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Question Number 14658    Answers: 1   Comments: 0

Given that: x = ((√2) + 1)^(1/3) − ((√2) − 1)^(1/3) Show that , x^3 + 3x = 2

$$\mathrm{Given}\:\mathrm{that}: \\ $$$$\mathrm{x}\:=\:\left(\sqrt{\mathrm{2}}\:+\:\mathrm{1}\right)^{\mathrm{1}/\mathrm{3}} \:−\:\left(\sqrt{\mathrm{2}}\:−\:\mathrm{1}\right)^{\mathrm{1}/\mathrm{3}} \\ $$$$\mathrm{Show}\:\mathrm{that}\:,\:\:\:\:\:\mathrm{x}^{\mathrm{3}} \:+\:\mathrm{3x}\:=\:\mathrm{2} \\ $$

Question Number 14660    Answers: 0   Comments: 2

Question Number 14646    Answers: 0   Comments: 0

Prove that: ∫_( 0) ^( 1) sin(x) cos^(−1) (x) dx > (1/e^2 )

$$\mathrm{Prove}\:\mathrm{that}:\:\:\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\mathrm{sin}\left(\mathrm{x}\right)\:\mathrm{cos}^{−\mathrm{1}} \left(\mathrm{x}\right)\:\mathrm{dx}\:>\:\frac{\mathrm{1}}{\mathrm{e}^{\mathrm{2}} } \\ $$

Question Number 14633    Answers: 3   Comments: 1

Solve the equation: (1 − tanθ)(1 + sin2θ) = 1 + tanθ

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{equation}: \\ $$$$\left(\mathrm{1}\:−\:\mathrm{tan}\theta\right)\left(\mathrm{1}\:+\:\mathrm{sin2}\theta\right)\:=\:\mathrm{1}\:+\:\mathrm{tan}\theta \\ $$

Question Number 14632    Answers: 2   Comments: 0

Find the general solution of the equation 3^(sin 2x + 2 cos^2 x) + 3^(1 − sin 2x + 2 sin^2 x) = 28

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{general}\:\mathrm{solution}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equation} \\ $$$$\mathrm{3}^{\mathrm{sin}\:\mathrm{2}{x}\:+\:\mathrm{2}\:\mathrm{cos}^{\mathrm{2}} \:{x}} \:+\:\mathrm{3}^{\mathrm{1}\:−\:\mathrm{sin}\:\mathrm{2}{x}\:+\:\mathrm{2}\:\mathrm{sin}^{\mathrm{2}} \:{x}} \:=\:\mathrm{28} \\ $$

Question Number 14630    Answers: 0   Comments: 2

solve the eqn dr/dθ=[r(a^2 −r^2 )/a^2 +r^2 ]cotθ hint. let a^2 +r^2 =a^2 −r^2 +2r^2 .

$$\mathrm{solve}\:\mathrm{the}\:\mathrm{eqn} \\ $$$$\mathrm{dr}/\mathrm{d}\theta=\left[\mathrm{r}\left(\mathrm{a}^{\mathrm{2}} −\mathrm{r}^{\mathrm{2}} \right)/\mathrm{a}^{\mathrm{2}} +\mathrm{r}^{\mathrm{2}} \right]\mathrm{cot}\theta \\ $$$$\mathrm{hint}.\:\mathrm{let}\:\mathrm{a}^{\mathrm{2}} +\mathrm{r}^{\mathrm{2}} =\mathrm{a}^{\mathrm{2}} −\mathrm{r}^{\mathrm{2}} +\mathrm{2r}^{\mathrm{2}} . \\ $$

