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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.

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{no}\:\mathrm{three}\:\mathrm{consecutive} \\ $$$$\mathrm{binomial}\:\mathrm{coefficient}\:\mathrm{can}\:\mathrm{be}\:\mathrm{in}\:\mathrm{G}.\mathrm{P}.\:\mathrm{or}\:\mathrm{H}.\mathrm{P}. \\ $$

Question Number 22407 Answers: 2 Comments: 0

Question Number 22404 Answers: 0 Comments: 16

Mr Tikutara,Do you study at AAKASH?

$${Mr}\:{Tikutara},{Do}\:{you}\: \\ $$$${study}\:{at}\:{AAKASH}? \\ $$

Question Number 22395 Answers: 0 Comments: 2

Question Number 22394 Answers: 0 Comments: 0

Prove that : ((^n C_0 )/n)−((^n C_1 )/(n+1))+((^n C_2 )/(n+2))−...+(−1)^n .((^n C_n )/(2n))=(1/(n.^(2n) C_n ))

$$\mathrm{Prove}\:\mathrm{that}\::\:\frac{\:^{{n}} {C}_{\mathrm{0}} }{{n}}−\frac{\:^{{n}} {C}_{\mathrm{1}} }{{n}+\mathrm{1}}+\frac{\:^{{n}} {C}_{\mathrm{2}} }{{n}+\mathrm{2}}−...+\left(−\mathrm{1}\right)^{{n}} .\frac{\:^{{n}} {C}_{{n}} }{\mathrm{2}{n}}=\frac{\mathrm{1}}{{n}.^{\mathrm{2}{n}} {C}_{{n}} } \\ $$

Question Number 22392 Answers: 0 Comments: 0

Show that the sum of odd coefficients in the expansion of (1 + 2x − 3x^2 )^(1025) is an even integer.

$$\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{odd}\:\mathrm{coefficients} \\ $$$$\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\:\left(\mathrm{1}\:+\:\mathrm{2}{x}\:−\:\mathrm{3}{x}^{\mathrm{2}} \right)^{\mathrm{1025}} \\ $$$$\mathrm{is}\:\mathrm{an}\:\mathrm{even}\:\mathrm{integer}. \\ $$

Question Number 22389 Answers: 0 Comments: 0

Simplify: ^(n−1) C_2 +2^(n−2) C_2 +3^(n−3) C_2 +...+(n−2)^2 C_2

$${Simplify}: \\ $$$$\:^{{n}−\mathrm{1}} {C}_{\mathrm{2}} +\mathrm{2}\:^{{n}−\mathrm{2}} {C}_{\mathrm{2}} +\mathrm{3}\:^{{n}−\mathrm{3}} {C}_{\mathrm{2}} +...+\left({n}−\mathrm{2}\right)\:^{\mathrm{2}} {C}_{\mathrm{2}} \\ $$

Question Number 22388 Answers: 1 Comments: 0

if t_1 ,t_2 are the extremeties of any focal chord of the parabola y^2 =4ax,then t_1 t_(2=)

$${if}\:{t}_{\mathrm{1}} \:,{t}_{\mathrm{2}} \:{are}\:{the}\:{extremeties}\:{of}\:{any}\:{focal}\:{chord}\:{of}\:{the}\:{parabola}\:{y}^{\mathrm{2}} =\mathrm{4}{ax},{then}\:{t}_{\mathrm{1}} {t}_{\mathrm{2}=} \\ $$

Question Number 22387 Answers: 0 Comments: 0

the arguments of n^(th) roots of a complex number differ by

$${the}\:{arguments}\:{of}\:{n}^{{th}} \:{roots}\:{of}\:{a}\:{complex}\:{number}\:{differ}\:{by} \\ $$

Question Number 22386 Answers: 0 Comments: 0

equivalent matrices are obtained by

$${equivalent}\:{matrices}\:{are}\:{obtained}\:{by} \\ $$

Question Number 22384 Answers: 0 Comments: 0

Question Number 22379 Answers: 1 Comments: 0

For each positive integer n, define a_n = 20 + n^2 , and d_n = gcd(a_n , a_(n+1) ). Find the set of all values that are taken by d_n and show by examples that each of these values are attained.

