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Question Number 25034    Answers: 2   Comments: 0

f(x)=((sin x+sec x)/(1+xtan x)) find f′(x)

$$\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{sin}\:\mathrm{x}+\mathrm{sec}\:\mathrm{x}}{\mathrm{1}+\mathrm{xtan}\:\mathrm{x}} \\ $$$$ \\ $$$$\mathrm{find}\:\mathrm{f}'\left(\mathrm{x}\right) \\ $$

Question Number 25026    Answers: 2   Comments: 0

lim_(x→1) (((x^(1/3) −1)/(x^(1/4) −1))) Evaluate this

$$\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\left(\frac{{x}^{\mathrm{1}/\mathrm{3}} −\mathrm{1}}{{x}^{\mathrm{1}/\mathrm{4}} −\mathrm{1}}\right) \\ $$$${Evaluate}\:{this} \\ $$$$ \\ $$

Question Number 25025    Answers: 2   Comments: 0

If a^4 + b^4 + c^4 + d^4 = 16, prove that: a^5 + b^5 + c^5 + d^5 ≤ 32 for a, b, c, d ∈ R

$$\mathrm{If}\:\:\:\:\mathrm{a}^{\mathrm{4}} \:+\:\mathrm{b}^{\mathrm{4}} \:+\:\mathrm{c}^{\mathrm{4}} \:+\:\mathrm{d}^{\mathrm{4}} \:=\:\mathrm{16},\:\:\mathrm{prove}\:\mathrm{that}:\:\:\mathrm{a}^{\mathrm{5}} \:+\:\mathrm{b}^{\mathrm{5}} \:+\:\mathrm{c}^{\mathrm{5}} \:+\:\mathrm{d}^{\mathrm{5}} \:\leqslant\:\mathrm{32} \\ $$$$\mathrm{for}\:\:\mathrm{a},\:\mathrm{b},\:\mathrm{c},\:\mathrm{d}\:\in\:\mathbb{R} \\ $$

Question Number 25023    Answers: 1   Comments: 0

Consider the function f(x) which satisfying the functional equation 2f(x) + f(1 − x) = x^2 + 1, ∀ x ∈ R and g(x) = 3f(x) + 1. The range of φ(x) = g(x) + (1/(g(x) + 1)) is

$$\mathrm{Consider}\:\mathrm{the}\:\mathrm{function}\:{f}\left({x}\right)\:\mathrm{which} \\ $$$$\mathrm{satisfying}\:\mathrm{the}\:\mathrm{functional}\:\mathrm{equation} \\ $$$$\mathrm{2}{f}\left({x}\right)\:+\:{f}\left(\mathrm{1}\:−\:{x}\right)\:=\:{x}^{\mathrm{2}} \:+\:\mathrm{1},\:\forall\:{x}\:\in\:{R} \\ $$$$\mathrm{and}\:{g}\left({x}\right)\:=\:\mathrm{3}{f}\left({x}\right)\:+\:\mathrm{1}.\:\mathrm{The}\:\mathrm{range}\:\mathrm{of} \\ $$$$\phi\left({x}\right)\:=\:{g}\left({x}\right)\:+\:\frac{\mathrm{1}}{{g}\left({x}\right)\:+\:\mathrm{1}}\:\mathrm{is} \\ $$

Question Number 25013    Answers: 0   Comments: 3

With reference to figure of a cube of edge a and mass m, state whether the following are true or false. (O is the centre of the cube.) (1) The moment of inertia of cube about z-axis is, I_z = I_x + I_y (2) The moment of inertia of cube about z′ is, I_(z′) = I_z + ((ma^2 )/2) (3) The moment of inertia of cube about z′′ is, I_(z′) = I_z + ((ma^2 )/2) (4) I_x = I_y

