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Question Number 22475 Answers: 0 Comments: 1

Question Number 22474 Answers: 0 Comments: 0

Let R = (5(√5) + 11)^(2n+1) and f = R − [R], then prove that Rf = 4^(2n+1) .

$$\mathrm{Let}\:{R}\:=\:\left(\mathrm{5}\sqrt{\mathrm{5}}\:+\:\mathrm{11}\right)^{\mathrm{2}{n}+\mathrm{1}} \:\mathrm{and}\:{f}\:=\:{R}\:−\:\left[{R}\right], \\ $$$$\mathrm{then}\:\mathrm{prove}\:\mathrm{that}\:{Rf}\:=\:\mathrm{4}^{\mathrm{2}{n}+\mathrm{1}} . \\ $$

Question Number 22473 Answers: 0 Comments: 1

Question Number 22472 Answers: 0 Comments: 0

If x^x ∙y^y ∙z^z = x^y ∙y^z ∙z^x = x^z ∙y^x ∙z^y such that x, y and z are positive integers greater than 1, then which of the following cannot be true for any of the possible value of x, y and z? (1) xyz = 27 (2) xyz = 1728 (3) x + y + z = 32 (4) x + y + z = 12

$$\mathrm{If}\:{x}^{{x}} \centerdot{y}^{{y}} \centerdot{z}^{{z}} \:=\:{x}^{{y}} \centerdot{y}^{{z}} \centerdot{z}^{{x}} \:=\:{x}^{{z}} \centerdot{y}^{{x}} \centerdot{z}^{{y}} \:\mathrm{such} \\ $$$$\mathrm{that}\:{x},\:{y}\:\mathrm{and}\:{z}\:\mathrm{are}\:\mathrm{positive}\:\mathrm{integers} \\ $$$$\mathrm{greater}\:\mathrm{than}\:\mathrm{1},\:\mathrm{then}\:\mathrm{which}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{following}\:\mathrm{cannot}\:\mathrm{be}\:\mathrm{true}\:\mathrm{for}\:\mathrm{any}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{possible}\:\mathrm{value}\:\mathrm{of}\:{x},\:{y}\:\mathrm{and}\:{z}? \\ $$$$\left(\mathrm{1}\right)\:{xyz}\:=\:\mathrm{27} \\ $$$$\left(\mathrm{2}\right)\:{xyz}\:=\:\mathrm{1728} \\ $$$$\left(\mathrm{3}\right)\:{x}\:+\:{y}\:+\:{z}\:=\:\mathrm{32} \\ $$$$\left(\mathrm{4}\right)\:{x}\:+\:{y}\:+\:{z}\:=\:\mathrm{12} \\ $$

Question Number 22470 Answers: 0 Comments: 0

If f(x)= determinant ((1,x,(x+1)),((2x),(x(x−1)),((x+1)x)),((3x(x−1)),(2(x−1)(x−2)),((x+1) x(x−1)))), then f(100) is equal to

$$\mathrm{If}\:{f}\left({x}\right)= \\ $$$$\begin{vmatrix}{\mathrm{1}}&{{x}}&{{x}+\mathrm{1}}\\{\mathrm{2}{x}}&{{x}\left({x}−\mathrm{1}\right)}&{\left({x}+\mathrm{1}\right){x}}\\{\mathrm{3}{x}\left({x}−\mathrm{1}\right)}&{\mathrm{2}\left({x}−\mathrm{1}\right)\left({x}−\mathrm{2}\right)}&{\left({x}+\mathrm{1}\right)\:{x}\left({x}−\mathrm{1}\right)}\end{vmatrix}, \\ $$$$\mathrm{then}\:{f}\left(\mathrm{100}\right)\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$

Question Number 22468 Answers: 0 Comments: 0

If a_r is the coefficient of x^r in the expansion (1 + x + x^2 )^n , then a_1 − 2a_2 + 3a_3 − ....... 2na_(2n) =

$$\mathrm{If}\:{a}_{{r}} \:\mathrm{is}\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:{x}^{{r}} \:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{expansion}\:\left(\mathrm{1}\:+\:{x}\:+\:{x}^{\mathrm{2}} \right)^{{n}} ,\:\mathrm{then} \\ $$$${a}_{\mathrm{1}} \:−\:\mathrm{2}{a}_{\mathrm{2}} \:+\:\mathrm{3}{a}_{\mathrm{3}} \:−\:.......\:\mathrm{2}{na}_{\mathrm{2}{n}} \:= \\ $$

Question Number 22463 Answers: 1 Comments: 0

Solve for real x: (1/([x])) + (1/([2x])) = (x) + (1/3), where [x] is the greatest integer less than or equal to x and (x) = x − [x], [e.g. [3.4] = 3 and (3.4) = 0.4].

