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AllQuestion and Answers: Page 667

Question Number 151495 Answers: 0 Comments: 4

Question Number 151493 Answers: 0 Comments: 2

((cos (((5π)/2)−6x)+sin (π+4x)+sin (3π−x))/(sin (((5π)/2)+6x)+cos (4x−2π)+cos (x+2π))) =?

$$\:\:\:\frac{\mathrm{cos}\:\left(\frac{\mathrm{5}\pi}{\mathrm{2}}−\mathrm{6}{x}\right)+\mathrm{sin}\:\left(\pi+\mathrm{4}{x}\right)+\mathrm{sin}\:\left(\mathrm{3}\pi−{x}\right)}{\mathrm{sin}\:\left(\frac{\mathrm{5}\pi}{\mathrm{2}}+\mathrm{6}{x}\right)+\mathrm{cos}\:\left(\mathrm{4}{x}−\mathrm{2}\pi\right)+\mathrm{cos}\:\left({x}+\mathrm{2}\pi\right)}\:=? \\ $$

Question Number 151492 Answers: 0 Comments: 0

∫_0 ^( 1) ((tan(x^2 +1))/(x^2 +1)) dx

$$\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\mathrm{tan}\left({x}^{\mathrm{2}} +\mathrm{1}\right)}{{x}^{\mathrm{2}} +\mathrm{1}}\:{dx} \\ $$$$\: \\ $$

Question Number 151490 Answers: 1 Comments: 0

{ ((a^2 =2a+b)),((b^2 =a+2b)) :} and a≠b find (√(a^2 +b^2 +1))

$$\begin{cases}{\mathrm{a}^{\mathrm{2}} =\mathrm{2a}+\mathrm{b}}\\{\mathrm{b}^{\mathrm{2}} =\mathrm{a}+\mathrm{2b}}\end{cases}\:\:\mathrm{and}\:\:\mathrm{a}\neq\mathrm{b}\:\:\mathrm{find}\:\:\sqrt{\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} +\mathrm{1}} \\ $$

Question Number 151486 Answers: 1 Comments: 0

∫ln(sinx)=? x ≠ 0 + 2kπ ; k ∈ Z

$$\int{ln}\left({sinx}\right)=?\: \\ $$$${x}\:\neq\:\mathrm{0}\:+\:\mathrm{2}{k}\pi\:;\:{k}\:\in\:\mathbb{Z} \\ $$

Question Number 151485 Answers: 1 Comments: 0

Question Number 151481 Answers: 0 Comments: 2

if x;y;z;t∈R and x+y+z ≤ 3t ; y+z+t ≤ 3x z+t+x ≤ 3y ; t+x+y ≤ 3z compare: x;y;z;t

$$\mathrm{if}\:\:\mathrm{x};\mathrm{y};\mathrm{z};\mathrm{t}\in\mathbb{R}\:\:\mathrm{and} \\ $$$$\mathrm{x}+\mathrm{y}+\mathrm{z}\:\leqslant\:\mathrm{3t}\:\:;\:\:\mathrm{y}+\mathrm{z}+\mathrm{t}\:\leqslant\:\mathrm{3x} \\ $$$$\mathrm{z}+\mathrm{t}+\mathrm{x}\:\leqslant\:\mathrm{3y}\:\:;\:\:\mathrm{t}+\mathrm{x}+\mathrm{y}\:\leqslant\:\mathrm{3z} \\ $$$$\mathrm{compare}:\:\:\mathrm{x};\mathrm{y};\mathrm{z};\mathrm{t} \\ $$

Question Number 151636 Answers: 1 Comments: 4

Question Number 151475 Answers: 2 Comments: 0

if x∈R prove that: x^6 - x + 1 > 0

$$\mathrm{if}\:\:\boldsymbol{\mathrm{x}}\in\mathbb{R}\:\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\mathrm{x}^{\mathrm{6}} \:-\:\mathrm{x}\:+\:\mathrm{1}\:>\:\mathrm{0} \\ $$

Question Number 151468 Answers: 2 Comments: 0

z^2 −7z+16=i(z−11)

$$\mathrm{z}^{\mathrm{2}} −\mathrm{7z}+\mathrm{16}=\mathrm{i}\left(\mathrm{z}−\mathrm{11}\right) \\ $$$$ \\ $$

Question Number 151461 Answers: 1 Comments: 0

∫((xdx)/(x^4 +4x^2 +5))

$$\int\frac{\mathrm{xdx}}{\mathrm{x}^{\mathrm{4}} +\mathrm{4x}^{\mathrm{2}} +\mathrm{5}} \\ $$

Question Number 151460 Answers: 1 Comments: 0

if f(x) = x^2 - 2x + 2 and g(x) = (√x) + 1 find [ f o g ] (x) = ?

$$\mathrm{if}\:\:\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\mathrm{x}^{\mathrm{2}} \:-\:\mathrm{2x}\:+\:\mathrm{2} \\ $$$$\mathrm{and}\:\:\mathrm{g}\left(\mathrm{x}\right)\:=\:\sqrt{\mathrm{x}}\:+\:\mathrm{1} \\ $$$$\mathrm{find}\:\:\:\left[\:\mathrm{f}\:{o}\:\mathrm{g}\:\right]\:\left(\mathrm{x}\right)\:=\:? \\ $$