$$\mathrm{With}\:\mathrm{reference}\:\mathrm{to}\:\mathrm{figure}\:\mathrm{of}\:\mathrm{a}\:\mathrm{cube}\:\mathrm{of} \\ $$$$\mathrm{edge}\:{a}\:\mathrm{and}\:\mathrm{mass}\:{m},\:\mathrm{state}\:\mathrm{whether}\:\mathrm{the} \\ $$$$\mathrm{following}\:\mathrm{are}\:\mathrm{true}\:\mathrm{or}\:\mathrm{false}.\:\left(\mathrm{O}\:\mathrm{is}\:\mathrm{the}\right. \\ $$$$\left.\mathrm{centre}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cube}.\right) \\ $$$$\left(\mathrm{1}\right)\:\mathrm{The}\:\mathrm{moment}\:\mathrm{of}\:\mathrm{inertia}\:\mathrm{of}\:\mathrm{cube} \\ $$$$\mathrm{about}\:{z}-\mathrm{axis}\:\mathrm{is},\:{I}_{{z}} \:=\:{I}_{{x}} \:+\:{I}_{{y}} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{The}\:\mathrm{moment}\:\mathrm{of}\:\mathrm{inertia}\:\mathrm{of}\:\mathrm{cube} \\ $$$$\mathrm{about}\:{z}'\:\mathrm{is},\:{I}_{{z}'} \:=\:{I}_{{z}} \:+\:\frac{{ma}^{\mathrm{2}} }{\mathrm{2}} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{The}\:\mathrm{moment}\:\mathrm{of}\:\mathrm{inertia}\:\mathrm{of}\:\mathrm{cube} \\ $$$$\mathrm{about}\:{z}''\:\mathrm{is},\:{I}_{{z}'} \:=\:{I}_{{z}} \:+\:\frac{{ma}^{\mathrm{2}} }{\mathrm{2}} \\ $$$$\left(\mathrm{4}\right)\:{I}_{{x}} \:=\:{I}_{{y}} \\ $$

Question Number 25002    Answers: 1   Comments: 1

Question Number 25001    Answers: 0   Comments: 5

If x, y > 0, then the minimum value of 2x^2 + (2/x) − 2x + 2y^2 + (2/y) − 2y + 2 is equal to

$$\mathrm{If}\:{x},\:{y}\:>\:\mathrm{0},\:\mathrm{then}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mathrm{2}{x}^{\mathrm{2}} \:+\:\frac{\mathrm{2}}{{x}}\:−\:\mathrm{2}{x}\:+\:\mathrm{2}{y}^{\mathrm{2}} \:+\:\frac{\mathrm{2}}{{y}}\:−\:\mathrm{2}{y}\:+\:\mathrm{2}\:\mathrm{is} \\ $$$$\mathrm{equal}\:\mathrm{to} \\ $$

Question Number 25000    Answers: 1   Comments: 0

find x,y from the equation: (1/2)x−yi+(1/(1+i))=((√(1+ω^8 ))+(√(1+ω^(10) )))^4

$${find}\:{x},{y}\:{from}\:{the}\:{equation}: \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}{x}−{yi}+\frac{\mathrm{1}}{\mathrm{1}+{i}}=\left(\sqrt{\mathrm{1}+\omega^{\mathrm{8}} }+\sqrt{\mathrm{1}+\omega^{\mathrm{10}} }\right)^{\mathrm{4}} \\ $$$$ \\ $$

Question Number 24997    Answers: 0   Comments: 0

Question Number 24998    Answers: 0   Comments: 2

A fair coin is tossed 100 times. The probability of getting tails an odd number of times is

$$\mathrm{A}\:\mathrm{fair}\:\mathrm{coin}\:\mathrm{is}\:\mathrm{tossed}\:\mathrm{100}\:\mathrm{times}.\:\mathrm{The} \\ $$$$\mathrm{probability}\:\mathrm{of}\:\mathrm{getting}\:\mathrm{tails}\:\mathrm{an}\:\mathrm{odd} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{times}\:\mathrm{is} \\ $$

Question Number 24985    Answers: 0   Comments: 0

Question Number 24981    Answers: 4   Comments: 1

Question Number 24974    Answers: 1   Comments: 0

If (x^(23) /x^m ) = x^(15) and (x^4 )^n =x^(20) , then mn=

$$\mathrm{If}\:\:\frac{{x}^{\mathrm{23}} }{{x}^{{m}} }\:=\:{x}^{\mathrm{15}} \:\mathrm{and}\:\left({x}^{\mathrm{4}} \right)^{{n}} ={x}^{\mathrm{20}} ,\:\mathrm{then}\:{mn}= \\ $$

