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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

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

Question Number 21493 Answers: 1 Comments: 0

Is always a÷b = (a/b) ?

$$\mathrm{Is}\:\mathrm{always}\:\mathrm{a}\boldsymbol{\div}\mathrm{b}\:=\:\frac{\mathrm{a}}{\mathrm{b}}\:? \\ $$

Question Number 21491 Answers: 0 Comments: 0

If Y is a non−void set, define Y^T to be the collection of all functions with domain T and range Y. Show that if T and Y are finite sets with m and n elements, then Y^T has n^m elements.

$$\mathrm{If}\:{Y}\:\mathrm{is}\:\mathrm{a}\:\mathrm{non}−\mathrm{void}\:\mathrm{set},\:\mathrm{define}\:{Y}^{{T}} \:\mathrm{to}\:\mathrm{be} \\ $$$$\mathrm{the}\:\mathrm{collection}\:\mathrm{of}\:\mathrm{all}\:\mathrm{functions}\:\mathrm{with} \\ $$$$\mathrm{domain}\:{T}\:\mathrm{and}\:\mathrm{range}\:{Y}.\:\mathrm{Show}\:\mathrm{that} \\ $$$$\mathrm{if}\:{T}\:\mathrm{and}\:{Y}\:\mathrm{are}\:\mathrm{finite}\:\mathrm{sets}\:\mathrm{with}\:{m}\:\mathrm{and} \\ $$$${n}\:\mathrm{elements},\:\mathrm{then}\:{Y}^{{T}} \:\mathrm{has}\:{n}^{{m}} \:\mathrm{elements}. \\ $$

Question Number 21490 Answers: 1 Comments: 0

The kinetic energy of a body increases by 100%.Find the % increase in its momentum. please solve with explanations where necessary. Thanks.

$$\mathrm{The}\:\mathrm{kinetic}\:\mathrm{energy}\:\mathrm{of}\:\mathrm{a}\:\mathrm{body} \\ $$$$\mathrm{increases}\:\mathrm{by}\:\mathrm{100\%}.\mathrm{Find}\:\mathrm{the}\:\% \\ $$$$\mathrm{increase}\:\mathrm{in}\:\mathrm{its}\:\mathrm{momentum}. \\ $$$$ \\ $$$$\mathrm{please}\:\mathrm{solve}\:\mathrm{with}\:\mathrm{explanations} \\ $$$$\mathrm{where}\:\mathrm{necessary}.\:\mathrm{Thanks}. \\ $$

Question Number 21486 Answers: 0 Comments: 0

Vapour pressure in a closed container can be changed by (1) Adding water vapours from outside at same temperature (2) Adding ice at same temperature (3) Adding water at same temperature (4) Increasing temperature

$$\mathrm{Vapour}\:\mathrm{pressure}\:\mathrm{in}\:\mathrm{a}\:\mathrm{closed}\:\mathrm{container} \\ $$$$\mathrm{can}\:\mathrm{be}\:\mathrm{changed}\:\mathrm{by} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{Adding}\:\mathrm{water}\:\mathrm{vapours}\:\mathrm{from}\:\mathrm{outside} \\ $$$$\mathrm{at}\:\mathrm{same}\:\mathrm{temperature} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{Adding}\:\mathrm{ice}\:\mathrm{at}\:\mathrm{same}\:\mathrm{temperature} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{Adding}\:\mathrm{water}\:\mathrm{at}\:\mathrm{same}\:\mathrm{temperature} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{Increasing}\:\mathrm{temperature} \\ $$

Question Number 21484 Answers: 0 Comments: 0

Positive deviation from ideal behaviour takes place because of (a) molecular interaction between atoms and PV/nRT > 1 (b) molecular interaction between atoms and PV/nRT < 1 (c) finite size of the atoms and PV/nRT > 1 (d) finite size of the atoms and PV/nRT < 1

