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Question Number 21471 Answers: 2 Comments: 0

If a + b + c = 0, then (((a + b)(b + c)(a + c))/(abc)) is equal to ...

$$\mathrm{If}\:\:{a}\:+\:{b}\:+\:{c}\:=\:\mathrm{0},\:\mathrm{then} \\ $$$$\frac{\left({a}\:+\:{b}\right)\left({b}\:+\:{c}\right)\left({a}\:+\:{c}\right)}{{abc}}\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:... \\ $$

Question Number 21470 Answers: 1 Comments: 1

2^x = 3^y = 6^(−z) Find the value of (((2017)/x) + ((2017)/y) + ((2017)/z))^(2017)

$$\mathrm{2}^{{x}} \:=\:\mathrm{3}^{{y}} \:=\:\mathrm{6}^{−{z}} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\: \\ $$$$\left(\frac{\mathrm{2017}}{{x}}\:+\:\frac{\mathrm{2017}}{{y}}\:+\:\frac{\mathrm{2017}}{{z}}\right)^{\mathrm{2017}} \\ $$

Question Number 21469 Answers: 1 Comments: 1

Three identical blocks, each having a mass M, are pushed by a force F on a frictionless table. What is the net force on the block A?

$$\mathrm{Three}\:\mathrm{identical}\:\mathrm{blocks},\:\mathrm{each}\:\mathrm{having}\:\mathrm{a} \\ $$$$\mathrm{mass}\:{M},\:\mathrm{are}\:\mathrm{pushed}\:\mathrm{by}\:\mathrm{a}\:\mathrm{force}\:{F}\:\mathrm{on}\:\mathrm{a} \\ $$$$\mathrm{frictionless}\:\mathrm{table}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{net}\:\mathrm{force} \\ $$$$\mathrm{on}\:\mathrm{the}\:\mathrm{block}\:{A}? \\ $$

Question Number 21467 Answers: 1 Comments: 0

∫_( 1) ^e log x dx =

$$\underset{\:\mathrm{1}} {\overset{{e}} {\int}}\:\:\mathrm{log}\:{x}\:{dx}\:= \\ $$

Question Number 21464 Answers: 2 Comments: 0

Question Number 21463 Answers: 0 Comments: 0

Let A be the collection of functions f : [0, 1] → R which have an infinite number of derivatives. Let A_0 ⊂ A be the subcollection of those functions f with f(0) = 0. Define D : A_0 → A by D(f) = df/dx. Use the mean value theorem to show that D is injective. Use the fundamental theorem of calculus to show that D is surjective.

$$\mathrm{Let}\:{A}\:\mathrm{be}\:\mathrm{the}\:\mathrm{collection}\:\mathrm{of}\:\mathrm{functions} \\ $$$${f}\::\:\left[\mathrm{0},\:\mathrm{1}\right]\:\rightarrow\:\mathbb{R}\:\mathrm{which}\:\mathrm{have}\:\mathrm{an}\:\mathrm{infinite} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{derivatives}.\:\mathrm{Let}\:{A}_{\mathrm{0}} \:\subset\:{A} \\ $$$$\mathrm{be}\:\mathrm{the}\:\mathrm{subcollection}\:\mathrm{of}\:\mathrm{those}\:\mathrm{functions} \\ $$$${f}\:\mathrm{with}\:{f}\left(\mathrm{0}\right)\:=\:\mathrm{0}.\:\mathrm{Define}\:{D}\::\:{A}_{\mathrm{0}} \:\rightarrow\:{A} \\ $$$$\mathrm{by}\:{D}\left({f}\right)\:=\:{df}/{dx}.\:\mathrm{Use}\:\mathrm{the}\:\mathrm{mean}\:\mathrm{value} \\ $$$$\mathrm{theorem}\:\mathrm{to}\:\mathrm{show}\:\mathrm{that}\:{D}\:\mathrm{is}\:\mathrm{injective}. \\ $$$$\mathrm{Use}\:\mathrm{the}\:\mathrm{fundamental}\:\mathrm{theorem}\:\mathrm{of} \\ $$$$\mathrm{calculus}\:\mathrm{to}\:\mathrm{show}\:\mathrm{that}\:{D}\:\mathrm{is}\:\mathrm{surjective}. \\ $$

