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Question Number 15292 Answers: 3 Comments: 0

Light version of Q#13724 Expansion of 100! has 24, 0′s at the end. Find the first non-zero digit from right. 100!=.....d000...00 What is the value of d?

$$\mathrm{Light}\:\mathrm{version}\:\mathrm{of}\:\mathcal{Q}#\mathrm{13724} \\ $$$$\mathcal{E}\mathrm{xpansion}\:\mathrm{of}\:\mathrm{100}!\:\mathrm{has}\:\mathrm{24},\:\mathrm{0}'\mathrm{s}\:\:\mathrm{at}\:\mathrm{the}\:\mathrm{end}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{first}\:\mathrm{non}-\mathrm{zero}\:\mathrm{digit}\:\mathrm{from}\:\mathrm{right}. \\ $$$$\mathrm{100}!=.....\mathrm{d000}...\mathrm{00} \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{d}? \\ $$

Question Number 15234 Answers: 0 Comments: 2

A question related to Q.15184 Find the maximum of f(x)=(ln x)^(1/x)

$$\mathrm{A}\:\mathrm{question}\:\mathrm{related}\:\mathrm{to}\:\mathrm{Q}.\mathrm{15184} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{of}\:\mathrm{f}\left(\mathrm{x}\right)=\left(\mathrm{ln}\:\mathrm{x}\right)^{\frac{\mathrm{1}}{\mathrm{x}}} \\ $$

Question Number 15233 Answers: 1 Comments: 0

Find the general solution of the following homogeneous system of equation x_1 + x_2 − 2x_3 = 0 3x_1 − x_2 − 6x_3 = 0 −2x_1 + 3x_(2 ) + 4x_3 = 0

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{general}\:\mathrm{solution}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{homogeneous}\:\mathrm{system}\:\mathrm{of} \\ $$$$\mathrm{equation}\:\: \\ $$$$\mathrm{x}_{\mathrm{1}} \:+\:\mathrm{x}_{\mathrm{2}} \:−\:\mathrm{2x}_{\mathrm{3}} \:=\:\mathrm{0} \\ $$$$\mathrm{3x}_{\mathrm{1}} \:−\:\mathrm{x}_{\mathrm{2}} \:−\:\mathrm{6x}_{\mathrm{3}} \:=\:\mathrm{0} \\ $$$$−\mathrm{2x}_{\mathrm{1}} \:+\:\mathrm{3x}_{\mathrm{2}\:} +\:\mathrm{4x}_{\mathrm{3}} \:=\:\mathrm{0} \\ $$

Question Number 15232 Answers: 1 Comments: 0

For what values of a does the following system have a non trivial solution ax_1 + 3x_2 − 2x_3 = 0 −x_1 + 4x_2 + ax_3 = 0 5x_1 − 6x_2 − 7x_3 = 0

$$\mathrm{For}\:\mathrm{what}\:\mathrm{values}\:\mathrm{of}\:\:\mathrm{a}\:\:\mathrm{does}\:\mathrm{the}\:\mathrm{following}\:\mathrm{system}\:\mathrm{have}\:\mathrm{a}\:\mathrm{non}\:\mathrm{trivial}\: \\ $$$$\mathrm{solution}\: \\ $$$$\mathrm{ax}_{\mathrm{1}} \:+\:\mathrm{3x}_{\mathrm{2}} \:−\:\mathrm{2x}_{\mathrm{3}} \:=\:\mathrm{0} \\ $$$$−\mathrm{x}_{\mathrm{1}} \:+\:\mathrm{4x}_{\mathrm{2}} \:+\:\mathrm{ax}_{\mathrm{3}} \:=\:\mathrm{0} \\ $$$$\mathrm{5x}_{\mathrm{1}} \:−\:\mathrm{6x}_{\mathrm{2}} \:−\:\mathrm{7x}_{\mathrm{3}} \:=\:\mathrm{0} \\ $$

