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Question Number 22090 Answers: 2 Comments: 0

Question Number 26917 Answers: 1 Comments: 2

Given a^2 + b^2 = 1 and c^2 + d^2 = 1 The minimum value of ac + bd − 2 is ...

$$\mathrm{Given}\:{a}^{\mathrm{2}} \:+\:{b}^{\mathrm{2}} \:=\:\mathrm{1}\:\mathrm{and}\:{c}^{\mathrm{2}} \:+\:{d}^{\mathrm{2}} \:=\:\mathrm{1} \\ $$$$\mathrm{The}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:{ac}\:+\:{bd}\:−\:\mathrm{2}\:\mathrm{is}\:... \\ $$

Question Number 22295 Answers: 1 Comments: 0

The ionization potential of hydrogen is 13.6 eV/mole. Calculate the energy in kJ required to produce 0.1 mole of H^+ ions. Given, 1 eV = 96.49 kJ mol^(−1) )

$$\mathrm{The}\:\mathrm{ionization}\:\mathrm{potential}\:\mathrm{of}\:\mathrm{hydrogen}\:\mathrm{is} \\ $$$$\mathrm{13}.\mathrm{6}\:\mathrm{eV}/\mathrm{mole}.\:\mathrm{Calculate}\:\mathrm{the}\:\mathrm{energy}\:\mathrm{in} \\ $$$$\mathrm{kJ}\:\mathrm{required}\:\mathrm{to}\:\mathrm{produce}\:\mathrm{0}.\mathrm{1}\:\mathrm{mole}\:\mathrm{of}\:\mathrm{H}^{+} \\ $$$$\left.\mathrm{ions}.\:\mathrm{Given},\:\mathrm{1}\:\mathrm{eV}\:=\:\mathrm{96}.\mathrm{49}\:\mathrm{kJ}\:\mathrm{mol}^{−\mathrm{1}} \right) \\ $$

Question Number 22276 Answers: 1 Comments: 0

Question Number 22082 Answers: 1 Comments: 0

If A is a fifty-element subset of the set {1, 2, 3, ...., 100} such that no two numbers from A add up to 100 show that A contains a square.

$$\mathrm{If}\:{A}\:\mathrm{is}\:\mathrm{a}\:\mathrm{fifty}-\mathrm{element}\:\mathrm{subset}\:\mathrm{of}\:\mathrm{the}\:\mathrm{set} \\ $$$$\left\{\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:....,\:\mathrm{100}\right\}\:\mathrm{such}\:\mathrm{that}\:\mathrm{no}\:\mathrm{two} \\ $$$$\mathrm{numbers}\:\mathrm{from}\:{A}\:\mathrm{add}\:\mathrm{up}\:\mathrm{to}\:\mathrm{100}\:\mathrm{show} \\ $$$$\mathrm{that}\:{A}\:\mathrm{contains}\:\mathrm{a}\:\mathrm{square}. \\ $$

Question Number 22080 Answers: 0 Comments: 1

Given any positive integer n show that there are two positive rational numbers a and b, a ≠ b, which are not integers and which are such that a − b, a^2 − b^2 , a^3 − b^3 , ....., a^n − b^n are all integers.

$$\mathrm{Given}\:\mathrm{any}\:\mathrm{positive}\:\mathrm{integer}\:{n}\:\mathrm{show} \\ $$$$\mathrm{that}\:\mathrm{there}\:\mathrm{are}\:\mathrm{two}\:\mathrm{positive}\:\mathrm{rational} \\ $$$$\mathrm{numbers}\:{a}\:\mathrm{and}\:{b},\:{a}\:\neq\:{b},\:\mathrm{which}\:\mathrm{are}\:\mathrm{not} \\ $$$$\mathrm{integers}\:\mathrm{and}\:\mathrm{which}\:\mathrm{are}\:\mathrm{such}\:\mathrm{that}\:{a}\:−\:{b}, \\ $$$${a}^{\mathrm{2}} \:−\:{b}^{\mathrm{2}} ,\:{a}^{\mathrm{3}} \:−\:{b}^{\mathrm{3}} ,\:.....,\:{a}^{{n}} \:−\:{b}^{{n}} \:\mathrm{are}\:\mathrm{all} \\ $$$$\mathrm{integers}. \\ $$