Question Number 21459 Answers: 1 Comments: 0

Solve (√(x + 1 − 4(√(x − 3)))) + (√(x + 22 + 10(√(x − 3)))) = 7

$$\mathrm{Solve} \\ $$$$\sqrt{{x}\:+\:\mathrm{1}\:−\:\mathrm{4}\sqrt{{x}\:−\:\mathrm{3}}}\:+\:\sqrt{{x}\:+\:\mathrm{22}\:+\:\mathrm{10}\sqrt{{x}\:−\:\mathrm{3}}}\:=\:\mathrm{7} \\ $$

Question Number 21453 Answers: 3 Comments: 0

If (1 + n)sin 2θ + (1 − n)cos 2θ = 1 + n find tan 2θ

$$\mathrm{If}\: \\ $$$$\left(\mathrm{1}\:+\:{n}\right)\mathrm{sin}\:\mathrm{2}\theta\:+\:\left(\mathrm{1}\:−\:{n}\right)\mathrm{cos}\:\mathrm{2}\theta\:=\:\mathrm{1}\:+\:{n} \\ $$$$\mathrm{find}\:\mathrm{tan}\:\mathrm{2}\theta \\ $$

Question Number 21450 Answers: 0 Comments: 0

Question Number 21507 Answers: 1 Comments: 1

The length of an ideal spring increases by 0.1 cm when a body of 1 kg is suspended from it. If this spring is laid on a frictionless horizontal table and bodies of 1 kg each are suspended from its ends, then what will be the increase in its length?

$$\mathrm{The}\:\mathrm{length}\:\mathrm{of}\:\mathrm{an}\:\mathrm{ideal}\:\mathrm{spring}\:\mathrm{increases} \\ $$$$\mathrm{by}\:\mathrm{0}.\mathrm{1}\:\mathrm{cm}\:\mathrm{when}\:\mathrm{a}\:\mathrm{body}\:\mathrm{of}\:\mathrm{1}\:\mathrm{kg}\:\mathrm{is} \\ $$$$\mathrm{suspended}\:\mathrm{from}\:\mathrm{it}.\:\mathrm{If}\:\mathrm{this}\:\mathrm{spring}\:\mathrm{is}\:\mathrm{laid} \\ $$$$\mathrm{on}\:\mathrm{a}\:\mathrm{frictionless}\:\mathrm{horizontal}\:\mathrm{table}\:\mathrm{and} \\ $$$$\mathrm{bodies}\:\mathrm{of}\:\mathrm{1}\:\mathrm{kg}\:\mathrm{each}\:\mathrm{are}\:\mathrm{suspended}\:\mathrm{from} \\ $$$$\mathrm{its}\:\mathrm{ends},\:\mathrm{then}\:\mathrm{what}\:\mathrm{will}\:\mathrm{be}\:\mathrm{the}\:\mathrm{increase} \\ $$$$\mathrm{in}\:\mathrm{its}\:\mathrm{length}? \\ $$

Question Number 21441 Answers: 1 Comments: 0

Find α in terms of θ using the equations: (i) u^2 sin^2 α = 2gd cos θ (ii) t = ((u cos α)/(g sin θ)) (iii) −d = ut sin α − ((gt^2 sin θ)/2)

$$\mathrm{Find}\:\alpha\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:\theta\:\mathrm{using}\:\mathrm{the}\:\mathrm{equations}: \\ $$$$\left({i}\right)\:{u}^{\mathrm{2}} \:\mathrm{sin}^{\mathrm{2}} \:\alpha\:=\:\mathrm{2}{gd}\:\mathrm{cos}\:\theta \\ $$$$\left({ii}\right)\:{t}\:=\:\frac{{u}\:\mathrm{cos}\:\alpha}{{g}\:\mathrm{sin}\:\theta} \\ $$$$\left({iii}\right)\:−{d}\:=\:{ut}\:\mathrm{sin}\:\alpha\:−\:\frac{{gt}^{\mathrm{2}} \:\mathrm{sin}\:\theta}{\mathrm{2}} \\ $$

