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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

Question Number 22348 Answers: 1 Comments: 0

The sum of all the solutions of the equation 1 + 2 cosec x = −((sec^2 (x/2))/2) in the interval [0, 4π] is nπ, where n is equal to

$$\mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{all}\:\mathrm{the}\:\mathrm{solutions}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equation}\:\mathrm{1}\:+\:\mathrm{2}\:\mathrm{cosec}\:{x}\:=\:−\frac{\mathrm{sec}^{\mathrm{2}} \:\frac{{x}}{\mathrm{2}}}{\mathrm{2}}\:\mathrm{in} \\ $$$$\mathrm{the}\:\mathrm{interval}\:\left[\mathrm{0},\:\mathrm{4}\pi\right]\:\mathrm{is}\:{n}\pi,\:\mathrm{where}\:{n}\:\mathrm{is} \\ $$$$\mathrm{equal}\:\mathrm{to} \\ $$

Question Number 22347 Answers: 0 Comments: 0

Total number of solutions of ∣cot x∣ = cot x + (1/(sin x)), x ∈ [0, 3π] is equal to

$$\mathrm{Total}\:\mathrm{number}\:\mathrm{of}\:\mathrm{solutions}\:\mathrm{of}\:\mid\mathrm{cot}\:{x}\mid\:= \\ $$$$\mathrm{cot}\:{x}\:+\:\frac{\mathrm{1}}{\mathrm{sin}\:{x}},\:{x}\:\in\:\left[\mathrm{0},\:\mathrm{3}\pi\right]\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$

Question Number 22345 Answers: 1 Comments: 1

6^(x+2) =2(3^x ), find x?

$$\mathrm{6}^{\mathrm{x}+\mathrm{2}} =\mathrm{2}\left(\mathrm{3}^{\mathrm{x}} \right),\:\mathrm{find}\:\mathrm{x}? \\ $$

Question Number 22332 Answers: 1 Comments: 0

Question Number 22325 Answers: 0 Comments: 0

If separate samples of argon, methane, nitrogen and carbon dioxide, all at the same initial temperature and pressure and expanded adiabatically reversibally to double their original volumes, then which one of these gases will have high temperature as final temperature?

$$\mathrm{If}\:\mathrm{separate}\:\mathrm{samples}\:\mathrm{of}\:\mathrm{argon},\:\mathrm{methane}, \\ $$$$\mathrm{nitrogen}\:\mathrm{and}\:\mathrm{carbon}\:\mathrm{dioxide},\:\mathrm{all}\:\mathrm{at}\:\mathrm{the} \\ $$$$\mathrm{same}\:\mathrm{initial}\:\mathrm{temperature}\:\mathrm{and}\:\mathrm{pressure} \\ $$$$\mathrm{and}\:\mathrm{expanded}\:\mathrm{adiabatically}\:\mathrm{reversibally} \\ $$$$\mathrm{to}\:\mathrm{double}\:\mathrm{their}\:\mathrm{original}\:\mathrm{volumes},\:\mathrm{then} \\ $$$$\mathrm{which}\:\mathrm{one}\:\mathrm{of}\:\mathrm{these}\:\mathrm{gases}\:\mathrm{will}\:\mathrm{have}\:\mathrm{high} \\ $$$$\mathrm{temperature}\:\mathrm{as}\:\mathrm{final}\:\mathrm{temperature}? \\ $$

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