Question Number 14882    Answers: 0   Comments: 6

Two vectors a^→ and b^→ are parallel and have same magnitude. Then they (1) have same direction, but they are not equal (2) are equal (3) are not equal (4) may or may not be equal

$$\mathrm{Two}\:\mathrm{vectors}\:\overset{\rightarrow} {{a}}\:\mathrm{and}\:\overset{\rightarrow} {{b}}\:\mathrm{are}\:\mathrm{parallel}\:\mathrm{and} \\ $$$$\mathrm{have}\:\mathrm{same}\:\mathrm{magnitude}.\:\mathrm{Then}\:\mathrm{they} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{have}\:\mathrm{same}\:\mathrm{direction},\:\mathrm{but}\:\mathrm{they}\:\mathrm{are} \\ $$$$\mathrm{not}\:\mathrm{equal} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{are}\:\mathrm{equal} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{are}\:\mathrm{not}\:\mathrm{equal} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{may}\:\mathrm{or}\:\mathrm{may}\:\mathrm{not}\:\mathrm{be}\:\mathrm{equal} \\ $$

Question Number 14879    Answers: 1   Comments: 0

Find the value of x, satisfying sin^2 x + sin x − 2 ≥ 0

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{x},\:\mathrm{satisfying} \\ $$$$\mathrm{sin}^{\mathrm{2}} \:{x}\:+\:\mathrm{sin}\:{x}\:−\:\mathrm{2}\:\geqslant\:\mathrm{0} \\ $$

Question Number 14878    Answers: 0   Comments: 3

Find the number of solution of equation x sin x = 2

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{solution}\:\mathrm{of}\:\mathrm{equation} \\ $$$${x}\:\mathrm{sin}\:{x}\:=\:\mathrm{2} \\ $$

Question Number 14876    Answers: 0   Comments: 0

Question Number 14614    Answers: 1   Comments: 2

If 5 doesn′t divide any of n,n+1, n+2,n+3 then prove that n(n+1)(n+2)(n+3)≡24(mod100)

$$\mathrm{If}\:\:\mathrm{5}\:\:\mathrm{doesn}'\mathrm{t}\:\mathrm{divide}\:\mathrm{any}\:\mathrm{of}\:\mathrm{n},\mathrm{n}+\mathrm{1}, \\ $$$$\mathrm{n}+\mathrm{2},\mathrm{n}+\mathrm{3}\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\mathrm{n}\left(\mathrm{n}+\mathrm{1}\right)\left(\mathrm{n}+\mathrm{2}\right)\left(\mathrm{n}+\mathrm{3}\right)\equiv\mathrm{24}\left(\mathrm{mod100}\right) \\ $$

Question Number 14758    Answers: 2   Comments: 4

Solve tan x + tan 2x + tan 3x = 0

$$\mathrm{Solve}\:\mathrm{tan}\:{x}\:+\:\mathrm{tan}\:\mathrm{2}{x}\:+\:\mathrm{tan}\:\mathrm{3}{x}\:=\:\mathrm{0} \\ $$

Question Number 14757    Answers: 1   Comments: 0

Solve: ((1000!)/(5×10×15×...1000))≡x(mod 10)

$$\mathrm{Solve}: \\ $$$$\frac{\mathrm{1000}!}{\mathrm{5}×\mathrm{10}×\mathrm{15}×...\mathrm{1000}}\equiv\mathrm{x}\left(\mathrm{mod}\:\mathrm{10}\right) \\ $$

Question Number 14759    Answers: 0   Comments: 2

Where is 123456 ? S/He was most senior of us and had a great knoledge of maths! S/He used to guide us when we were wrong.

$$\:\:\:\:\:\:\:\:\mathrm{Where}\:\mathrm{is} \\ $$$$\:\:\:\:\:\:\:\:\mathrm{123456} \\ $$$$\:\:\:\:\:\:\:\:\:? \\ $$$$\mathrm{S}/\mathrm{He}\:\mathrm{was}\:\mathrm{most}\:\mathrm{senior}\:\mathrm{of}\:\mathrm{us}\: \\ $$$$\mathrm{and} \\ $$$$\mathrm{had}\:\mathrm{a}\:\mathrm{great}\:\mathrm{knoledge}\:\mathrm{of}\:\mathrm{maths}! \\ $$$$\mathrm{S}/\mathrm{He}\:\mathrm{used}\:\mathrm{to}\:\mathrm{guide}\:\mathrm{us}\:\mathrm{when} \\ $$$$\mathrm{we}\:\mathrm{were}\:\mathrm{wrong}. \\ $$$$ \\ $$

