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Question Number 21582    Answers: 1   Comments: 0

∫_(π/2) ^(π/4) (3x+7)

$$\int_{\pi/\mathrm{2}} ^{\pi/\mathrm{4}} \left(\mathrm{3}{x}+\mathrm{7}\right) \\ $$

Question Number 21581    Answers: 0   Comments: 0

Question Number 21580    Answers: 1   Comments: 0

Question Number 21578    Answers: 0   Comments: 1

Find the whole part of A? A=(1/(√2))+(1/(√3))+(1/(√4))+......+(1/(√(9999)))+(1/(√(10000))).

$$\boldsymbol{\mathrm{Find}}\:\:\boldsymbol{\mathrm{the}}\:\:\boldsymbol{\mathrm{whole}}\:\:\boldsymbol{\mathrm{part}}\:\:\boldsymbol{\mathrm{of}}\:\:\boldsymbol{\mathrm{A}}? \\ $$$$\boldsymbol{\mathrm{A}}=\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}+\frac{\mathrm{1}}{\sqrt{\mathrm{3}}}+\frac{\mathrm{1}}{\sqrt{\mathrm{4}}}+......+\frac{\mathrm{1}}{\sqrt{\mathrm{9999}}}+\frac{\mathrm{1}}{\sqrt{\mathrm{10000}}}. \\ $$

Question Number 21574    Answers: 1   Comments: 0

The cyclic octagon ABCDEFGH has sides a, a, a, a, b, b, b, b respectively. Find the radius of the circle that circumscribes ABCDEFGH in terms of a and b.

$$\mathrm{The}\:\mathrm{cyclic}\:\mathrm{octagon}\:{ABCDEFGH}\:\mathrm{has} \\ $$$$\mathrm{sides}\:{a},\:{a},\:{a},\:{a},\:{b},\:{b},\:{b},\:{b}\:\mathrm{respectively}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{radius}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circle}\:\mathrm{that} \\ $$$$\mathrm{circumscribes}\:{ABCDEFGH}\:\mathrm{in}\:\mathrm{terms} \\ $$$$\mathrm{of}\:{a}\:\mathrm{and}\:{b}. \\ $$

Question Number 21573    Answers: 0   Comments: 0

Prove that 1 < (1/(1001)) + (1/(1002)) + (1/(1003)) + ... + (1/(3001)) < 1(1/3).

$$\mathrm{Prove}\:\mathrm{that} \\ $$$$\mathrm{1}\:<\:\frac{\mathrm{1}}{\mathrm{1001}}\:+\:\frac{\mathrm{1}}{\mathrm{1002}}\:+\:\frac{\mathrm{1}}{\mathrm{1003}}\:+\:...\:+\:\frac{\mathrm{1}}{\mathrm{3001}}\:<\:\mathrm{1}\frac{\mathrm{1}}{\mathrm{3}}. \\ $$

Question Number 21572    Answers: 0   Comments: 0

Determine the largest 3-digit prime factor of the integer^(2000) C_(1000) .

$$\mathrm{Determine}\:\mathrm{the}\:\mathrm{largest}\:\mathrm{3}-\mathrm{digit}\:\mathrm{prime} \\ $$$$\mathrm{factor}\:\mathrm{of}\:\mathrm{the}\:\mathrm{integer}\:^{\mathrm{2000}} {C}_{\mathrm{1000}} . \\ $$

Question Number 21566    Answers: 0   Comments: 2

Question Number 21564    Answers: 0   Comments: 1

A block of mass M is placed on smooth ground. Its upper surface is smooth and vertical surface is rough with coefficient of friction μ. A block of mass m_1 is placed on its horizontal surface and tied with a massless inextensible string passing over smooth pulley. Its other end is connected to another block of mass m_2 , which touches the vertical surface of block M. Now a horizontal force F starts acting on it. Q1. Which of the following is incorrect about above system? (1) There exists a value of F at which friction force is equal to zero (2) When F = 0, the blocks cannot remain stationary (3) There exists two limiting values of F at which the blocks m_1 and m_2 will remain stationary w.r.t. block of mass M (4) The limiting friction between m_2 and M is independent of F Q2. In the above case, let m_1 − μm_2 be greater than 1. Choose the incorrect value of F for which the blocks m_1 and m_2 remain stationary with respect to M (1) (M + m_1 + m_2 )((m_2 g)/m_1 ) (2) ((m_2 (M + m_1 + m_2 ))/((m_1 − μm_2 )))g (3) (((M + m_1 + m_2 )m_2 g)/((m_1 + μm_2 ))) (4) (M + m_1 + m_2 )(g/μ) Q3. Let vertical part of block M be smooth. Choose the correct alternative (1) There exist two limiting values for system to remain relatively at rest (2) For one unique value of F, the blocks m_1 and m_2 remain stationary with respect to block M (3) The blocks m_1 and m_2 cannot be in equilibrium for any value of F (4) There exists a range of mass M, for which system remains stationary

