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Question Number 27027 Answers: 0 Comments: 0

Find max and min of the function: f(x)=x^2 cos((1/x))

$${Find}\:{max}\:{and}\:{min}\:{of}\:{the}\:{function}: \\ $$$${f}\left({x}\right)={x}^{\mathrm{2}} {cos}\left(\frac{\mathrm{1}}{{x}}\right) \\ $$

Question Number 26904 Answers: 1 Comments: 1

y^((2)) +y=sec^3 x

$${y}^{\left(\mathrm{2}\right)} +{y}=\mathrm{sec}\:^{\mathrm{3}} {x} \\ $$

Question Number 26909 Answers: 1 Comments: 0

∫((e^x ((1/x)−(2/x^3 )))/(−2))dx

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

Question Number 26889 Answers: 1 Comments: 0

Question Number 26888 Answers: 1 Comments: 0

If α, β are the roots of ax^2 +bx+c=0 then find the quadratic equation whose roots are α+β, αβ.

$$\mathrm{If}\:\alpha,\:\beta\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\:{ax}^{\mathrm{2}} +{bx}+{c}=\mathrm{0} \\ $$$$\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{quadratic}\:\mathrm{equation}\: \\ $$$$\mathrm{whose}\:\mathrm{roots}\:\mathrm{are}\:\alpha+\beta,\:\:\alpha\beta. \\ $$

Question Number 26864 Answers: 1 Comments: 0

Question Number 27000 Answers: 0 Comments: 1

P is a polynomial havng n roots (x_i )_(1≤i≤n) with x_i ≠ x_j for i≠ j find the values of Σ_(k1) ^(k=n) (1/(x−x_k )) and Σ_(k=1) ^n (1/((x−x_k )^2 )) .

$${P}\:{is}\:{a}\:{polynomial}\:{havng}\:{n}\:{roots}\:\left({x}_{{i}} \right)_{\mathrm{1}\leqslant{i}\leqslant{n}} \:\:{with}\:{x}_{{i}} \neq\:{x}_{{j}} \:{for}\:{i}\neq\:{j} \\ $$$${find}\:{the}\:{values}\:{of}\:\sum_{{k}\mathrm{1}} ^{{k}={n}} \frac{\mathrm{1}}{{x}−{x}_{{k}} }\:\:{and}\:\:\sum_{{k}=\mathrm{1}} ^{{n}} \frac{\mathrm{1}}{\left({x}−{x}_{{k}} \right)^{\mathrm{2}} }\:. \\ $$

Question Number 26858 Answers: 2 Comments: 0

Prove that the internal bisector of an angle of a triangle divides the opposite sides in the ratio of the sides containing the angle.

$${Prove}\:{that}\:{the}\:{internal}\:{bisector}\:{of}\:{an}\:{angle}\: \\ $$$${of}\:{a}\:{triangle}\:{divides}\:{the}\:{opposite} \\ $$$${sides}\:{in}\:{the}\:{ratio}\:{of}\:{the}\:{sides}\:{containing} \\ $$$${the}\:{angle}. \\ $$

Question Number 26853 Answers: 1 Comments: 1

Question Number 26839 Answers: 1 Comments: 0

a=3 b=6 a−b=?

$$\mathrm{a}=\mathrm{3}\:\mathrm{b}=\mathrm{6} \\ $$$$\mathrm{a}−\mathrm{b}=? \\ $$

Question Number 26837 Answers: 1 Comments: 0

xy^((2)) =y^((1)) ×ln (y^((1)) /x)

$${xy}^{\left(\mathrm{2}\right)} ={y}^{\left(\mathrm{1}\right)} ×\mathrm{ln}\:\frac{{y}^{\left(\mathrm{1}\right)} }{{x}} \\ $$

Question Number 26824 Answers: 3 Comments: 0

Question Number 26819 Answers: 0 Comments: 0

Question Number 26812 Answers: 1 Comments: 0

sum of infinite seris tan^(−1) (2/n^2 )

$${sum}\:{of}\:{infinite}\:{seris} \\ $$$$\mathrm{tan}^{−\mathrm{1}} \left(\mathrm{2}/{n}^{\mathrm{2}} \right) \\ $$

Question Number 26799 Answers: 1 Comments: 0

Question Number 26801 Answers: 0 Comments: 0

find expansion of α^3 +β^3 +γ^3

$${find}\:{expansion}\:{of}\:\alpha^{\mathrm{3}} +\beta^{\mathrm{3}} +\gamma^{\mathrm{3}} \\ $$

