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circle of centre P touches externally both the circle x^2 +y^2 −4x+3=0 and x^2 +y^2 −6y+5=0 . The locus of P is (3/4)x^2 −3xy+2y^2 = λ(y−x) where λ is __

$$\:{circle}\:{of}\:{centre}\:{P}\:\:{touches}\:{externally}\:{both} \\ $$$${the}\:{circle}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} −\mathrm{4}{x}+\mathrm{3}=\mathrm{0}\:{and}\: \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} −\mathrm{6}{y}+\mathrm{5}=\mathrm{0}\:.\:{The}\:{locus}\:{of}\:{P}\:{is} \\ $$$$\frac{\mathrm{3}}{\mathrm{4}}{x}^{\mathrm{2}} −\mathrm{3}{xy}+\mathrm{2}{y}^{\mathrm{2}} \:=\:\lambda\left({y}−{x}\right)\:{where}\:\lambda\:{is}\:\_\_ \\ $$

Question Number 115030 Answers: 2 Comments: 0

∫_(−(π/2)) ^(π/2) (√(sec x−cos x)) dx =?

$$\underset{−\frac{\pi}{\mathrm{2}}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\sqrt{\mathrm{sec}\:{x}−\mathrm{cos}\:{x}}\:{dx}\:=? \\ $$

Question Number 115027 Answers: 1 Comments: 0

If lim_(x→3) ((17 ((ax+3))^(1/(3 )) +b)/(x−3)) = ((136)/(27)) then 8a+b = ?

$$\mathrm{If}\:\underset{{x}\rightarrow\mathrm{3}} {\mathrm{lim}}\:\frac{\mathrm{17}\:\sqrt[{\mathrm{3}\:}]{\mathrm{ax}+\mathrm{3}}\:+\mathrm{b}}{\mathrm{x}−\mathrm{3}}\:=\:\frac{\mathrm{136}}{\mathrm{27}} \\ $$$$\mathrm{then}\:\mathrm{8a}+\mathrm{b}\:=\:? \\ $$

Question Number 115026 Answers: 2 Comments: 0

Solve: ∫_(1/π) ^(1/2) ln ⌊(1/x)⌋dx

$$\mathrm{Solve}:\:\:\int_{\mathrm{1}/\pi} ^{\mathrm{1}/\mathrm{2}} \mathrm{ln}\:\lfloor\frac{\mathrm{1}}{{x}}\rfloor{dx} \\ $$

Question Number 115023 Answers: 2 Comments: 0

solve { ((7x−5y+3z=6)),((2x+4y−5z=−5)),((9x−8y+2z=−1)) :}. Find x+y+z

$${solve}\:\begin{cases}{\mathrm{7}{x}−\mathrm{5}{y}+\mathrm{3}{z}=\mathrm{6}}\\{\mathrm{2}{x}+\mathrm{4}{y}−\mathrm{5}{z}=−\mathrm{5}}\\{\mathrm{9}{x}−\mathrm{8}{y}+\mathrm{2}{z}=−\mathrm{1}}\end{cases}.\:{Find}\:{x}+{y}+{z}\: \\ $$

Question Number 115022 Answers: 1 Comments: 0

[Q.1 ] Find the domain of f(x)=((⌊cos^(−1) (x^4 )⌋+∣⌊x−2tan^(−1) (x)⌋∣+(√(sin(lnx))))/({3x^2 −7}+a^(√(sin(x)+3cos(x))) +ln cos((1/( (√(−x^2 ))))))) where {x} reprents the fractionare part of x: A) ]−2, (√2)[ B) ]0,2[ C) ]−1, 1 [ D) ]−2, −(√2) Explanation please!

$$\left[\boldsymbol{{Q}}.\mathrm{1}\:\right]\:\:\:{Find}\:{the}\:{domain}\:{of} \\ $$$$\:\:\:\: \\ $$$$\:\:\:\:\:\:\:{f}\left({x}\right)=\frac{\lfloor{cos}^{−\mathrm{1}} \left({x}^{\mathrm{4}} \right)\rfloor+\mid\lfloor{x}−\mathrm{2}{tan}^{−\mathrm{1}} \left({x}\right)\rfloor\mid+\sqrt{{sin}\left({lnx}\right)}}{\left\{\mathrm{3}{x}^{\mathrm{2}} −\mathrm{7}\right\}+{a}^{\sqrt{{sin}\left({x}\right)+\mathrm{3}{cos}\left({x}\right)}} +{ln}\:{cos}\left(\frac{\mathrm{1}}{\:\sqrt{−{x}^{\mathrm{2}} }}\right)} \\ $$$${where}\:\left\{{x}\right\}\:{reprents}\:\:{the}\:{fractionare}\:{part}\:{of}\:{x}: \\ $$$$ \\ $$$$\left.{A}\left.\right)\left.\:\left.\right]\left.−\mathrm{2},\:\sqrt{\mathrm{2}}\left[\:\:\:\:\:\:\:{B}\right)\:\:\right]\mathrm{0},\mathrm{2}\left[\:\:\:\:\:\:{C}\right)\:\right]−\mathrm{1},\:\mathrm{1}\:\left[\:\:\:\:{D}\right)\:\right]−\mathrm{2},\:−\sqrt{\mathrm{2}} \\ $$$$ \\ $$$$\:\:\:\:\:\:{Explanation}\:{please}! \\ $$

