Question and Answers Forum

AllQuestion and Answers: Page 1817

Question Number 20559 Answers: 1 Comments: 0

Question Number 20558 Answers: 1 Comments: 1

Question Number 20552 Answers: 1 Comments: 0

The roots of the equation (3−x)^4 +(2−x)^4 =(5−2x)^4 are (a) all real (b) all imaginary (c) two real and two imaginary (d)none of the above .

$${The}\:{roots}\:{of}\:{the}\:{equation}\: \\ $$$$\:\left(\mathrm{3}−{x}\right)^{\mathrm{4}} +\left(\mathrm{2}−{x}\right)^{\mathrm{4}} =\left(\mathrm{5}−\mathrm{2}{x}\right)^{\mathrm{4}} \:{are} \\ $$$$\left({a}\right)\:{all}\:{real}\:\:\:\:\left({b}\right)\:{all}\:{imaginary} \\ $$$$\left({c}\right)\:{two}\:{real}\:{and}\:{two}\:{imaginary} \\ $$$$\left({d}\right){none}\:{of}\:{the}\:{above}\:. \\ $$

Question Number 20574 Answers: 1 Comments: 0

(1 + tan 1°)(1 + tan 2°)...(1 + tan 45°) = 2^(n+1) find n !

$$\left(\mathrm{1}\:+\:\mathrm{tan}\:\mathrm{1}°\right)\left(\mathrm{1}\:+\:\mathrm{tan}\:\mathrm{2}°\right)...\left(\mathrm{1}\:+\:\mathrm{tan}\:\mathrm{45}°\right)\:=\:\mathrm{2}^{{n}+\mathrm{1}} \\ $$$$\mathrm{find}\:{n}\:! \\ $$

Question Number 20541 Answers: 1 Comments: 0

∫((xdx)/((x−1)(x^2 +1)))

$$\int\frac{{xdx}}{\left({x}−\mathrm{1}\right)\left({x}^{\mathrm{2}} +\mathrm{1}\right)} \\ $$

Question Number 20540 Answers: 1 Comments: 0

∫((√x)/(1+x^(1/3) ))dx

$$\int\frac{\sqrt{{x}}}{\mathrm{1}+{x}^{\frac{\mathrm{1}}{\mathrm{3}}} }{dx} \\ $$

Question Number 20539 Answers: 0 Comments: 0

∫(dx/(x^(1/3) −x^(1/6) ))

$$\int\frac{{dx}}{{x}^{\frac{\mathrm{1}}{\mathrm{3}}} −{x}^{\frac{\mathrm{1}}{\mathrm{6}}} } \\ $$

Question Number 20538 Answers: 0 Comments: 0

∫(dx/(e^(2x) −3e^x ))

$$\int\frac{{dx}}{{e}^{\mathrm{2}{x}} −\mathrm{3}{e}^{{x}} } \\ $$

Question Number 20537 Answers: 0 Comments: 0

∫((2e^x +5)/(5e^x +2))dx

$$\int\frac{\mathrm{2}{e}^{{x}} +\mathrm{5}}{\mathrm{5}{e}^{{x}} +\mathrm{2}}{dx} \\ $$

Question Number 20550 Answers: 1 Comments: 0

Find the equation of circle in complex form which touches iz + z^ + 1 + i = 0 and for which the lines (1 − i)z = (1 + i)z^ and (1 + i)z + (i − 1)z^ − 4i = 0 are normals.

$${Find}\:{the}\:{equation}\:{of}\:{circle}\:{in}\:{complex} \\ $$$${form}\:{which}\:{touches}\:{iz}\:+\:\bar {{z}}\:+\:\mathrm{1}\:+\:{i}\:=\:\mathrm{0} \\ $$$${and}\:{for}\:{which}\:{the}\:{lines}\:\left(\mathrm{1}\:−\:{i}\right){z}\:= \\ $$$$\left(\mathrm{1}\:+\:{i}\right)\bar {{z}}\:{and}\:\left(\mathrm{1}\:+\:{i}\right){z}\:+\:\left({i}\:−\:\mathrm{1}\right)\bar {{z}}\:−\:\mathrm{4}{i}\:=\:\mathrm{0} \\ $$$${are}\:{normals}. \\ $$

Question Number 20549 Answers: 1 Comments: 1

Show that if z_1 z_2 + z_3 z_4 = 0 and z_1 + z_2 = 0, then the complex numbers z_1 , z_2 , z_3 , z_4 are concyclic.

$${Show}\:{that}\:{if}\:{z}_{\mathrm{1}} {z}_{\mathrm{2}} \:+\:{z}_{\mathrm{3}} {z}_{\mathrm{4}} \:=\:\mathrm{0}\:{and}\:{z}_{\mathrm{1}} \:+ \\ $$$${z}_{\mathrm{2}} \:=\:\mathrm{0},\:{then}\:{the}\:{complex}\:{numbers}\:{z}_{\mathrm{1}} , \\ $$$${z}_{\mathrm{2}} ,\:{z}_{\mathrm{3}} ,\:{z}_{\mathrm{4}} \:{are}\:{concyclic}. \\ $$

Question Number 20545 Answers: 0 Comments: 1

Let ABC be an acute-angled triangle with AC ≠ BC and let O be the circumcenter and F be the foot of altitude through C. Further, let X and Y be the feet of perpendiculars dropped from A and B respectively to (the extension of) CO. Prove that FY ⊥ CA using that ∠CFY = ∠CBY = ∠CAF.