$$\mathrm{Positive}\:\mathrm{deviation}\:\mathrm{from}\:\mathrm{ideal}\:\mathrm{behaviour} \\ $$$$\mathrm{takes}\:\mathrm{place}\:\mathrm{because}\:\mathrm{of} \\ $$$$\left({a}\right)\:\mathrm{molecular}\:\mathrm{interaction}\:\mathrm{between} \\ $$$$\mathrm{atoms}\:\mathrm{and}\:\mathrm{PV}/{n}\mathrm{RT}\:>\:\mathrm{1} \\ $$$$\left({b}\right)\:\mathrm{molecular}\:\mathrm{interaction}\:\mathrm{between} \\ $$$$\mathrm{atoms}\:\mathrm{and}\:\mathrm{PV}/{n}\mathrm{RT}\:<\:\mathrm{1} \\ $$$$\left({c}\right)\:\mathrm{finite}\:\mathrm{size}\:\mathrm{of}\:\mathrm{the}\:\mathrm{atoms}\:\mathrm{and}\:\mathrm{PV}/{n}\mathrm{RT}\:>\:\mathrm{1} \\ $$$$\left({d}\right)\:\mathrm{finite}\:\mathrm{size}\:\mathrm{of}\:\mathrm{the}\:\mathrm{atoms}\:\mathrm{and}\:\mathrm{PV}/{n}\mathrm{RT}\:<\:\mathrm{1} \\ $$

Question Number 21482 Answers: 0 Comments: 0

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Question Number 21471 Answers: 2 Comments: 0

If a + b + c = 0, then (((a + b)(b + c)(a + c))/(abc)) is equal to ...

$$\mathrm{If}\:\:{a}\:+\:{b}\:+\:{c}\:=\:\mathrm{0},\:\mathrm{then} \\ $$$$\frac{\left({a}\:+\:{b}\right)\left({b}\:+\:{c}\right)\left({a}\:+\:{c}\right)}{{abc}}\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:... \\ $$

Question Number 21470 Answers: 1 Comments: 1

2^x = 3^y = 6^(−z) Find the value of (((2017)/x) + ((2017)/y) + ((2017)/z))^(2017)

$$\mathrm{2}^{{x}} \:=\:\mathrm{3}^{{y}} \:=\:\mathrm{6}^{−{z}} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\: \\ $$$$\left(\frac{\mathrm{2017}}{{x}}\:+\:\frac{\mathrm{2017}}{{y}}\:+\:\frac{\mathrm{2017}}{{z}}\right)^{\mathrm{2017}} \\ $$

Question Number 21469 Answers: 1 Comments: 1

Three identical blocks, each having a mass M, are pushed by a force F on a frictionless table. What is the net force on the block A?

$$\mathrm{Three}\:\mathrm{identical}\:\mathrm{blocks},\:\mathrm{each}\:\mathrm{having}\:\mathrm{a} \\ $$$$\mathrm{mass}\:{M},\:\mathrm{are}\:\mathrm{pushed}\:\mathrm{by}\:\mathrm{a}\:\mathrm{force}\:{F}\:\mathrm{on}\:\mathrm{a} \\ $$$$\mathrm{frictionless}\:\mathrm{table}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{net}\:\mathrm{force} \\ $$$$\mathrm{on}\:\mathrm{the}\:\mathrm{block}\:{A}? \\ $$

Question Number 21467 Answers: 1 Comments: 0

∫_( 1) ^e log x dx =

$$\underset{\:\mathrm{1}} {\overset{{e}} {\int}}\:\:\mathrm{log}\:{x}\:{dx}\:= \\ $$

Question Number 21464 Answers: 2 Comments: 0

Question Number 21463 Answers: 0 Comments: 0

Let A be the collection of functions f : [0, 1] → R which have an infinite number of derivatives. Let A_0 ⊂ A be the subcollection of those functions f with f(0) = 0. Define D : A_0 → A by D(f) = df/dx. Use the mean value theorem to show that D is injective. Use the fundamental theorem of calculus to show that D is surjective.

$$\mathrm{Let}\:{A}\:\mathrm{be}\:\mathrm{the}\:\mathrm{collection}\:\mathrm{of}\:\mathrm{functions} \\ $$$${f}\::\:\left[\mathrm{0},\:\mathrm{1}\right]\:\rightarrow\:\mathbb{R}\:\mathrm{which}\:\mathrm{have}\:\mathrm{an}\:\mathrm{infinite} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{derivatives}.\:\mathrm{Let}\:{A}_{\mathrm{0}} \:\subset\:{A} \\ $$$$\mathrm{be}\:\mathrm{the}\:\mathrm{subcollection}\:\mathrm{of}\:\mathrm{those}\:\mathrm{functions} \\ $$$${f}\:\mathrm{with}\:{f}\left(\mathrm{0}\right)\:=\:\mathrm{0}.\:\mathrm{Define}\:{D}\::\:{A}_{\mathrm{0}} \:\rightarrow\:{A} \\ $$$$\mathrm{by}\:{D}\left({f}\right)\:=\:{df}/{dx}.\:\mathrm{Use}\:\mathrm{the}\:\mathrm{mean}\:\mathrm{value} \\ $$$$\mathrm{theorem}\:\mathrm{to}\:\mathrm{show}\:\mathrm{that}\:{D}\:\mathrm{is}\:\mathrm{injective}. \\ $$$$\mathrm{Use}\:\mathrm{the}\:\mathrm{fundamental}\:\mathrm{theorem}\:\mathrm{of} \\ $$$$\mathrm{calculus}\:\mathrm{to}\:\mathrm{show}\:\mathrm{that}\:{D}\:\mathrm{is}\:\mathrm{surjective}. \\ $$

