{3}}}\: \\ $$$$ \mathrm{x}^{\mathrm{2}} −{a}\mathrm{x}+\:{b}\:=\mathrm{0}\: {a}\: {b}\: \\ $$$$ \\ $$$$ {a}\:+\:{b}\:.\: \\ $$

Question Number 211657 Answers: 1 Comments: 0

sin(9x) = sin(5x) + sin(3x) find: x = ?

$$\mathrm{sin}\left(\mathrm{9x}\right)\:=\:\mathrm{sin}\left(\mathrm{5x}\right)\:+\:\mathrm{sin}\left(\mathrm{3x}\right) \\ $$$$\mathrm{find}:\:\:\mathrm{x}\:=\:? \\ $$

Question Number 211651 Answers: 1 Comments: 0

In triangle ABC, ∠C = 60°. If the length of opposite sides of ∠A, ∠B and ∠C are a, b and c respectively then prove that (1/(a + c)) + (1/(b + c)) = (3/(a + b + c)) .

$$\mathrm{In}\:\mathrm{triangle}\:\mathrm{ABC},\:\angle\mathrm{C}\:=\:\mathrm{60}°.\:\mathrm{If}\:\mathrm{the}\:\mathrm{length} \\ $$$$\mathrm{of}\:\mathrm{opposite}\:\mathrm{sides}\:\mathrm{of}\:\angle\mathrm{A},\:\angle\mathrm{B}\:\mathrm{and}\:\angle\mathrm{C}\:\mathrm{are} \\ $$$${a},\:{b}\:\mathrm{and}\:{c}\:\mathrm{respectively}\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\frac{\mathrm{1}}{{a}\:+\:{c}}\:+\:\frac{\mathrm{1}}{{b}\:+\:{c}}\:=\:\frac{\mathrm{3}}{{a}\:+\:{b}\:+\:{c}}\:. \\ $$

Question Number 211643 Answers: 1 Comments: 0

Let A={x ∈ R∣x^2 <4}and B={y ∈ Q∣y>−3}find A∩B

$$\mathrm{Let}\:\boldsymbol{{A}}=\left\{\boldsymbol{{x}}\:\in\:\mathbb{R}\mid\boldsymbol{{x}}^{\mathrm{2}} <\mathrm{4}\right\}\mathrm{and} \\ $$$$\boldsymbol{{B}}=\left\{\boldsymbol{{y}}\:\in\:\mathbb{Q}\mid\boldsymbol{{y}}>−\mathrm{3}\right\}\mathrm{find}\:\boldsymbol{{A}}\cap\boldsymbol{{B}} \\ $$

Question Number 211637 Answers: 1 Comments: 0

f(x) = sin x − e^x + 1. Prove that f(x) has only 2 zeros in −π≤x≤0.

$${f}\left({x}\right)\:=\:\mathrm{sin}\:{x}\:−\:\mathrm{e}^{{x}} \:+\:\mathrm{1}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:{f}\left({x}\right)\:\mathrm{has}\:\mathrm{only}\:\mathrm{2}\:\mathrm{zeros}\:\mathrm{in}\:−\pi\leqslant{x}\leqslant\mathrm{0}. \\ $$

Question Number 211636 Answers: 1 Comments: 0

Question Number 211635 Answers: 1 Comments: 0

Question Number 211634 Answers: 1 Comments: 0

Question Number 211633 Answers: 1 Comments: 1

Question Number 211632 Answers: 1 Comments: 0

Question Number 211631 Answers: 0 Comments: 0

Question Number 211630 Answers: 0 Comments: 0

Question Number 211627 Answers: 0 Comments: 0

certificate: ∣∣h∣∣_^L^p ^p =∫_0 ^∞ ∣h(x)∣^p dx.

$$ \\ $$$$\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{certificate}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\mid\mid\boldsymbol{{h}}\mid\mid_{\:^{\boldsymbol{{L}}^{\boldsymbol{{p}}} } } ^{\boldsymbol{{p}}} =\int_{\mathrm{0}} ^{\infty} \mid\boldsymbol{{h}}\left(\boldsymbol{{x}}\right)\mid^{\boldsymbol{{p}}} \boldsymbol{{dx}}. \\ $$

Question Number 211614 Answers: 2 Comments: 1

A group of people are standing in a circle. Every second person is removed from the circle and this process continues until only one person remains in the circle. If there are 100 people in the circle, what will be the number of the last person left?

$$ \\ $$A group of people are standing in a circle. Every second person is removed from the circle and this process continues until only one person remains in the circle. If there are 100 people in the circle, what will be the number of the last person left?

Question Number 211617 Answers: 1 Comments: 0

Ato starts a business with $1,250.00. Ama joins the business later with a capital of 1,875.00. At the end of the first year, profits are shared equally between Ato and Ama. When did Ama join the business?

$$\mathrm{Ato}\:\:\mathrm{starts}\:\mathrm{a}\:\mathrm{business}\:\mathrm{with}\:\$\mathrm{1},\mathrm{250}.\mathrm{00}.\:\mathrm{Ama}\:\mathrm{joins}\:\mathrm{the}\:\mathrm{business} \\ $$$$\mathrm{later}\:\mathrm{with}\:\mathrm{a}\:\mathrm{capital}\:\mathrm{of}\:\mathrm{1},\mathrm{875}.\mathrm{00}.\:\mathrm{At}\:\mathrm{the}\: \\ $$$$\mathrm{end}\:\mathrm{of}\:\mathrm{the}\:\mathrm{first}\:\mathrm{year},\:\mathrm{profits}\:\mathrm{are}\:\mathrm{shared}\:\mathrm{equally} \\ $$$$\mathrm{between}\:\mathrm{Ato}\:\mathrm{and}\:\mathrm{Ama}.\:\mathrm{When}\:\mathrm{did}\:\mathrm{Ama} \\ $$$$\mathrm{join}\:\mathrm{the}\:\mathrm{business}? \\ $$

Question Number 211609 Answers: 2 Comments: 0

Prove that, in a triangle the ratios of the sides and the sine of the opposite angles are equal. Also prove that each ratio is equal to the diameter of the circum circle of the triangle.

$$\mathrm{Prove}\:\mathrm{that},\:\mathrm{in}\:\mathrm{a}\:\mathrm{triangle}\:\mathrm{the}\:\mathrm{ratios}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{sides}\:\mathrm{and}\:\mathrm{the}\:\mathrm{sine}\:\mathrm{of}\:\mathrm{the}\:\mathrm{opposite}\:\mathrm{angles} \\ $$$$\mathrm{are}\:\mathrm{equal}.\:\mathrm{Also}\:\mathrm{prove}\:\mathrm{that}\:\mathrm{each}\:\mathrm{ratio}\:\mathrm{is} \\ $$$$\mathrm{equal}\:\mathrm{to}\:\mathrm{the}\:\mathrm{diameter}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circum}\:\mathrm{circle} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{triangle}. \\ $$

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