Question Number 24973    Answers: 1   Comments: 2

If a sin^2 x+b cos^2 x=c, b sin^2 y+a cos^2 y=d and a tan x= b tan y then (a^2 /b^2 ) =? (in terms of a,b,c,d)

$$\mathrm{If}\:{a}\:\mathrm{sin}^{\mathrm{2}} {x}+{b}\:\mathrm{cos}^{\mathrm{2}} {x}={c},\:{b}\:\mathrm{sin}^{\mathrm{2}} {y}+{a}\:\mathrm{cos}^{\mathrm{2}} {y}={d} \\ $$$$\mathrm{and}\:\:{a}\:\mathrm{tan}\:{x}=\:{b}\:\mathrm{tan}\:{y}\:\mathrm{then}\:\frac{{a}^{\mathrm{2}} }{{b}^{\mathrm{2}} }\:=? \\ $$$$\left(\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:{a},{b},{c},{d}\right) \\ $$

Question Number 24972    Answers: 1   Comments: 0

If α,β and γ are conected by the relation 2tan^2 α tan^2 β tan^2 γ+tan^2 α tan^2 β + tan^2 β tan^2 γ+tan^2 γ tan^2 α=1 then which of these are correct(multi correct) (A)sin^2 α+sin^2 β+ sin^2 γ=1 (B)cos^2 α+cos^2 β+cos^2 γ=2 (C)cos2α+ cos2β+ cos2γ=1 (D)cos(α+β) cos(α−β)= cos^2 γ

$$\mathrm{If}\:\alpha,\beta\:\mathrm{and}\:\gamma\:\mathrm{are}\:\mathrm{conected}\:\mathrm{by}\:\mathrm{the}\:\mathrm{relation} \\ $$$$\mathrm{2tan}^{\mathrm{2}} \alpha\:\mathrm{tan}^{\mathrm{2}} \beta\:\mathrm{tan}^{\mathrm{2}} \gamma+\mathrm{tan}^{\mathrm{2}} \alpha\:\mathrm{tan}^{\mathrm{2}} \beta\:+ \\ $$$$\:\:\:\mathrm{tan}^{\mathrm{2}} \beta\:\mathrm{tan}^{\mathrm{2}} \gamma+\mathrm{tan}^{\mathrm{2}} \gamma\:\mathrm{tan}^{\mathrm{2}} \alpha=\mathrm{1}\:\mathrm{then} \\ $$$$\mathrm{which}\:\mathrm{of}\:\mathrm{these}\:\mathrm{are}\:\mathrm{correct}\left(\mathrm{multi}\:\mathrm{correct}\right) \\ $$$$\left(\mathrm{A}\right)\mathrm{sin}^{\mathrm{2}} \alpha+\mathrm{sin}^{\mathrm{2}} \beta+\:\mathrm{sin}^{\mathrm{2}} \gamma=\mathrm{1}\: \\ $$$$\left(\mathrm{B}\right)\mathrm{cos}^{\mathrm{2}} \alpha+\mathrm{cos}^{\mathrm{2}} \beta+\mathrm{cos}^{\mathrm{2}} \gamma=\mathrm{2} \\ $$$$\left(\mathrm{C}\right)\mathrm{cos2}\alpha+\:\mathrm{cos2}\beta+\:\mathrm{cos2}\gamma=\mathrm{1}\: \\ $$$$\left(\mathrm{D}\right)\mathrm{cos}\left(\alpha+\beta\right)\:\mathrm{cos}\left(\alpha−\beta\right)=\:\mathrm{cos}^{\mathrm{2}} \gamma \\ $$

Question Number 24971    Answers: 1   Comments: 0

Show that ((sin x)/(cos 3x))+((sin 3x)/(cos 9x))+((cos 9x)/(cos 27x))=(1/2)(tan 27x−tan x).