Question Number 15235 Answers: 1 Comments: 0

∫(log (√(x )) )^2 dx=?

$$\int\left(\mathrm{log}\:\sqrt{\mathrm{x}\:}\:\right)^{\mathrm{2}} \mathrm{dx}=? \\ $$

Question Number 15217 Answers: 1 Comments: 0

Question Number 15216 Answers: 0 Comments: 0

[((z−2)/(z+2))]=6

$$\left[\frac{\mathrm{z}−\mathrm{2}}{\mathrm{z}+\mathrm{2}}\right]=\mathrm{6} \\ $$

Question Number 15194 Answers: 0 Comments: 2

The equation determinant (((x−a),(x−b),(x−c)),((x−b),(x−c),(x−a)),((x−c),(x−a),(x−b)))=0, where a, b, c are different, is satisfied by

$$\mathrm{The}\:\mathrm{equation}\:\begin{vmatrix}{{x}−{a}}&{{x}−{b}}&{{x}−{c}}\\{{x}−{b}}&{{x}−{c}}&{{x}−{a}}\\{{x}−{c}}&{{x}−{a}}&{{x}−{b}}\end{vmatrix}=\mathrm{0}, \\ $$$$\mathrm{where}\:{a},\:{b},\:{c}\:\mathrm{are}\:\mathrm{different},\:\mathrm{is}\:\mathrm{satisfied}\:\mathrm{by} \\ $$

Question Number 15193 Answers: 0 Comments: 0

The value of the determinant △= determinant ((( 2a_1 b_1 ),(a_1 b_2 +a_2 b_1 ),(a_1 b_3 +a_3 b_1 )),((a_1 b_2 +a_2 b_1 ),( 2a_2 b_2 ),(a_2 b_3 +a_3 b_2 )),((a_1 b_3 +a_3 b_1 ),(a_3 b_2 +a_2 b_3 ),( 2a_3 b_3 )))is

$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{determinant} \\ $$$$\bigtriangleup=\begin{vmatrix}{\:\:\:\mathrm{2}{a}_{\mathrm{1}} {b}_{\mathrm{1}} }&{{a}_{\mathrm{1}} {b}_{\mathrm{2}} +{a}_{\mathrm{2}} {b}_{\mathrm{1}} }&{{a}_{\mathrm{1}} {b}_{\mathrm{3}} +{a}_{\mathrm{3}} {b}_{\mathrm{1}} }\\{{a}_{\mathrm{1}} {b}_{\mathrm{2}} +{a}_{\mathrm{2}} {b}_{\mathrm{1}} }&{\:\:\:\:\mathrm{2}{a}_{\mathrm{2}} {b}_{\mathrm{2}} }&{{a}_{\mathrm{2}} {b}_{\mathrm{3}} +{a}_{\mathrm{3}} {b}_{\mathrm{2}} }\\{{a}_{\mathrm{1}} {b}_{\mathrm{3}} +{a}_{\mathrm{3}} {b}_{\mathrm{1}} }&{{a}_{\mathrm{3}} {b}_{\mathrm{2}} +{a}_{\mathrm{2}} {b}_{\mathrm{3}} }&{\:\:\:\:\mathrm{2}{a}_{\mathrm{3}} {b}_{\mathrm{3}} }\end{vmatrix}\mathrm{is} \\ $$

Question Number 15192 Answers: 0 Comments: 0

Let D_r = determinant ((2^(r−1) ,(2 ∙ 3^(r−1) ),(4 ∙ 5^(r−1) )),(( α),( β),( γ)),((2^n −1),(3^n −1),( 5^n −1))). Then the value of Σ_(r=1) ^n D_r is

$$\mathrm{Let}\:{D}_{{r}} =\begin{vmatrix}{\mathrm{2}^{{r}−\mathrm{1}} }&{\mathrm{2}\:\centerdot\:\mathrm{3}^{{r}−\mathrm{1}} }&{\mathrm{4}\:\centerdot\:\mathrm{5}^{{r}−\mathrm{1}} }\\{\:\:\:\alpha}&{\:\:\:\beta}&{\:\:\:\:\:\gamma}\\{\mathrm{2}^{{n}} −\mathrm{1}}&{\mathrm{3}^{{n}} −\mathrm{1}}&{\:\:\mathrm{5}^{{n}} −\mathrm{1}}\end{vmatrix}. \\ $$$$\mathrm{Then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}{D}_{{r}} \:\:\mathrm{is} \\ $$

