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Question Number 54473    Answers: 1   Comments: 0

Solve for x: 2^(2x − 4) = x^2

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{x}:\:\:\:\:\mathrm{2}^{\mathrm{2x}\:−\:\mathrm{4}} \:\:=\:\:\mathrm{x}^{\mathrm{2}} \\ $$

Question Number 54472    Answers: 1   Comments: 0

If A, B, C are angle of a triangle, show that tanA + tanB + tanC = tanA tanB tanC

$$\mathrm{If}\:\mathrm{A},\:\mathrm{B},\:\mathrm{C}\:\:\mathrm{are}\:\mathrm{angle}\:\mathrm{of}\:\mathrm{a}\:\mathrm{triangle},\:\:\mathrm{show}\:\mathrm{that} \\ $$$$\:\:\:\mathrm{tanA}\:+\:\mathrm{tanB}\:+\:\mathrm{tanC}\:\:=\:\:\mathrm{tanA}\:\mathrm{tanB}\:\mathrm{tanC} \\ $$

Question Number 54468    Answers: 0   Comments: 0

First three terms of the sequence given by a_1 =1, a_n =a_(n−1) +2a_(n−2) are in

$$\mathrm{First}\:\mathrm{three}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sequence}\:\mathrm{given} \\ $$$$\mathrm{by}\:\:{a}_{\mathrm{1}} =\mathrm{1},\:{a}_{{n}} ={a}_{{n}−\mathrm{1}} +\mathrm{2}{a}_{{n}−\mathrm{2}} \:\mathrm{are}\:\mathrm{in} \\ $$

Question Number 54466    Answers: 1   Comments: 1

((6+(√((6)^2 −4(1)(10))))/2)

$$\frac{\mathrm{6}+\sqrt{\left(\mathrm{6}\right)^{\mathrm{2}} −\mathrm{4}\left(\mathrm{1}\right)\left(\mathrm{10}\right)}}{\mathrm{2}} \\ $$$$ \\ $$

Question Number 54462    Answers: 1   Comments: 1

Question Number 54460    Answers: 1   Comments: 0

Solve (dy/dx)+3x=5

$${Solve}\:\frac{{dy}}{{dx}}+\mathrm{3}{x}=\mathrm{5} \\ $$

Question Number 54457    Answers: 0   Comments: 0

Question Number 54447    Answers: 2   Comments: 1

Question Number 54487    Answers: 0   Comments: 0

Question Number 54422    Answers: 1   Comments: 4

The median AD of the triangle ABC is bisected at E, BE meets AC in F, then AF : AC =

$$\mathrm{The}\:\mathrm{median}\:{AD}\:\mathrm{of}\:\mathrm{the}\:\mathrm{triangle}\:{ABC} \\ $$$$\mathrm{is}\:\mathrm{bisected}\:\mathrm{at}\:{E},\:{BE}\:\mathrm{meets}\:{AC}\:\mathrm{in}\:{F}, \\ $$$$\mathrm{then}\:{AF}\::\:{AC}\:= \\ $$

Question Number 54421    Answers: 2   Comments: 2

If the vector c, a=xi+yj+zk and b=j are such that a, c and b form a right handed system, then c is

$$\mathrm{If}\:\mathrm{the}\:\mathrm{vector}\:\boldsymbol{\mathrm{c}},\:\boldsymbol{\mathrm{a}}={x}\boldsymbol{\mathrm{i}}+{y}\boldsymbol{\mathrm{j}}+{z}\boldsymbol{\mathrm{k}}\:\mathrm{and}\:\boldsymbol{\mathrm{b}}=\boldsymbol{\mathrm{j}} \\ $$$$\mathrm{are}\:\mathrm{such}\:\mathrm{that}\:\boldsymbol{\mathrm{a}},\:\boldsymbol{\mathrm{c}}\:\mathrm{and}\:\boldsymbol{\mathrm{b}}\:\mathrm{form}\:\mathrm{a}\:\mathrm{right} \\ $$$$\mathrm{handed}\:\mathrm{system},\:\mathrm{then}\:\boldsymbol{\mathrm{c}}\:\mathrm{is} \\ $$

Question Number 54420    Answers: 1   Comments: 0

Let a=i+j and b=2i−k, the point of intersection of the lines r×a=b×a and r×b=a×b is

$$\mathrm{Let}\:\boldsymbol{\mathrm{a}}=\boldsymbol{\mathrm{i}}+\boldsymbol{\mathrm{j}}\:\mathrm{and}\:\boldsymbol{\mathrm{b}}=\mathrm{2}\boldsymbol{\mathrm{i}}−\boldsymbol{\mathrm{k}},\:\mathrm{the}\:\mathrm{point}\:\mathrm{of} \\ $$$$\mathrm{intersection}\:\mathrm{of}\:\mathrm{the}\:\mathrm{lines}\:\boldsymbol{\mathrm{r}}×\boldsymbol{\mathrm{a}}=\boldsymbol{\mathrm{b}}×\boldsymbol{\mathrm{a}} \\ $$$$\mathrm{and}\:\boldsymbol{\mathrm{r}}×\boldsymbol{\mathrm{b}}=\boldsymbol{\mathrm{a}}×\boldsymbol{\mathrm{b}}\:\mathrm{is} \\ $$