Question Number 21459 Answers: 1 Comments: 0

Solve (√(x + 1 − 4(√(x − 3)))) + (√(x + 22 + 10(√(x − 3)))) = 7

$$\mathrm{Solve} \\ $$$$\sqrt{{x}\:+\:\mathrm{1}\:−\:\mathrm{4}\sqrt{{x}\:−\:\mathrm{3}}}\:+\:\sqrt{{x}\:+\:\mathrm{22}\:+\:\mathrm{10}\sqrt{{x}\:−\:\mathrm{3}}}\:=\:\mathrm{7} \\ $$

Question Number 21453 Answers: 3 Comments: 0

If (1 + n)sin 2θ + (1 − n)cos 2θ = 1 + n find tan 2θ

$$\mathrm{If}\: \\ $$$$\left(\mathrm{1}\:+\:{n}\right)\mathrm{sin}\:\mathrm{2}\theta\:+\:\left(\mathrm{1}\:−\:{n}\right)\mathrm{cos}\:\mathrm{2}\theta\:=\:\mathrm{1}\:+\:{n} \\ $$$$\mathrm{find}\:\mathrm{tan}\:\mathrm{2}\theta \\ $$

Question Number 21450 Answers: 0 Comments: 0

Question Number 21507 Answers: 1 Comments: 1

The length of an ideal spring increases by 0.1 cm when a body of 1 kg is suspended from it. If this spring is laid on a frictionless horizontal table and bodies of 1 kg each are suspended from its ends, then what will be the increase in its length?

$$\mathrm{The}\:\mathrm{length}\:\mathrm{of}\:\mathrm{an}\:\mathrm{ideal}\:\mathrm{spring}\:\mathrm{increases} \\ $$$$\mathrm{by}\:\mathrm{0}.\mathrm{1}\:\mathrm{cm}\:\mathrm{when}\:\mathrm{a}\:\mathrm{body}\:\mathrm{of}\:\mathrm{1}\:\mathrm{kg}\:\mathrm{is} \\ $$$$\mathrm{suspended}\:\mathrm{from}\:\mathrm{it}.\:\mathrm{If}\:\mathrm{this}\:\mathrm{spring}\:\mathrm{is}\:\mathrm{laid} \\ $$$$\mathrm{on}\:\mathrm{a}\:\mathrm{frictionless}\:\mathrm{horizontal}\:\mathrm{table}\:\mathrm{and} \\ $$$$\mathrm{bodies}\:\mathrm{of}\:\mathrm{1}\:\mathrm{kg}\:\mathrm{each}\:\mathrm{are}\:\mathrm{suspended}\:\mathrm{from} \\ $$$$\mathrm{its}\:\mathrm{ends},\:\mathrm{then}\:\mathrm{what}\:\mathrm{will}\:\mathrm{be}\:\mathrm{the}\:\mathrm{increase} \\ $$$$\mathrm{in}\:\mathrm{its}\:\mathrm{length}? \\ $$

Question Number 21441 Answers: 1 Comments: 0

Find α in terms of θ using the equations: (i) u^2 sin^2 α = 2gd cos θ (ii) t = ((u cos α)/(g sin θ)) (iii) −d = ut sin α − ((gt^2 sin θ)/2)

$$\mathrm{Find}\:\alpha\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:\theta\:\mathrm{using}\:\mathrm{the}\:\mathrm{equations}: \\ $$$$\left({i}\right)\:{u}^{\mathrm{2}} \:\mathrm{sin}^{\mathrm{2}} \:\alpha\:=\:\mathrm{2}{gd}\:\mathrm{cos}\:\theta \\ $$$$\left({ii}\right)\:{t}\:=\:\frac{{u}\:\mathrm{cos}\:\alpha}{{g}\:\mathrm{sin}\:\theta} \\ $$$$\left({iii}\right)\:−{d}\:=\:{ut}\:\mathrm{sin}\:\alpha\:−\:\frac{{gt}^{\mathrm{2}} \:\mathrm{sin}\:\theta}{\mathrm{2}} \\ $$

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