Question Number 15191 Answers: 0 Comments: 1

If m is a positive integer and △_r = determinant ((( 2r−1),(^m C_r ),( 1)),(( m^2 −1),( 2^m ),( m+1)),((sin^2 (m^2 )),(sin^2 (m)),(sin^2 (m+1)))) then the value of Σ_(r=0) ^m △_r is

$$\mathrm{If}\:{m}\:\mathrm{is}\:\mathrm{a}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{and} \\ $$$$\bigtriangleup_{{r}} =\begin{vmatrix}{\:\:\:\mathrm{2}{r}−\mathrm{1}}&{\:^{{m}} {C}_{{r}} }&{\:\:\:\:\:\:\:\:\mathrm{1}}\\{\:\:{m}^{\mathrm{2}} −\mathrm{1}}&{\:\:\:\:\mathrm{2}^{{m}} }&{\:\:\:{m}+\mathrm{1}}\\{\mathrm{sin}^{\mathrm{2}} \left({m}^{\mathrm{2}} \right)}&{\mathrm{sin}^{\mathrm{2}} \left({m}\right)}&{\mathrm{sin}^{\mathrm{2}} \left({m}+\mathrm{1}\right)}\end{vmatrix} \\ $$$$\mathrm{then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\:\:\underset{{r}=\mathrm{0}} {\overset{{m}} {\sum}}\bigtriangleup_{{r}} \:\:\:\mathrm{is} \\ $$

Question Number 15190 Answers: 0 Comments: 0

If I_3 is the identity matrix of order 3, then (I_3 )^(−1) =

$$\mathrm{If}\:\:\:{I}_{\mathrm{3}} \:\mathrm{is}\:\mathrm{the}\:\mathrm{identity}\:\mathrm{matrix}\:\mathrm{of}\:\mathrm{order}\:\mathrm{3},\: \\ $$$$\mathrm{then}\:\:\left({I}_{\mathrm{3}} \right)^{−\mathrm{1}} = \\ $$

Question Number 15189 Answers: 0 Comments: 0

If A= [(a_(ij) ) ]_(m×n) is a matrix and B is a non−singular square submatrix of order r , then

$$\mathrm{If}\:{A}=\begin{bmatrix}{{a}_{{ij}} }\end{bmatrix}_{{m}×{n}} \:\mathrm{is}\:\mathrm{a}\:\mathrm{matrix}\:\mathrm{and}\:{B}\:\mathrm{is}\:\mathrm{a} \\ $$$$\mathrm{non}−\mathrm{singular}\:\mathrm{square}\:\mathrm{submatrix}\:\mathrm{of} \\ $$$$\mathrm{order}\:\:{r}\:,\:\mathrm{then}\: \\ $$

Question Number 15186 Answers: 0 Comments: 3

lim_(x→∞) (((x + 3)/(x −1)))^x

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left(\frac{{x}\:+\:\mathrm{3}}{{x}\:−\mathrm{1}}\right)^{{x}} \\ $$