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:{x}: \\ $$$$\frac{\mathrm{1}}{\left[{x}\right]}\:+\:\frac{\mathrm{1}}{\left[\mathrm{2}{x}\right]}\:=\:\left({x}\right)\:+\:\frac{\mathrm{1}}{\mathrm{3}}, \\ $$$$\mathrm{where}\:\left[{x}\right]\:\mathrm{is}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{integer}\:\mathrm{less} \\ $$$$\mathrm{than}\:\mathrm{or}\:\mathrm{equal}\:\mathrm{to}\:{x}\:\mathrm{and}\:\left({x}\right)\:=\:{x}\:−\:\left[{x}\right], \\ $$$$\left[\mathrm{e}.\mathrm{g}.\:\left[\mathrm{3}.\mathrm{4}\right]\:=\:\mathrm{3}\:\mathrm{and}\:\left(\mathrm{3}.\mathrm{4}\right)\:=\:\mathrm{0}.\mathrm{4}\right]. \\ $$

Question Number 22457 Answers: 0 Comments: 1

Prove the inequality cos (sin x) > sin (cos x) .

$${Prove}\:{the}\:{inequality} \\ $$$$\:\:\mathrm{cos}\:\left(\mathrm{sin}\:{x}\right)\:>\:\mathrm{sin}\:\left(\mathrm{cos}\:{x}\right)\:. \\ $$

Question Number 22456 Answers: 1 Comments: 0

A long plank begins to move at t = 0 and accelerates along a straight track with a speed given by v = 2t^2 for 0 ≤ t ≤ 2. After 2 s, the plank continues to move at the constant speed acquired. A small block initially at rest on the plank begins to slip at t = 1 s and stops sliding at t = 3 s. Find the coefficient of static and kinetic friction between the block and the plank.

$$\mathrm{A}\:\mathrm{long}\:\mathrm{plank}\:\mathrm{begins}\:\mathrm{to}\:\mathrm{move}\:\mathrm{at}\:{t}\:=\:\mathrm{0} \\ $$$$\mathrm{and}\:\mathrm{accelerates}\:\mathrm{along}\:\mathrm{a}\:\mathrm{straight}\:\mathrm{track} \\ $$$$\mathrm{with}\:\mathrm{a}\:\mathrm{speed}\:\mathrm{given}\:\mathrm{by}\:{v}\:=\:\mathrm{2}{t}^{\mathrm{2}} \:\mathrm{for}\:\mathrm{0}\:\leqslant\:{t} \\ $$$$\leqslant\:\mathrm{2}.\:\mathrm{After}\:\mathrm{2}\:\mathrm{s},\:\mathrm{the}\:\mathrm{plank}\:\mathrm{continues}\:\mathrm{to} \\ $$$$\mathrm{move}\:\mathrm{at}\:\mathrm{the}\:\mathrm{constant}\:\mathrm{speed}\:\mathrm{acquired}. \\ $$$$\mathrm{A}\:\mathrm{small}\:\mathrm{block}\:\mathrm{initially}\:\mathrm{at}\:\mathrm{rest}\:\mathrm{on}\:\mathrm{the} \\ $$$$\mathrm{plank}\:\mathrm{begins}\:\mathrm{to}\:\mathrm{slip}\:\mathrm{at}\:{t}\:=\:\mathrm{1}\:\mathrm{s}\:\mathrm{and}\:\mathrm{stops} \\ $$$$\mathrm{sliding}\:\mathrm{at}\:{t}\:=\:\mathrm{3}\:\mathrm{s}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of} \\ $$$$\mathrm{static}\:\mathrm{and}\:\mathrm{kinetic}\:\mathrm{friction}\:\mathrm{between}\:\mathrm{the} \\ $$$$\mathrm{block}\:\mathrm{and}\:\mathrm{the}\:\mathrm{plank}. \\ $$

Question Number 29184 Answers: 1 Comments: 0

{ (((√(x^2 −4xy))+(√(y^2 +2xy+9))=10)),((x−y=7)) :} How many real roots of the equtions system?

$$\begin{cases}{\sqrt{\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\mathrm{4}\boldsymbol{\mathrm{xy}}}+\sqrt{\boldsymbol{\mathrm{y}}^{\mathrm{2}} +\mathrm{2}\boldsymbol{\mathrm{xy}}+\mathrm{9}}=\mathrm{10}}\\{\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{y}}=\mathrm{7}}\end{cases} \\ $$$$\boldsymbol{\mathrm{How}}\:\boldsymbol{\mathrm{many}}\:\:\boldsymbol{\mathrm{real}}\:\boldsymbol{\mathrm{roots}}\:\boldsymbol{\mathrm{of}}\: \\ $$$$\boldsymbol{\mathrm{the}}\:\:\boldsymbol{\mathrm{equtions}}\:\boldsymbol{\mathrm{system}}? \\ $$