Question Number 26781 Answers: 1 Comments: 4

Question Number 26778 Answers: 1 Comments: 1

If sin(A) + sin(B) = p and cos(A) + cos(B) = q show that tan(A) + tan(B) = ((8pq)/((p^2 + q^2 )^2 − 4q^2 ))

$$\mathrm{If}\:\:\:\:\:\:\:\mathrm{sin}\left(\mathrm{A}\right)\:+\:\mathrm{sin}\left(\mathrm{B}\right)\:=\:\mathrm{p}\:\:\:\:\:\mathrm{and}\:\:\:\:\mathrm{cos}\left(\mathrm{A}\right)\:+\:\mathrm{cos}\left(\mathrm{B}\right)\:=\:\mathrm{q} \\ $$$$\mathrm{show}\:\mathrm{that}\:\:\:\:\mathrm{tan}\left(\mathrm{A}\right)\:+\:\mathrm{tan}\left(\mathrm{B}\right)\:=\:\frac{\mathrm{8pq}}{\left(\mathrm{p}^{\mathrm{2}} \:+\:\mathrm{q}^{\mathrm{2}} \right)^{\mathrm{2}} \:−\:\mathrm{4q}^{\mathrm{2}} } \\ $$

Question Number 26776 Answers: 0 Comments: 2

Question Number 26774 Answers: 0 Comments: 5

If one side of a square reduces by 10% and the adjacent side increases by 30%.Using the application of differentiationfind the % change in the area of the square.

$${If}\:{one}\:{side}\:{of}\:{a}\:{square}\:{reduces}\:{by} \\ $$$$\mathrm{10\%}\:{and}\:{the}\:{adjacent}\:{side}\:{increases} \\ $$$${by}\:\mathrm{30\%}.{Using}\:{the}\:{application}\:{of} \\ $$$${differentiationfind}\:{the}\:\%\:{change} \\ $$$${in}\:{the}\:{area}\:{of}\:{the}\:{square}. \\ $$

Question Number 26772 Answers: 1 Comments: 0

With a center on a given circle of radius r ,an arc has been drawn in order to divide the circle in two equal (in area) parts. What is the radius of the arc in terms of r (radius of given circle)?

$$\mathrm{With}\:\mathrm{a}\:\mathrm{center}\:\boldsymbol{\mathrm{on}}\:\mathrm{a}\:\mathrm{given}\:\mathrm{circle}\:\mathrm{of} \\ $$$$\mathrm{radius}\:\mathrm{r}\:,\mathrm{an}\:\mathrm{arc}\:\mathrm{has}\:\mathrm{been}\:\mathrm{drawn}\:\mathrm{in}\:\mathrm{order} \\ $$$$\mathrm{to}\:\mathrm{divide}\:\mathrm{the}\:\mathrm{circle}\:\mathrm{in}\:\mathrm{two}\:\mathrm{equal} \\ $$$$\left(\mathrm{in}\:\mathrm{area}\right)\:\mathrm{parts}. \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{radius}\:\mathrm{of}\:\mathrm{the}\:\mathrm{arc}\:\mathrm{in}\:\mathrm{terms} \\ $$$$\mathrm{of}\:\:\mathrm{r}\:\left(\mathrm{radius}\:\mathrm{of}\:\mathrm{given}\:\mathrm{circle}\right)?\: \\ $$

Question Number 26771 Answers: 0 Comments: 1

∫_( 0) ^( [x]) (2^x /2^([x]) ) dx =

$$\underset{\:\mathrm{0}} {\overset{\:\:\:\left[{x}\right]} {\int}}\frac{\mathrm{2}^{{x}} }{\mathrm{2}^{\left[{x}\right]} }\:{dx}\:= \\ $$

Question Number 26770 Answers: 0 Comments: 1

lim_(n→∞) (((n !)^(1/n) )/n) =

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\left({n}\:!\right)^{\mathrm{1}/{n}} }{{n}}\:= \\ $$

Question Number 26768 Answers: 0 Comments: 0

find the nature of U_n =Σ_(p=0) ^(p=n) (1/C_n ^p ) .

$${find}\:{the}\:{nature}\:{of}\:{U}_{{n}} \:\:=\sum_{{p}=\mathrm{0}} ^{{p}={n}} \:\:\frac{\mathrm{1}}{{C}_{{n}} ^{{p}} }\:\:. \\ $$

Question Number 26769 Answers: 1 Comments: 1

∫_( 0) ^∞ (dx/([x+(√(x^2 +1)) ]^3 )) =

$$\underset{\:\mathrm{0}} {\overset{\infty} {\int}}\:\:\frac{{dx}}{\left[{x}+\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}\:\right]^{\mathrm{3}} }\:=\: \\ $$

Question Number 26764 Answers: 0 Comments: 3

find lim_(n−>∝) (1/n) ln( Π_(k=1) ^(n−1) (1− (k/n))).

$${find}\:\:{lim}_{{n}−>\propto} \:\frac{\mathrm{1}}{{n}}\:{ln}\left(\:\prod_{{k}=\mathrm{1}} ^{{n}−\mathrm{1}} \:\left(\mathrm{1}−\:\frac{{k}}{{n}}\right)\right). \\ $$

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