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Question Number 19122 Answers: 1 Comments: 0

Prove that r_1 + r_2 + r_3 = 4R + r

$$\mathrm{Prove}\:\mathrm{that}\:{r}_{\mathrm{1}} \:+\:{r}_{\mathrm{2}} \:+\:{r}_{\mathrm{3}} \:=\:\mathrm{4}{R}\:+\:{r} \\ $$

Question Number 19121 Answers: 0 Comments: 0

Question Number 19118 Answers: 0 Comments: 0

Question Number 19104 Answers: 1 Comments: 1

Let ABC be an acute-angled triangle with AC ≠ BC and let O be the circumcenter and F be the foot of altitude through C. Further, let X and Y be the feet of perpendiculars dropped from A and B respectively to (the extension of) CO. The line FO intersects the circumcircle of ΔFXY, second time at P. Prove that OP < OF.

$$\mathrm{Let}\:\mathrm{ABC}\:\mathrm{be}\:\mathrm{an}\:\mathrm{acute}-\mathrm{angled}\:\mathrm{triangle} \\ $$$$\mathrm{with}\:\mathrm{AC}\:\neq\:\mathrm{BC}\:\mathrm{and}\:\mathrm{let}\:\mathrm{O}\:\mathrm{be}\:\mathrm{the} \\ $$$$\mathrm{circumcenter}\:\mathrm{and}\:\mathrm{F}\:\mathrm{be}\:\mathrm{the}\:\mathrm{foot}\:\mathrm{of} \\ $$$$\mathrm{altitude}\:\mathrm{through}\:\mathrm{C}.\:\mathrm{Further},\:\mathrm{let}\:\mathrm{X}\:\mathrm{and}\:\mathrm{Y} \\ $$$$\mathrm{be}\:\mathrm{the}\:\mathrm{feet}\:\mathrm{of}\:\mathrm{perpendiculars}\:\mathrm{dropped} \\ $$$$\mathrm{from}\:\mathrm{A}\:\mathrm{and}\:\mathrm{B}\:\mathrm{respectively}\:\mathrm{to}\:\left(\mathrm{the}\right. \\ $$$$\left.\mathrm{extension}\:\mathrm{of}\right)\:\mathrm{CO}.\:\mathrm{The}\:\mathrm{line}\:\mathrm{FO}\:\mathrm{intersects} \\ $$$$\mathrm{the}\:\mathrm{circumcircle}\:\mathrm{of}\:\Delta\mathrm{FXY},\:\mathrm{second}\:\mathrm{time} \\ $$$$\mathrm{at}\:\mathrm{P}.\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{OP}\:<\:\mathrm{OF}. \\ $$

Question Number 19101 Answers: 0 Comments: 3

A polynomial f(x) with rational coefficients leaves remainder 15, when divided by x − 3 and remainder 2x + 1, when divided by (x − 1)^2 . Find the remainder when f(x) is divided by (x − 3)(x − 1)^2 .

$$\mathrm{A}\:\mathrm{polynomial}\:{f}\left({x}\right)\:\mathrm{with}\:\mathrm{rational} \\ $$$$\mathrm{coefficients}\:\mathrm{leaves}\:\mathrm{remainder}\:\mathrm{15},\:\mathrm{when} \\ $$$$\mathrm{divided}\:\mathrm{by}\:{x}\:−\:\mathrm{3}\:\mathrm{and}\:\mathrm{remainder}\:\mathrm{2}{x}\:+\:\mathrm{1}, \\ $$$$\mathrm{when}\:\mathrm{divided}\:\mathrm{by}\:\left({x}\:−\:\mathrm{1}\right)^{\mathrm{2}} .\:\mathrm{Find}\:\mathrm{the} \\ $$$$\mathrm{remainder}\:\mathrm{when}\:{f}\left({x}\right)\:\mathrm{is}\:\mathrm{divided}\:\mathrm{by} \\ $$$$\left({x}\:−\:\mathrm{3}\right)\left({x}\:−\:\mathrm{1}\right)^{\mathrm{2}} . \\ $$

Question Number 19097 Answers: 1 Comments: 0

If tan ((π/4) + x) = tan^3 ((π/4) + α) then prove that cosec 2x = ((1 + 3 sin^2 2α)/(3 sin 2α + sin^3 2α))

$$\mathrm{If}\:\mathrm{tan}\:\left(\frac{\pi}{\mathrm{4}}\:+\:{x}\right)\:=\:\mathrm{tan}^{\mathrm{3}} \:\left(\frac{\pi}{\mathrm{4}}\:+\:\alpha\right)\:\mathrm{then} \\ $$$$\mathrm{prove}\:\mathrm{that}\:\mathrm{cosec}\:\mathrm{2}{x}\:=\:\frac{\mathrm{1}\:+\:\mathrm{3}\:\mathrm{sin}^{\mathrm{2}} \:\mathrm{2}\alpha}{\mathrm{3}\:\mathrm{sin}\:\mathrm{2}\alpha\:+\:\mathrm{sin}^{\mathrm{3}} \:\mathrm{2}\alpha} \\ $$

Question Number 19095 Answers: 0 Comments: 3

Question Number 19085 Answers: 0 Comments: 0

f_n (x)=(√(f_(n−1) (x)×(f_(n−1) (x))′)) f_1 (x)=x^(2017) +x^8 +x^4 lim_(n→∞) f_n (x)=?

$$\mathrm{f}_{\mathrm{n}} \left(\mathrm{x}\right)=\sqrt{\mathrm{f}_{\mathrm{n}−\mathrm{1}} \left(\mathrm{x}\right)×\left(\mathrm{f}_{\mathrm{n}−\mathrm{1}} \left(\mathrm{x}\right)\right)'} \\ $$$$\mathrm{f}_{\mathrm{1}} \left(\mathrm{x}\right)=\mathrm{x}^{\mathrm{2017}} +\mathrm{x}^{\mathrm{8}} +\mathrm{x}^{\mathrm{4}} \\ $$$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}f}_{\mathrm{n}} \left(\mathrm{x}\right)=? \\ $$

