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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?

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

Question Number 15093 Answers: 2 Comments: 0

The range of f(x) = (({x}^2 − {x} + 1)/({x}^2 + {x} + 1)); (where {∙} denotes fractional function) is?

$$\mathrm{The}\:\mathrm{range}\:\mathrm{of}\:{f}\left({x}\right)\:=\:\frac{\left\{{x}\right\}^{\mathrm{2}} \:−\:\left\{{x}\right\}\:+\:\mathrm{1}}{\left\{{x}\right\}^{\mathrm{2}} \:+\:\left\{{x}\right\}\:+\:\mathrm{1}}; \\ $$$$\left(\mathrm{where}\:\left\{\centerdot\right\}\:\mathrm{denotes}\:\mathrm{fractional}\:\mathrm{function}\right) \\ $$$$\mathrm{is}? \\ $$

Question Number 15095 Answers: 1 Comments: 0

Solve: (1 − x)(dy/dx) = y(1 + x)

$$\mathrm{Solve}: \\ $$$$\left(\mathrm{1}\:−\:\mathrm{x}\right)\frac{\mathrm{dy}}{\mathrm{dx}}\:=\:\mathrm{y}\left(\mathrm{1}\:+\:\mathrm{x}\right) \\ $$

Question Number 15086 Answers: 1 Comments: 0

The range of f(x) = (√((10^x − 10^4 )/(10^x + 10^2 ))) is?

$$\mathrm{The}\:\mathrm{range}\:\mathrm{of}\:{f}\left({x}\right)\:=\:\sqrt{\frac{\mathrm{10}^{{x}} \:−\:\mathrm{10}^{\mathrm{4}} }{\mathrm{10}^{{x}} \:+\:\mathrm{10}^{\mathrm{2}} }}\:\mathrm{is}? \\ $$

Question Number 15084 Answers: 1 Comments: 0

The domain of f(x) = (1/(√(−x^2 + {x}))); (where {∙} denotes fractional part of x) is?

$$\mathrm{The}\:\mathrm{domain}\:\mathrm{of}\:{f}\left({x}\right)\:=\:\frac{\mathrm{1}}{\sqrt{−{x}^{\mathrm{2}} \:+\:\left\{{x}\right\}}}; \\ $$$$\left(\mathrm{where}\:\left\{\centerdot\right\}\:\mathrm{denotes}\:\mathrm{fractional}\:\mathrm{part}\:\mathrm{of}\:{x}\right) \\ $$$$\mathrm{is}? \\ $$

Question Number 15082 Answers: 1 Comments: 0

The domain of f(x) = (√(x − 2{x})). (where {∙} denotes fractional part of x) is?

$$\mathrm{The}\:\mathrm{domain}\:\mathrm{of}\:{f}\left({x}\right)\:=\:\sqrt{{x}\:−\:\mathrm{2}\left\{{x}\right\}}.\:\left(\mathrm{where}\right. \\ $$$$\left.\left\{\centerdot\right\}\:\mathrm{denotes}\:\mathrm{fractional}\:\mathrm{part}\:\mathrm{of}\:{x}\right)\:\mathrm{is}? \\ $$

Question Number 15052 Answers: 2 Comments: 4

Gravitational potential of a ring of radius R and mass M on the axis at a distance x from the center is given by v(x) = − ((GM)/(√(R^2 + x^2 ))) Nm/kg Using the above expression find the gravitational potential of the disc of mass M and radius R on the axis at a distance x from the center of the disc.

$$\mathrm{Gravitational}\:\mathrm{potential}\:\mathrm{of}\:\mathrm{a}\:\mathrm{ring}\:\mathrm{of} \\ $$$$\mathrm{radius}\:{R}\:\mathrm{and}\:\mathrm{mass}\:{M}\:\mathrm{on}\:\mathrm{the}\:\mathrm{axis}\:\mathrm{at}\:\mathrm{a} \\ $$$$\mathrm{distance}\:{x}\:\mathrm{from}\:\mathrm{the}\:\mathrm{center}\:\mathrm{is}\:\mathrm{given}\:\mathrm{by} \\ $$$${v}\left({x}\right)\:=\:−\:\frac{{GM}}{\sqrt{{R}^{\mathrm{2}} \:+\:{x}^{\mathrm{2}} }}\:\mathrm{Nm}/\mathrm{kg} \\ $$$$\mathrm{Using}\:\mathrm{the}\:\mathrm{above}\:\mathrm{expression}\:\mathrm{find}\:\mathrm{the} \\ $$$$\mathrm{gravitational}\:\mathrm{potential}\:\mathrm{of}\:\mathrm{the}\:\mathrm{disc}\:\mathrm{of} \\ $$$$\mathrm{mass}\:{M}\:\mathrm{and}\:\mathrm{radius}\:{R}\:\mathrm{on}\:\mathrm{the}\:\mathrm{axis}\:\mathrm{at}\:\mathrm{a} \\ $$$$\mathrm{distance}\:{x}\:\mathrm{from}\:\mathrm{the}\:\mathrm{center}\:\mathrm{of}\:\mathrm{the}\:\mathrm{disc}. \\ $$

