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Question Number 23269 Answers: 1 Comments: 0

Prove that 3k+2 is not perfect square for all k∈{0,1,2,3,...}.

$$\mathbb{P}\mathrm{rove}\:\mathrm{that} \\ $$$$\:\mathrm{3k}+\mathrm{2}\:\mathrm{is}\:\mathrm{not}\:\mathrm{perfect}\:\mathrm{square}\:\mathrm{for} \\ $$$$\mathrm{all}\:\mathrm{k}\in\left\{\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3},...\right\}. \\ $$

Question Number 23262 Answers: 0 Comments: 2

Question Number 23251 Answers: 0 Comments: 1

Question Number 23250 Answers: 0 Comments: 0

If (1 + x)^n = C_0 + C_1 x + C_2 x^2 + C_3 x^3 + ... + C_n x^n , then prove that ΣΣ_(0≤i<j≤n) ((i/(^n C_i )) + (j/(^n C_j ))) = (n^2 /2)(Σ_(r=0) ^n (1/(^n C_r ))).

$${If}\:\left(\mathrm{1}\:+\:{x}\right)^{{n}} \:=\:{C}_{\mathrm{0}} \:+\:{C}_{\mathrm{1}} {x}\:+\:{C}_{\mathrm{2}} {x}^{\mathrm{2}} \:+\:{C}_{\mathrm{3}} {x}^{\mathrm{3}} \\ $$$$+\:...\:+\:{C}_{{n}} {x}^{{n}} ,\:{then}\:{prove}\:{that} \\ $$$$\underset{\mathrm{0}\leqslant{i}<{j}\leqslant{n}} {\Sigma\Sigma}\left(\frac{{i}}{\:^{{n}} {C}_{{i}} }\:+\:\frac{{j}}{\:^{{n}} {C}_{{j}} }\right)\:=\:\frac{{n}^{\mathrm{2}} }{\mathrm{2}}\left(\underset{{r}=\mathrm{0}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{\:^{{n}} {C}_{{r}} }\right). \\ $$

Question Number 23243 Answers: 0 Comments: 2

2^(n−1) sin a×sin 2a×sin 3a×......×sin (n−1)a=n why?

$$\mathrm{2}^{\mathrm{n}−\mathrm{1}} \mathrm{sin}\:\mathrm{a}×\mathrm{sin}\:\mathrm{2a}×\mathrm{sin}\:\mathrm{3a}×......×\mathrm{sin}\:\left(\mathrm{n}−\mathrm{1}\right)\mathrm{a}=\mathrm{n}\:\mathrm{why}? \\ $$

Question Number 23237 Answers: 0 Comments: 6

Question Number 23231 Answers: 1 Comments: 0

The number of solution(s) of the equation x^3 + x^2 + 4x + 2sinx = 0 in [0, 2π], is/are

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{solution}\left(\mathrm{s}\right)\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$${x}^{\mathrm{3}} \:+\:{x}^{\mathrm{2}} \:+\:\mathrm{4}{x}\:+\:\mathrm{2sin}{x}\:=\:\mathrm{0}\:\mathrm{in}\:\left[\mathrm{0},\:\mathrm{2}\pi\right],\:\mathrm{is}/\mathrm{are} \\ $$

Question Number 23226 Answers: 1 Comments: 4

Question Number 23224 Answers: 0 Comments: 4

Consider the system shown in the figure. Initially the system was in rest. (i) Find the acceleration of block if man climbs the rod with acceleration a (w.r.t. rod) (ii) If the man climb to the top of the rod then find the distance moved by the block.

$$\mathrm{Consider}\:\mathrm{the}\:\mathrm{system}\:\mathrm{shown}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{figure}.\:\mathrm{Initially}\:\mathrm{the}\:\mathrm{system}\:\mathrm{was}\:\mathrm{in}\:\mathrm{rest}. \\ $$$$\left(\mathrm{i}\right)\:\mathrm{Find}\:\mathrm{the}\:\mathrm{acceleration}\:\mathrm{of}\:\mathrm{block}\:\mathrm{if}\:\mathrm{man} \\ $$$$\mathrm{climbs}\:\mathrm{the}\:\mathrm{rod}\:\mathrm{with}\:\mathrm{acceleration}\:{a}\:\left(\mathrm{w}.\mathrm{r}.\mathrm{t}.\right. \\ $$$$\left.\mathrm{rod}\right) \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{If}\:\mathrm{the}\:\mathrm{man}\:\mathrm{climb}\:\mathrm{to}\:\mathrm{the}\:\mathrm{top}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{rod}\:\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{distance}\:\mathrm{moved}\:\mathrm{by}\:\mathrm{the} \\ $$$$\mathrm{block}. \\ $$

