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GeometryQuestion and Answers: Page 113

Question Number 3055 Answers: 2 Comments: 0

Given a set S={a_1 , ..., a_n } a_k ∈Z a_1 =a, a_n =a_(n−1) +1, a_n =b ∴S contains all integers between a and b. Are there more even or odd values?

$$\mathrm{Given}\:\mathrm{a}\:\mathrm{set}\:\:{S}=\left\{{a}_{\mathrm{1}} ,\:...,\:{a}_{{n}} \right\} \\ $$$${a}_{{k}} \in\mathbb{Z} \\ $$$$ \\ $$$${a}_{\mathrm{1}} ={a},\:\:\:{a}_{{n}} ={a}_{{n}−\mathrm{1}} +\mathrm{1},\:\:\:{a}_{{n}} ={b} \\ $$$$ \\ $$$$\therefore{S}\:\mathrm{contains}\:\mathrm{all}\:\mathrm{integers}\:\mathrm{between} \\ $$$${a}\:\mathrm{and}\:{b}. \\ $$$$ \\ $$$$\mathrm{Are}\:\mathrm{there}\:\mathrm{more}\:\mathrm{even}\:\mathrm{or}\:\mathrm{odd}\:\mathrm{values}? \\ $$

Question Number 2795 Answers: 2 Comments: 0

Prove that (1+x+x^2 +...)(1+2x+3x^2 +...) =(1/2)(1.2+2.3x+3.4x^2 +...)

$${Prove}\:{that} \\ $$$$\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} +...\right)\left(\mathrm{1}+\mathrm{2}{x}+\mathrm{3}{x}^{\mathrm{2}} +...\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}.\mathrm{2}+\mathrm{2}.\mathrm{3}{x}+\mathrm{3}.\mathrm{4}{x}^{\mathrm{2}} +...\right) \\ $$

Question Number 2499 Answers: 1 Comments: 2

ABCD.EFGH is cube, X is midpoint EF if AB = 6 cm, how distance AX to BD??

$${ABCD}.{EFGH}\:{is}\:{cube},\:{X}\:{is}\:{midpoint}\:{EF} \\ $$$${if}\:{AB}\:=\:\mathrm{6}\:{cm},\:{how}\:{distance}\:{AX}\:{to}\:{BD}?? \\ $$

Question Number 2466 Answers: 3 Comments: 0

(√(−(1/5) ))−(1/(√(−5)))=?

$$\sqrt{−\frac{\mathrm{1}}{\mathrm{5}}\:}−\frac{\mathrm{1}}{\sqrt{−\mathrm{5}}}=? \\ $$

Question Number 2458 Answers: 1 Comments: 0

The medians of a triangle are m_1 , m_2 , m_3 . Find the length of each sides the triangle.

$${The}\:{medians}\:{of}\:{a}\:{triangle} \\ $$$${are}\:{m}_{\mathrm{1}} ,\:{m}_{\mathrm{2}} ,\:{m}_{\mathrm{3}} . \\ $$$${Find}\:{the}\:{length}\:{of}\:{each}\:{sides}\: \\ $$$${the}\:{triangle}. \\ $$

Question Number 2329 Answers: 0 Comments: 3

a_n =(−1)^n (⌊(n/9)⌋+1) Σ_(n≥0) a_n =? b_n =(((a_n −1)(9−a_n ))/(n+1)) Σ_(n≥0) b_n =?

$${a}_{{n}} =\left(−\mathrm{1}\right)^{{n}} \left(\lfloor\frac{{n}}{\mathrm{9}}\rfloor+\mathrm{1}\right) \\ $$$$\underset{{n}\geqslant\mathrm{0}} {\sum}{a}_{{n}} =? \\ $$$${b}_{{n}} =\frac{\left({a}_{{n}} −\mathrm{1}\right)\left(\mathrm{9}−{a}_{{n}} \right)}{{n}+\mathrm{1}} \\ $$$$\underset{{n}\geqslant\mathrm{0}} {\sum}{b}_{{n}} =? \\ $$

Question Number 2196 Answers: 1 Comments: 0

(a+b)^2 =a^2 +b^2 +2ab (a+b+c)^2 =a^2 +b^2 +c^2 +2ab+2bc+2ca (a+b+c+d)^2 =? (a_1 +a_2 +...+a_n )^2 =? Derive a formula or give a technique.

