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Question Number 31868 Answers: 0 Comments: 0

Let a>b>1 be positive integers with b odd. Let n be a positive integer as well. If b^n divides a^n −1, prove that a^b > (3^n /n). Solution please. Thanks in advance!!

$${Let}\:{a}>{b}>\mathrm{1}\:{be}\:{positive}\:{integers}\:{with}\:{b}\:{odd}. \\ $$$${Let}\:{n}\:{be}\:{a}\:{positive}\:{integer}\:{as}\:{well}.\:{If}\:\:{b}^{{n}} \:{divides} \\ $$$${a}^{{n}} −\mathrm{1},\:{prove}\:{that}\:{a}^{{b}} \:>\:\frac{\mathrm{3}^{{n}} }{{n}}. \\ $$$${Solution}\:{please}.\:{Thanks}\:{in}\:{advance}!! \\ $$

Question Number 31865 Answers: 1 Comments: 6

Range of function : f(x)= 6^x +3^x +6^(−x) +3^(−x) +2.

$${Range}\:{of}\:{function}\:: \\ $$$${f}\left({x}\right)=\:\mathrm{6}^{{x}} +\mathrm{3}^{{x}} +\mathrm{6}^{−{x}} +\mathrm{3}^{−{x}} +\mathrm{2}. \\ $$

Question Number 31864 Answers: 1 Comments: 0

Let S_n = Σ_(k=1) ^(4n) (−1)^((k(k+1))/2) k^2 . Then S_n can take the value(s) 1) 1056 2) 1088 3) 1120 4) 1332.

$${Let}\:{S}_{{n}} =\:\underset{{k}=\mathrm{1}} {\overset{\mathrm{4}{n}} {\sum}}\left(−\mathrm{1}\right)^{\frac{{k}\left({k}+\mathrm{1}\right)}{\mathrm{2}}} {k}^{\mathrm{2}} . \\ $$$${Then}\:{S}_{{n}} \:{can}\:{take}\:{the}\:{value}\left({s}\right) \\ $$$$\left.\mathrm{1}\right)\:\mathrm{1056} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{1088} \\ $$$$\left.\mathrm{3}\right)\:\mathrm{1120} \\ $$$$\left.\mathrm{4}\right)\:\mathrm{1332}. \\ $$

Question Number 31863 Answers: 0 Comments: 9

Question Number 31858 Answers: 0 Comments: 1

∫((sinx)/x)dx

$$\int\frac{{sinx}}{{x}}{dx} \\ $$

Question Number 31846 Answers: 1 Comments: 2

25[(x−2)^2 +(y−3)^2 ]=(3x−4y+7)^2 is the equation of parabola.Find length of latus rectum

$$\mathrm{25}\left[\left({x}−\mathrm{2}\right)^{\mathrm{2}} +\left({y}−\mathrm{3}\right)^{\mathrm{2}} \right]=\left(\mathrm{3}{x}−\mathrm{4}{y}+\mathrm{7}\right)^{\mathrm{2}} \\ $$$${is}\:{the}\:{equation}\:{of}\:{parabola}.{Find} \\ $$$${length}\:{of}\:{latus}\:{rectum} \\ $$

Question Number 31839 Answers: 0 Comments: 1

I = ∫ (√(x + (√(x^2 − 1)))) dx

$${I}\:=\:\int\:\sqrt{{x}\:+\:\sqrt{{x}^{\mathrm{2}} \:−\:\mathrm{1}}}\:{dx} \\ $$

