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Question Number 15262 Answers: 0 Comments: 7

Solve : x^2 − 6x + [x] + 7 = 0.

$$\mathrm{Solve}\::\:{x}^{\mathrm{2}} \:−\:\mathrm{6}{x}\:+\:\left[{x}\right]\:+\:\mathrm{7}\:=\:\mathrm{0}. \\ $$

Question Number 15267 Answers: 2 Comments: 6

In an acute angled ΔABC, the minimum value of tan A tan B tan C is?

$$\mathrm{In}\:\mathrm{an}\:\mathrm{acute}\:\mathrm{angled}\:\Delta{ABC},\:\mathrm{the} \\ $$$$\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:\mathrm{tan}\:{A}\:\mathrm{tan}\:{B}\:\mathrm{tan}\:{C}\:\mathrm{is}? \\ $$

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

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