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Question Number 111734 Answers: 1 Comments: 0

A chord which is a perpendicular bisector of radius of length 18cm in a circle, has length.

$$\mathrm{A}\:\mathrm{chord}\:\mathrm{which}\:\mathrm{is}\:\mathrm{a}\:\mathrm{perpendicular} \\ $$$$\mathrm{bisector}\:\mathrm{of}\:\mathrm{radius}\:\mathrm{of}\:\mathrm{length}\:\mathrm{18cm}\:\mathrm{in}\:\mathrm{a} \\ $$$$\mathrm{circle},\:\mathrm{has}\:\mathrm{length}. \\ $$$$ \\ $$

Question Number 111732 Answers: 1 Comments: 0

A blind man is to place 5 letters into 5 pigeon holes, how many ways can 4 of the letters be wrongly placed? (note that only one letter must be in a pigeon hole)

$$\mathrm{A}\:\mathrm{blind}\:\mathrm{man}\:\mathrm{is}\:\mathrm{to}\:\mathrm{place}\:\mathrm{5}\:\mathrm{letters}\:\mathrm{into}\:\mathrm{5} \\ $$$$\mathrm{pigeon}\:\mathrm{holes},\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{4}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{letters}\:\mathrm{be}\:\mathrm{wrongly}\:\mathrm{placed}? \\ $$$$\left(\mathrm{note}\:\mathrm{that}\:\mathrm{only}\:\mathrm{one}\:\mathrm{letter}\:\mathrm{must}\:\mathrm{be}\:\mathrm{in}\:\mathrm{a}\right. \\ $$$$\left.\mathrm{pigeon}\:\mathrm{hole}\right) \\ $$

Question Number 111730 Answers: 0 Comments: 9

How many triples of positive integers (x,y,z) satisfy 79x+80y+81z =2016

$$\mathrm{How}\:\mathrm{many}\:\mathrm{triples}\:\mathrm{of}\:\mathrm{positive}\:\mathrm{integers} \\ $$$$\left(\mathrm{x},\mathrm{y},\mathrm{z}\right)\:\mathrm{satisfy}\: \\ $$$$\mathrm{79x}+\mathrm{80y}+\mathrm{81z}\:=\mathrm{2016} \\ $$

Question Number 112813 Answers: 0 Comments: 2

A binary operation has the property a∗(b∗c) = (a∗b)•c and that a∗a=1 for all non−zero real numbers a,b and c. (′•′ here represent multiplication). The solution of the equation 2016∗(6∗x)=100 can be written as (p/q) where p and q are relatively prime positive integers. What is q−p?

$$\mathrm{A}\:\mathrm{binary}\:\mathrm{operation}\:\mathrm{has}\:\mathrm{the}\:\mathrm{property} \\ $$$$\mathrm{a}\ast\left(\mathrm{b}\ast\mathrm{c}\right)\:=\:\left(\mathrm{a}\ast\mathrm{b}\right)\bullet\mathrm{c}\:\mathrm{and}\:\mathrm{that}\:\mathrm{a}\ast\mathrm{a}=\mathrm{1}\:\mathrm{for} \\ $$$$\mathrm{all}\:\mathrm{non}−\mathrm{zero}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{a},\mathrm{b}\:\mathrm{and}\:\mathrm{c}. \\ $$$$\left('\bullet'\:\mathrm{here}\:\mathrm{represent}\:\mathrm{multiplication}\right). \\ $$$$\mathrm{The}\:\mathrm{solution}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\mathrm{2016}\ast\left(\mathrm{6}\ast\mathrm{x}\right)=\mathrm{100}\:\mathrm{can}\:\mathrm{be}\:\mathrm{written}\:\mathrm{as}\:\frac{\mathrm{p}}{\mathrm{q}} \\ $$$$\mathrm{where}\:\mathrm{p}\:\mathrm{and}\:\mathrm{q}\:\mathrm{are}\:\mathrm{relatively}\:\mathrm{prime} \\ $$$$\mathrm{positive}\:\mathrm{integers}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{q}−\mathrm{p}? \\ $$