Question Number 31838 Answers: 0 Comments: 0

Given f(x) = (3/(16)) (∫_0 ^1 f(x)dx)x^2 − (9/(10))(∫_0 ^2 f(x)dx)x + 2(∫_0 ^3 f(x)dx) + 4 Solve lim_(t→0) ((2t + (∫_(f(2) + 2) ^(f^(−1) (t)) [f ′(x)]^2 dx))/(1 − cos t cosh 2t cos 3t))

$$\mathrm{Given} \\ $$$${f}\left({x}\right)\:=\:\frac{\mathrm{3}}{\mathrm{16}}\:\left(\int_{\mathrm{0}} ^{\mathrm{1}} \:{f}\left({x}\right){dx}\right){x}^{\mathrm{2}} \:−\:\frac{\mathrm{9}}{\mathrm{10}}\left(\int_{\mathrm{0}} ^{\mathrm{2}} \:{f}\left({x}\right){dx}\right){x}\:+\:\mathrm{2}\left(\int_{\mathrm{0}} ^{\mathrm{3}} \:{f}\left({x}\right){dx}\right)\:+\:\mathrm{4} \\ $$$$\mathrm{Solve} \\ $$$$\underset{\mathrm{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{2}{t}\:+\:\left(\int_{{f}\left(\mathrm{2}\right)\:+\:\mathrm{2}} ^{{f}^{−\mathrm{1}} \left({t}\right)} \left[{f}\:'\left({x}\right)\right]^{\mathrm{2}} \:{dx}\right)}{\mathrm{1}\:−\:\mathrm{cos}\:{t}\:\mathrm{cosh}\:\mathrm{2}{t}\:\mathrm{cos}\:\mathrm{3}{t}} \\ $$

Question Number 31835 Answers: 0 Comments: 5

Question Number 31833 Answers: 1 Comments: 0

Question Number 31822 Answers: 1 Comments: 0

Question Number 31820 Answers: 1 Comments: 0

Question Number 31812 Answers: 2 Comments: 0

Find the equation of the line that is tangent to the curve y=x^3 and is parallel to the line 3x−y+1=0

$${Find}\:{the}\:{equation}\:{of}\:{the}\:{line} \\ $$$${that}\:{is}\:{tangent}\:{to}\:{the}\:{curve}\:{y}={x}^{\mathrm{3}} \\ $$$${and}\:{is}\:{parallel}\:{to}\:{the}\:{line} \\ $$$$\mathrm{3}{x}−{y}+\mathrm{1}=\mathrm{0} \\ $$

Question Number 31811 Answers: 1 Comments: 2

find the domain of f(x)=((2x+7)/([2−x^2 ]))

$${find}\:{the}\:{domain}\:{of} \\ $$$${f}\left({x}\right)=\frac{\mathrm{2}{x}+\mathrm{7}}{\left[\mathrm{2}−{x}^{\mathrm{2}} \right]} \\ $$

Question Number 31809 Answers: 0 Comments: 1

Find the domain and range of f(x)=((x−1)/([x]))

$${Find}\:{the}\:{domain}\:{and}\:{range}\:{of} \\ $$$${f}\left({x}\right)=\frac{{x}−\mathrm{1}}{\left[{x}\right]} \\ $$

Question Number 31808 Answers: 0 Comments: 2

The numeric value of the expression is: ((Sec 1320°)/2) − 2 ∙ cos (((53π)/3)) + (tg 2220°)^2

$$\mathrm{The}\:\mathrm{numeric}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{expression}\:\mathrm{is}: \\ $$$$ \\ $$$$\frac{\mathrm{Sec}\:\mathrm{1320}°}{\mathrm{2}}\:−\:\mathrm{2}\:\centerdot\:\mathrm{cos}\:\left(\frac{\mathrm{53}\pi}{\mathrm{3}}\right)\:+\:\left(\mathrm{tg}\:\mathrm{2220}°\right)^{\mathrm{2}} \\ $$

Question Number 31804 Answers: 1 Comments: 0

A quadratic equation p(x)=0 having coefficient of x^2 unity is such that p(x)=0 and p(p(p(x)))=0 have a common root then, prove that : p(0)×p(1)=0.