Question Number 14596    Answers: 0   Comments: 3

Question Number 14594    Answers: 1   Comments: 0

Question Number 14593    Answers: 1   Comments: 0

Question Number 14592    Answers: 0   Comments: 2

Question Number 14591    Answers: 1   Comments: 0

Question Number 14590    Answers: 1   Comments: 0

Question Number 14572    Answers: 0   Comments: 1

The number of possible pairs of successive prime numbers such that each of them is greater than 40 and their sum is atmost 100 is

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{possible}\:\mathrm{pairs}\:\mathrm{of}\: \\ $$$$\mathrm{successive}\:\mathrm{prime}\:\mathrm{numbers}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{each}\:\mathrm{of}\:\mathrm{them}\:\mathrm{is}\:\mathrm{greater}\:\mathrm{than}\:\mathrm{40}\:\mathrm{and}\: \\ $$$$\mathrm{their}\:\mathrm{sum}\:\mathrm{is}\:\mathrm{atmost}\:\mathrm{100}\:\mathrm{is} \\ $$

Question Number 14588    Answers: 1   Comments: 0

Determine the fourth roots of − 16, giving the results in the form a + jb.

$$\mathrm{Determine}\:\mathrm{the}\:\mathrm{fourth}\:\mathrm{roots}\:\mathrm{of}\:\:\:−\:\mathrm{16},\:\:\:\mathrm{giving}\:\mathrm{the}\:\mathrm{results}\:\mathrm{in}\:\mathrm{the}\:\mathrm{form}\:\:\mathrm{a}\:+\:\mathrm{jb}. \\ $$

Question Number 14587    Answers: 1   Comments: 0

Determine the roots of the equation x^3 + 64 = 0 in the polar form a + jb, Where a and b are real.

$$\mathrm{Determine}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation}\:\:\mathrm{x}^{\mathrm{3}} \:+\:\mathrm{64}\:=\:\mathrm{0}\:\:\mathrm{in}\:\mathrm{the}\:\mathrm{polar}\:\mathrm{form}\:\:\mathrm{a}\:+\:\mathrm{jb}, \\ $$$$\mathrm{Where}\:\:\mathrm{a}\:\:\mathrm{and}\:\:\mathrm{b}\:\:\mathrm{are}\:\mathrm{real}. \\ $$

Question Number 14564    Answers: 0   Comments: 9

What is the last 2 digits of 2^(613)

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{last}\:\mathrm{2}\:\mathrm{digits}\:\mathrm{of}\:\:\:\:\:\:\mathrm{2}^{\mathrm{613}} \\ $$

Question Number 14560    Answers: 0   Comments: 0

Question Number 14559    Answers: 1   Comments: 0

Solve for x ((6x + 2a + 3b + c )/(6x + 2a − 3b − c)) = ((2x + 6a + b + 3c)/(2x + 6a − b − 3c))

$$\mathrm{Solve}\:\mathrm{for}\:\:\mathrm{x} \\ $$$$\frac{\mathrm{6x}\:+\:\mathrm{2a}\:+\:\mathrm{3b}\:+\:\mathrm{c}\:}{\mathrm{6x}\:+\:\mathrm{2a}\:−\:\mathrm{3b}\:−\:\mathrm{c}}\:=\:\frac{\mathrm{2x}\:+\:\mathrm{6a}\:+\:\mathrm{b}\:+\:\mathrm{3c}}{\mathrm{2x}\:+\:\mathrm{6a}\:−\:\mathrm{b}\:−\:\mathrm{3c}} \\ $$

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