Question Number 22079 Answers: 0 Comments: 1

Let ABC be a triangle and h_a the altitude through A. Prove that (b + c)^2 ≥ a^2 + 4h_a ^2 . (As usual a, b, c denote the sides BC, CA, AB respectively.)

$$\mathrm{Let}\:{ABC}\:\mathrm{be}\:\mathrm{a}\:\mathrm{triangle}\:\mathrm{and}\:{h}_{{a}} \:\mathrm{the} \\ $$$$\mathrm{altitude}\:\mathrm{through}\:{A}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\left({b}\:+\:{c}\right)^{\mathrm{2}} \:\geqslant\:{a}^{\mathrm{2}} \:+\:\mathrm{4}{h}_{{a}} ^{\mathrm{2}} . \\ $$$$\left(\mathrm{As}\:\mathrm{usual}\:{a},\:{b},\:{c}\:\mathrm{denote}\:\mathrm{the}\:\mathrm{sides}\:{BC},\right. \\ $$$$\left.{CA},\:{AB}\:\mathrm{respectively}.\right) \\ $$

Question Number 22076 Answers: 0 Comments: 3

find approximately and quickly without calculator ((54329)/(2467)) .

$${find}\:{approximately}\:{and}\:{quickly} \\ $$$${without}\:{calculator}\:\frac{\mathrm{54329}}{\mathrm{2467}}\:. \\ $$

Question Number 22161 Answers: 0 Comments: 1

The students were asked whether they had dictionary(D) or thesau rus(T) in their room.the results showed that 650 students had dict ionary,150 did not had dictionary, 175 had a thesaurus,and 50 had neither a dictionary nor a thesaur us,fimd the number of student who (i)live in domitory ( ii)have both dictionary and thesaurus (iii)have only thesaurus

$${The}\:{students}\:{were}\:{asked}\:{whether} \\ $$$${they}\:{had}\:{dictionary}\left({D}\right)\:{or}\:{thesau} \\ $$$${rus}\left({T}\right)\:{in}\:{their}\:{room}.{the}\:{results}\: \\ $$$${showed}\:{that}\:\mathrm{650}\:{students}\:{had}\:{dict} \\ $$$${ionary},\mathrm{150}\:{did}\:{not}\:{had}\:{dictionary}, \\ $$$$\mathrm{175}\:{had}\:{a}\:{thesaurus},{and}\:\mathrm{50}\:{had} \\ $$$${neither}\:{a}\:{dictionary}\:{nor}\:{a}\:{thesaur} \\ $$$${us},{fimd}\:{the}\:{number}\:{of}\:{student}\:{who} \\ $$$$\:\:\left({i}\right){live}\:{in}\:{domitory} \\ $$$$\:\:\:\left(\:{ii}\right){have}\:{both}\:{dictionary}\:{and}\:{thesaurus} \\ $$$$\:\:\left({iii}\right){have}\:{only}\:{thesaurus} \\ $$$$ \\ $$

Question Number 22071 Answers: 1 Comments: 4

STATEMENT-1 : If an object is at rest then there should not be any friction on it. STATEMENT-2 : If an object is moving then the friction acting on it has to be kinetic. STATEMENT-3 : If an object is at rest then kinetic friction cannot act on it.

$$\mathrm{STATEMENT}-\mathrm{1}\::\:\mathrm{If}\:\mathrm{an}\:\mathrm{object}\:\mathrm{is}\:\mathrm{at} \\ $$$$\mathrm{rest}\:\mathrm{then}\:\mathrm{there}\:\mathrm{should}\:\mathrm{not}\:\mathrm{be}\:\mathrm{any}\:\mathrm{friction} \\ $$$$\mathrm{on}\:\mathrm{it}. \\ $$$$\mathrm{STATEMENT}-\mathrm{2}\::\:\mathrm{If}\:\mathrm{an}\:\mathrm{object}\:\mathrm{is}\:\mathrm{moving} \\ $$$$\mathrm{then}\:\mathrm{the}\:\mathrm{friction}\:\mathrm{acting}\:\mathrm{on}\:\mathrm{it}\:\mathrm{has}\:\mathrm{to}\:\mathrm{be} \\ $$$$\mathrm{kinetic}. \\ $$$$\mathrm{STATEMENT}-\mathrm{3}\::\:\mathrm{If}\:\mathrm{an}\:\mathrm{object}\:\mathrm{is}\:\mathrm{at}\:\mathrm{rest} \\ $$$$\mathrm{then}\:\mathrm{kinetic}\:\mathrm{friction}\:\mathrm{cannot}\:\mathrm{act}\:\mathrm{on}\:\mathrm{it}. \\ $$

