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AlgebraQuestion and Answers: Page 34

Question Number 209318 Answers: 2 Comments: 0

Question Number 209309 Answers: 0 Comments: 0

m , n ∈ N m ≥ 2 and n ≥ 2 p > 0 and q > 0 p + q = 1 Prove that: (1−q^n )^m + (1−p^m )^n ≥ 1

$$\mathrm{m}\:,\:\mathrm{n}\:\in\:\mathbb{N} \\ $$$$\mathrm{m}\:\geqslant\:\mathrm{2}\:\:\:\mathrm{and}\:\:\:\mathrm{n}\:\geqslant\:\mathrm{2} \\ $$$$\mathrm{p}\:>\:\mathrm{0}\:\:\:\mathrm{and}\:\:\:\mathrm{q}\:>\:\mathrm{0} \\ $$$$\mathrm{p}\:+\:\mathrm{q}\:=\:\mathrm{1} \\ $$$$\mathrm{Prove}\:\mathrm{that}:\:\:\:\left(\mathrm{1}−\mathrm{q}^{\boldsymbol{\mathrm{n}}} \right)^{\boldsymbol{\mathrm{m}}} \:+\:\left(\mathrm{1}−\mathrm{p}^{\boldsymbol{\mathrm{m}}} \right)^{\boldsymbol{\mathrm{n}}} \:\geqslant\:\mathrm{1} \\ $$

Question Number 209290 Answers: 0 Comments: 1

a^2 −a−^(1000) (√((1+8000a)))=1000 find a

$$\boldsymbol{\mathrm{a}}^{\mathrm{2}} −\boldsymbol{\mathrm{a}}−^{\mathrm{1000}} \sqrt{\left(\mathrm{1}+\mathrm{8000}\boldsymbol{\mathrm{a}}\right)}=\mathrm{1000} \\ $$$$\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{a}} \\ $$

Question Number 209263 Answers: 0 Comments: 5

6 different letters were written to 6 different people and 6 different envelopes were prepared with the addresses of these people written on them. In how many different ways can you put a letter in each envelope without putting a letter written to this person in the envelope with the name of any person?

$$ \\ $$6 different letters were written to 6 different people and 6 different envelopes were prepared with the addresses of these people written on them. In how many different ways can you put a letter in each envelope without putting a letter written to this person in the envelope with the name of any person?

Question Number 209240 Answers: 1 Comments: 0

If x + ((49)/(x + 48)) = − 34 find (2x + 83)^3 + (1/((2x + 83)^3 ))

$${If}\:\:{x}\:\:+\:\:\frac{\mathrm{49}}{{x}\:+\:\mathrm{48}}\:\:=\:\:−\:\mathrm{34} \\ $$$${find}\:\:\left(\mathrm{2}{x}\:+\:\mathrm{83}\right)^{\mathrm{3}} \:+\:\frac{\mathrm{1}}{\left(\mathrm{2}{x}\:+\:\mathrm{83}\right)^{\mathrm{3}} } \\ $$

Question Number 209234 Answers: 2 Comments: 0

Arrange in descending order: (√5) − (√2), (√7) − (√5) , (√(13)) − (√(11)) , (√(19)) − (√(17))

$$\mathrm{Arrange}\:\mathrm{in}\:\mathrm{descending}\:\mathrm{order}: \\ $$$$\:\:\:\:\sqrt{\mathrm{5}}\:\:−\:\:\sqrt{\mathrm{2}},\:\:\:\:\:\sqrt{\mathrm{7}}\:\:−\:\:\sqrt{\mathrm{5}}\:,\:\:\:\sqrt{\mathrm{13}}\:\:−\:\:\sqrt{\mathrm{11}}\:,\:\:\:\sqrt{\mathrm{19}}\:\:−\:\:\sqrt{\mathrm{17}} \\ $$

Question Number 209223 Answers: 2 Comments: 0

Question Number 209187 Answers: 3 Comments: 0

:: α , β and γ are roots of the following equation . Find the value of ” F ” : Equation : x^( 3) −2x −1=0 F := α^( 5) + β^( 5) + γ^( 5)

$$ \\ $$$$\:\:\:::\:\:\:\alpha\:,\:\beta\:\:{and}\:\:\gamma\:\:{are}\:{roots}\:{of}\:{the} \\ $$$$\:\:\:\:\:{following}\:\:{equation}\:.\:{Find}\:{the} \\ $$$$\:\:\:\:\:{value}\:\:{of}\:\:\:''\:\:\mathrm{F}\:\:''\::\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{E}{quation}\::\:\:\:\:\:\:{x}^{\:\mathrm{3}} \:−\mathrm{2}{x}\:\:−\mathrm{1}=\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{F}\::=\:\alpha^{\:\mathrm{5}} \:+\:\beta^{\:\mathrm{5}} \:+\:\gamma^{\:\mathrm{5}} \\ $$$$\:\:\:\:\:\:\:\:\: \\ $$

