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Question Number 44128 Answers: 1 Comments: 0

The number of all possible 5−tuples (a_1 , a_2 , a_3 , a_4 , a_5 ) such that a_1 +a_2 sin x+a_3 cos x+a_4 sin 2x+a_5 cos 2x=0 holds for all x is

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{all}\:\mathrm{possible}\:\mathrm{5}−\mathrm{tuples} \\ $$$$\left({a}_{\mathrm{1}} ,\:{a}_{\mathrm{2}} ,\:{a}_{\mathrm{3}} ,\:{a}_{\mathrm{4}} ,\:{a}_{\mathrm{5}} \right)\:\mathrm{such}\:\mathrm{that}\: \\ $$$${a}_{\mathrm{1}} +{a}_{\mathrm{2}} \mathrm{sin}\:{x}+{a}_{\mathrm{3}} \mathrm{cos}\:{x}+{a}_{\mathrm{4}} \mathrm{sin}\:\mathrm{2}{x}+{a}_{\mathrm{5}} \mathrm{cos}\:\mathrm{2}{x}=\mathrm{0} \\ $$$$\mathrm{holds}\:\mathrm{for}\:\mathrm{all}\:\:\:{x}\:\:\mathrm{is} \\ $$

Question Number 44120 Answers: 1 Comments: 0

Question Number 44117 Answers: 0 Comments: 0

Question Number 44103 Answers: 1 Comments: 1

Question Number 44098 Answers: 3 Comments: 0

Question Number 44097 Answers: 3 Comments: 0

Question Number 44096 Answers: 2 Comments: 0

Question Number 44095 Answers: 1 Comments: 0

Question Number 44094 Answers: 2 Comments: 2

Question Number 44092 Answers: 0 Comments: 1

Question Number 44091 Answers: 1 Comments: 0

Question Number 44089 Answers: 0 Comments: 0

Question Number 44088 Answers: 1 Comments: 0

Question Number 44085 Answers: 0 Comments: 6

lim x→0 [((sin ∣x∣)/x)]

$${lim}\:\mathrm{x}\rightarrow\mathrm{0}\:\left[\frac{\mathrm{sin}\:\mid{x}\mid}{{x}}\right] \\ $$

Question Number 44069 Answers: 1 Comments: 1

∫(dx/((x+1)(√(x^2 +2)))) = ?

$$\int\frac{{dx}}{\left({x}+\mathrm{1}\right)\sqrt{{x}^{\mathrm{2}} +\mathrm{2}}}\:=\:? \\ $$

Question Number 44068 Answers: 2 Comments: 0

Question Number 44064 Answers: 0 Comments: 0

Question Number 44063 Answers: 0 Comments: 0

Question Number 44062 Answers: 1 Comments: 0

Question Number 44059 Answers: 1 Comments: 0

Prove that any integer can be expressed as in the form of 4k or4k+_− 1 or 4k+_− 2.

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{any}\:\mathrm{integer}\:\mathrm{can}\:\mathrm{be}\:\mathrm{expressed}\: \\ $$$$\mathrm{as}\:\mathrm{in}\:\mathrm{the}\:\mathrm{form}\:\mathrm{of}\:\mathrm{4k}\:\mathrm{or4k}\underset{−} {+}\mathrm{1}\:\mathrm{or}\:\mathrm{4k}\underset{−} {+}\mathrm{2}. \\ $$

Question Number 44051 Answers: 1 Comments: 1

Question Number 44038 Answers: 2 Comments: 0

How many times does the digit 6 appear when writing from 6 to 400 ?

$$\mathrm{How}\:\mathrm{many}\:\mathrm{times}\:\mathrm{does}\:\mathrm{the}\:\mathrm{digit}\:\mathrm{6}\:\mathrm{appear}\:\mathrm{when}\:\mathrm{writing}\:\mathrm{from}\:\:\mathrm{6}\:\mathrm{to}\:\mathrm{400}\:? \\ $$

Question Number 44035 Answers: 0 Comments: 3

Question Number 44034 Answers: 0 Comments: 1

Let I_1 = ∫_( 1) ^2 (1/(√(1+x^2 ))) dx and I_2 = ∫_( 1) ^2 (1/x) dx. Then

$$\mathrm{Let}\:\:{I}_{\mathrm{1}} =\:\underset{\:\mathrm{1}} {\overset{\mathrm{2}} {\int}}\:\:\frac{\mathrm{1}}{\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\:{dx}\:\mathrm{and}\:{I}_{\mathrm{2}} =\:\underset{\:\mathrm{1}} {\overset{\mathrm{2}} {\int}}\:\frac{\mathrm{1}}{{x}}\:{dx}. \\ $$$$\mathrm{Then} \\ $$

Question Number 44172 Answers: 1 Comments: 1

The number of solutions of the equation cos^(−1) ((x^2 −1)/(x^2 +1)) + sin^(−1) ((2x)/(x^2 +1)) +tan^(−1) ((2x)/(x^2 −1))=((2π)/3).

$${The}\:{number}\:{of}\:{solutions}\:{of}\:{the}\:{equation} \\ $$$$\mathrm{cos}^{−\mathrm{1}} \frac{{x}^{\mathrm{2}} −\mathrm{1}}{{x}^{\mathrm{2}} +\mathrm{1}}\:+\:\mathrm{sin}^{−\mathrm{1}} \frac{\mathrm{2}{x}}{{x}^{\mathrm{2}} +\mathrm{1}}\:+\mathrm{tan}^{−\mathrm{1}} \frac{\mathrm{2}{x}}{{x}^{\mathrm{2}} −\mathrm{1}}=\frac{\mathrm{2}\pi}{\mathrm{3}}. \\ $$

Question Number 44029 Answers: 0 Comments: 0

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