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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.

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

Question Number 14971 Answers: 0 Comments: 0

Question Number 14970 Answers: 0 Comments: 0

Question Number 14965 Answers: 2 Comments: 2

Question Number 14991 Answers: 2 Comments: 0

Question Number 14964 Answers: 0 Comments: 0

proof that ∀ x,y ∈N ∃ a,b,c ∈N ∍ (4/(x^2 +y^2 ))=(1/a) + (1/b) + (1/c)

$$\mathrm{proof}\:\mathrm{that}\: \\ $$$$\forall\:{x},{y}\:\in\mathbb{N}\:\:\exists\:{a},{b},{c}\:\in\mathbb{N}\:\backepsilon\:\frac{\mathrm{4}}{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }=\frac{\mathrm{1}}{{a}}\:+\:\frac{\mathrm{1}}{{b}}\:+\:\frac{\mathrm{1}}{{c}} \\ $$

Question Number 14963 Answers: 1 Comments: 0

A resistor R is connected in series with a parallel combination of two resistors of 24 and 8 ohms . The total power disipated in the circuit is 64 watt when the applied voltage is 24 volt.Find R

$$\mathrm{A}\:\mathrm{resistor}\:\mathrm{R}\:\mathrm{is}\:\mathrm{connected}\:\mathrm{in}\:\mathrm{series}\:\mathrm{with}\:\mathrm{a}\:\mathrm{parallel}\:\mathrm{combination}\:\mathrm{of}\:\mathrm{two}\:\mathrm{resistors} \\ $$$$\mathrm{of}\:\mathrm{24}\:\mathrm{and}\:\mathrm{8}\:\mathrm{ohms}\:.\:\mathrm{The}\:\mathrm{total}\:\mathrm{power}\:\mathrm{disipated}\:\mathrm{in}\:\mathrm{the}\:\mathrm{circuit}\:\mathrm{is}\:\mathrm{64}\:\mathrm{watt}\:\mathrm{when}\:\mathrm{the} \\ $$$$\mathrm{applied}\:\mathrm{voltage}\:\mathrm{is}\:\mathrm{24}\:\mathrm{volt}.\mathrm{Find}\:\mathrm{R} \\ $$

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