Question Number 54419    Answers: 1   Comments: 0

If a=i+j−k, b=i−j+k and c is a unit vector ⊥ to the vector a and coplanar with a and b, then a unit vector d ⊥ to both a and c is

$$\mathrm{If}\:\boldsymbol{\mathrm{a}}=\boldsymbol{\mathrm{i}}+\boldsymbol{\mathrm{j}}−\boldsymbol{\mathrm{k}},\:\boldsymbol{\mathrm{b}}=\boldsymbol{\mathrm{i}}−\boldsymbol{\mathrm{j}}+\boldsymbol{\mathrm{k}}\:\mathrm{and}\:\boldsymbol{\mathrm{c}}\:\mathrm{is}\:\mathrm{a}\:\mathrm{unit} \\ $$$$\mathrm{vector}\:\bot\:\mathrm{to}\:\mathrm{the}\:\mathrm{vector}\:\boldsymbol{\mathrm{a}}\:\mathrm{and}\:\mathrm{coplanar} \\ $$$$\mathrm{with}\:\boldsymbol{\mathrm{a}}\:\mathrm{and}\:\boldsymbol{\mathrm{b}},\:\mathrm{then}\:\mathrm{a}\:\mathrm{unit}\:\mathrm{vector}\:\boldsymbol{\mathrm{d}}\:\bot\:\mathrm{to} \\ $$$$\mathrm{both}\:\boldsymbol{\mathrm{a}}\:\mathrm{and}\:\boldsymbol{\mathrm{c}}\:\mathrm{is} \\ $$

Question Number 54418    Answers: 1   Comments: 0

The projection of the vector a=4i−3j+2k on the axis making equal acute angles with the coordinate axes is

$$\mathrm{The}\:\mathrm{projection}\:\mathrm{of}\:\mathrm{the}\:\mathrm{vector}\:\boldsymbol{\mathrm{a}}=\mathrm{4}\boldsymbol{\mathrm{i}}−\mathrm{3}\boldsymbol{\mathrm{j}}+\mathrm{2}\boldsymbol{\mathrm{k}} \\ $$$$\mathrm{on}\:\mathrm{the}\:\mathrm{axis}\:\mathrm{making}\:\mathrm{equal}\:\mathrm{acute}\:\mathrm{angles} \\ $$$$\mathrm{with}\:\mathrm{the}\:\mathrm{coordinate}\:\mathrm{axes}\:\mathrm{is} \\ $$

Question Number 54417    Answers: 0   Comments: 0

A force of 39 kg weight is acting at a point P(−4, 2, 5) in the direction 12i−4j−3k. The moment of this force about a line through the origin having the direction of 2i−2j+k is

$$\mathrm{A}\:\mathrm{force}\:\mathrm{of}\:\mathrm{39}\:\mathrm{kg}\:\mathrm{weight}\:\mathrm{is}\:\mathrm{acting}\:\mathrm{at}\:\mathrm{a} \\ $$$$\mathrm{point}\:{P}\left(−\mathrm{4},\:\mathrm{2},\:\mathrm{5}\right)\:\mathrm{in}\:\mathrm{the}\:\mathrm{direction}\: \\ $$$$\mathrm{12}\boldsymbol{\mathrm{i}}−\mathrm{4}\boldsymbol{\mathrm{j}}−\mathrm{3}\boldsymbol{\mathrm{k}}.\:\mathrm{The}\:\mathrm{moment}\:\mathrm{of}\:\mathrm{this}\:\mathrm{force} \\ $$$$\mathrm{about}\:\mathrm{a}\:\mathrm{line}\:\mathrm{through}\:\mathrm{the}\:\mathrm{origin}\:\mathrm{having} \\ $$$$\mathrm{the}\:\mathrm{direction}\:\mathrm{of}\:\mathrm{2}\boldsymbol{\mathrm{i}}−\mathrm{2}\boldsymbol{\mathrm{j}}+\boldsymbol{\mathrm{k}}\:\mathrm{is} \\ $$

Question Number 54416    Answers: 0   Comments: 0

Question Number 54413    Answers: 0   Comments: 8

Question Number 54409    Answers: 2   Comments: 1

lim_(x→∞) (√((x^2 +x+1)))−x=? pls solve this

$${li}\underset{{x}\rightarrow\infty} {{m}}\sqrt{\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right)}−{x}=? \\ $$$${pls}\:{solve}\:{this} \\ $$$$ \\ $$