$$\mathrm{For}\:\mathrm{each}\:\mathrm{positive}\:\mathrm{integer}\:{n},\:\mathrm{define}\:{a}_{{n}} \:= \\ $$$$\mathrm{20}\:+\:{n}^{\mathrm{2}} ,\:\mathrm{and}\:{d}_{{n}} \:=\:{gcd}\left({a}_{{n}} ,\:{a}_{{n}+\mathrm{1}} \right).\:\mathrm{Find} \\ $$$$\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{all}\:\mathrm{values}\:\mathrm{that}\:\mathrm{are}\:\mathrm{taken}\:\mathrm{by} \\ $$$${d}_{{n}} \:\mathrm{and}\:\mathrm{show}\:\mathrm{by}\:\mathrm{examples}\:\mathrm{that}\:\mathrm{each}\:\mathrm{of} \\ $$$$\mathrm{these}\:\mathrm{values}\:\mathrm{are}\:\mathrm{attained}. \\ $$

Question Number 22372 Answers: 1 Comments: 4

Question Number 22370 Answers: 1 Comments: 3

A uniform flexible chain of length (3/2) m rests on a fixed smooth sphere of radius R = (2/π) m such that one end A of chain is on the top of the sphere while the other end B is hanging freely. Chain is held stationary by a horizontal thread PA. Calculate the acceleration of chain when the horizontal string PA is burnt. (g = 10 m/s^2 )

$$\mathrm{A}\:\mathrm{uniform}\:\mathrm{flexible}\:\mathrm{chain}\:\mathrm{of}\:\mathrm{length}\:\frac{\mathrm{3}}{\mathrm{2}}\:\mathrm{m} \\ $$$$\mathrm{rests}\:\mathrm{on}\:\mathrm{a}\:\mathrm{fixed}\:\mathrm{smooth}\:\mathrm{sphere}\:\mathrm{of} \\ $$$$\mathrm{radius}\:{R}\:=\:\frac{\mathrm{2}}{\pi}\:\mathrm{m}\:\mathrm{such}\:\mathrm{that}\:\mathrm{one}\:\mathrm{end}\:{A}\:\mathrm{of} \\ $$$$\mathrm{chain}\:\mathrm{is}\:\mathrm{on}\:\mathrm{the}\:\mathrm{top}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sphere}\:\mathrm{while} \\ $$$$\mathrm{the}\:\mathrm{other}\:\mathrm{end}\:{B}\:\mathrm{is}\:\mathrm{hanging}\:\mathrm{freely}.\:\mathrm{Chain} \\ $$$$\mathrm{is}\:\mathrm{held}\:\mathrm{stationary}\:\mathrm{by}\:\mathrm{a}\:\mathrm{horizontal} \\ $$$$\mathrm{thread}\:{PA}.\:\mathrm{Calculate}\:\mathrm{the}\:\mathrm{acceleration} \\ $$$$\mathrm{of}\:\mathrm{chain}\:\mathrm{when}\:\mathrm{the}\:\mathrm{horizontal}\:\mathrm{string}\:{PA} \\ $$$$\mathrm{is}\:\mathrm{burnt}.\:\left({g}\:=\:\mathrm{10}\:\mathrm{m}/\mathrm{s}^{\mathrm{2}} \right) \\ $$

Question Number 22368 Answers: 0 Comments: 0

The first and second ionization potentials of helium atoms are 24.58 eV and 54.4 eV per mole respectively. Calculate the energy in kJ required to produce 1 mole of He^(2+) ions.

$$\mathrm{The}\:\mathrm{first}\:\mathrm{and}\:\mathrm{second}\:\mathrm{ionization} \\ $$$$\mathrm{potentials}\:\mathrm{of}\:\mathrm{helium}\:\mathrm{atoms}\:\mathrm{are}\:\mathrm{24}.\mathrm{58}\:\mathrm{eV} \\ $$$$\mathrm{and}\:\mathrm{54}.\mathrm{4}\:\mathrm{eV}\:\mathrm{per}\:\mathrm{mole}\:\mathrm{respectively}. \\ $$$$\mathrm{Calculate}\:\mathrm{the}\:\mathrm{energy}\:\mathrm{in}\:\mathrm{kJ}\:\mathrm{required}\:\mathrm{to} \\ $$$$\mathrm{produce}\:\mathrm{1}\:\mathrm{mole}\:\mathrm{of}\:\mathrm{He}^{\mathrm{2}+} \:\mathrm{ions}. \\ $$

Question Number 22365 Answers: 1 Comments: 0

Question Number 22361 Answers: 0 Comments: 0

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