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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?

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

Question Number 15115 Answers: 1 Comments: 0

Drops are falling regularly from a water tap at a height of 9 m from the ground. The 4^(th) drop is about to fall from the tap when the 1^(st) hit the ground. Find the distance between 2^(nd) and 3^(rd) drop.

$$\mathrm{Drops}\:\mathrm{are}\:\mathrm{falling}\:\mathrm{regularly}\:\mathrm{from}\:\mathrm{a}\:\mathrm{water} \\ $$$$\mathrm{tap}\:\mathrm{at}\:\mathrm{a}\:\mathrm{height}\:\mathrm{of}\:\mathrm{9}\:\mathrm{m}\:\mathrm{from}\:\mathrm{the}\:\mathrm{ground}. \\ $$$$\mathrm{The}\:\mathrm{4}^{\mathrm{th}} \:\mathrm{drop}\:\mathrm{is}\:\mathrm{about}\:\mathrm{to}\:\mathrm{fall}\:\mathrm{from}\:\mathrm{the} \\ $$$$\mathrm{tap}\:\mathrm{when}\:\mathrm{the}\:\mathrm{1}^{\mathrm{st}} \:\mathrm{hit}\:\mathrm{the}\:\mathrm{ground}.\:\mathrm{Find} \\ $$$$\mathrm{the}\:\mathrm{distance}\:\mathrm{between}\:\mathrm{2}^{\mathrm{nd}} \:\mathrm{and}\:\mathrm{3}^{\mathrm{rd}} \:\mathrm{drop}. \\ $$

Question Number 15102 Answers: 2 Comments: 1

Small steel balls falls from rest through the opening at A, at the steady rate of n balls per second. Find the vertical separation h of two consecutive balls when the lower one has dropped d meters.

$$\mathrm{Small}\:\mathrm{steel}\:\mathrm{balls}\:\mathrm{falls}\:\mathrm{from}\:\mathrm{rest}\:\mathrm{through} \\ $$$$\mathrm{the}\:\mathrm{opening}\:\mathrm{at}\:{A},\:\mathrm{at}\:\mathrm{the}\:\mathrm{steady}\:\mathrm{rate}\:\mathrm{of} \\ $$$${n}\:\mathrm{balls}\:\mathrm{per}\:\mathrm{second}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{vertical} \\ $$$$\mathrm{separation}\:{h}\:\mathrm{of}\:\mathrm{two}\:\mathrm{consecutive}\:\mathrm{balls} \\ $$$$\mathrm{when}\:\mathrm{the}\:\mathrm{lower}\:\mathrm{one}\:\mathrm{has}\:\mathrm{dropped}\:{d} \\ $$$$\mathrm{meters}. \\ $$

Question Number 15100 Answers: 1 Comments: 2

A car A is travelling with a speed of 72 km/h on a straight horizontal road. It is followed by another car B which is moving with a velocity of 36 km/h. When the distance between them is 25 km, the car A is given a deceleration of 2 ms^(−2) . After how much time will B catch up with A?

$$\mathrm{A}\:\mathrm{car}\:{A}\:\mathrm{is}\:\mathrm{travelling}\:\mathrm{with}\:\mathrm{a}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{72} \\ $$$$\mathrm{km}/\mathrm{h}\:\mathrm{on}\:\mathrm{a}\:\mathrm{straight}\:\mathrm{horizontal}\:\mathrm{road}.\:\mathrm{It} \\ $$$$\mathrm{is}\:\mathrm{followed}\:\mathrm{by}\:\mathrm{another}\:\mathrm{car}\:{B}\:\mathrm{which}\:\mathrm{is} \\ $$$$\mathrm{moving}\:\mathrm{with}\:\mathrm{a}\:\mathrm{velocity}\:\mathrm{of}\:\mathrm{36}\:\mathrm{km}/\mathrm{h}. \\ $$$$\mathrm{When}\:\mathrm{the}\:\mathrm{distance}\:\mathrm{between}\:\mathrm{them}\:\mathrm{is}\:\mathrm{25} \\ $$$$\mathrm{km},\:\mathrm{the}\:\mathrm{car}\:{A}\:\mathrm{is}\:\mathrm{given}\:\mathrm{a}\:\mathrm{deceleration}\:\mathrm{of} \\ $$$$\mathrm{2}\:\mathrm{ms}^{−\mathrm{2}} .\:\mathrm{After}\:\mathrm{how}\:\mathrm{much}\:\mathrm{time}\:\mathrm{will}\:{B} \\ $$$$\mathrm{catch}\:\mathrm{up}\:\mathrm{with}\:{A}? \\ $$

Question Number 15098 Answers: 2 Comments: 0

∫ sin^8 (x) dx

$$\int\:\mathrm{sin}^{\mathrm{8}} \left(\mathrm{x}\right)\:\mathrm{dx} \\ $$

Question Number 15097 Answers: 2 Comments: 2

If log_4 log_(1/2) log_3 (x) > 0 then x belongs to (1, a), then the value of a^2 is?

$$\mathrm{If}\:\mathrm{log}_{\mathrm{4}} \:\mathrm{log}_{\frac{\mathrm{1}}{\mathrm{2}}} \:\mathrm{log}_{\mathrm{3}} \:\left({x}\right)\:>\:\mathrm{0}\:\mathrm{then}\:{x}\:\mathrm{belongs} \\ $$$$\mathrm{to}\:\left(\mathrm{1},\:{a}\right),\:\mathrm{then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{a}^{\mathrm{2}} \:\mathrm{is}? \\ $$

Question Number 15094 Answers: 1 Comments: 0

Number of integers in the range of y = ((7^x − 7^(−x) )/(7^x + 7^(−x) )) are?

$$\mathrm{Number}\:\mathrm{of}\:\mathrm{integers}\:\mathrm{in}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of} \\ $$$${y}\:=\:\frac{\mathrm{7}^{{x}} \:−\:\mathrm{7}^{−{x}} }{\mathrm{7}^{{x}} \:+\:\mathrm{7}^{−{x}} }\:\mathrm{are}? \\ $$

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