Question Number 54396    Answers: 1   Comments: 1

Question Number 54388    Answers: 0   Comments: 0

Question Number 54378    Answers: 0   Comments: 3

∫_(π/3) ^(3π/2) [ 2 cos x ] dx =

$$\underset{\pi/\mathrm{3}} {\overset{\mathrm{3}\pi/\mathrm{2}} {\int}}\:\:\left[\:\mathrm{2}\:\mathrm{cos}\:{x}\:\right]\:{dx}\:= \\ $$

Question Number 54377    Answers: 0   Comments: 0

∫_(π/3) ^(3π/2) [ 2 cos x ] dx =

$$\underset{\pi/\mathrm{3}} {\overset{\mathrm{3}\pi/\mathrm{2}} {\int}}\:\:\left[\:\mathrm{2}\:\mathrm{cos}\:{x}\:\right]\:{dx}\:= \\ $$

Question Number 54376    Answers: 3   Comments: 3

1) calculate f(a) =∫_(−∞) ^(+∞) (dx/(x^2 +ax +1)) with ∣a∣<2 2) calculate g(a) =∫_(−∞) ^(+∞) (x/((x^2 +ax+1)^2 )) 3)find values of integrals ∫_(−∞) ^(+∞) (dx/(x^2 +(√2)x +1)) and ∫_(−∞) ^(+∞) (x/((x^2 +(√2)x +1)^2 )) 4) calculate A(θ) = ∫_(−∞) ^(+∞) (dx/(x^2 +2cosθ +1)) θ is a given real.

$$\left.\mathrm{1}\right)\:{calculate}\:\:{f}\left({a}\right)\:=\int_{−\infty} ^{+\infty} \:\:\:\frac{{dx}}{{x}^{\mathrm{2}} \:+{ax}\:\:+\mathrm{1}} \\ $$$${with}\:\:\:\mid{a}\mid<\mathrm{2} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{g}\left({a}\right)\:=\int_{−\infty} ^{+\infty} \:\frac{{x}}{\left({x}^{\mathrm{2}} \:+{ax}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right){find}\:{values}\:{of}\:{integrals}\:\int_{−\infty} ^{+\infty} \:\:\frac{{dx}}{{x}^{\mathrm{2}} \:+\sqrt{\mathrm{2}}{x}\:+\mathrm{1}} \\ $$$${and}\:\int_{−\infty} ^{+\infty} \:\frac{{x}}{\left({x}^{\mathrm{2}} \:+\sqrt{\mathrm{2}}{x}\:+\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:{A}\left(\theta\right)\:=\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{dx}}{{x}^{\mathrm{2}} \:+\mathrm{2}{cos}\theta\:+\mathrm{1}} \\ $$$$\theta\:{is}\:{a}\:{given}\:{real}. \\ $$

Question Number 54374    Answers: 0   Comments: 1

calculate h(a) =∫_(−∞) ^(+∞) ((cos(at))/(ch((t/2))))dt .

$${calculate}\:{h}\left({a}\right)\:=\int_{−\infty} ^{+\infty} \:\:\frac{{cos}\left({at}\right)}{{ch}\left(\frac{{t}}{\mathrm{2}}\right)}{dt}\:. \\ $$

Question Number 54372    Answers: 1   Comments: 3

let f(x) =∫_0 ^(2π) ((sint)/(x+sint))dt 1) calculate f(x) 2) calculate g(x) =∫_0 ^(2π) ((sint)/((x+sint)^2 )) dt 3) calculste for n∈N ∫_0 ^(2π) ((sint)/((x+sint)^n ))dt 4) calculate ∫_0 ^(2π) ((sint)/(2+sint))dt and ∫_0 ^(2π) ((sint)/((2+sint)^2 ))dt .

$${let}\:\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{sint}}{{x}+{sint}}{dt}\:\:\: \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{sint}}{\left({x}+{sint}\right)^{\mathrm{2}} }\:{dt}\: \\ $$$$\left.\mathrm{3}\right)\:{calculste}\:{for}\:{n}\in{N}\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\frac{{sint}}{\left({x}+{sint}\right)^{{n}} }{dt}\: \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{sint}}{\mathrm{2}+{sint}}{dt}\:\:{and}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{sint}}{\left(\mathrm{2}+{sint}\right)^{\mathrm{2}} }{dt}\:. \\ $$

Question Number 54371    Answers: 1   Comments: 1

prove that ln(z) = ∫_0 ^1 ((z−1)/(1+t(z−1)))dt .

$${prove}\:{that}\:{ln}\left({z}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{z}−\mathrm{1}}{\mathrm{1}+{t}\left({z}−\mathrm{1}\right)}{dt}\:. \\ $$

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