$$\left({a}+{b}\right)^{\mathrm{2}} ={a}^{\mathrm{2}} +{b}^{\mathrm{2}} +\mathrm{2}{ab} \\ $$$$\left({a}+{b}+{c}\right)^{\mathrm{2}} ={a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +\mathrm{2}{ab}+\mathrm{2}{bc}+\mathrm{2}{ca} \\ $$$$\left({a}+{b}+{c}+{d}\right)^{\mathrm{2}} =? \\ $$$$\left({a}_{\mathrm{1}} +{a}_{\mathrm{2}} +...+{a}_{{n}} \right)^{\mathrm{2}} =? \\ $$$${Derive}\:{a}\:{formula}\:{or}\:{give}\:{a}\:{technique}. \\ $$

Question Number 2138 Answers: 0 Comments: 8

Is the following proof correct? Δ=Σ_(i=0) ^∞ (−1)^i 2^i =1−2+4−8+16−32+... Let: Δ_1 =1−2+4−8+16−32+... Δ_2 = 1−2+4−8+16−32+... Δ_1 +Δ_2 =1+(−2+1)+(4−2)+(−8+4)+... ∴Δ_1 +Δ_2 =1−(1−2+4−8+...) ∴Δ_1 +Δ_2 =1−Δ_1 Δ_1 =Δ_2 =Δ 3Δ=1 ∴Δ=(1/3) ∴Δ=Σ_(i=0) ^∞ (−1)^i 2^i =(1/3)

$$\mathrm{Is}\:\mathrm{the}\:\mathrm{following}\:\mathrm{proof}\:\mathrm{correct}? \\ $$$$ \\ $$$$\Delta=\underset{{i}=\mathrm{0}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{{i}} \mathrm{2}^{{i}} =\mathrm{1}−\mathrm{2}+\mathrm{4}−\mathrm{8}+\mathrm{16}−\mathrm{32}+... \\ $$$$ \\ $$$$\mathrm{Let}: \\ $$$$\Delta_{\mathrm{1}} =\mathrm{1}−\mathrm{2}+\mathrm{4}−\mathrm{8}+\mathrm{16}−\mathrm{32}+... \\ $$$$\Delta_{\mathrm{2}} =\:\:\:\:\:\:\:\mathrm{1}−\mathrm{2}+\mathrm{4}−\mathrm{8}+\mathrm{16}−\mathrm{32}+... \\ $$$$ \\ $$$$\Delta_{\mathrm{1}} +\Delta_{\mathrm{2}} =\mathrm{1}+\left(−\mathrm{2}+\mathrm{1}\right)+\left(\mathrm{4}−\mathrm{2}\right)+\left(−\mathrm{8}+\mathrm{4}\right)+... \\ $$$$\therefore\Delta_{\mathrm{1}} +\Delta_{\mathrm{2}} =\mathrm{1}−\left(\mathrm{1}−\mathrm{2}+\mathrm{4}−\mathrm{8}+...\right) \\ $$$$\therefore\Delta_{\mathrm{1}} +\Delta_{\mathrm{2}} =\mathrm{1}−\Delta_{\mathrm{1}} \\ $$$$\Delta_{\mathrm{1}} =\Delta_{\mathrm{2}} =\Delta \\ $$$$\mathrm{3}\Delta=\mathrm{1} \\ $$$$\therefore\Delta=\frac{\mathrm{1}}{\mathrm{3}} \\ $$$$ \\ $$$$\therefore\Delta=\underset{{i}=\mathrm{0}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{{i}} \mathrm{2}^{{i}} =\frac{\mathrm{1}}{\mathrm{3}} \\ $$$$ \\ $$

Question Number 1999 Answers: 1 Comments: 0

if (1 + tan 1°)(1 + tan 2°)(1 + tan 3°)...... ....)(1 + tan 44°)(1 + tan 45°) ^ = { ((((√(50)) + 7)))^(1/3) − ((((√(50))−7)))^(1/3) }^((x − 7)) find x = ...?

$${if}\: \\ $$$$\left(\mathrm{1}\:+\:{tan}\:\mathrm{1}°\right)\left(\mathrm{1}\:+\:{tan}\:\mathrm{2}°\right)\left(\mathrm{1}\:+\:{tan}\:\mathrm{3}°\right)...... \\ $$$$\left.....\right)\left(\mathrm{1}\:+\:{tan}\:\mathrm{44}°\right)\left(\mathrm{1}\:+\:{tan}\:\mathrm{45}°\right)\overset{} {\:}=\:\:\left\{\:\:\:\sqrt[{\mathrm{3}}]{\left(\sqrt{\mathrm{50}}\:+\:\mathrm{7}\right)}\:−\:\sqrt[{\mathrm{3}}]{\left(\sqrt{\mathrm{50}}−\mathrm{7}\right)}\:\overset{\left({x}\:−\:\mathrm{7}\right)} {\right\}} \\ $$$$\: \\ $$$${find}\:{x}\:=\:...? \\ $$