Question Number 23223 Answers: 1 Comments: 0

Question Number 23253 Answers: 1 Comments: 8

Question Number 23190 Answers: 1 Comments: 1

lim_(x→∞) (((x−2)/(3x+10)))^(5x)

$$\underset{\boldsymbol{\mathrm{x}}\rightarrow\infty} {\boldsymbol{\mathrm{lim}}}\left(\frac{\boldsymbol{\mathrm{x}}−\mathrm{2}}{\mathrm{3}\boldsymbol{\mathrm{x}}+\mathrm{10}}\right)^{\mathrm{5}\boldsymbol{\mathrm{x}}} \\ $$

Question Number 23208 Answers: 1 Comments: 0

Is it possible to find how many real roots exist in the equation x^4 + ∣x∣ = 3 without find all the value of x?

$$\mathrm{Is}\:\mathrm{it}\:\mathrm{possible}\:\mathrm{to}\:\mathrm{find}\:\mathrm{how}\:\mathrm{many}\:\mathrm{real}\:\mathrm{roots}\: \\ $$$$\mathrm{exist}\:\mathrm{in}\:\mathrm{the}\:\mathrm{equation} \\ $$$${x}^{\mathrm{4}} \:+\:\mid{x}\mid\:=\:\mathrm{3} \\ $$$$\mathrm{without}\:\mathrm{find}\:\mathrm{all}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\:{x}? \\ $$

Question Number 23205 Answers: 0 Comments: 0

thank u.....

$${thank}\:{u}..... \\ $$

Question Number 23187 Answers: 0 Comments: 0

plz anyone answer the question 23181...plz plz plz...

$${plz}\:\:\:{anyone}\:{answer}\:{the}\:{question}\: \\ $$$$\mathrm{23181}...{plz}\:{plz}\:{plz}... \\ $$

Question Number 23181 Answers: 0 Comments: 2

show that the curve with parametric equcations x=t^2 −3t+5, y=t^3 +t^2 −10t+9 intersect at the point (3,1).

$${show}\:{that}\:{the}\:{curve}\:{with}\:{parametric}\: \\ $$$${equcations}\:\:{x}={t}^{\mathrm{2}} \:−\mathrm{3}{t}+\mathrm{5}, \\ $$$${y}={t}^{\mathrm{3}} \:+{t}^{\mathrm{2}} \:−\mathrm{10}{t}+\mathrm{9}\:{intersect}\:{at}\:{the}\: \\ $$$${point}\:\left(\mathrm{3},\mathrm{1}\right). \\ $$

Question Number 23179 Answers: 2 Comments: 1

Question Number 23212 Answers: 1 Comments: 1

Question Number 23170 Answers: 0 Comments: 2

Question Number 23215 Answers: 0 Comments: 0

Assertion: The element with electronic configuration [Xe]^(54) 4f^1 5d^1 6s^2 is a d- block element. Reason: The last electron enters the d- orbital.

$$\boldsymbol{\mathrm{Assertion}}:\:\mathrm{The}\:\mathrm{element}\:\mathrm{with}\:\mathrm{electronic} \\ $$$$\mathrm{configuration}\:\left[\mathrm{Xe}\right]^{\mathrm{54}} \:\mathrm{4}{f}^{\mathrm{1}} \:\mathrm{5}{d}^{\mathrm{1}} \:\mathrm{6}{s}^{\mathrm{2}} \:\mathrm{is}\:\mathrm{a}\:{d}- \\ $$$$\mathrm{block}\:\mathrm{element}. \\ $$$$\boldsymbol{\mathrm{Reason}}:\:\mathrm{The}\:\mathrm{last}\:\mathrm{electron}\:\mathrm{enters}\:\mathrm{the}\:{d}- \\ $$$$\mathrm{orbital}. \\ $$

Question Number 23146 Answers: 0 Comments: 5

Square planar complex is formed by hybridisation of which atomic orbitals? (1) s, p_x , p_y , p_z (2) s, p_x , p_y , d_z^2 (3) s, p_x , p_y , d_(x^2 −y^2 ) (4) s, p_x , p_y , d_z^3

$$\mathrm{Square}\:\mathrm{planar}\:\mathrm{complex}\:\mathrm{is}\:\mathrm{formed}\:\mathrm{by} \\ $$$$\mathrm{hybridisation}\:\mathrm{of}\:\mathrm{which}\:\mathrm{atomic}\:\mathrm{orbitals}? \\ $$$$\left(\mathrm{1}\right)\:{s},\:{p}_{{x}} ,\:{p}_{{y}} ,\:{p}_{{z}} \\ $$$$\left(\mathrm{2}\right)\:{s},\:{p}_{{x}} ,\:{p}_{{y}} ,\:{d}_{{z}^{\mathrm{2}} } \\ $$$$\left(\mathrm{3}\right)\:{s},\:{p}_{{x}} ,\:{p}_{{y}} ,\:{d}_{{x}^{\mathrm{2}} −{y}^{\mathrm{2}} } \\ $$$$\left(\mathrm{4}\right)\:{s},\:{p}_{{x}} ,\:{p}_{{y}} ,\:{d}_{{z}^{\mathrm{3}} } \\ $$