Question Number 19083 Answers: 1 Comments: 4

Question Number 19080 Answers: 0 Comments: 1

Question Number 19064 Answers: 1 Comments: 2

Question Number 19063 Answers: 2 Comments: 0

find the possible values of x if ((8^x +27^x )/(12^x +18^x ))=(7/6)

$$\mathrm{find}\:\mathrm{the}\:\mathrm{possible}\:\mathrm{values}\:\mathrm{of}\:\mathrm{x}\:\mathrm{if} \\ $$$$\frac{\mathrm{8}^{\mathrm{x}} +\mathrm{27}^{\mathrm{x}} }{\mathrm{12}^{\mathrm{x}} +\mathrm{18}^{\mathrm{x}} }=\frac{\mathrm{7}}{\mathrm{6}} \\ $$

Question Number 19058 Answers: 0 Comments: 0

Question Number 19055 Answers: 1 Comments: 0

Find the cubic equation whose roots are the radius of three escribed circles in term of inradius, circumradius and semiperimeter.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{cubic}\:\mathrm{equation}\:\mathrm{whose}\:\mathrm{roots} \\ $$$$\mathrm{are}\:\mathrm{the}\:\mathrm{radius}\:\mathrm{of}\:\mathrm{three}\:\mathrm{escribed}\:\mathrm{circles} \\ $$$$\mathrm{in}\:\mathrm{term}\:\mathrm{of}\:\mathrm{inradius},\:\mathrm{circumradius}\:\mathrm{and} \\ $$$$\mathrm{semiperimeter}. \\ $$

Question Number 19033 Answers: 2 Comments: 1

Question Number 19073 Answers: 0 Comments: 0

Question Number 19025 Answers: 0 Comments: 0

Question Number 19077 Answers: 1 Comments: 0

∫ sec^3 x dx

$$\int\:\mathrm{sec}^{\mathrm{3}} \:{x}\:{dx} \\ $$

Question Number 19021 Answers: 0 Comments: 4

Question Number 19060 Answers: 1 Comments: 2

Question Number 19009 Answers: 1 Comments: 0

The sum of four consecutive 2−digit numbers when divided by 10 becomes a perfect square.Which of the following can possibly be one of these four numbers? (a)21(b)25(c)41(d)67(e)73 please show workings

$$\mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{four}\:\mathrm{consecutive} \\ $$$$\mathrm{2}−\mathrm{digit}\:\mathrm{numbers}\:\mathrm{when}\:\mathrm{divided}\:\mathrm{by} \\ $$$$\mathrm{10}\:\mathrm{becomes}\:\mathrm{a}\:\mathrm{perfect}\:\mathrm{square}.\mathrm{Which} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{can}\:\mathrm{possibly}\:\mathrm{be} \\ $$$$\mathrm{one}\:\mathrm{of}\:\mathrm{these}\:\mathrm{four}\:\mathrm{numbers}? \\ $$$$\left(\mathrm{a}\right)\mathrm{21}\left(\mathrm{b}\right)\mathrm{25}\left(\mathrm{c}\right)\mathrm{41}\left(\mathrm{d}\right)\mathrm{67}\left(\mathrm{e}\right)\mathrm{73} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$\mathrm{please}\:\mathrm{show}\:\mathrm{workings} \\ $$

Question Number 19002 Answers: 1 Comments: 0

what is the maximum number of time three divides 333^(505)

$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{number}\:\mathrm{of}\:\:\mathrm{time}\:\mathrm{three}\:\mathrm{divides}\:\mathrm{333}^{\mathrm{505}} \\ $$

Question Number 19117 Answers: 0 Comments: 0

Σ_(n=1) ^7 (n/((n^2 +1))) = ? please help

$$\underset{\mathrm{n}=\mathrm{1}} {\overset{\mathrm{7}} {\sum}}\:\frac{\mathrm{n}}{\left(\mathrm{n}^{\mathrm{2}} +\mathrm{1}\right)}\:=\:? \\ $$$$\mathrm{please}\:\mathrm{help} \\ $$

Question Number 19053 Answers: 1 Comments: 0

Find the sum: (1/2) − (1/3) + (1/4) − (1/5) + (1/6) − (1/7) + (1/8) − (1/9) + ...

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}:\:\frac{\mathrm{1}}{\mathrm{2}}\:−\:\frac{\mathrm{1}}{\mathrm{3}}\:+\:\frac{\mathrm{1}}{\mathrm{4}}\:−\:\frac{\mathrm{1}}{\mathrm{5}}\:+\:\frac{\mathrm{1}}{\mathrm{6}} \\ $$$$−\:\frac{\mathrm{1}}{\mathrm{7}}\:+\:\frac{\mathrm{1}}{\mathrm{8}}\:−\:\frac{\mathrm{1}}{\mathrm{9}}\:+\:... \\ $$

Question Number 18979 Answers: 0 Comments: 0

Question Number 18969 Answers: 1 Comments: 0

If ((a/b))+((b/a))=2, then find ((a/b))^(10) − ((b/a))^(10) .

$$\mathrm{If}\:\left(\frac{{a}}{{b}}\right)+\left(\frac{{b}}{{a}}\right)=\mathrm{2},\:\mathrm{then}\:\mathrm{find}\:\left(\frac{{a}}{{b}}\right)^{\mathrm{10}} −\:\left(\frac{{b}}{{a}}\right)^{\mathrm{10}} . \\ $$

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