Question Number 2001 Answers: 0 Comments: 0

Prove that : (1/(15)) < (1/2) ∙ (3/4) ∙ (5/6) ∙ ∙ ∙ ∙ ∙ ((99)/(100)) < (1/(10))

$${Prove}\:{that}\:: \\ $$$$\frac{\mathrm{1}}{\mathrm{15}}\:<\:\frac{\mathrm{1}}{\mathrm{2}}\:\centerdot\:\frac{\mathrm{3}}{\mathrm{4}}\:\centerdot\:\frac{\mathrm{5}}{\mathrm{6}}\:\centerdot\:\centerdot\:\centerdot\:\centerdot\:\centerdot\:\frac{\mathrm{99}}{\mathrm{100}}\:<\:\frac{\mathrm{1}}{\mathrm{10}} \\ $$

Question Number 1928 Answers: 2 Comments: 10

Prove that, if p>q>0 and x≥0, then (1/p)((x^p /(p+1))−1)≥(1/q)((x^q /(q+1))−1).

$${Prove}\:{that},\:{if}\:{p}>{q}>\mathrm{0}\:{and}\:{x}\geqslant\mathrm{0},\:{then} \\ $$$$\:\:\:\:\:\frac{\mathrm{1}}{{p}}\left(\frac{{x}^{{p}} }{{p}+\mathrm{1}}−\mathrm{1}\right)\geqslant\frac{\mathrm{1}}{{q}}\left(\frac{{x}^{{q}} }{{q}+\mathrm{1}}−\mathrm{1}\right).\: \\ $$

Question Number 1793 Answers: 0 Comments: 0

Evaluate ∫_(1/2) ^2 ((sinx)/(x(sinx+sin(1/x))))dx .

$${Evaluate}\: \\ $$$$\:\:\:\:\:\:\:\:\int_{\mathrm{1}/\mathrm{2}} ^{\mathrm{2}} \frac{{sinx}}{{x}\left({sinx}+{sin}\frac{\mathrm{1}}{{x}}\right)}{dx}\:. \\ $$

Question Number 1740 Answers: 1 Comments: 1

how 0!=1

$${how}\:\:\mathrm{0}!=\mathrm{1} \\ $$

Question Number 1750 Answers: 1 Comments: 0

(x+3)^3

$$\left({x}+\mathrm{3}\right)^{\mathrm{3}} \\ $$

Question Number 1611 Answers: 0 Comments: 1

A right angled triangle has fixed hypotenuse measuring h units. What are the measures of its legs, for maximum perimeter P units. Will the area be also maximum, when the perimeter be maximum?

$$\mathrm{A}\:\mathrm{right}\:\mathrm{angled}\:\mathrm{triangle}\:\mathrm{has}\:\mathrm{fixed}\:\mathrm{hypotenuse}\:\mathrm{measuring} \\ $$$$\mathrm{h}\:\mathrm{units}.\:\mathrm{What}\:\mathrm{are}\:\mathrm{the}\:\mathrm{measures}\:\:\mathrm{of}\:\mathrm{its}\:\mathrm{legs}, \\ $$$$\mathrm{for}\:\boldsymbol{\mathrm{maximum}}\:\boldsymbol{\mathrm{perimeter}}\:\mathrm{P}\:\mathrm{units}. \\ $$$$\mathrm{Will}\:\mathrm{the}\:\mathrm{area}\:\mathrm{be}\:\mathrm{also}\:\mathrm{maximum},\:\mathrm{when}\:\mathrm{the}\:\mathrm{perimeter}\:\mathrm{be} \\ $$$$\mathrm{maximum}? \\ $$

Question Number 1592 Answers: 1 Comments: 1

I have a loop of string of length(perimeter) p units. I want to make a triangle of largest area from the loop. What will be the dimensions of that triangle?

$$\mathrm{I}\:\mathrm{have}\:\mathrm{a}\:\mathrm{loop}\:\mathrm{of}\:\mathrm{string}\:\mathrm{of}\:\mathrm{length}\left(\mathrm{perimeter}\right)\:\:\mathrm{p}\:\mathrm{units}.\: \\ $$$$\mathrm{I}\:\mathrm{want}\:\mathrm{to}\:\mathrm{make}\:\mathrm{a}\:\mathrm{triangle}\:\mathrm{of}\:\mathrm{largest}\:\mathrm{area}\:\mathrm{from}\:\mathrm{the} \\ $$$$\mathrm{loop}.\:\mathrm{What}\:\mathrm{will}\:\mathrm{be}\:\mathrm{the}\:\mathrm{dimensions}\:\mathrm{of}\:\mathrm{that}\:\mathrm{triangle}? \\ $$