Question Number 21431 Answers: 3 Comments: 0

In any ΔABC, a(b cos C − c cos B) =

$$\mathrm{In}\:\mathrm{any}\:\Delta{ABC},\:{a}\left({b}\:\mathrm{cos}\:{C}\:−\:{c}\:\mathrm{cos}\:{B}\right)\:= \\ $$

Question Number 21430 Answers: 1 Comments: 0

In any ΔABC, the value of 2ac sin (((A − B + C)/2)) is

$$\mathrm{In}\:\mathrm{any}\:\Delta{ABC},\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mathrm{2}{ac}\:\mathrm{sin}\:\left(\frac{{A}\:−\:{B}\:+\:{C}}{\mathrm{2}}\right)\:\mathrm{is} \\ $$

Question Number 21429 Answers: 1 Comments: 0

In any ΔABC, Σa^2 (sin B − sin C) =

$$\mathrm{In}\:\mathrm{any}\:\Delta{ABC},\:\Sigma{a}^{\mathrm{2}} \left(\mathrm{sin}\:{B}\:−\:\mathrm{sin}\:{C}\right)\:= \\ $$

Question Number 21423 Answers: 1 Comments: 0

Prove that n^4 + 4^n is composite for all integer values of n greater than 1.

$$\mathrm{Prove}\:\mathrm{that}\:{n}^{\mathrm{4}} \:+\:\mathrm{4}^{{n}} \:\mathrm{is}\:\mathrm{composite}\:\mathrm{for}\:\mathrm{all} \\ $$$$\mathrm{integer}\:\mathrm{values}\:\mathrm{of}\:{n}\:\mathrm{greater}\:\mathrm{than}\:\mathrm{1}. \\ $$

Question Number 21412 Answers: 0 Comments: 0

The atomic masses of ′He′ and ′Ne′ are 4 and 20 a.m.u., respectively. The value of the de Broglie wavelength of ′He′ gas at −73°C is “M” times that of the de Broglie wavelength of ′Ne′ at 727°C ′M′ is

$$\mathrm{The}\:\mathrm{atomic}\:\mathrm{masses}\:\mathrm{of}\:'\mathrm{He}'\:\mathrm{and}\:'\mathrm{Ne}'\:\mathrm{are} \\ $$$$\mathrm{4}\:\mathrm{and}\:\mathrm{20}\:\mathrm{a}.\mathrm{m}.\mathrm{u}.,\:\mathrm{respectively}.\:\mathrm{The} \\ $$$$\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{de}\:\mathrm{Broglie}\:\mathrm{wavelength}\:\mathrm{of} \\ $$$$'\mathrm{He}'\:\mathrm{gas}\:\mathrm{at}\:−\mathrm{73}°\mathrm{C}\:\mathrm{is}\:``\mathrm{M}''\:\mathrm{times}\:\mathrm{that}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{de}\:\mathrm{Broglie}\:\mathrm{wavelength}\:\mathrm{of}\:'\mathrm{Ne}'\:\mathrm{at} \\ $$$$\mathrm{727}°\mathrm{C}\:'\mathrm{M}'\:\mathrm{is} \\ $$

Question Number 21411 Answers: 0 Comments: 0

The critical temperature of water is higher than that of O_2 because the H_2 O molecule has (a) fewer electrons than O_2 (b) two covalent bonds (c) V-shape (d) dipole moment.

