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AllQuestion and Answers: Page 1080

Question Number 110365 Answers: 3 Comments: 0

Question Number 110359 Answers: 1 Comments: 0

Let f(x)=∣x−2∣+∣x−4∣−∣2x−6∣, for 2≤x≤8. The sum of the largest and smallest values of f(x) is

$$\mathrm{Let} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)=\mid\mathrm{x}−\mathrm{2}\mid+\mid\mathrm{x}−\mathrm{4}\mid−\mid\mathrm{2x}−\mathrm{6}\mid, \\ $$$$\mathrm{for}\:\mathrm{2}\leqslant\mathrm{x}\leqslant\mathrm{8}.\:\mathrm{The}\:\mathrm{sum}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{largest}\:\mathrm{and} \\ $$$$\mathrm{smallest}\:\mathrm{values}\:\mathrm{of}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{is} \\ $$

Question Number 110358 Answers: 1 Comments: 0

if positive integer x satisfies x^2 −4x+56 ≡14 (mod 17) , what is the minimum value of x.

$${if}\:{positive}\:{integer}\:{x}\:{satisfies}\:{x}^{\mathrm{2}} −\mathrm{4}{x}+\mathrm{56}\:\equiv\mathrm{14}\:\left({mod}\:\mathrm{17}\right)\: \\ $$$$,\:{what}\:{is}\:{the}\:{minimum}\:{value}\:{of}\:{x}. \\ $$

Question Number 110357 Answers: 2 Comments: 1

Given that p,q are primes and pq divides p^2 +q^2 −4. How many possible values does ∣p−q∣ have?

$$\mathrm{Given}\:\mathrm{that}\:\mathrm{p},\mathrm{q}\:\mathrm{are}\:\mathrm{primes}\:\mathrm{and}\:\mathrm{pq} \\ $$$$\mathrm{divides}\:\mathrm{p}^{\mathrm{2}} +\mathrm{q}^{\mathrm{2}} −\mathrm{4}.\:\mathrm{How}\:\mathrm{many} \\ $$$$\mathrm{possible}\:\mathrm{values}\:\mathrm{does}\:\mid\mathrm{p}−\mathrm{q}\mid\:\mathrm{have}? \\ $$

Question Number 110354 Answers: 0 Comments: 1

The Diophantine equation x^2 +y^2 +1 =N(xy+1) has infinitely many integer solutions if N equals?

$$\mathrm{The}\:\mathrm{Diophantine}\:\mathrm{equation} \\ $$$$\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{1}\:=\mathrm{N}\left(\mathrm{xy}+\mathrm{1}\right)\:\mathrm{has} \\ $$$$\mathrm{infinitely}\:\mathrm{many}\:\mathrm{integer} \\ $$$$\mathrm{solutions}\:\mathrm{if}\:\mathrm{N}\:\mathrm{equals}? \\ $$

Question Number 110320 Answers: 3 Comments: 0

find the point of intersection of the line r^→ =(1−2t,3+4t,t) and the plane 3x−2y+5z=15

$${find}\:{the}\:{point}\:{of}\:{intersection} \\ $$$${of}\:{the}\:{line}\:\overset{\rightarrow} {{r}}=\left(\mathrm{1}−\mathrm{2}{t},\mathrm{3}+\mathrm{4}{t},{t}\right) \\ $$$${and}\:{the}\:{plane}\:\mathrm{3}{x}−\mathrm{2}{y}+\mathrm{5}{z}=\mathrm{15}\: \\ $$

Question Number 110318 Answers: 1 Comments: 0

(a+b−c)^2 =??

$$\left({a}+{b}−{c}\right)^{\mathrm{2}} =?? \\ $$

Question Number 110307 Answers: 6 Comments: 0

(1)lim_(x→−∞) ((3−3x)/( (√(x^2 −4x+1)))) ? (2) ∫_0 ^1 arctan (((2x−1)/(1+x−x^2 ))) dx (3)how many integer solution sets exist for the equation x^2 +y^2 =2

$$\left(\mathrm{1}\right)\underset{{x}\rightarrow−\infty} {\mathrm{lim}}\:\frac{\mathrm{3}−\mathrm{3}{x}}{\:\sqrt{{x}^{\mathrm{2}} −\mathrm{4}{x}+\mathrm{1}}}\:? \\ $$$$\left(\mathrm{2}\right)\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\mathrm{arctan}\:\left(\frac{\mathrm{2}{x}−\mathrm{1}}{\mathrm{1}+{x}−{x}^{\mathrm{2}} }\right)\:{dx} \\ $$$$\left(\mathrm{3}\right){how}\:{many}\:{integer}\:{solution}\:{sets} \\ $$$${exist}\:{for}\:{the}\:{equation}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{2} \\ $$

