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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 .

$$\mathrm{Prove}\:\mathrm{that}\:\left(\mathrm{6}{n}\right)!\:\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{2}^{\mathrm{2}{n}} .\mathrm{3}^{{n}} . \\ $$

Question Number 21401 Answers: 0 Comments: 0

∫_0 ^∞ ((xdx)/(e^x +1))=?

$$\underset{\mathrm{0}} {\overset{\infty} {\int}}\frac{{xdx}}{{e}^{{x}} +\mathrm{1}}=? \\ $$

Question Number 21398 Answers: 1 Comments: 0

∫_( 0) ^∞ (1/(1+e^x )) dx =

$$\underset{\:\mathrm{0}} {\overset{\infty} {\int}}\:\:\:\frac{\mathrm{1}}{\mathrm{1}+{e}^{{x}} }\:{dx}\:= \\ $$

Question Number 21394 Answers: 2 Comments: 0

Question Number 21392 Answers: 1 Comments: 0

Question Number 21390 Answers: 0 Comments: 0

∫_0 ^1 ((x^7 − 1)/(ln x)) dx

$$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\:\frac{{x}^{\mathrm{7}} \:−\:\mathrm{1}}{\mathrm{ln}\:{x}}\:{dx} \\ $$

Question Number 21388 Answers: 0 Comments: 4

A block of mass m is connected with another block of mass 2m by a light spring. 2m is connected with a hanging mass 3m by an inextensible light string. At the time of release of block 3m, find tension in the string and acceleration of all the masses.

$$\mathrm{A}\:\mathrm{block}\:\mathrm{of}\:\mathrm{mass}\:{m}\:\mathrm{is}\:\mathrm{connected}\:\mathrm{with} \\ $$$$\mathrm{another}\:\mathrm{block}\:\mathrm{of}\:\mathrm{mass}\:\mathrm{2}{m}\:\mathrm{by}\:\mathrm{a}\:\mathrm{light} \\ $$$$\mathrm{spring}.\:\mathrm{2}{m}\:\mathrm{is}\:\mathrm{connected}\:\mathrm{with}\:\mathrm{a}\:\mathrm{hanging} \\ $$$$\mathrm{mass}\:\mathrm{3}{m}\:\mathrm{by}\:\mathrm{an}\:\mathrm{inextensible}\:\mathrm{light}\:\mathrm{string}. \\ $$$$\mathrm{At}\:\mathrm{the}\:\mathrm{time}\:\mathrm{of}\:\mathrm{release}\:\mathrm{of}\:\mathrm{block}\:\mathrm{3}{m},\:\mathrm{find} \\ $$$$\mathrm{tension}\:\mathrm{in}\:\mathrm{the}\:\mathrm{string}\:\mathrm{and}\:\mathrm{acceleration} \\ $$$$\mathrm{of}\:\mathrm{all}\:\mathrm{the}\:\mathrm{masses}. \\ $$

Question Number 21377 Answers: 0 Comments: 0

Balls are dropped from the roof of a tower at a fixed interval of time. At the moment when 9th ball reaches the ground the nth ball is (3/4)th height of the tower. What is the value of n?

$$\mathrm{Balls}\:\mathrm{are}\:\mathrm{dropped}\:\mathrm{from}\:\mathrm{the}\:\mathrm{roof}\:\mathrm{of}\:\mathrm{a} \\ $$$$\mathrm{tower}\:\mathrm{at}\:\mathrm{a}\:\mathrm{fixed}\:\mathrm{interval}\:\mathrm{of}\:\mathrm{time}.\:\mathrm{At}\:\mathrm{the} \\ $$$$\mathrm{moment}\:\mathrm{when}\:\mathrm{9th}\:\mathrm{ball}\:\mathrm{reaches}\:\mathrm{the} \\ $$$$\mathrm{ground}\:\mathrm{the}\:{n}\mathrm{th}\:\mathrm{ball}\:\mathrm{is}\:\left(\mathrm{3}/\mathrm{4}\right)\mathrm{th}\:\mathrm{height} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{tower}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{n}? \\ $$

Question Number 21366 Answers: 3 Comments: 0