Question and Answers Forum

All Questions   Topic List

AllQuestion and Answers: Page 1421

Question Number 68825    Answers: 1   Comments: 0

Question Number 68808    Answers: 1   Comments: 0

Question Number 69018    Answers: 0   Comments: 1

Question Number 68783    Answers: 1   Comments: 5

Question Number 68782    Answers: 0   Comments: 0

Let d_n be the determinant of the n×n matrix whose entries, from left to right and then from top to bottom, are cos 1, cos 2, ..., cos n^2 . (For example, d_3 = determinant (((cos 1 cos 2 cos 3)),((cos 4 cos 5 cos 6)),((cos 7 cos 8 cos 9))). The argument of cos is always in radians not degrees.) Evalue lim_(n→∞) d_(n.)

$$\mathrm{Let}\:{d}_{{n}} \:\mathrm{be}\:\mathrm{the}\:\mathrm{determinant}\:\mathrm{of}\:\mathrm{the}\:{n}×{n} \\ $$$$\mathrm{matrix}\:\mathrm{whose}\:\mathrm{entries},\:\mathrm{from}\:\mathrm{left}\:\mathrm{to}\:\mathrm{right} \\ $$$$\mathrm{and}\:\mathrm{then}\:\mathrm{from}\:\mathrm{top}\:\mathrm{to}\:\mathrm{bottom},\:\mathrm{are} \\ $$$${cos}\:\mathrm{1},\:{cos}\:\mathrm{2},\:...,\:{cos}\:{n}^{\mathrm{2}} .\:\left(\mathrm{For}\:\mathrm{example},\right. \\ $$$${d}_{\mathrm{3}} =\begin{vmatrix}{{cos}\:\mathrm{1}\:\:{cos}\:\mathrm{2}\:\:{cos}\:\mathrm{3}}\\{{cos}\:\mathrm{4}\:\:{cos}\:\mathrm{5}\:\:{cos}\:\mathrm{6}}\\{{cos}\:\mathrm{7}\:\:{cos}\:\mathrm{8}\:\:{cos}\:\mathrm{9}}\end{vmatrix}. \\ $$$$\mathrm{The}\:\mathrm{argument}\:\mathrm{of}\:{cos}\:\mathrm{is}\:\mathrm{always}\:\mathrm{in}\:\mathrm{radians} \\ $$$$\left.\mathrm{not}\:\mathrm{degrees}.\right)\: \\ $$$$\mathrm{Evalue}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:{d}_{{n}.} \\ $$

Question Number 68775    Answers: 1   Comments: 0

Question Number 68774    Answers: 0   Comments: 0

Question Number 68773    Answers: 0   Comments: 0

Question Number 68768    Answers: 1   Comments: 1

((2x−1))^(1/3) +((x−1))^(1/3) = 1

$$\sqrt[{\mathrm{3}}]{\mathrm{2}{x}−\mathrm{1}}\:+\sqrt[{\mathrm{3}}]{{x}−\mathrm{1}}\:=\:\mathrm{1} \\ $$

Question Number 68767    Answers: 0   Comments: 1

Question Number 68761    Answers: 1   Comments: 1

Two arcs having their centers on a circle are cutting each other at a single point inside the circle and thus dividing the circle in four regions. If the arcs cut each other in a:b & c:d ratios what is the ratio between four regions of the circle when the circle has radius R,the arc divided in a:b has radius r_1 and the arc divided in c:d has radius r_2 .