$$\mathrm{A}\:\mathrm{block}\:\mathrm{of}\:\mathrm{mass}\:{M}\:\mathrm{is}\:\mathrm{placed}\:\mathrm{on}\:\mathrm{smooth} \\ $$$$\mathrm{ground}.\:\mathrm{Its}\:\mathrm{upper}\:\mathrm{surface}\:\mathrm{is}\:\mathrm{smooth}\:\mathrm{and} \\ $$$$\mathrm{vertical}\:\mathrm{surface}\:\mathrm{is}\:\mathrm{rough}\:\mathrm{with}\:\mathrm{coefficient} \\ $$$$\mathrm{of}\:\mathrm{friction}\:\mu.\:\mathrm{A}\:\mathrm{block}\:\mathrm{of}\:\mathrm{mass}\:{m}_{\mathrm{1}} \:\mathrm{is}\:\mathrm{placed} \\ $$$$\mathrm{on}\:\mathrm{its}\:\mathrm{horizontal}\:\mathrm{surface}\:\mathrm{and}\:\mathrm{tied}\:\mathrm{with} \\ $$$$\mathrm{a}\:\mathrm{massless}\:\mathrm{inextensible}\:\mathrm{string}\:\mathrm{passing} \\ $$$$\mathrm{over}\:\mathrm{smooth}\:\mathrm{pulley}.\:\mathrm{Its}\:\mathrm{other}\:\mathrm{end}\:\mathrm{is} \\ $$$$\mathrm{connected}\:\mathrm{to}\:\mathrm{another}\:\mathrm{block}\:\mathrm{of}\:\mathrm{mass}\:{m}_{\mathrm{2}} , \\ $$$$\mathrm{which}\:\mathrm{touches}\:\mathrm{the}\:\mathrm{vertical}\:\mathrm{surface}\:\mathrm{of} \\ $$$$\mathrm{block}\:{M}.\:\mathrm{Now}\:\mathrm{a}\:\mathrm{horizontal}\:\mathrm{force}\:{F} \\ $$$$\mathrm{starts}\:\mathrm{acting}\:\mathrm{on}\:\mathrm{it}. \\ $$$$\boldsymbol{\mathrm{Q}}\mathrm{1}.\:\mathrm{Which}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{is}\:\mathrm{incorrect} \\ $$$$\mathrm{about}\:\mathrm{above}\:\mathrm{system}? \\ $$$$\left(\mathrm{1}\right)\:\mathrm{There}\:\mathrm{exists}\:\mathrm{a}\:\mathrm{value}\:\mathrm{of}\:{F}\:\mathrm{at}\:\mathrm{which} \\ $$$$\mathrm{friction}\:\mathrm{force}\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:\mathrm{zero} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{When}\:{F}\:=\:\mathrm{0},\:\mathrm{the}\:\mathrm{blocks}\:\mathrm{cannot} \\ $$$$\mathrm{remain}\:\mathrm{stationary} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{There}\:\mathrm{exists}\:\mathrm{two}\:\mathrm{limiting}\:\mathrm{values}\:\mathrm{of} \\ $$$${F}\:\mathrm{at}\:\mathrm{which}\:\mathrm{the}\:\mathrm{blocks}\:{m}_{\mathrm{1}} \:\mathrm{and}\:{m}_{\mathrm{2}} \:\mathrm{will} \\ $$$$\mathrm{remain}\:\mathrm{stationary}\:\mathrm{w}.\mathrm{r}.\mathrm{t}.\:\mathrm{block}\:\mathrm{of}\:\mathrm{mass} \\ $$$${M} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{The}\:\mathrm{limiting}\:\mathrm{friction}\:\mathrm{between}\:{m}_{\mathrm{2}} \\ $$$$\mathrm{and}\:{M}\:\mathrm{is}\:\mathrm{independent}\:\mathrm{of}\:{F} \\ $$$$\boldsymbol{\mathrm{Q}}\mathrm{2}.