Question Number 151454 Answers: 0 Comments: 2

when a die is rolled 42 times it is so happened that a face having the digit i times occured 2i times. then find the mean deviation from the mean of this discrete frequency distribution. ans is ((80)/(63)) sol pls

$${when}\:{a}\:{die}\:{is}\:{rolled}\:\mathrm{42}\:{times}\:{it}\:{is}\:{so} \\ $$$${happened}\:{that}\:{a}\:{face}\:{having}\:{the}\:{digit}\:{i} \\ $$$${times}\:{occured}\:\mathrm{2}{i}\:{times}.\:{then}\:{find}\:{the} \\ $$$${mean}\:{deviation}\:{from}\:{the}\:{mean}\:{of}\:{this} \\ $$$${discrete}\:{frequency}\:{distribution}. \\ $$$${ans}\:{is}\:\frac{\mathrm{80}}{\mathrm{63}} \\ $$$${sol}\:{pls} \\ $$

Question Number 151453 Answers: 0 Comments: 7

find all continous functions f : R→R such that: f(x^2 + 1) = f((√(1 + x^4 ))) ; ∀x∈R

$$\mathrm{find}\:\mathrm{all}\:\mathrm{continous}\:\mathrm{functions}\:\mathrm{f}\::\:\mathbb{R}\rightarrow\mathbb{R} \\ $$$$\mathrm{such}\:\mathrm{that}: \\ $$$$\mathrm{f}\left(\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{1}\right)\:=\:\mathrm{f}\left(\sqrt{\mathrm{1}\:+\:\mathrm{x}^{\mathrm{4}} }\right)\:;\:\forall\mathrm{x}\in\mathbb{R} \\ $$

Question Number 151449 Answers: 2 Comments: 0

Ω =∫_( 0) ^( ∞) sin(x^2 ) dx = ?

$$\Omega\:=\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\:\mathrm{sin}\left(\mathrm{x}^{\mathrm{2}} \right)\:\mathrm{dx}\:=\:? \\ $$

Question Number 151448 Answers: 1 Comments: 2

Prove that: Artimetric mean ≥ Geometric mean ((a + b)/2) ≥ (√(ab))

$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\boldsymbol{\mathrm{A}}\mathrm{rtimetric}\:\mathrm{mean}\:\geqslant\:\boldsymbol{\mathrm{G}}\mathrm{eometric}\:\mathrm{mean} \\ $$$$\frac{\mathrm{a}\:+\:\mathrm{b}}{\mathrm{2}}\:\geqslant\:\sqrt{\mathrm{ab}} \\ $$

Question Number 151447 Answers: 1 Comments: 0

prove that... I:= ∫_0 ^( ∞) (( ln (x ))/(1+ e^( x) )) dx= ((−1)/2) ln^( 2) (2) ..■

$$ \\ $$$$\:\:\:\:\:\:{prove}\:{that}... \\ $$$$\:\:\:\mathrm{I}:=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{ln}\:\left({x}\:\right)}{\mathrm{1}+\:{e}^{\:{x}} }\:{dx}=\:\frac{−\mathrm{1}}{\mathrm{2}}\:{ln}^{\:\mathrm{2}} \left(\mathrm{2}\right)\:..\blacksquare \\ $$

Question Number 151444 Answers: 2 Comments: 0

Question Number 151436 Answers: 0 Comments: 0

Question Number 151435 Answers: 1 Comments: 4

Question Number 151429 Answers: 1 Comments: 0

∫_0 ^π ((xdx)/((a^2 cos^2 x+b^2 sin^2 x)^2 ))

$$\int_{\mathrm{0}} ^{\pi} \frac{\mathrm{xdx}}{\left(\mathrm{a}^{\mathrm{2}} \mathrm{cos}\:^{\mathrm{2}} \mathrm{x}+\mathrm{b}^{\mathrm{2}} \mathrm{sin}\:^{\mathrm{2}} \mathrm{x}\right)^{\mathrm{2}} } \\ $$

Question Number 151428 Answers: 1 Comments: 0

∫((x+2)/((x^2 +3x+3)(√(x+1))))dx

$$\int\frac{\mathrm{x}+\mathrm{2}}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{3x}+\mathrm{3}\right)\sqrt{\mathrm{x}+\mathrm{1}}}\mathrm{dx} \\ $$$$ \\ $$

Question Number 151426 Answers: 3 Comments: 0

Express (1/(5×9)) in partial fraction

$$\mathrm{Express}\:\frac{\mathrm{1}}{\mathrm{5}×\mathrm{9}}\:\mathrm{in}\:\mathrm{partial}\:\mathrm{fraction} \\ $$

Question Number 151425 Answers: 2 Comments: 0

∫_0 ^(π/2) (dx/((cos x+(√3) sin x)^2 ))dx=(1/( (√3)))

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{dx}}{\left(\mathrm{cos}\:\mathrm{x}+\sqrt{\mathrm{3}}\:\mathrm{sin}\:\mathrm{x}\right)^{\mathrm{2}} }\mathrm{dx}=\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}} \\ $$

Question Number 151423 Answers: 1 Comments: 0

∫_0 ^(π/2) (dx/(1+tan^5 x))

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{dx}}{\mathrm{1}+\mathrm{tan}\:^{\mathrm{5}} \mathrm{x}} \\ $$

Question Number 151421 Answers: 2 Comments: 2

∫_0 ^(π/4) log (1+tan x)dx

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \mathrm{log}\:\left(\mathrm{1}+\mathrm{tan}\:\mathrm{x}\right)\mathrm{dx} \\ $$

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