$$\mathrm{Let}\:\mathrm{ABC}\:\mathrm{be}\:\mathrm{an}\:\mathrm{acute}-\mathrm{angled}\:\mathrm{triangle} \\ $$$$\mathrm{with}\:\mathrm{AC}\:\neq\:\mathrm{BC}\:\mathrm{and}\:\mathrm{let}\:\mathrm{O}\:\mathrm{be}\:\mathrm{the} \\ $$$$\mathrm{circumcenter}\:\mathrm{and}\:\mathrm{F}\:\mathrm{be}\:\mathrm{the}\:\mathrm{foot}\:\mathrm{of} \\ $$$$\mathrm{altitude}\:\mathrm{through}\:\mathrm{C}.\:\mathrm{Further},\:\mathrm{let}\:\mathrm{X}\:\mathrm{and} \\ $$$$\mathrm{Y}\:\mathrm{be}\:\mathrm{the}\:\mathrm{feet}\:\mathrm{of}\:\mathrm{perpendiculars}\:\mathrm{dropped} \\ $$$$\mathrm{from}\:\mathrm{A}\:\mathrm{and}\:\mathrm{B}\:\mathrm{respectively}\:\mathrm{to}\:\left(\mathrm{the}\right. \\ $$$$\left.\mathrm{extension}\:\mathrm{of}\right)\:\mathrm{CO}.\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{FY}\:\bot\:\mathrm{CA} \\ $$$$\mathrm{using}\:\mathrm{that}\:\angle\mathrm{CFY}\:=\:\angle\mathrm{CBY}\:=\:\angle\mathrm{CAF}. \\ $$

Question Number 20523 Answers: 1 Comments: 0

Let the sum Σ_(n=1) ^9 (1/(n(n + 1)(n + 2))) written in its lowest terms be (p/q). Find the value of q − p.

$$\mathrm{Let}\:\mathrm{the}\:\mathrm{sum}\:\underset{{n}=\mathrm{1}} {\overset{\mathrm{9}} {\sum}}\frac{\mathrm{1}}{{n}\left({n}\:+\:\mathrm{1}\right)\left({n}\:+\:\mathrm{2}\right)}\:\mathrm{written} \\ $$$$\mathrm{in}\:\mathrm{its}\:\mathrm{lowest}\:\mathrm{terms}\:\mathrm{be}\:\frac{{p}}{{q}}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{value} \\ $$$$\mathrm{of}\:{q}\:−\:{p}. \\ $$

Question Number 20524 Answers: 1 Comments: 0

∫x^x dx

$$\int{x}^{{x}} {dx} \\ $$

Question Number 20511 Answers: 0 Comments: 0

Given a sphere of unit radius. Find the expression of a circular spot on the sphere′s surface given the latitude β and the longitude λ of its center and its angular radius r.

$$\mathrm{Given}\:\mathrm{a}\:\mathrm{sphere}\:\mathrm{of}\:\mathrm{unit}\:\mathrm{radius}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{expression}\:\mathrm{of}\:\mathrm{a}\:\mathrm{circular} \\ $$$$\mathrm{spot}\:\mathrm{on}\:\mathrm{the}\:\mathrm{sphere}'\mathrm{s}\:\mathrm{surface}\:\mathrm{given} \\ $$$$\mathrm{the}\:\mathrm{latitude}\:\beta\:\mathrm{and}\:\mathrm{the}\:\mathrm{longitude}\:\lambda \\ $$$$\mathrm{of}\:\mathrm{its}\:\mathrm{center}\:\mathrm{and}\:\mathrm{its}\:\mathrm{angular}\:\mathrm{radius}\:{r}. \\ $$

Question Number 20509 Answers: 0 Comments: 0

Simplify: cos^(−1) (((sin x + cos x)/(√2))), ((5π)/4) < x < ((9π)/4)

$${Simplify}: \\ $$$$\mathrm{cos}^{−\mathrm{1}} \:\left(\frac{\mathrm{sin}\:{x}\:+\:\mathrm{cos}\:{x}}{\sqrt{\mathrm{2}}}\right),\:\frac{\mathrm{5}\pi}{\mathrm{4}}\:<\:{x}\:<\:\frac{\mathrm{9}\pi}{\mathrm{4}} \\ $$

Question Number 20506 Answers: 1 Comments: 0

Simplify: cos^(−1) (((sin x + cos x)/(√2))), (π/4) < x < ((5π)/4)

$${Simplify}: \\ $$$$\mathrm{cos}^{−\mathrm{1}} \:\left(\frac{\mathrm{sin}\:{x}\:+\:\mathrm{cos}\:{x}}{\sqrt{\mathrm{2}}}\right),\:\frac{\pi}{\mathrm{4}}\:<\:{x}\:<\:\frac{\mathrm{5}\pi}{\mathrm{4}} \\ $$

Question Number 20505 Answers: 1 Comments: 0

Simplify: cos^(−1) ((3/5) cos x + (4/5) sin x), where −((3π)/4) ≤ x ≤ (π/4)

$${Simplify}: \\ $$$$\mathrm{cos}^{−\mathrm{1}} \:\left(\frac{\mathrm{3}}{\mathrm{5}}\:\mathrm{cos}\:{x}\:+\:\frac{\mathrm{4}}{\mathrm{5}}\:\mathrm{sin}\:{x}\right),\:{where} \\ $$$$−\frac{\mathrm{3}\pi}{\mathrm{4}}\:\leqslant\:{x}\:\leqslant\:\frac{\pi}{\mathrm{4}} \\ $$

Question Number 20501 Answers: 0 Comments: 5

The force acting on the block is given by F = 5 − 2t. The frictional force acting on the block at t = 2 s. (The block is at rest at t = 0)