Question Number 115018 Answers: 2 Comments: 0

If 9^x +9^(−x) = 3^(2+x) +3^(2−x) −20, then 27^x +27^(−x) =?

$${If}\:\mathrm{9}^{{x}} +\mathrm{9}^{−{x}} \:=\:\mathrm{3}^{\mathrm{2}+{x}} +\mathrm{3}^{\mathrm{2}−{x}} \:−\mathrm{20},\:{then}\: \\ $$$$\mathrm{27}^{{x}} +\mathrm{27}^{−{x}} \:=? \\ $$

Question Number 115014 Answers: 1 Comments: 0

if I_n =∫_x ^(π/2) xcos^n xdx,where n≻1 show that I_n =((n(n−1)I_(n−2) −1)/n^2 ) and then evaluate ∫_x ^(π/2) xcos^8 xdx

$${if}\:{I}_{{n}} =\int_{{x}} ^{\frac{\pi}{\mathrm{2}}} {x}\mathrm{cos}^{{n}} {xdx},{where}\:{n}\succ\mathrm{1}\:{show} \\ $$$${that}\:{I}_{{n}} =\frac{{n}\left({n}−\mathrm{1}\right){I}_{{n}−\mathrm{2}} −\mathrm{1}}{{n}^{\mathrm{2}} }\:{and}\:{then} \\ $$$${evaluate}\:\:\int_{{x}} ^{\frac{\pi}{\mathrm{2}}} {x}\mathrm{cos}^{\mathrm{8}} {xdx} \\ $$

Question Number 115013 Answers: 0 Comments: 0

consider the change in the direction of a curve W=f(θ) between point A and B. Derive from first principle an expression for the radius of curvature R for the hyperbola

$${consider}\:{the}\:{change}\:{in}\:{the}\:{direction}\:{of} \\ $$$${a}\:{curve}\:{W}={f}\left(\theta\right)\:{between}\:{point}\:{A}\:{and} \\ $$$${B}.\:{Derive}\:{from}\:{first}\:{principle}\:{an} \\ $$$${expression}\:{for}\:{the}\:{radius}\:{of}\:{curvature} \\ $$$${R}\:{for}\:{the}\:{hyperbola} \\ $$

Question Number 115011 Answers: 0 Comments: 0

given the quadratic function f(x)=x+2+px+q where p and q are?integer.s let a,b, and c distinc integers such that 2^(2020) evenly divides f(a),f(b),and f(c),but 2^(1000) does not divide b−a and also does not divide c−a.show that 2^(1021) just divide b−c?

$${given}\:{the}\:{quadratic}\:{function}\: \\ $$$${f}\left({x}\right)={x}+\mathrm{2}+{px}+{q}\:{where}\:{p}\:{and}\:{q}\:{are}?{integer}.{s} \\ $$$${let}\:{a},{b},\:{and}\:{c}\:{distinc}\:{integers}\:{such}\:{that} \\ $$$$\mathrm{2}^{\mathrm{2020}} \:\:{evenly}\:{divides}\:{f}\left({a}\right),{f}\left({b}\right),{and}\:{f}\left({c}\right),{but} \\ $$$$\mathrm{2}^{\mathrm{1000}} \:{does}\:{not}\:{divide}\:{b}−{a}\:{and}\:{also}\:{does}\:{not} \\ $$$${divide}\:{c}−{a}.{show}\:{that}\:\mathrm{2}^{\mathrm{1021}} \:{just}\:{divide}\:{b}−{c}? \\ $$

Question Number 115009 Answers: 2 Comments: 0

∫_C (e^z /(1−cos z))dz ; C:∣z∣=1

$$\int_{\mathrm{C}} \frac{{e}^{{z}} }{\mathrm{1}−\mathrm{cos}\:{z}}{dz}\:;\:\mathrm{C}:\mid{z}\mid=\mathrm{1} \\ $$

Question Number 115000 Answers: 2 Comments: 0

....nice math... if y =(cos(2x))^(−(1/2)) then prove :: y+y^(′′) = 3y^5 ...m.n.july.1970...

$$\:\:\:\:\:\:\:\:\:\:\:\:....{nice}\:\:\:{math}... \\ $$$$ \\ $$$$\:{if}\:\:{y}\:=\left({cos}\left(\mathrm{2}{x}\right)\right)^{−\frac{\mathrm{1}}{\mathrm{2}}} \:{then} \\ $$$${prove}\:::\:\:\:{y}+{y}^{''} =\:\mathrm{3}{y}^{\mathrm{5}} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:...{m}.{n}.{july}.\mathrm{1970}... \\ $$

Question Number 115095 Answers: 2 Comments: 0