\:\mathrm{In}\:\mathrm{the}\:\mathrm{above}\:\mathrm{case},\:\mathrm{let}\:{m}_{\mathrm{1}} \:−\:\mu{m}_{\mathrm{2}} \:\mathrm{be} \\ $$$$\mathrm{greater}\:\mathrm{than}\:\mathrm{1}.\:\mathrm{Choose}\:\mathrm{the}\:\mathrm{incorrect} \\ $$$$\mathrm{value}\:\mathrm{of}\:{F}\:\mathrm{for}\:\mathrm{which}\:\mathrm{the}\:\mathrm{blocks}\:{m}_{\mathrm{1}} \:\mathrm{and} \\ $$$${m}_{\mathrm{2}} \:\mathrm{remain}\:\mathrm{stationary}\:\mathrm{with}\:\mathrm{respect}\:\mathrm{to} \\ $$$${M} \\ $$$$\left(\mathrm{1}\right)\:\left({M}\:+\:{m}_{\mathrm{1}} \:+\:{m}_{\mathrm{2}} \right)\frac{{m}_{\mathrm{2}} {g}}{{m}_{\mathrm{1}} } \\ $$$$\left(\mathrm{2}\right)\:\frac{{m}_{\mathrm{2}} \left({M}\:+\:{m}_{\mathrm{1}} \:+\:{m}_{\mathrm{2}} \right)}{\left({m}_{\mathrm{1}} \:−\:\mu{m}_{\mathrm{2}} \right)}{g} \\ $$$$\left(\mathrm{3}\right)\:\frac{\left({M}\:+\:{m}_{\mathrm{1}} \:+\:{m}_{\mathrm{2}} \right){m}_{\mathrm{2}} {g}}{\left({m}_{\mathrm{1}} \:+\:\mu{m}_{\mathrm{2}} \right)} \\ $$$$\left(\mathrm{4}\right)\:\left({M}\:+\:{m}_{\mathrm{1}} \:+\:{m}_{\mathrm{2}} \right)\frac{{g}}{\mu} \\ $$$$\boldsymbol{\mathrm{Q}}\mathrm{3}.\:\mathrm{Let}\:\mathrm{vertical}\:\mathrm{part}\:\mathrm{of}\:\mathrm{block}\:{M}\:\mathrm{be} \\ $$$$\mathrm{smooth}.\:\mathrm{Choose}\:\mathrm{the}\:\mathrm{correct}\:\mathrm{alternative} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{There}\:\mathrm{exist}\:\mathrm{two}\:\mathrm{limiting}\:\mathrm{values}\:\mathrm{for} \\ $$$$\mathrm{system}\:\mathrm{to}\:\mathrm{remain}\:\mathrm{relatively}\:\mathrm{at}\:\mathrm{rest} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{For}\:\mathrm{one}\:\mathrm{unique}\:\mathrm{value}\:\mathrm{of}\:{F},\:\mathrm{the}\:\mathrm{blocks} \\ $$$${m}_{\mathrm{1}} \:\mathrm{and}\:{m}_{\mathrm{2}} \:\mathrm{remain}\:\mathrm{stationary}\:\mathrm{with} \\ $$$$\mathrm{respect}\:\mathrm{to}\:\mathrm{block}\:{M} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{The}\:\mathrm{blocks}\:{m}_{\mathrm{1}} \:\mathrm{and}\:{m}_{\mathrm{2}} \:\mathrm{cannot}\:\mathrm{be}\:\mathrm{in} \\ $$$$\mathrm{equilibrium}\:\mathrm{for}\:\mathrm{any}\:\mathrm{value}\:\mathrm{of}\:{F} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{There}\:\mathrm{exists}\:\mathrm{a}\:\mathrm{range}\:\mathrm{of}\:\mathrm{mass}\:{M},\:\mathrm{for} \\ $$$$\mathrm{which}\:\mathrm{system}\:\mathrm{remains}\:\mathrm{stationary} \\ $$