Question Number 110306 Answers: 1 Comments: 2

Question Number 110920 Answers: 1 Comments: 1

Question Number 110919 Answers: 1 Comments: 0

lim_(n→∞) (Σ_(r=1) ^n (1/(3^r r!))(Π_(k=1) ^r (2k−1)))

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left(\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{\mathrm{3}^{{r}} {r}!}\left(\underset{{k}=\mathrm{1}} {\overset{{r}} {\prod}}\left(\mathrm{2}{k}−\mathrm{1}\right)\right)\right) \\ $$

Question Number 110301 Answers: 0 Comments: 1

Question Number 110299 Answers: 3 Comments: 0

Question Number 110294 Answers: 1 Comments: 0

∣2x+1∣−∣x−2∣ < 4 find the solution set

$$\mid\mathrm{2}{x}+\mathrm{1}\mid−\mid{x}−\mathrm{2}\mid\:<\:\mathrm{4}\: \\ $$$${find}\:{the}\:{solution}\:{set} \\ $$

Question Number 110293 Answers: 0 Comments: 0

Simplify: ((tan((3π)/7) − 4sin(π/7))/(tan((6π)/7) + 4sin((2π)/7)))

$$\mathrm{Simplify}:\:\:\:\frac{\mathrm{tan}\frac{\mathrm{3}\pi}{\mathrm{7}}\:\:\:−\:\:\mathrm{4sin}\frac{\pi}{\mathrm{7}}}{\mathrm{tan}\frac{\mathrm{6}\pi}{\mathrm{7}}\:\:+\:\:\mathrm{4sin}\frac{\mathrm{2}\pi}{\mathrm{7}}} \\ $$

Question Number 110288 Answers: 1 Comments: 0

Question Number 110287 Answers: 3 Comments: 2

Question Number 110286 Answers: 1 Comments: 0

Question Number 110285 Answers: 1 Comments: 0

Question Number 110281 Answers: 1 Comments: 0

Question Number 110280 Answers: 0 Comments: 0

Question Number 110269 Answers: 1 Comments: 4

Question Number 110268 Answers: 0 Comments: 3

Question Number 110265 Answers: 0 Comments: 0

Question Number 110262 Answers: 0 Comments: 0

Solve for X(x,y,z), Y(x,y,z), Z(x,y,z) { (((∂Z/∂y)−(∂Y/∂z)=1−x^2 )),(((∂Z/∂x)−(∂X/∂z)=−(y^2 /2))),(((∂Y/∂x)−(∂X/∂y)=z(2x−y))) :} where { ((X(x,y,0)=0)),((Y(x,y,0)=0)),((Z(x,y,0)=0)) :}

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{X}\left(\mathrm{x},\mathrm{y},\mathrm{z}\right),\:\mathrm{Y}\left(\mathrm{x},\mathrm{y},\mathrm{z}\right),\:\mathrm{Z}\left(\mathrm{x},\mathrm{y},\mathrm{z}\right) \\ $$$$\begin{cases}{\frac{\partial\mathrm{Z}}{\partial\mathrm{y}}−\frac{\partial\mathrm{Y}}{\partial\mathrm{z}}=\mathrm{1}−\mathrm{x}^{\mathrm{2}} }\\{\frac{\partial\mathrm{Z}}{\partial\mathrm{x}}−\frac{\partial\mathrm{X}}{\partial\mathrm{z}}=−\frac{\mathrm{y}^{\mathrm{2}} }{\mathrm{2}}}\\{\frac{\partial\mathrm{Y}}{\partial\mathrm{x}}−\frac{\partial\mathrm{X}}{\partial\mathrm{y}}=\mathrm{z}\left(\mathrm{2x}−\mathrm{y}\right)}\end{cases}\:\mathrm{where}\:\begin{cases}{\mathrm{X}\left(\mathrm{x},\mathrm{y},\mathrm{0}\right)=\mathrm{0}}\\{\mathrm{Y}\left(\mathrm{x},\mathrm{y},\mathrm{0}\right)=\mathrm{0}}\\{\mathrm{Z}\left(\mathrm{x},\mathrm{y},\mathrm{0}\right)=\mathrm{0}}\end{cases} \\ $$

Question Number 110260 Answers: 1 Comments: 0

lim_(x→∞) x cos ((1/x)) ?

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{x}\:\mathrm{cos}\:\left(\frac{\mathrm{1}}{{x}}\right)\:? \\ $$

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