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Question Number 10195    Answers: 1   Comments: 1

The number of integral solutions of the equation 7(y+(1/y))−2(y^2 +(1/y^2 ))=9 are?

$${The}\:{number}\:{of}\:{integral}\:{solutions}\:{of}\:{the}\:{equation}\: \\ $$$$\mathrm{7}\left({y}+\frac{\mathrm{1}}{{y}}\right)−\mathrm{2}\left({y}^{\mathrm{2}} +\frac{\mathrm{1}}{{y}^{\mathrm{2}} }\right)=\mathrm{9}\:\mathrm{are}? \\ $$

Question Number 10193    Answers: 1   Comments: 0

x=^3 (√(11+(√(57)))) +^3 (√(11−(√(57)))) ⇒ x^3 −12x=?

$$\mathrm{x}=^{\mathrm{3}} \sqrt{\mathrm{11}+\sqrt{\mathrm{57}}}\:\:+\:^{\mathrm{3}} \sqrt{\mathrm{11}−\sqrt{\mathrm{57}}} \\ $$$$\Rightarrow\:\mathrm{x}^{\mathrm{3}} −\mathrm{12x}=? \\ $$

Question Number 10191    Answers: 1   Comments: 0

∫_0 ^( π) sin(x)^(cos(x)) dx

$$\int_{\mathrm{0}} ^{\:\pi} \mathrm{sin}\left({x}\right)^{\mathrm{cos}\left({x}\right)} {dx} \\ $$

Question Number 10190    Answers: 0   Comments: 3

Question Number 10188    Answers: 0   Comments: 1

Question Number 10186    Answers: 0   Comments: 0

Prove ln (1+(1/n))^n =[1−(1/(2(n+1)))+(1/(2∙3(n+1)^2 ))−(1/(3∙4(n+1)^3 ))+..]

$$\mathrm{Prove} \\ $$$$\mathrm{ln}\:\left(\mathrm{1}+\frac{\mathrm{1}}{{n}}\right)^{{n}} =\left[\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}\left({n}+\mathrm{1}\right)}+\frac{\mathrm{1}}{\mathrm{2}\centerdot\mathrm{3}\left({n}+\mathrm{1}\right)^{\mathrm{2}} }−\frac{\mathrm{1}}{\mathrm{3}\centerdot\mathrm{4}\left({n}+\mathrm{1}\right)^{\mathrm{3}} }+..\right] \\ $$

Question Number 10181    Answers: 0   Comments: 0

Let A be a subset of {0;1;∙∙∙;1997} containing more than 1000 elements. Prove that A contains either a power of 2 or two distinct integers whose sum is a power of 2.

$${Let}\:{A}\:{be}\:{a}\:{subset}\:{of}\:\left\{\mathrm{0};\mathrm{1};\centerdot\centerdot\centerdot;\mathrm{1997}\right\}\: \\ $$$${containing}\:{more}\:{than}\:\mathrm{1000}\:{elements}. \\ $$$${Prove}\:{that}\:{A}\:{contains}\:{either}\:{a}\:{power} \\ $$$${of}\:\mathrm{2}\:{or}\:{two}\:{distinct}\:{integers}\:{whose}\: \\ $$$${sum}\:{is}\:{a}\:{power}\:{of}\:\mathrm{2}. \\ $$

Question Number 10175    Answers: 1   Comments: 0

Question Number 10172    Answers: 1   Comments: 0

x^x =2^(24) , y^y =3^(18) ⇒((2y+3x)/(y−x))=?

$$\mathrm{x}^{\mathrm{x}} =\mathrm{2}^{\mathrm{24}} \:,\:\mathrm{y}^{\mathrm{y}} =\mathrm{3}^{\mathrm{18}} \Rightarrow\frac{\mathrm{2y}+\mathrm{3x}}{\mathrm{y}−\mathrm{x}}=? \\ $$

