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Question Number 31951 Answers: 1 Comments: 0

Evaluate ∫sin (√x)dx

$${Evaluate}\:\int\mathrm{sin}\:\sqrt{{x}}{dx} \\ $$

Question Number 31949 Answers: 1 Comments: 0

Question Number 31946 Answers: 0 Comments: 0

Calculate Σ_(j≤k≤i) (−1)^k ((i),(k) ) ((k),(j) )

$$\mathrm{Calculate}\:\underset{{j}\leqslant{k}\leqslant{i}} {\Sigma}\:\left(−\mathrm{1}\right)^{{k}} \begin{pmatrix}{{i}}\\{{k}}\end{pmatrix}\begin{pmatrix}{{k}}\\{{j}}\end{pmatrix} \\ $$

Question Number 31941 Answers: 1 Comments: 1

Question Number 31935 Answers: 0 Comments: 0

The rigid body of mass 10kg rotates such that the radius of the circular path it describes is 5cm.It it moves from the point P to Q in 5s covering an angular distance of 30°,Calculate : a)the final angular velocity if the angular acceleration is 4m/s^2 b)The moment of inertia of the body. c)angular momentum d)rotational kinetic energy

$${The}\:{rigid}\:{body}\:{of}\:{mass}\:\mathrm{10}{kg} \\ $$$${rotates}\:{such}\:{that}\:{the}\:{radius}\:{of}\:{the} \\ $$$${circular}\:{path}\:{it}\:{describes}\:{is}\:\mathrm{5}{cm}.{It} \\ $$$${it}\:{moves}\:{from}\:{the}\:{point}\:{P}\:{to}\:{Q}\:{in} \\ $$$$\mathrm{5}{s}\:{covering}\:{an}\:{angular}\:{distance} \\ $$$${of}\:\mathrm{30}°,{Calculate}\:: \\ $$$$\left.{a}\right){the}\:{final}\:{angular}\:{velocity}\:{if} \\ $$$${the}\:{angular}\:{acceleration}\:{is}\:\mathrm{4}{m}/{s}^{\mathrm{2}} \\ $$$$\left.{b}\right){The}\:{moment}\:{of}\:{inertia}\:{of}\:{the} \\ $$$${body}. \\ $$$$\left.{c}\right){angular}\:{momentum} \\ $$$$\left.{d}\right){rotational}\:{kinetic}\:{energy} \\ $$

Question Number 31934 Answers: 1 Comments: 0

The flywheel of a punch press has a moment of inertia of 16kgm^2 and runs at 300rev/min.The flywheel supplies all the energy needed in a quick punching operation. a)Find the speed in rev/min to which the flywheel will be reduced by a sudden punching operation requiring 5000J of work. b)What must be the constant power supply to the flywheel to bring it back to its initial speed in a time of 5s?

$${The}\:{flywheel}\:{of}\:{a}\:{punch}\:{press}\:{has} \\ $$$${a}\:{moment}\:{of}\:{inertia}\:{of}\:\mathrm{16}{kgm}^{\mathrm{2}} \\ $$$${and}\:{runs}\:{at}\:\mathrm{300}{rev}/{min}.{The} \\ $$$${flywheel}\:{supplies}\:{all}\:{the}\:{energy} \\ $$$${needed}\:{in}\:{a}\:{quick}\:{punching} \\ $$$${operation}. \\ $$$$\left.{a}\right){Find}\:{the}\:{speed}\:{in}\:{rev}/{min}\:{to} \\ $$$${which}\:{the}\:{flywheel}\:{will}\:{be}\:{reduced} \\ $$$${by}\:{a}\:{sudden}\:{punching}\:{operation} \\ $$$${requiring}\:\mathrm{5000}{J}\:{of}\:{work}. \\ $$$$\left.{b}\right){What}\:{must}\:{be}\:{the}\:{constant} \\ $$$${power}\:{supply}\:{to}\:{the}\:{flywheel}\:{to} \\ $$$${bring}\:{it}\:{back}\:{to}\:{its}\:{initial}\:{speed} \\ $$$${in}\:{a}\:{time}\:{of}\:\mathrm{5}{s}? \\ $$

Question Number 31933 Answers: 1 Comments: 0

The sides of a cubical container are given by the vectors a_1 =2i+3j−4k,a_2 =i+2j−3k, a_3 =3i+6j−4k.What is the volumr of the container?

$${The}\:{sides}\:{of}\:{a}\:{cubical}\:{container} \\ $$$${are}\:{given}\:{by}\:{the}\:{vectors} \\ $$$${a}_{\mathrm{1}} =\mathrm{2}{i}+\mathrm{3}{j}−\mathrm{4}{k},{a}_{\mathrm{2}} ={i}+\mathrm{2}{j}−\mathrm{3}{k}, \\ $$$${a}_{\mathrm{3}} =\mathrm{3}{i}+\mathrm{6}{j}−\mathrm{4}{k}.{What}\:{is}\:{the}\:{volumr} \\ $$$${of}\:{the}\:{container}? \\ $$

Question Number 31927 Answers: 1 Comments: 0

The number of distinct real roots of equation x^4 −4x^3 +12x^2 +x−1=0.

$${The}\:{number}\:{of}\:{distinct}\:{real}\:{roots} \\ $$$${of}\:{equation}\:\boldsymbol{{x}}^{\mathrm{4}} −\mathrm{4}\boldsymbol{{x}}^{\mathrm{3}} +\mathrm{12}\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{x}}−\mathrm{1}=\mathrm{0}. \\ $$

Question Number 31915 Answers: 1 Comments: 3

If : (x^2 +x+2)^2 −(a−3)(x^2 +x+1)(x^2 +x+2) + (a−4)(x^2 +x+1)^2 =0 has at least one root , then find complete set of values of a.

$$\boldsymbol{{If}}\::\: \\ $$$$\left({x}^{\mathrm{2}} +{x}+\mathrm{2}\right)^{\mathrm{2}} −\left({a}−\mathrm{3}\right)\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right)\left({x}^{\mathrm{2}} +{x}+\mathrm{2}\right) \\ $$$$+\:\left({a}−\mathrm{4}\right)\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right)^{\mathrm{2}} =\mathrm{0}\:{has}\:{at}\:{least}\: \\ $$$${one}\:{root}\:,\:{then}\:{find}\:{complete}\:{set}\:{of}\: \\ $$$${values}\:{of}\:{a}. \\ $$

Question Number 31914 Answers: 0 Comments: 0

Question Number 31899 Answers: 1 Comments: 0

Question Number 31898 Answers: 0 Comments: 0

Question Number 31895 Answers: 0 Comments: 3

Question Number 31894 Answers: 0 Comments: 9

Question Number 31919 Answers: 1 Comments: 3

Question Number 32372 Answers: 1 Comments: 0

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

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