$$\mathrm{Two}\:\boldsymbol{\mathrm{arcs}}\:\mathrm{having}\:\mathrm{their}\:\mathrm{centers}\:\mathrm{on}\:\mathrm{a} \\ $$$$\boldsymbol{\mathrm{circle}}\:\mathrm{are}\:\mathrm{cutting}\:\mathrm{each}\:\mathrm{other}\:\mathrm{at}\:\mathrm{a}\: \\ $$$$\mathrm{single}\:\mathrm{point}\:\mathrm{inside}\:\mathrm{the}\:\mathrm{circle}\:\mathrm{and}\:\mathrm{thus} \\ $$$$\:\mathrm{dividing}\:\mathrm{the}\:\mathrm{circle}\:\mathrm{in}\:\mathrm{four}\:\mathrm{regions}. \\ $$$$ \\ $$$$\mathrm{If}\:\mathrm{the}\:\mathrm{arcs}\:\mathrm{cut}\:\mathrm{each}\:\mathrm{other}\:\mathrm{in}\:\boldsymbol{\mathrm{a}}:\boldsymbol{\mathrm{b}}\:\&\:\boldsymbol{\mathrm{c}}:\boldsymbol{\mathrm{d}}\: \\ $$$$\mathrm{ratios}\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{ratio}\:\mathrm{between}\:\mathrm{four} \\ $$$$\mathrm{regions}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circle}\:\mathrm{when}\:\mathrm{the}\:\mathrm{circle} \\ $$$$\mathrm{has}\:\mathrm{radius}\:\boldsymbol{\mathrm{R}},\mathrm{the}\:\mathrm{arc}\:\mathrm{divided}\:\mathrm{in}\:\mathrm{a}:\mathrm{b} \\ $$$$\:\mathrm{has}\:\mathrm{radius}\:\boldsymbol{\mathrm{r}}_{\mathrm{1}} \:\mathrm{and}\:\mathrm{the}\:\mathrm{arc}\:\mathrm{divided}\:\mathrm{in} \\ $$$$\mathrm{c}:\mathrm{d}\:\mathrm{has}\:\mathrm{radius}\:\boldsymbol{\mathrm{r}}_{\mathrm{2}} . \\ $$

Question Number 68740    Answers: 1   Comments: 1

Question Number 68721    Answers: 1   Comments: 1

Question Number 68732    Answers: 0   Comments: 0

$$ \\ $$

Question Number 68714    Answers: 1   Comments: 0

Question Number 68712    Answers: 0   Comments: 3

given that x and y are two numbers other one. given that a>0 and b>0 and a^x = b^y = (ab)^(xy) show that x + y =0

$${given}\:{that}\:{x}\:{and}\:{y}\:{are}\:{two}\:{numbers}\:{other}\:{one}.\: \\ $$$${given}\:{that}\:\:{a}>\mathrm{0}\:{and}\:{b}>\mathrm{0} \\ $$$${and}\:\:{a}^{{x}} \:=\:{b}^{{y}} \:=\:\left({ab}\right)^{{xy}} \:\:{show}\:{that}\:\:{x}\:+\:{y}\:=\mathrm{0} \\ $$

Question Number 68710    Answers: 0   Comments: 3

Question Number 68703    Answers: 0   Comments: 5

(d/dx)(ln((√((x^2 −1)/(x^2 +1)))))=?

$$\frac{{d}}{{dx}}\left({ln}\left(\sqrt{\frac{{x}^{\mathrm{2}} −\mathrm{1}}{{x}^{\mathrm{2}} +\mathrm{1}}}\right)\right)=? \\ $$

Question Number 68699    Answers: 1   Comments: 2

∫ ln(x + 4) dx =

$$\int\:{ln}\left({x}\:+\:\mathrm{4}\right)\:{dx}\:= \\ $$

Question Number 68695    Answers: 1   Comments: 0

pour 1<k<n montrer que k(n+1−k)<(n+1/2)^2

$${pour}\:\mathrm{1}<{k}<{n}\:\:\:\:\:{montrer}\:{que} \\ $$$${k}\left({n}+\mathrm{1}−{k}\right)<\left({n}+\mathrm{1}/\mathrm{2}\right)^{\mathrm{2}} \\ $$