Question Number 112812 Answers: 0 Comments: 2

There are 2016 straight lines drawn on a board such that (1/2) of the lines are parallel to one another. (3/8) of them meet at a point and each of the remaining ones intersect with all other lines on the board. Determine the total number of intersections possible.

$$\mathrm{There}\:\mathrm{are}\:\mathrm{2016}\:\mathrm{straight}\:\mathrm{lines}\:\mathrm{drawn}\:\mathrm{on} \\ $$$$\mathrm{a}\:\mathrm{board}\:\mathrm{such}\:\mathrm{that}\:\frac{\mathrm{1}}{\mathrm{2}}\:\mathrm{of}\:\mathrm{the}\:\mathrm{lines}\:\mathrm{are} \\ $$$$\mathrm{parallel}\:\mathrm{to}\:\mathrm{one}\:\mathrm{another}.\:\frac{\mathrm{3}}{\mathrm{8}}\:\mathrm{of}\:\mathrm{them} \\ $$$$\mathrm{meet}\:\mathrm{at}\:\mathrm{a}\:\mathrm{point}\:\mathrm{and}\:\mathrm{each}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{remaining}\:\mathrm{ones}\:\mathrm{intersect}\:\mathrm{with}\:\mathrm{all} \\ $$$$\mathrm{other}\:\mathrm{lines}\:\mathrm{on}\:\mathrm{the}\:\mathrm{board}.\:\mathrm{Determine} \\ $$$$\mathrm{the}\:\mathrm{total}\:\mathrm{number}\:\mathrm{of}\:\mathrm{intersections} \\ $$$$\mathrm{possible}. \\ $$

Question Number 111725 Answers: 1 Comments: 0

Find the positive integer n such that tan^(−1) ((1/3))+tan^(−1) ((1/4))+tan^(−1) ((1/5))+tan^(−1) ((1/n))=(π/4)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{n}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{3}}\right)+\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{4}}\right)+\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{5}}\right)+\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{n}}\right)=\frac{\pi}{\mathrm{4}} \\ $$

Question Number 111724 Answers: 0 Comments: 6

How many real numbers x satisfy the equation 3^(2x+2) −3^(x+3) −3^x +3=0 ?

$$\mathrm{How}\:\mathrm{many}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{x}\:\mathrm{satisfy}\:\mathrm{the} \\ $$$$\mathrm{equation}\:\mathrm{3}^{\mathrm{2x}+\mathrm{2}} −\mathrm{3}^{\mathrm{x}+\mathrm{3}} −\mathrm{3}^{\mathrm{x}} +\mathrm{3}=\mathrm{0}\:? \\ $$

Question Number 111719 Answers: 2 Comments: 0

....advanced mathematics.... please demonstrate that:: Φ =∫_0 ^( 1) xlog(1−x).log(1+x)= (1/4) − log(2) ... m.n.july 1970 #

$$\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:....{advanced}\:\:{mathematics}....\: \\ $$$$ \\ $$$${please}\:\:{demonstrate}\:{that}:: \\ $$$$\: \\ $$$$\Phi\:=\int_{\mathrm{0}} ^{\:\mathrm{1}} {xlog}\left(\mathrm{1}−{x}\right).{log}\left(\mathrm{1}+{x}\right)=\:\frac{\mathrm{1}}{\mathrm{4}}\:−\:{log}\left(\mathrm{2}\right)\:\:... \\ $$$$ \\ $$$$\:\:\:\:\:\:{m}.{n}.{july}\:\mathrm{1970}\:# \\ $$$$ \\ $$

Question Number 111704 Answers: 1 Comments: 0

Assuming FLT, prove Fermat−Euler theorem: (a,n) =1,n≥2⇒a^(∅(n)) ≡1(mod n)