Question Number 1581 Answers: 0 Comments: 2

Find a function f(x) satisfying the following equation. ∫_a ^( b) [(1/2){f(x)}^2 −(√({f(x)}^2 +{(d/dx)(f(x))}^2 ))]dx=0 b>0,a>0 , b≠a.

$${Find}\:{a}\:{function}\:{f}\left({x}\right)\:{satisfying} \\ $$$${the}\:{following}\:{equation}. \\ $$$$\int_{{a}} ^{\:{b}} \left[\frac{\mathrm{1}}{\mathrm{2}}\left\{{f}\left({x}\right)\right\}^{\mathrm{2}} −\sqrt{\left\{{f}\left({x}\right)\right\}^{\mathrm{2}} +\left\{\frac{{d}}{{dx}}\left({f}\left({x}\right)\right)\right\}^{\mathrm{2}} }\right]{dx}=\mathrm{0} \\ $$$${b}>\mathrm{0},{a}>\mathrm{0}\:,\:{b}\neq{a}.\:\: \\ $$

Question Number 1387 Answers: 0 Comments: 1

Q. is there any angle in a circle?

$${Q}.\:{is}\:{there}\:{any}\:{angle}\:{in}\:{a}\:{circle}? \\ $$

Question Number 1395 Answers: 0 Comments: 5

Consider quadrilateral ABCD same as in Q 1378 with same conditions/restrictions (Pl refer the Question again). • What could be possible minimum and maximum area of the quadrilateral? •When qusdrilateral has minimum area what is the value/s of m∠A? Similarly what is value/s of m∠A in case of maximum area?

Question Number 1378 Answers: 0 Comments: 4

Four sides mAB^(−) , mBC^(−) , mCD^(−) and mDA^(−) of a quadrilateral ABCD have measurement a , b , c and d units respectively. Let the sum of any adjacent sides is not equal to the sum of remaining adjacent sides and measurement of all the sides is positive and real. What could be the possible minimum and maximum values of its one angle m∠A ?

$$\:\:\:\:\:{Four}\:{sides}\:{m}\overline {\boldsymbol{\mathrm{AB}}}\:,\:{m}\overline {\boldsymbol{\mathrm{BC}}}\:,\:{m}\overline {\boldsymbol{\mathrm{CD}}}\:\:{and}\:\:{m}\overline {\boldsymbol{\mathrm{DA}}}\:{of}\:{a}\:\boldsymbol{\mathrm{quadrilateral}}\: \\ $$$$\boldsymbol{\mathrm{ABCD}}\:\:{have}\:{measurement}\:\boldsymbol{{a}}\:,\:\boldsymbol{{b}}\:,\:\boldsymbol{{c}}\:\:{and}\:\boldsymbol{{d}}\:{units}\:{respectively}. \\ $$$$\:\:\:\:\:{Let}\:{the}\:{sum}\:{of}\:{any}\:{adjacent}\:{sides}\:{is}\:{not}\:{equal}\:{to}\:{the}\:{sum}\:{of} \\ $$$${remaining}\:{adjacent}\:{sides}\:\:{and}\:{measurement}\:{of}\:{all}\:{the}\:{sides}\: \\ $$$${is}\:{positive}\:{and}\:{real}. \\ $$$$\:\:\:\:\:\:{What}\:{could}\:{be}\:{the}\:{possible}\:{minimum}\:{and}\:{maximum}\:{values} \\ $$$${of}\:{its}\:{one}\:{angle}\:\:{m}\angle\boldsymbol{\mathrm{A}}\:? \\ $$

Question Number 1304 Answers: 1 Comments: 0

If 3^(log 3x) =4^(log 4x) , find x.

$${If}\:\mathrm{3}^{{log}\:\mathrm{3}{x}} =\mathrm{4}^{{log}\:\mathrm{4}{x}} ,\:{find}\:{x}. \\ $$$$ \\ $$

Question Number 1343 Answers: 1 Comments: 0

3^(log 3x+4) =4^(log 4x+3)

$$\mathrm{3}^{{log}\:\mathrm{3}{x}+\mathrm{4}} =\mathrm{4}^{{log}\:\mathrm{4}{x}+\mathrm{3}} \\ $$

Question Number 1268 Answers: 1 Comments: 0

Prove that AM > HM.

$$\boldsymbol{\mathrm{Prove}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{AM}}\:>\:\boldsymbol{\mathrm{HM}}. \\ $$

Question Number 1157 Answers: 1 Comments: 0

show that tan^(−1) ((((√(1+x^2 ))−1)/x))=(1/2)tan^(−1) x