Question Number 15051 Answers: 1 Comments: 0

Solve simultaneously x + y + z = 6 ............ equation (i) x^3 + y^3 + z^3 = 92 .......... equation (ii) x − y = z ........... equation (iii)

$$\mathrm{Solve}\:\mathrm{simultaneously} \\ $$$$ \\ $$$$\mathrm{x}\:+\:\mathrm{y}\:+\:\mathrm{z}\:=\:\mathrm{6}\:\:\:\:\:\:\:\:\:\:\:\:\:\:............\:\mathrm{equation}\:\left(\mathrm{i}\right) \\ $$$$\mathrm{x}^{\mathrm{3}} \:+\:\mathrm{y}^{\mathrm{3}} \:+\:\mathrm{z}^{\mathrm{3}} \:=\:\mathrm{92}\:\:\:\:\:\:\:\:\:..........\:\mathrm{equation}\:\left(\mathrm{ii}\right) \\ $$$$\mathrm{x}\:−\:\mathrm{y}\:=\:\mathrm{z}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...........\:\mathrm{equation}\:\left(\mathrm{iii}\right) \\ $$

Question Number 15045 Answers: 1 Comments: 0

In a curve the x and y co-ordinate is function of t is given by the equation x = cos t and y = sin t, then find the length of the curve for t = 0 to t = (π/2).

$$\mathrm{In}\:\mathrm{a}\:\mathrm{curve}\:\mathrm{the}\:{x}\:\mathrm{and}\:{y}\:\mathrm{co}-\mathrm{ordinate}\:\mathrm{is} \\ $$$$\mathrm{function}\:\mathrm{of}\:{t}\:\mathrm{is}\:\mathrm{given}\:\mathrm{by}\:\mathrm{the}\:\mathrm{equation} \\ $$$${x}\:=\:\mathrm{cos}\:{t}\:\mathrm{and}\:{y}\:=\:\mathrm{sin}\:{t},\:\mathrm{then}\:\mathrm{find}\:\mathrm{the} \\ $$$$\mathrm{length}\:\mathrm{of}\:\mathrm{the}\:\mathrm{curve}\:\mathrm{for}\:{t}\:=\:\mathrm{0}\:\mathrm{to}\:{t}\:=\:\frac{\pi}{\mathrm{2}}. \\ $$

Question Number 15039 Answers: 2 Comments: 0

The acceleration of an object is given by a(t) = cos(πt) ms^(−2) and its velocity at time t = 0 is (1/(2π)) m/s at origin. Its velocity at t = (3/2) s is? The object′s position at t = (3/2) s is?

$$\mathrm{The}\:\mathrm{acceleration}\:\mathrm{of}\:\mathrm{an}\:\mathrm{object}\:\mathrm{is}\:\mathrm{given} \\ $$$$\mathrm{by}\:{a}\left({t}\right)\:=\:\mathrm{cos}\left(\pi{t}\right)\:\mathrm{ms}^{−\mathrm{2}} \:\mathrm{and}\:\mathrm{its}\:\mathrm{velocity} \\ $$$$\mathrm{at}\:\mathrm{time}\:{t}\:=\:\mathrm{0}\:\mathrm{is}\:\frac{\mathrm{1}}{\mathrm{2}\pi}\:\mathrm{m}/\mathrm{s}\:\mathrm{at}\:\mathrm{origin}.\:\mathrm{Its} \\ $$$$\mathrm{velocity}\:\mathrm{at}\:{t}\:=\:\frac{\mathrm{3}}{\mathrm{2}}\:\mathrm{s}\:\mathrm{is}? \\ $$$$\mathrm{The}\:\mathrm{object}'\mathrm{s}\:\mathrm{position}\:\mathrm{at}\:{t}\:=\:\frac{\mathrm{3}}{\mathrm{2}}\:\mathrm{s}\:\mathrm{is}? \\ $$

Question Number 15036 Answers: 1 Comments: 2

Rotating a curve y = (√x) about the x- axis produces a “head light” as shown below. What is the area of disc at any x?

$$\mathrm{Rotating}\:\mathrm{a}\:\mathrm{curve}\:{y}\:=\:\sqrt{{x}}\:\mathrm{about}\:\mathrm{the}\:{x}- \\ $$$$\mathrm{axis}\:\mathrm{produces}\:\mathrm{a}\:``\mathrm{head}\:\mathrm{light}''\:\mathrm{as}\:\mathrm{shown} \\ $$$$\mathrm{below}. \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{disc}\:\mathrm{at}\:\mathrm{any}\:{x}? \\ $$

Question Number 15019 Answers: 0 Comments: 0

1.00 Mol of a monoatomic gas initially at 3.00 × 10^2 K and occupying 2.00 × 10^(−3 ) m^3 is heated to 3.25 × 10^2 K and the final volume is 4.00 × 10^(−3) m^3 . Assuming ideal behaviour , Calculate the entropy change for the system.