Question Number 209131 Answers: 1 Comments: 0

prove : curve { ((x(t)=((a+r.cos(t))/(a^2 +r^2 +2ar.cos(t))))),((y(t)=((r.sin(t))/(a^2 +r^2 +2ar.cos(t))))) :} 0≤t≤2π is circle , find center & radius

$${prove}\:: \\ $$$${curve}\:\begin{cases}{{x}\left({t}\right)=\frac{{a}+{r}.{cos}\left({t}\right)}{{a}^{\mathrm{2}} +{r}^{\mathrm{2}} +\mathrm{2}{ar}.{cos}\left({t}\right)}}\\{{y}\left({t}\right)=\frac{{r}.{sin}\left({t}\right)}{{a}^{\mathrm{2}} +{r}^{\mathrm{2}} +\mathrm{2}{ar}.{cos}\left({t}\right)}}\end{cases}\:\:\:\:\mathrm{0}\leqslant{t}\leqslant\mathrm{2}\pi \\ $$$${is}\:{circle}\:,\:{find}\:{center}\:\&\:{radius} \\ $$

Question Number 209072 Answers: 0 Comments: 8

Question Number 209065 Answers: 1 Comments: 0

The cost of maintaining a school is partly constant and partly varies as the number of students. With 50 students the cost is $15705 and with 40 students the cost is $13305. If the fee per student is $360.00, what is the least number of students for which the school can be run without loss?

$${The}\:{cost}\:{of}\:{maintaining}\:{a}\:{school}\:{is} \\ $$$${partly}\:{constant}\:{and}\:{partly}\:{varies}\:{as} \\ $$$${the}\:{number}\:{of}\:{students}.\:{With}\:\mathrm{50}\:{students} \\ $$$${the}\:{cost}\:{is}\:\$\mathrm{15705}\:{and}\:{with}\:\mathrm{40}\:{students} \\ $$$${the}\:{cost}\:{is}\:\$\mathrm{13305}.\:{If}\:{the}\:{fee}\:{per}\:{student} \\ $$$${is}\:\$\mathrm{360}.\mathrm{00},\:{what}\:{is}\:{the}\:{least}\:{number}\:{of} \\ $$$${students}\:{for}\:{which}\:{the}\:{school}\:{can}\:{be} \\ $$$${run}\:{without}\:{loss}? \\ $$

Question Number 209062 Answers: 1 Comments: 0

Question Number 209059 Answers: 0 Comments: 3

Compare: 8! and 8!!

$$\mathrm{Compare}: \\ $$$$\mathrm{8}!\:\:\:\mathrm{and}\:\:\:\mathrm{8}!! \\ $$

Question Number 209055 Answers: 0 Comments: 5

Question Number 209021 Answers: 1 Comments: 7

Hello, I present to you an interesting combinatorics question: A group of people from k families should be seated around a round table, with a_{i} number of people in the i family. Each family member must sit together (i.e. no family member can sit between other family members). There are l spaces around the table. There are seats (l>k). How many ways can we seat k number of families around a round table under these conditions.

$$ \\ $$Hello, I present to you an interesting combinatorics question: A group of people from k families should be seated around a round table, with a_{i} number of people in the i family. Each family member must sit together (i.e. no family member can sit between other family members). There are l spaces around the table. There are seats (l>k). How many ways can we seat k number of families around a round table under these conditions.

Question Number 208959 Answers: 5 Comments: 0

If z = − (1/2) + ((√3)/2) i Find (z^4 + 2z)∙(z^3 + z) = ?

$$\mathrm{If}\:\:\:\boldsymbol{\mathrm{z}}\:=\:−\:\frac{\mathrm{1}}{\mathrm{2}}\:\:+\:\:\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\:\boldsymbol{\mathrm{i}} \\ $$$$\mathrm{Find}\:\:\:\left(\mathrm{z}^{\mathrm{4}} \:+\:\mathrm{2z}\right)\centerdot\left(\mathrm{z}^{\mathrm{3}} \:+\:\mathrm{z}\right)\:=\:? \\ $$

Question Number 208951 Answers: 1 Comments: 0

2^(2024) : 2024 = ... (Remainder = ?)

$$\mathrm{2}^{\mathrm{2024}} \::\:\mathrm{2024}\:=\:...\:\left(\mathrm{Remainder}\:=\:?\right) \\ $$

Question Number 208945 Answers: 1 Comments: 0

4 cos^2 x − 4 cos^2 3x cos x + cos^2 3x = 0 [ 0 ; (π/2) ] Find: x = ?

$$\mathrm{4}\:\mathrm{cos}^{\mathrm{2}} \:\mathrm{x}\:−\:\mathrm{4}\:\mathrm{cos}^{\mathrm{2}} \:\mathrm{3x}\:\mathrm{cos}\:\mathrm{x}\:+\:\mathrm{cos}^{\mathrm{2}} \:\mathrm{3x}\:=\:\mathrm{0} \\ $$$$\left[\:\mathrm{0}\:;\:\frac{\pi}{\mathrm{2}}\:\right] \\ $$$$\mathrm{Find}:\:\:\mathrm{x}\:=\:? \\ $$

Question Number 208891 Answers: 0 Comments: 0

(√( ( )^(1/ ) −( )^(1/ ) ))