Question Number 22447 Answers: 1 Comments: 0

The standard heats of formation of at 298 K for CCl_4 (g), H_2 O(g), CO_2 (g) and HCl(g) are −25.5, −57.8, −94.1 and −22.1 kcal mol^(−1) respectively. Calculate Δ_r H^⊝ for the reaction CCl_4 (g) + 2H_2 O(g) → CO_2 (g) + 4HCl(g)

Question Number 22446 Answers: 0 Comments: 0

Select the species which has the smallest radius stating appropriate reason. K^+ , Sr^(2+) , Ar

$$\mathrm{Select}\:\mathrm{the}\:\mathrm{species}\:\mathrm{which}\:\mathrm{has}\:\mathrm{the}\:\mathrm{smallest} \\ $$$$\mathrm{radius}\:\mathrm{stating}\:\mathrm{appropriate}\:\mathrm{reason}. \\ $$$$\mathrm{K}^{+} ,\:\mathrm{Sr}^{\mathrm{2}+} ,\:\mathrm{Ar} \\ $$

Question Number 22437 Answers: 0 Comments: 11

A cubical block is held stationary against a rough wall by applying force ′F′ then incorrect statement among the following is (1) frictional force, f = Mg (2) f = N, N is normal reaction (3) F does not apply any torque (4) N does not apply any torque

$$\mathrm{A}\:\mathrm{cubical}\:\mathrm{block}\:\mathrm{is}\:\mathrm{held}\:\mathrm{stationary} \\ $$$$\mathrm{against}\:\mathrm{a}\:\mathrm{rough}\:\mathrm{wall}\:\mathrm{by}\:\mathrm{applying}\:\mathrm{force} \\ $$$$'\mathrm{F}'\:\mathrm{then}\:\boldsymbol{{incorrect}}\:\mathrm{statement}\:\mathrm{among} \\ $$$$\mathrm{the}\:\mathrm{following}\:\mathrm{is} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{frictional}\:\mathrm{force},\:\mathrm{f}\:=\:\mathrm{Mg} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{f}\:=\:\mathrm{N},\:\mathrm{N}\:\mathrm{is}\:\mathrm{normal}\:\mathrm{reaction} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{F}\:\mathrm{does}\:\mathrm{not}\:\mathrm{apply}\:\mathrm{any}\:\mathrm{torque} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{N}\:\mathrm{does}\:\mathrm{not}\:\mathrm{apply}\:\mathrm{any}\:\mathrm{torque} \\ $$

Question Number 22435 Answers: 0 Comments: 0

C_0 ^(2n) C_n −C_1 ^(2n−2) C_n +C_2 ^(2n−4) C_n .... equals to

$${C}_{\mathrm{0}} \:^{\mathrm{2}{n}} {C}_{{n}} −{C}_{\mathrm{1}} \:^{\mathrm{2}{n}−\mathrm{2}} {C}_{{n}} +{C}_{\mathrm{2}} \:^{\mathrm{2}{n}−\mathrm{4}} {C}_{{n}} \:.... \\ $$$$\mathrm{equals}\:\mathrm{to} \\ $$

Question Number 22434 Answers: 1 Comments: 0

Im(2z+1)/(iz−1)=2

$$\boldsymbol{\mathrm{I}}\mathrm{m}\left(\mathrm{2z}+\mathrm{1}\right)/\left(\mathrm{iz}−\mathrm{1}\right)=\mathrm{2} \\ $$

Question Number 22432 Answers: 0 Comments: 0

(((5 3)),((3 2)) )A + (((2 5)),((5 1)) ) = (((4 7)),((6 2)) ) Find ∣4A^(−1) ∣

$$\begin{pmatrix}{\mathrm{5}\:\:\:\mathrm{3}}\\{\mathrm{3}\:\:\:\mathrm{2}}\end{pmatrix}{A}\:+\:\begin{pmatrix}{\mathrm{2}\:\:\:\mathrm{5}}\\{\mathrm{5}\:\:\:\mathrm{1}}\end{pmatrix}\:=\:\begin{pmatrix}{\mathrm{4}\:\:\:\:\mathrm{7}}\\{\mathrm{6}\:\:\:\:\mathrm{2}}\end{pmatrix} \\ $$$$\mathrm{Find}\:\mid\mathrm{4}{A}^{−\mathrm{1}} \mid \\ $$

Question Number 22423 Answers: 0 Comments: 0

Prove that no three consecutive binomial coefficient can be in G.P. or H.P.