$$\mathrm{Show}\:\mathrm{that} \\ $$$$\frac{\mathrm{sin}\:{x}}{\mathrm{cos}\:\mathrm{3}{x}}+\frac{\mathrm{sin}\:\mathrm{3}{x}}{\mathrm{cos}\:\mathrm{9}{x}}+\frac{\mathrm{cos}\:\mathrm{9}{x}}{\mathrm{cos}\:\mathrm{27}{x}}=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{tan}\:\mathrm{27}{x}−\mathrm{tan}\:{x}\right). \\ $$

Question Number 24966    Answers: 0   Comments: 1

find the focus of the hyperbola x^2 −16xy−11y^2 −12x+6y+21=0 ?

$$\mathrm{find}\:\mathrm{the}\:\mathrm{focus}\:\mathrm{of}\:\mathrm{the}\:\mathrm{hyperbola}\: \\ $$$${x}^{\mathrm{2}} −\mathrm{16}{xy}−\mathrm{11}{y}^{\mathrm{2}} −\mathrm{12}{x}+\mathrm{6}{y}+\mathrm{21}=\mathrm{0}\:? \\ $$

Question Number 24961    Answers: 0   Comments: 0

Question Number 24959    Answers: 2   Comments: 0

In a △ABC, ∠B=π/6, ∠C=π/4 and D divides BC internally in the ratio 1 : 3 then , ((sin ∠BAD)/(sin ∠CAD)) equal to__

$$\mathrm{In}\:\mathrm{a}\:\bigtriangleup\mathrm{ABC},\:\angle\mathrm{B}=\pi/\mathrm{6},\:\angle\mathrm{C}=\pi/\mathrm{4}\:\mathrm{and} \\ $$$$\mathrm{D}\:\mathrm{divides}\:\mathrm{BC}\:\mathrm{internally}\:\mathrm{in}\:\mathrm{the}\:\mathrm{ratio}\: \\ $$$$\mathrm{1}\::\:\mathrm{3}\:\mathrm{then}\:,\:\:\frac{\mathrm{sin}\:\angle\mathrm{BAD}}{\mathrm{sin}\:\angle\mathrm{CAD}}\:\:\:\mathrm{equal}\:\mathrm{to\_\_} \\ $$

Question Number 24948    Answers: 1   Comments: 3

Assuming that the moon′s diameter subtends and angle (1/2)° at the eye of an observer, find how far from the eye of a coin of 10 cm diameter must be held so as just to hide moon ?

$$\mathrm{Assuming}\:\mathrm{that}\:\mathrm{the}\:\mathrm{moon}'\mathrm{s}\:\mathrm{diameter}\: \\ $$$$\mathrm{subtends}\:\mathrm{and}\:\mathrm{angle}\:\left(\mathrm{1}/\mathrm{2}\right)°\:\mathrm{at}\:\mathrm{the}\:\mathrm{eye}\: \\ $$$$\mathrm{of}\:\mathrm{an}\:\mathrm{observer},\:\mathrm{find}\:\mathrm{how}\:\mathrm{far}\:\mathrm{from}\:\mathrm{the} \\ $$$$\mathrm{eye}\:\mathrm{of}\:\mathrm{a}\:\mathrm{coin}\:\mathrm{of}\:\mathrm{10}\:\mathrm{cm}\:\mathrm{diameter}\:\mathrm{must}\: \\ $$$$\mathrm{be}\:\mathrm{held}\:\mathrm{so}\:\mathrm{as}\:\mathrm{just}\:\mathrm{to}\:\mathrm{hide}\:\mathrm{moon}\:? \\ $$

Question Number 24944    Answers: 0   Comments: 4

Question Number 24945    Answers: 0   Comments: 4

A particle of mass m moving with a speed v hits elastically another stationary particle of mass 2m on a smooth horizontal circular tube of radius r. The time in which the next collision will take place is equal to