Question Number 15184 Answers: 0 Comments: 4

If y = ln x and y = a^x have a solution then find the range of a. (1) (0, 1) (2) ((1/e), e) (3) (1, e) (4) (0, 1]

$$\mathrm{If}\:{y}\:=\:\mathrm{ln}\:{x}\:\mathrm{and}\:{y}\:=\:{a}^{{x}} \:\mathrm{have}\:\mathrm{a}\:\mathrm{solution} \\ $$$$\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of}\:{a}. \\ $$$$\left(\mathrm{1}\right)\:\left(\mathrm{0},\:\mathrm{1}\right) \\ $$$$\left(\mathrm{2}\right)\:\left(\frac{\mathrm{1}}{{e}},\:{e}\right) \\ $$$$\left(\mathrm{3}\right)\:\left(\mathrm{1},\:{e}\right) \\ $$$$\left(\mathrm{4}\right)\:\left(\mathrm{0},\:\mathrm{1}\right] \\ $$

Question Number 15181 Answers: 1 Comments: 0

f(x) = (((px + q) . sin 2x)/(ax + b)) lim_(x→0) f(x) = 2 and lim_(x→∞) f(x) = 0 Find a,b,p,q

$${f}\left({x}\right)\:=\:\frac{\left({px}\:+\:{q}\right)\:.\:\mathrm{sin}\:\mathrm{2}{x}}{{ax}\:+\:{b}} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{f}\left({x}\right)\:=\:\mathrm{2}\:\:\:\mathrm{and}\:\:\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{f}\left({x}\right)\:=\:\mathrm{0} \\ $$$$\mathrm{Find}\:{a},{b},{p},{q}\:\: \\ $$

Question Number 15180 Answers: 1 Comments: 0

If ω is a cube root of unity, then a root of the following polynomial is determinant (((x+1),( ω),( ω^2 )),(( ω),(x+ω^2 ),( 1)),(( ω^2 ),( 1),(x+ω)))

$$\mathrm{If}\:\:\:\omega\:\mathrm{is}\:\mathrm{a}\:\mathrm{cube}\:\mathrm{root}\:\mathrm{of}\:\mathrm{unity},\:\mathrm{then}\:\mathrm{a}\:\mathrm{root} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{polynomial}\:\mathrm{is} \\ $$$$\begin{vmatrix}{{x}+\mathrm{1}}&{\:\:\omega}&{\:\:\omega^{\mathrm{2}} }\\{\:\:\omega}&{{x}+\omega^{\mathrm{2}} }&{\:\:\mathrm{1}}\\{\:\:\omega^{\mathrm{2}} }&{\:\:\mathrm{1}}&{{x}+\omega}\end{vmatrix} \\ $$

Question Number 15179 Answers: 1 Comments: 0

If (d^2 y/dx^2 ) + 4y = 0 y(0) = 1 and y((π/6)) = ((√3)/2) + (1/2) How to find y(x) ?

$$\mathrm{If}\:\:\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }\:+\:\mathrm{4}{y}\:=\:\mathrm{0}\: \\ $$$${y}\left(\mathrm{0}\right)\:=\:\mathrm{1}\:\mathrm{and}\:{y}\left(\frac{\pi}{\mathrm{6}}\right)\:=\:\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\:+\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\mathrm{How}\:\mathrm{to}\:\mathrm{find}\:{y}\left({x}\right)\:? \\ $$

Question Number 15175 Answers: 2 Comments: 1

Question Number 15172 Answers: 0 Comments: 0

If z = x + jy, show that the locus arg{((z − 1)/(z − j))} = (π/6) is a circle . Find its centre and radius.

$$\mathrm{If}\:\:\mathrm{z}\:=\:\mathrm{x}\:+\:\mathrm{jy},\:\mathrm{show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{locus}\:\mathrm{arg}\left\{\frac{\mathrm{z}\:−\:\mathrm{1}}{\mathrm{z}\:−\:\mathrm{j}}\right\}\:=\:\frac{\pi}{\mathrm{6}}\:\mathrm{is}\:\mathrm{a}\:\mathrm{circle}\:.\:\mathrm{Find}\:\mathrm{its}\:\mathrm{centre} \\ $$$$\mathrm{and}\:\mathrm{radius}.\: \\ $$