$$\mathrm{1}.\mathrm{00}\:\mathrm{Mol}\:\mathrm{of}\:\mathrm{a}\:\mathrm{monoatomic}\:\mathrm{gas}\:\mathrm{initially}\:\mathrm{at}\:\mathrm{3}.\mathrm{00}\:×\:\mathrm{10}^{\mathrm{2}} \mathrm{K}\:\mathrm{and}\:\mathrm{occupying}\:\: \\ $$$$\mathrm{2}.\mathrm{00}\:×\:\mathrm{10}^{−\mathrm{3}\:} \mathrm{m}^{\mathrm{3}} \:\mathrm{is}\:\mathrm{heated}\:\mathrm{to}\:\mathrm{3}.\mathrm{25}\:×\:\mathrm{10}^{\mathrm{2}} \mathrm{K}\:\mathrm{and}\:\mathrm{the}\:\mathrm{final}\:\mathrm{volume}\:\mathrm{is}\:\mathrm{4}.\mathrm{00}\:×\:\mathrm{10}^{−\mathrm{3}} \mathrm{m}^{\mathrm{3}} .\: \\ $$$$\mathrm{Assuming}\:\mathrm{ideal}\:\mathrm{behaviour}\:,\:\mathrm{Calculate}\:\mathrm{the}\:\mathrm{entropy}\:\mathrm{change}\:\mathrm{for}\:\mathrm{the}\:\mathrm{system}. \\ $$

Question Number 15017 Answers: 0 Comments: 0

Calculate the heat neccessary to raise the temperature of 5.00 mol of butane from 290K to 593K at a constant pressure. where Cp(19.41 + 0.233T)J/mol/K

$$\mathrm{Calculate}\:\mathrm{the}\:\mathrm{heat}\:\mathrm{neccessary}\:\mathrm{to}\:\mathrm{raise}\:\mathrm{the}\:\mathrm{temperature}\:\mathrm{of}\:\mathrm{5}.\mathrm{00}\:\mathrm{mol}\:\mathrm{of}\:\mathrm{butane} \\ $$$$\mathrm{from}\:\mathrm{290K}\:\mathrm{to}\:\mathrm{593K}\:\mathrm{at}\:\mathrm{a}\:\mathrm{constant}\:\mathrm{pressure}.\:\mathrm{where}\:\mathrm{Cp}\left(\mathrm{19}.\mathrm{41}\:+\:\mathrm{0}.\mathrm{233T}\right)\mathrm{J}/\mathrm{mol}/\mathrm{K} \\ $$

Question Number 15006 Answers: 1 Comments: 8

Question Number 14988 Answers: 0 Comments: 2

Solve on Z_4 ax+b=[0]_4 a,b∈Z_4 ax^2 +bx+c=[0]_4 a,b,c∈Z_4

$${Solve}\:{on}\:\mathbb{Z}_{\mathrm{4}} \: \\ $$$${ax}+{b}=\left[\mathrm{0}\right]_{\mathrm{4}} \:\:{a},{b}\in\mathbb{Z}_{\mathrm{4}} \\ $$$${ax}^{\mathrm{2}} +{bx}+{c}=\left[\mathrm{0}\right]_{\mathrm{4}} \:\:{a},{b},{c}\in\mathbb{Z}_{\mathrm{4}} \\ $$

Question Number 14977 Answers: 2 Comments: 2

Find the real roots of the equation cos^4 x + sin^7 x = 1 in the interval [−π, π].

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{real}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\mathrm{cos}^{\mathrm{4}} \:{x}\:+\:\mathrm{sin}^{\mathrm{7}} \:{x}\:=\:\mathrm{1}\:\mathrm{in}\:\mathrm{the}\:\mathrm{interval}\:\left[−\pi,\:\pi\right]. \\ $$

Question Number 14999 Answers: 1 Comments: 0

A 20 kg box is released from the top of an inclined plane that is 5 m long and makes an angle of 20° to the horizontal. A 60N friction force impedes the motion of the box . How long will it take to reach the bottom of the box.

$$\mathrm{A}\:\mathrm{20}\:\mathrm{kg}\:\mathrm{box}\:\mathrm{is}\:\mathrm{released}\:\mathrm{from}\:\mathrm{the}\:\mathrm{top}\:\mathrm{of}\:\mathrm{an}\:\mathrm{inclined}\:\mathrm{plane}\:\mathrm{that}\:\mathrm{is}\:\mathrm{5}\:\mathrm{m}\:\mathrm{long}\:\mathrm{and} \\ $$$$\mathrm{makes}\:\mathrm{an}\:\mathrm{angle}\:\mathrm{of}\:\mathrm{20}°\:\mathrm{to}\:\mathrm{the}\:\mathrm{horizontal}.\:\mathrm{A}\:\mathrm{60N}\:\mathrm{friction}\:\mathrm{force}\:\mathrm{impedes}\:\mathrm{the}\:\mathrm{motion}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{box}\:.\:\mathrm{How}\:\mathrm{long}\:\mathrm{will}\:\mathrm{it}\:\mathrm{take}\:\mathrm{to}\:\mathrm{reach}\:\mathrm{the}\:\mathrm{bottom}\:\mathrm{of}\:\mathrm{the}\:\mathrm{box}. \\ $$

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