Question Number 10184    Answers: 2   Comments: 0

if a_− =2i+3j . b_− =19−15j and c_− =5i−7j. find the value of x such that xa_− + yc_− =b

$${if}\:\underset{−} {{a}}\:=\mathrm{2}{i}+\mathrm{3}{j}\:.\:\underset{−} {{b}}=\mathrm{19}−\mathrm{15}{j}\:{and}\: \\ $$$$\underset{−} {{c}}\:=\mathrm{5}{i}−\mathrm{7}{j}.\:{find}\:{the}\:{value}\:{of}\:{x}\:{such} \\ $$$${that}\:{x}\underset{−} {{a}}\:+\:{y}\underset{−} {{c}}\:={b} \\ $$$$ \\ $$

Question Number 10170    Answers: 1   Comments: 0

y . f(xy) = f(x) x,y ∈ R If f(4) = 1006, so f(2012) = ?

$${y}\:.\:{f}\left({xy}\right)\:=\:{f}\left({x}\right)\:\:\:\:\:\:\:\:{x},\mathrm{y}\:\in\:\mathbb{R} \\ $$$$\mathrm{If}\:{f}\left(\mathrm{4}\right)\:=\:\mathrm{1006},\:\mathrm{so}\:{f}\left(\mathrm{2012}\right)\:=\:? \\ $$

Question Number 10169    Answers: 1   Comments: 0

((2013)/1) + ((2013)/(1+2)) + ((2013)/(1+2+3)) + ... + ((2013)/(1+2+3+...+2012)) = ?

$$\frac{\mathrm{2013}}{\mathrm{1}}\:+\:\frac{\mathrm{2013}}{\mathrm{1}+\mathrm{2}}\:+\:\frac{\mathrm{2013}}{\mathrm{1}+\mathrm{2}+\mathrm{3}}\:+\:...\:+\:\frac{\mathrm{2013}}{\mathrm{1}+\mathrm{2}+\mathrm{3}+...+\mathrm{2012}}\:=\:? \\ $$

Question Number 10168    Answers: 0   Comments: 0

If the roots of quadratic equation ax^2 + bx + c = 0 were within the interval [0,1], the maximum value from (((2a−b)(a−b))/(a(a−b+c))) is ...

$$\mathrm{If}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{quadratic}\:\mathrm{equation} \\ $$$${ax}^{\mathrm{2}} \:+\:{bx}\:+\:{c}\:=\:\mathrm{0} \\ $$$$\mathrm{were}\:\mathrm{within}\:\mathrm{the}\:\mathrm{interval}\:\left[\mathrm{0},\mathrm{1}\right], \\ $$$$\mathrm{the}\:\mathrm{maximum}\:\mathrm{value}\:\mathrm{from} \\ $$$$\frac{\left(\mathrm{2}{a}−{b}\right)\left({a}−{b}\right)}{{a}\left({a}−{b}+{c}\right)}\:\:\:\:\mathrm{is}\:... \\ $$

Question Number 10166    Answers: 1   Comments: 1

Question Number 10159    Answers: 1   Comments: 2

Question Number 10157    Answers: 1   Comments: 0

the difference of two number is 3. if the sum of their reciprocal is (7/(10)) . find the numbres

$${the}\:{difference}\:{of}\:{two}\:{number}\:{is}\:\mathrm{3}. \\ $$$${if}\:{the}\:{sum}\:{of}\:{their}\:{reciprocal}\:{is}\: \\ $$$$\frac{\mathrm{7}}{\mathrm{10}}\:.\:{find}\:{the}\:{numbres} \\ $$

Question Number 10155    Answers: 1   Comments: 0

solve for x sin(x) − (√(3cos(x))) = 1

$$\mathrm{solve}\:\mathrm{for}\:\mathrm{x} \\ $$$$\mathrm{sin}\left(\mathrm{x}\right)\:−\:\sqrt{\mathrm{3cos}\left(\mathrm{x}\right)}\:=\:\mathrm{1} \\ $$