Question Number 68693    Answers: 0   Comments: 2

Question Number 68788    Answers: 0   Comments: 1

∫1/(1+x^2 )^n dx

$$\int\mathrm{1}/\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{{n}} \:{dx} \\ $$

Question Number 68676    Answers: 0   Comments: 2

Solve the equation tanh^(−1) (((x−2)/(x+1))) = ln 2 show that the set {1,2,4,8} under ×_(15) ,multiplication mod 15 forms a group.

$${Solve}\:{the}\:{equation} \\ $$$${tanh}^{−\mathrm{1}} \left(\frac{{x}−\mathrm{2}}{{x}+\mathrm{1}}\right)\:=\:{ln}\:\mathrm{2} \\ $$$${show}\:{that}\:{the}\:{set}\:\left\{\mathrm{1},\mathrm{2},\mathrm{4},\mathrm{8}\right\}\:\:{under}\:×_{\mathrm{15}} \:,{multiplication}\:{mod}\:\mathrm{15}\:\:{forms}\:{a}\:{group}. \\ $$

Question Number 68675    Answers: 0   Comments: 3

Express in partial fraction f(x) ≡ ((2x^3 + x + 2)/((x^2 +1)(x+1)(x−2))) x ≠ −1,2 Hence or otherwise show that ∫_0 ^1 f(x) dx = −(1/(12))[ 13ln 2 + π]

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:{Express}\:{in}\:{partial}\:{fraction}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:{f}\left({x}\right)\:\equiv\:\frac{\mathrm{2}{x}^{\mathrm{3}} \:+\:{x}\:+\:\mathrm{2}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left({x}+\mathrm{1}\right)\left({x}−\mathrm{2}\right)}\:{x}\:\neq\:−\mathrm{1},\mathrm{2} \\ $$$${Hence}\:{or}\:{otherwise}\:\:{show}\:{that}\:\: \\ $$$$\:\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right)\:{dx}\:=\:−\frac{\mathrm{1}}{\mathrm{12}}\left[\:\mathrm{13}{ln}\:\mathrm{2}\:+\:\pi\right] \\ $$$$ \\ $$

Question Number 68673    Answers: 0   Comments: 8

find sin 20°=?

$${find}\:\boldsymbol{\mathrm{sin}}\:\mathrm{20}°=? \\ $$

Question Number 68664    Answers: 1   Comments: 0

In a equilateral triangle ABC whose side is a, the points M and N are taken on the side BC, such that the triangles ABM, AMN and ANC have the same perimeter. Calculate the distances from vertex A to points M and N. (solve in detail.)

$$\mathrm{In}\:\mathrm{a}\:\mathrm{equilateral}\:\mathrm{triangle}\:{ABC}\:\mathrm{whose} \\ $$$$\mathrm{side}\:\mathrm{is}\:\boldsymbol{{a}},\:\mathrm{the}\:\mathrm{points}\:{M}\:\mathrm{and}\:{N}\:\mathrm{are}\:\mathrm{taken} \\ $$$$\mathrm{on}\:\mathrm{the}\:\mathrm{side}\:{BC},\:\mathrm{such}\:\mathrm{that}\:\mathrm{the}\:\mathrm{triangles} \\ $$$${ABM},\:{AMN}\:\mathrm{and}\:{ANC}\:\mathrm{have}\:\mathrm{the}\:\mathrm{same}\: \\ $$$$\mathrm{perimeter}.\:\mathrm{Calculate}\:\mathrm{the}\:\mathrm{distances}\:\mathrm{from} \\ $$$$\mathrm{vertex}\:{A}\:\mathrm{to}\:\mathrm{points}\:{M}\:\mathrm{and}\:{N}. \\ $$$$\left(\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{detail}}.\right) \\ $$

  Pg 1416      Pg 1417      Pg 1418      Pg 1419      Pg 1420      Pg 1421      Pg 1422      Pg 1423      Pg 1424      Pg 1425   

Terms of Service

Privacy Policy

Contact: info@tinkutara.com