Question Number 23135 Answers: 1 Comments: 0

The standard enthalpy of formation of gaseous H_2 O at 298 K is −241.82 kJ mol^(−1) . Estimate its value of 100°C given the following values of the molar heat capacities at constant pressure: H_2 O(g) : 35.58 JK^(−1) mol^(−1) , H_2 (g) : 28.84 J mol^(−1) K^(−1) and O_2 (g) : 29.37 J mol^(−1) K^(−1) . Assume heat capacity to be independent of temperature.

$$\mathrm{The}\:\mathrm{standard}\:\mathrm{enthalpy}\:\mathrm{of}\:\mathrm{formation}\:\mathrm{of} \\ $$$$\mathrm{gaseous}\:\mathrm{H}_{\mathrm{2}} \mathrm{O}\:\mathrm{at}\:\mathrm{298}\:\mathrm{K}\:\mathrm{is}\:−\mathrm{241}.\mathrm{82}\:\mathrm{kJ} \\ $$$$\mathrm{mol}^{−\mathrm{1}} .\:\mathrm{Estimate}\:\mathrm{its}\:\mathrm{value}\:\mathrm{of}\:\mathrm{100}°\mathrm{C}\:\mathrm{given} \\ $$$$\mathrm{the}\:\mathrm{following}\:\mathrm{values}\:\mathrm{of}\:\mathrm{the}\:\mathrm{molar}\:\mathrm{heat} \\ $$$$\mathrm{capacities}\:\mathrm{at}\:\mathrm{constant}\:\mathrm{pressure}: \\ $$$$\mathrm{H}_{\mathrm{2}} \mathrm{O}\left(\mathrm{g}\right)\::\:\mathrm{35}.\mathrm{58}\:\mathrm{JK}^{−\mathrm{1}} \:\mathrm{mol}^{−\mathrm{1}} ,\:\mathrm{H}_{\mathrm{2}} \left(\mathrm{g}\right)\:: \\ $$$$\mathrm{28}.\mathrm{84}\:\mathrm{J}\:\mathrm{mol}^{−\mathrm{1}} \:\mathrm{K}^{−\mathrm{1}} \:\mathrm{and}\:\mathrm{O}_{\mathrm{2}} \left(\mathrm{g}\right)\::\:\mathrm{29}.\mathrm{37}\:\mathrm{J} \\ $$$$\mathrm{mol}^{−\mathrm{1}} \:\mathrm{K}^{−\mathrm{1}} .\:\mathrm{Assume}\:\mathrm{heat}\:\mathrm{capacity}\:\mathrm{to}\:\mathrm{be} \\ $$$$\mathrm{independent}\:\mathrm{of}\:\mathrm{temperature}. \\ $$

Question Number 23138 Answers: 1 Comments: 1

Question Number 23133 Answers: 1 Comments: 0

Find the minimum surface area of a solid circular cylinder , if its volume is 16π cm^3 (leave your answer in terms of π)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{surface}\:\mathrm{area}\:\mathrm{of}\:\mathrm{a}\:\mathrm{solid}\:\mathrm{circular}\:\mathrm{cylinder}\:,\:\:\mathrm{if}\:\mathrm{its}\:\mathrm{volume}\:\mathrm{is} \\ $$$$\mathrm{16}\pi\:\mathrm{cm}^{\mathrm{3}} \:\:\:\left(\mathrm{leave}\:\mathrm{your}\:\mathrm{answer}\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:\pi\right) \\ $$

Question Number 23131 Answers: 0 Comments: 7

Question Number 23130 Answers: 0 Comments: 1

A baloon filled with helium rises against gravity increasing its potential energy. The speed of the baloon also increases as it rises. How do you reconcile this with the law of conservation of mechanical energy? You can neglect viscous drag of air and assume that density of air is constant.

$$\mathrm{A}\:\mathrm{baloon}\:\mathrm{filled}\:\mathrm{with}\:\mathrm{helium}\:\mathrm{rises}\:\mathrm{against} \\ $$$$\mathrm{gravity}\:\mathrm{increasing}\:\mathrm{its}\:\mathrm{potential}\:\mathrm{energy}. \\ $$$$\mathrm{The}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{the}\:\mathrm{baloon}\:\mathrm{also}\:\mathrm{increases} \\ $$$$\mathrm{as}\:\mathrm{it}\:\mathrm{rises}.\:\mathrm{How}\:\mathrm{do}\:\mathrm{you}\:\mathrm{reconcile}\:\mathrm{this} \\ $$$$\mathrm{with}\:\mathrm{the}\:\mathrm{law}\:\mathrm{of}\:\mathrm{conservation}\:\mathrm{of} \\ $$$$\mathrm{mechanical}\:\mathrm{energy}?\:\mathrm{You}\:\mathrm{can}\:\mathrm{neglect} \\ $$$$\mathrm{viscous}\:\mathrm{drag}\:\mathrm{of}\:\mathrm{air}\:\mathrm{and}\:\mathrm{assume}\:\mathrm{that} \\ $$$$\mathrm{density}\:\mathrm{of}\:\mathrm{air}\:\mathrm{is}\:\mathrm{constant}. \\ $$

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