Question Number 10154    Answers: 0   Comments: 0

The sum of the first term of sequence is given by S_n = 5n^2 − 2n. A sequence U_1 , U_2 , U_3 .... is defined by U_t = S_t − S_(t − 1) . Express U_t in terms of it simplest form. and show that sequences is linear (AP). (a) Find the sum S_n of the n terms of the sequence r^(th ) term is 4 × 2^(−1) (b) The value of n for which the difference between S_n and less than 10^(−4)

$$\mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{first}\:\mathrm{term}\:\mathrm{of}\:\mathrm{sequence}\:\mathrm{is}\: \\ $$$$\mathrm{given}\:\mathrm{by}\:\:\mathrm{S}_{\mathrm{n}} \:=\:\mathrm{5n}^{\mathrm{2}} \:−\:\mathrm{2n}.\:\mathrm{A}\:\mathrm{sequence}\:\: \\ $$$$\mathrm{U}_{\mathrm{1}} ,\:\mathrm{U}_{\mathrm{2}} ,\:\mathrm{U}_{\mathrm{3}} \:....\:\mathrm{is}\:\mathrm{defined}\:\mathrm{by}\:\mathrm{U}_{\mathrm{t}} \:=\:\mathrm{S}_{\mathrm{t}} \:−\:\mathrm{S}_{\mathrm{t}\:−\:\mathrm{1}} . \\ $$$$\mathrm{Express}\:\mathrm{U}_{\mathrm{t}} \:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{it}\:\mathrm{simplest}\:\mathrm{form}.\:\mathrm{and} \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{sequences}\:\mathrm{is}\:\mathrm{linear}\:\left(\mathrm{AP}\right). \\ $$$$\left(\mathrm{a}\right)\:\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{S}_{\mathrm{n}} \:\mathrm{of}\:\mathrm{the}\:\mathrm{n}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{sequence}\:\mathrm{r}^{\mathrm{th}\:} \:\mathrm{term}\:\mathrm{is}\:\mathrm{4}\:×\:\mathrm{2}^{−\mathrm{1}} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\mathrm{n}\:\mathrm{for}\:\mathrm{which}\:\mathrm{the}\:\mathrm{difference} \\ $$$$\mathrm{between}\:\mathrm{S}_{\mathrm{n}} \:\mathrm{and}\:\mathrm{less}\:\mathrm{than}\:\mathrm{10}^{−\mathrm{4}} \\ $$

Question Number 10152    Answers: 0   Comments: 0

Question Number 10147    Answers: 0   Comments: 1

∫(√(p/(p−1)))dp=...?

$$\int\sqrt{\frac{\mathrm{p}}{\mathrm{p}−\mathrm{1}}}\mathrm{dp}=...? \\ $$

Question Number 10143    Answers: 2   Comments: 0

x∈Z (x+4)^(x^2 −16) =1 ⇒Σx=?

$$\mathrm{x}\in\mathrm{Z} \\ $$$$\left(\mathrm{x}+\mathrm{4}\right)^{\mathrm{x}^{\mathrm{2}} −\mathrm{16}} =\mathrm{1}\:\:\:\Rightarrow\Sigma\mathrm{x}=? \\ $$

Question Number 10142    Answers: 0   Comments: 3

Question Number 10138    Answers: 1   Comments: 0

2^(x+y) =0.125 (0.125)^(x−y) =2 ⇒ x×y=?

$$\mathrm{2}^{\mathrm{x}+\mathrm{y}} =\mathrm{0}.\mathrm{125} \\ $$$$\left(\mathrm{0}.\mathrm{125}\right)^{\mathrm{x}−\mathrm{y}} =\mathrm{2}\:\Rightarrow\:\mathrm{x}×\mathrm{y}=? \\ $$

Question Number 10137    Answers: 1   Comments: 0

lim_(x→0^+ ) ((sin x^m )/(sin^n x)) n;m ∈Z

$$\underset{{x}\rightarrow\mathrm{0}^{+} } {{lim}}\:\frac{{sin}\:{x}^{{m}} }{{sin}^{{n}} \:{x}}\:\:{n};{m}\:\in\mathbb{Z} \\ $$

Question Number 10136    Answers: 0   Comments: 0

Question Number 10133    Answers: 1   Comments: 0

((1+2^(x−y) )/(1+2^(y−x) ))=4⇒x−y=?

$$\frac{\mathrm{1}+\mathrm{2}^{\mathrm{x}−\mathrm{y}} }{\mathrm{1}+\mathrm{2}^{\mathrm{y}−\mathrm{x}} }=\mathrm{4}\Rightarrow\mathrm{x}−\mathrm{y}=? \\ $$

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