Question Number 21571    Answers: 1   Comments: 0

ABCD is a cyclic quadrilateral; x, y, z are the distances of A from the lines BD, BC, CD respectively. Prove that ((BD)/x) = ((BC)/y) + ((CD)/z).

$${ABCD}\:\mathrm{is}\:\mathrm{a}\:\mathrm{cyclic}\:\mathrm{quadrilateral};\:{x},\:{y},\:{z} \\ $$$$\mathrm{are}\:\mathrm{the}\:\mathrm{distances}\:\mathrm{of}\:{A}\:\mathrm{from}\:\mathrm{the}\:\mathrm{lines} \\ $$$${BD},\:{BC},\:{CD}\:\mathrm{respectively}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\frac{{BD}}{{x}}\:=\:\frac{{BC}}{{y}}\:+\:\frac{{CD}}{{z}}. \\ $$

Question Number 22554    Answers: 1   Comments: 0

find the eqution of normal to the circle 2x^2 +2y^2 −3x+4y−32=0 at(2,3)

$${find}\:{the}\:{eqution}\:{of}\:{normal}\:{to} \\ $$$${the}\:{circle} \\ $$$$\mathrm{2}{x}^{\mathrm{2}} +\mathrm{2}{y}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{4}{y}−\mathrm{32}=\mathrm{0}\:{at}\left(\mathrm{2},\mathrm{3}\right) \\ $$

Question Number 21557    Answers: 0   Comments: 0

A man of mass 85 kg stands on a lift of mass 30 kg. When he pulls on the rope, he exerts a force of 400 N on the floor of the lift. Calculate acceleration of the lift. Given g = 10 m/s^2 .

$$\mathrm{A}\:\mathrm{man}\:\mathrm{of}\:\mathrm{mass}\:\mathrm{85}\:\mathrm{kg}\:\mathrm{stands}\:\mathrm{on}\:\mathrm{a}\:\mathrm{lift}\:\mathrm{of} \\ $$$$\mathrm{mass}\:\mathrm{30}\:\mathrm{kg}.\:\mathrm{When}\:\mathrm{he}\:\mathrm{pulls}\:\mathrm{on}\:\mathrm{the}\:\mathrm{rope}, \\ $$$$\mathrm{he}\:\mathrm{exerts}\:\mathrm{a}\:\mathrm{force}\:\mathrm{of}\:\mathrm{400}\:\mathrm{N}\:\mathrm{on}\:\mathrm{the}\:\mathrm{floor}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{lift}.\:\mathrm{Calculate}\:\mathrm{acceleration}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{lift}.\:\mathrm{Given}\:{g}\:=\:\mathrm{10}\:\mathrm{m}/\mathrm{s}^{\mathrm{2}} . \\ $$

Question Number 21549    Answers: 2   Comments: 0

The value of the expression 3(sin θ−cos θ)^4 +6(sin θ+cos θ)^2 +4(sin^6 θ+cos^6 θ) is

$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{expression}\: \\ $$$$\mathrm{3}\left(\mathrm{sin}\:\theta−\mathrm{cos}\:\theta\right)^{\mathrm{4}} +\mathrm{6}\left(\mathrm{sin}\:\theta+\mathrm{cos}\:\theta\right)^{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\mathrm{4}\left(\mathrm{sin}^{\mathrm{6}} \theta+\mathrm{cos}^{\mathrm{6}} \theta\right)\:\:\mathrm{is} \\ $$

Question Number 21545    Answers: 0   Comments: 0

Question Number 21541    Answers: 0   Comments: 0

Question Number 21535    Answers: 0   Comments: 0

Which forces of attraction are responsible for liquefaction of H_2 ? (a) Coulombic forces (b) Dipole forces (c) Hydrogen bonding (d) Van der Waal′s forces.

$$\mathrm{Which}\:\mathrm{forces}\:\mathrm{of}\:\mathrm{attraction}\:\mathrm{are}\:\mathrm{responsible} \\ $$$$\mathrm{for}\:\mathrm{liquefaction}\:\mathrm{of}\:\mathrm{H}_{\mathrm{2}} ? \\ $$$$\left({a}\right)\:\mathrm{Coulombic}\:\mathrm{forces} \\ $$$$\left({b}\right)\:\mathrm{Dipole}\:\mathrm{forces} \\ $$$$\left({c}\right)\:\mathrm{Hydrogen}\:\mathrm{bonding} \\ $$$$\left({d}\right)\:\mathrm{Van}\:\mathrm{der}\:\mathrm{Waal}'\mathrm{s}\:\mathrm{forces}. \\ $$

Question Number 21531    Answers: 1   Comments: 2

Factorise the equation by factor theorem 12x^ 3 + 4x^ 2−3x−1

$${Factorise}\:{the}\:{equation}\:{by}\:{factor}\:{theorem} \\ $$$$\mathrm{12}\hat {{x}}\mathrm{3}\:+\:\mathrm{4}\hat {{x}}\mathrm{2}−\mathrm{3}{x}−\mathrm{1} \\ $$