$$\mathrm{The}\:\mathrm{critical}\:\mathrm{temperature}\:\mathrm{of}\:\mathrm{water}\:\mathrm{is} \\ $$$$\mathrm{higher}\:\mathrm{than}\:\mathrm{that}\:\mathrm{of}\:\mathrm{O}_{\mathrm{2}} \:\mathrm{because}\:\mathrm{the} \\ $$$$\mathrm{H}_{\mathrm{2}} \mathrm{O}\:\mathrm{molecule}\:\mathrm{has} \\ $$$$\left({a}\right)\:\mathrm{fewer}\:\mathrm{electrons}\:\mathrm{than}\:\mathrm{O}_{\mathrm{2}} \\ $$$$\left({b}\right)\:\mathrm{two}\:\mathrm{covalent}\:\mathrm{bonds} \\ $$$$\left({c}\right)\:\mathrm{V}-\mathrm{shape} \\ $$$$\left({d}\right)\:\mathrm{dipole}\:\mathrm{moment}. \\ $$

Question Number 21422 Answers: 1 Comments: 3

Find all integer values of a such that the quadratic expression (x + a)(x + 1991) + 1 can be factored as a product (x + b)(x + c) where b and c are integers.

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{integer}\:\mathrm{values}\:\mathrm{of}\:{a}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{the}\:\mathrm{quadratic}\:\mathrm{expression} \\ $$$$\left({x}\:+\:{a}\right)\left({x}\:+\:\mathrm{1991}\right)\:+\:\mathrm{1}\:\mathrm{can}\:\mathrm{be}\:\mathrm{factored} \\ $$$$\mathrm{as}\:\mathrm{a}\:\mathrm{product}\:\left({x}\:+\:{b}\right)\left({x}\:+\:{c}\right)\:\mathrm{where}\:{b}\:\mathrm{and} \\ $$$${c}\:\mathrm{are}\:\mathrm{integers}. \\ $$

Question Number 21426 Answers: 1 Comments: 0

given (√(2017x^2 + 4034x + 17)) + (√(2017x^2 + 4034x − 3)) = 10, then find the value x^2 + 2x.

$$\mathrm{given}\:\sqrt{\mathrm{2017}{x}^{\mathrm{2}} \:+\:\mathrm{4034}{x}\:+\:\mathrm{17}}\:+\:\sqrt{\mathrm{2017}{x}^{\mathrm{2}} \:+\:\mathrm{4034}{x}\:−\:\mathrm{3}}\:=\:\mathrm{10}, \\ $$$$\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:{x}^{\mathrm{2}} \:+\:\mathrm{2}{x}. \\ $$

Question Number 21406 Answers: 0 Comments: 2

Find the number of ways in which n distinct balls can be put into three boxes so that no two boxes remain empty.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ways}\:\mathrm{in}\:\mathrm{which}\:{n} \\ $$$$\mathrm{distinct}\:\mathrm{balls}\:\mathrm{can}\:\mathrm{be}\:\mathrm{put}\:\mathrm{into}\:\mathrm{three} \\ $$$$\mathrm{boxes}\:\mathrm{so}\:\mathrm{that}\:\mathrm{no}\:\mathrm{two}\:\mathrm{boxes}\:\mathrm{remain} \\ $$$$\mathrm{empty}. \\ $$

Question Number 21405 Answers: 1 Comments: 0

Four dice are rolled. The number of ways in which at least one die shows 3, is

$$\mathrm{Four}\:\mathrm{dice}\:\mathrm{are}\:\mathrm{rolled}.\:\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ways} \\ $$$$\mathrm{in}\:\mathrm{which}\:\mathrm{at}\:\mathrm{least}\:\mathrm{one}\:\mathrm{die}\:\mathrm{shows}\:\mathrm{3},\:\mathrm{is} \\ $$

Question Number 21404 Answers: 1 Comments: 0

Prove that (6n)! is divisible by 2^(2n) .3^n .