$${A}\:{quadratic}\:{equation}\:{p}\left({x}\right)=\mathrm{0}\:{having} \\ $$$${coefficient}\:{of}\:{x}^{\mathrm{2}} \:{unity}\:{is}\:{such}\:{that} \\ $$$${p}\left({x}\right)=\mathrm{0}\:{and}\:{p}\left({p}\left({p}\left({x}\right)\right)\right)=\mathrm{0}\:{have}\:{a}\: \\ $$$${common}\:{root}\:{then}, \\ $$$${prove}\:{that}\::\:\:{p}\left(\mathrm{0}\right)×{p}\left(\mathrm{1}\right)=\mathrm{0}. \\ $$

Question Number 31794 Answers: 0 Comments: 2

Question Number 31792 Answers: 0 Comments: 4

Question Number 31787 Answers: 2 Comments: 0

∫((4x−3)/(x^2 +3x+8))dx

$$\int\frac{\mathrm{4}{x}−\mathrm{3}}{{x}^{\mathrm{2}} +\mathrm{3}{x}+\mathrm{8}}{dx} \\ $$

Question Number 31785 Answers: 1 Comments: 0

A= [((1 5)),((6 7)) ]find A^k

$${A}=\:\begin{bmatrix}{\mathrm{1}\:\mathrm{5}}\\{\mathrm{6}\:\mathrm{7}}\end{bmatrix}{find}\:{A}^{{k}} \\ $$

Question Number 31772 Answers: 0 Comments: 0

Question Number 31771 Answers: 1 Comments: 1

Consider a sequence in the form of groups (1),(2,2),(3,3,3),(4,4,4,4), (5,5,5,5,5),............ then the 2000th term of the above sequence is : ?

$${Consider}\:{a}\:{sequence}\:{in}\:{the}\:{form}\:{of} \\ $$$${groups}\:\left(\mathrm{1}\right),\left(\mathrm{2},\mathrm{2}\right),\left(\mathrm{3},\mathrm{3},\mathrm{3}\right),\left(\mathrm{4},\mathrm{4},\mathrm{4},\mathrm{4}\right), \\ $$$$\left(\mathrm{5},\mathrm{5},\mathrm{5},\mathrm{5},\mathrm{5}\right),............ \\ $$$${then}\:{the}\:\mathrm{2000}{th}\:{term}\:{of}\:{the}\:{above}\: \\ $$$${sequence}\:{is}\::\:? \\ $$

Question Number 31768 Answers: 0 Comments: 0

Please help What is the kinetic energy of the earth?Please prove the result. Thanks in advance!

$${Please}\:{help} \\ $$$$ \\ $$$${What}\:{is}\:{the}\:{kinetic}\:{energy}\:{of}\:{the} \\ $$$${earth}?{Please}\:{prove}\:{the}\:{result}. \\ $$$$ \\ $$$${Thanks}\:{in}\:{advance}! \\ $$

Question Number 31767 Answers: 0 Comments: 0

calculate the angular velocity of the earth about its axis and its angular velocity about the axis and the sun.

$${calculate}\:{the}\:{angular}\:{velocity}\:{of} \\ $$$${the}\:{earth}\:{about}\:{its}\:{axis}\:{and}\:{its} \\ $$$${angular}\:{velocity}\:{about}\:{the}\:{axis}\: \\ $$$${and}\:{the}\:{sun}. \\ $$

Question Number 31766 Answers: 0 Comments: 1

Calculate the moment of inertia of the earth abouth its axis.If the mass of the earth is 6.0×10^(34) kg and radius 6.4×10^6 m. (Assume the earth to be a perfect sphere)

$${Calculate}\:{the}\:{moment}\:{of}\:{inertia} \\ $$$${of}\:{the}\:{earth}\:{abouth}\:{its}\:{axis}.{If} \\ $$$${the}\:{mass}\:{of}\:{the}\:{earth}\:{is}\:\mathrm{6}.\mathrm{0}×\mathrm{10}^{\mathrm{34}} {kg} \\ $$$${and}\:{radius}\:\mathrm{6}.\mathrm{4}×\mathrm{10}^{\mathrm{6}} {m}. \\ $$$$\left({Assume}\:{the}\:{earth}\:{to}\:{be}\:{a}\:{perfect}\right. \\ $$$$\left.{sphere}\right) \\ $$

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