Question Number 21526    Answers: 1   Comments: 3

Question Number 21692    Answers: 1   Comments: 0

Let f(x) is a quadratic equation and x^2 − 2x + 3 ≤ f(x) ≤ 2x^2 − 4x + 4 for every x ∈ R If f(5) = 26, then f(7) is equal to ... (A) 38 (D) 74 (B) 50 (E) 92 (C) 56

$$\mathrm{Let}\:{f}\left({x}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{quadratic}\:\mathrm{equation}\:\mathrm{and} \\ $$$${x}^{\mathrm{2}} \:−\:\mathrm{2}{x}\:+\:\mathrm{3}\:\leqslant\:{f}\left({x}\right)\:\leqslant\:\mathrm{2}{x}^{\mathrm{2}} \:−\:\mathrm{4}{x}\:+\:\mathrm{4} \\ $$$$\mathrm{for}\:\mathrm{every}\:\:{x}\:\in\:\mathbb{R} \\ $$$$\mathrm{If}\:{f}\left(\mathrm{5}\right)\:=\:\mathrm{26},\:\mathrm{then}\:{f}\left(\mathrm{7}\right)\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:... \\ $$$$ \\ $$$$\left({A}\right)\:\mathrm{38}\:\:\:\:\:\:\:\:\:\left({D}\right)\:\mathrm{74} \\ $$$$\left({B}\right)\:\mathrm{50}\:\:\:\:\:\:\:\:\:\left({E}\right)\:\mathrm{92} \\ $$$$\left({C}\right)\:\mathrm{56} \\ $$

Question Number 21519    Answers: 1   Comments: 0

If th roots of the equation x^2 +2ax+b=0 are real and disinct and they differ by at most 2m, then b lies in the interval

$$\mathrm{If}\:\mathrm{th}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation}\:{x}^{\mathrm{2}} +\mathrm{2}{ax}+{b}=\mathrm{0} \\ $$$$\mathrm{are}\:\mathrm{real}\:\mathrm{and}\:\mathrm{disinct}\:\mathrm{and}\:\mathrm{they}\:\mathrm{differ}\:\mathrm{by} \\ $$$$\mathrm{at}\:\mathrm{most}\:\mathrm{2}{m},\:\mathrm{then}\:\:{b}\:\mathrm{lies}\:\mathrm{in}\:\mathrm{the}\:\mathrm{interval} \\ $$

Question Number 21516    Answers: 2   Comments: 0

A 5-kg body is suspended from a spring- balance, and an identical body is balanced on a pan of a physical balance. If both the balances are kept in an elevator, then what would happen in each case when the elevator is moving with an upward acceleration?

$$\mathrm{A}\:\mathrm{5}-\mathrm{kg}\:\mathrm{body}\:\mathrm{is}\:\mathrm{suspended}\:\mathrm{from}\:\mathrm{a}\:\mathrm{spring}- \\ $$$$\mathrm{balance},\:\mathrm{and}\:\mathrm{an}\:\mathrm{identical}\:\mathrm{body}\:\mathrm{is} \\ $$$$\mathrm{balanced}\:\mathrm{on}\:\mathrm{a}\:\mathrm{pan}\:\mathrm{of}\:\mathrm{a}\:\mathrm{physical} \\ $$$$\mathrm{balance}.\:\mathrm{If}\:\mathrm{both}\:\mathrm{the}\:\mathrm{balances}\:\mathrm{are}\:\mathrm{kept} \\ $$$$\mathrm{in}\:\mathrm{an}\:\mathrm{elevator},\:\mathrm{then}\:\mathrm{what}\:\mathrm{would}\:\mathrm{happen} \\ $$$$\mathrm{in}\:\mathrm{each}\:\mathrm{case}\:\mathrm{when}\:\mathrm{the}\:\mathrm{elevator}\:\mathrm{is}\:\mathrm{moving} \\ $$$$\mathrm{with}\:\mathrm{an}\:\mathrm{upward}\:\mathrm{acceleration}? \\ $$

Question Number 21510    Answers: 0   Comments: 0

Question Number 21504    Answers: 1   Comments: 0

Question Number 21503    Answers: 1   Comments: 0

Question Number 21502    Answers: 1   Comments: 0

In an A.P, the sum of p terms is q and the sum of q terms is p. Prove that the sum of (p+q) terms is −(p+q).

$$\mathrm{In}\:\mathrm{an}\:\mathrm{A}.\mathrm{P},\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{p}\:\mathrm{terms}\:\mathrm{is}\:\mathrm{q}\:\mathrm{and}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{q}\:\mathrm{terms}\:\mathrm{is}\:\mathrm{p}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\left(\mathrm{p}+\mathrm{q}\right)\:\mathrm{terms}\:\mathrm{is}\:−\left(\mathrm{p}+\mathrm{q}\right). \\ $$

Question Number 21498    Answers: 0   Comments: 0

a^(−2 ) + b^3 + c^(−4) = ((433)/(499)) Find a + b + c

$${a}^{−\mathrm{2}\:} +\:{b}^{\mathrm{3}} \:+\:{c}^{−\mathrm{4}} \:=\:\frac{\mathrm{433}}{\mathrm{499}} \\ $$$$\mathrm{Find}\:{a}\:+\:{b}\:+\:{c} \\ $$

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