Question Number 15182 Answers: 0 Comments: 0

If a piece of iron gains 10% of its mass due to partial rusting into Fe_2 O_3 the percentage of total iron that has rusted is? [Answer is 23.3]

$$\mathrm{If}\:\mathrm{a}\:\mathrm{piece}\:\mathrm{of}\:\mathrm{iron}\:\mathrm{gains}\:\mathrm{10\%}\:\mathrm{of}\:\mathrm{its}\:\mathrm{mass} \\ $$$$\mathrm{due}\:\mathrm{to}\:\mathrm{partial}\:\mathrm{rusting}\:\mathrm{into}\:\mathrm{Fe}_{\mathrm{2}} \mathrm{O}_{\mathrm{3}} \:\mathrm{the} \\ $$$$\mathrm{percentage}\:\mathrm{of}\:\mathrm{total}\:\mathrm{iron}\:\mathrm{that}\:\mathrm{has}\:\mathrm{rusted} \\ $$$$\mathrm{is}?\:\left[\mathrm{Answer}\:\mathrm{is}\:\mathrm{23}.\mathrm{3}\right] \\ $$

Question Number 15170 Answers: 2 Comments: 2

Question Number 15623 Answers: 1 Comments: 0

Question Number 15158 Answers: 0 Comments: 3

In a third order determinant, each element of the first column consists of sum of 2 terms, each element of the second column consists of sum of 3 terms and each element of the third column consists of sum of 4 terms. Then it can be decomposed into n determinants, where n has the value Ans= 24

Question Number 15117 Answers: 1 Comments: 0

A Juggler is maintaining 4 balls in motion making each in term to rise to height of 20 m. Which of following position is not possible for the balls, when one ball is just leaving his hand? (1) 5 m (2) All position are possible (3) 15 m (4) 20 m

$$\mathrm{A}\:\mathrm{Juggler}\:\mathrm{is}\:\mathrm{maintaining}\:\mathrm{4}\:\mathrm{balls}\:\mathrm{in} \\ $$$$\mathrm{motion}\:\mathrm{making}\:\mathrm{each}\:\mathrm{in}\:\mathrm{term}\:\mathrm{to}\:\mathrm{rise}\:\mathrm{to} \\ $$$$\mathrm{height}\:\mathrm{of}\:\mathrm{20}\:\mathrm{m}.\:\mathrm{Which}\:\mathrm{of}\:\mathrm{following} \\ $$$$\mathrm{position}\:\mathrm{is}\:\mathrm{not}\:\mathrm{possible}\:\mathrm{for}\:\mathrm{the}\:\mathrm{balls}, \\ $$$$\mathrm{when}\:\mathrm{one}\:\mathrm{ball}\:\mathrm{is}\:\mathrm{just}\:\mathrm{leaving}\:\mathrm{his}\:\mathrm{hand}? \\ $$$$\left(\mathrm{1}\right)\:\mathrm{5}\:\mathrm{m} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{All}\:\mathrm{position}\:\mathrm{are}\:\mathrm{possible} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{15}\:\mathrm{m} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{20}\:\mathrm{m} \\ $$

Question Number 15116 Answers: 2 Comments: 0

A balloon ascends vertically with a constant speed for 5 seconds, when a pebble falls from it reaching the ground in 5 s. The speed of the balloon is?

$$\mathrm{A}\:\mathrm{balloon}\:\mathrm{ascends}\:\mathrm{vertically}\:\mathrm{with}\:\mathrm{a} \\ $$$$\mathrm{constant}\:\mathrm{speed}\:\mathrm{for}\:\mathrm{5}\:\mathrm{seconds},\:\mathrm{when}\:\mathrm{a} \\ $$$$\mathrm{pebble}\:\mathrm{falls}\:\mathrm{from}\:\mathrm{it}\:\mathrm{reaching}\:\mathrm{the}\:\mathrm{ground} \\ $$$$\mathrm{in}\:\mathrm{5}\:\mathrm{s}.\:\mathrm{The}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{the}\:\mathrm{balloon}\:\mathrm{is}? \\ $$

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