$$\mathrm{A}\:\mathrm{particle}\:\mathrm{of}\:\mathrm{mass}\:{m}\:\mathrm{moving}\:\mathrm{with}\:\mathrm{a} \\ $$$$\mathrm{speed}\:{v}\:\mathrm{hits}\:\mathrm{elastically}\:\mathrm{another} \\ $$$$\mathrm{stationary}\:\mathrm{particle}\:\mathrm{of}\:\mathrm{mass}\:\mathrm{2}{m}\:\mathrm{on}\:\mathrm{a} \\ $$$$\mathrm{smooth}\:\mathrm{horizontal}\:\mathrm{circular}\:\mathrm{tube}\:\mathrm{of} \\ $$$$\mathrm{radius}\:{r}.\:\mathrm{The}\:\mathrm{time}\:\mathrm{in}\:\mathrm{which}\:\mathrm{the}\:\mathrm{next} \\ $$$$\mathrm{collision}\:\mathrm{will}\:\mathrm{take}\:\mathrm{place}\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$

Question Number 24938    Answers: 0   Comments: 1

3^y =2 2^x =3 4^(xy+1) ×9^(xy) = 24^(xy+1) =

$$\mathrm{3}^{{y}} =\mathrm{2} \\ $$$$\mathrm{2}^{{x}} =\mathrm{3} \\ $$$$\mathrm{4}^{{xy}+\mathrm{1}} ×\mathrm{9}^{{xy}} = \\ $$$$\mathrm{24}^{{xy}+\mathrm{1}} = \\ $$

Question Number 24937    Answers: 0   Comments: 1

Question Number 24933    Answers: 0   Comments: 2

if 6sin^4 θ+3cos^4 θ=2 then find the value of (7cosec^6 θ+8sec^6 θ)^(1/3)

$$\mathrm{if}\:\mathrm{6sin}^{\mathrm{4}} \theta+\mathrm{3cos}^{\mathrm{4}} \theta=\mathrm{2}\:\mathrm{then}\:\mathrm{find}\:\mathrm{the} \\ $$$$\mathrm{value}\:\mathrm{of}\:\left(\mathrm{7cosec}^{\mathrm{6}} \theta+\mathrm{8sec}^{\mathrm{6}} \theta\right)^{\mathrm{1}/\mathrm{3}} \\ $$

Question Number 24934    Answers: 0   Comments: 3

A uniform circular disc of mass 1.5 kg and radius 0.5 m is initially at rest on a horizontal frictionless surface. Three forces of equal magnitude F = 0.5 N are applied simultaneously along the three sides of an equilateral triangle xyz with its vertices on the perimeter of the disc. One second after applying the forces, the angular speed of the disc in rad/s is :

$$\mathrm{A}\:\mathrm{uniform}\:\mathrm{circular}\:\mathrm{disc}\:\mathrm{of}\:\mathrm{mass}\:\mathrm{1}.\mathrm{5}\:\mathrm{kg} \\ $$$$\mathrm{and}\:\mathrm{radius}\:\mathrm{0}.\mathrm{5}\:\mathrm{m}\:\mathrm{is}\:\mathrm{initially}\:\mathrm{at}\:\mathrm{rest}\:\mathrm{on}\:\mathrm{a} \\ $$$$\mathrm{horizontal}\:\mathrm{frictionless}\:\mathrm{surface}.\:\mathrm{Three} \\ $$$$\mathrm{forces}\:\mathrm{of}\:\mathrm{equal}\:\mathrm{magnitude}\:{F}\:=\:\mathrm{0}.\mathrm{5}\:\mathrm{N} \\ $$$$\mathrm{are}\:\mathrm{applied}\:\mathrm{simultaneously}\:\mathrm{along}\:\mathrm{the} \\ $$$$\mathrm{three}\:\mathrm{sides}\:\mathrm{of}\:\mathrm{an}\:\mathrm{equilateral}\:\mathrm{triangle} \\ $$$${xyz}\:\mathrm{with}\:\mathrm{its}\:\mathrm{vertices}\:\mathrm{on}\:\mathrm{the}\:\mathrm{perimeter} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{disc}.\:\mathrm{One}\:\mathrm{second}\:\mathrm{after}\:\mathrm{applying} \\ $$$$\mathrm{the}\:\mathrm{forces},\:\mathrm{the}\:\mathrm{angular}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{the}\:\mathrm{disc} \\ $$$$\mathrm{in}\:\mathrm{rad}/\mathrm{s}\:\mathrm{is}\:: \\ $$

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