Question Number 26924 Answers: 1 Comments: 0

Question Number 22062 Answers: 1 Comments: 1

Question Number 22061 Answers: 1 Comments: 0

If 9x^2 +6xy+4y^2 is a factor of 27x^3 −8y^3 . find the other factor.

$$\mathrm{If}\:\mathrm{9x}^{\mathrm{2}} +\mathrm{6xy}+\mathrm{4y}^{\mathrm{2}} \:\mathrm{is}\:\mathrm{a}\:\mathrm{factor}\:\mathrm{of} \\ $$$$\mathrm{27x}^{\mathrm{3}} −\mathrm{8y}^{\mathrm{3}} .\:\mathrm{find}\:\mathrm{the}\:\mathrm{other}\:\mathrm{factor}. \\ $$

Question Number 22059 Answers: 1 Comments: 0

A flywheel whose diameter is 1.5m decrease uniformly from 240rad/min until it came to rest 10s. Find the number of revolution made.

$$\mathrm{A}\:\mathrm{flywheel}\:\mathrm{whose}\:\mathrm{diameter}\:\mathrm{is}\:\mathrm{1}.\mathrm{5m}\:\mathrm{decrease}\:\mathrm{uniformly}\:\mathrm{from}\:\mathrm{240rad}/\mathrm{min} \\ $$$$\mathrm{until}\:\mathrm{it}\:\mathrm{came}\:\mathrm{to}\:\mathrm{rest}\:\mathrm{10s}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{revolution}\:\mathrm{made}. \\ $$

Question Number 22058 Answers: 1 Comments: 0

Two balls of mass 500g and 750g moving with 15m/s and 10m/s towards each other collides. Find the velocities of the ball after collision, if the coefficient of restitution is 0.8

$$\mathrm{Two}\:\mathrm{balls}\:\mathrm{of}\:\mathrm{mass}\:\mathrm{500g}\:\mathrm{and}\:\mathrm{750g}\:\mathrm{moving}\:\mathrm{with}\:\mathrm{15m}/\mathrm{s}\:\mathrm{and} \\ $$$$\mathrm{10m}/\mathrm{s}\:\mathrm{towards}\:\mathrm{each}\:\mathrm{other}\:\mathrm{collides}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{velocities}\:\mathrm{of}\:\mathrm{the}\:\mathrm{ball}\:\mathrm{after} \\ $$$$\mathrm{collision},\:\mathrm{if}\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:\mathrm{restitution}\:\mathrm{is}\:\mathrm{0}.\mathrm{8} \\ $$

Question Number 22057 Answers: 0 Comments: 0

∫f(x)dx=((d/dx))^2 f(x)=???

$$\int{f}\left({x}\right){dx}=\left(\frac{{d}}{{dx}}\right)^{\mathrm{2}} \\ $$$${f}\left({x}\right)=??? \\ $$

Question Number 22055 Answers: 0 Comments: 1

Question Number 22052 Answers: 0 Comments: 2

A hockey player is moving northward and suddenly turns westward with the same speed to avoid an opponent. The force that acts on the player is (a) frictional force along westward (b) muscle force along southward (c) frictional force along south-west (d) muscle force along south-west

$$\mathrm{A}\:\mathrm{hockey}\:\mathrm{player}\:\mathrm{is}\:\mathrm{moving}\:\mathrm{northward} \\ $$$$\mathrm{and}\:\mathrm{suddenly}\:\mathrm{turns}\:\mathrm{westward}\:\mathrm{with} \\ $$$$\mathrm{the}\:\mathrm{same}\:\mathrm{speed}\:\mathrm{to}\:\mathrm{avoid}\:\mathrm{an}\:\mathrm{opponent}. \\ $$$$\mathrm{The}\:\mathrm{force}\:\mathrm{that}\:\mathrm{acts}\:\mathrm{on}\:\mathrm{the}\:\mathrm{player}\:\mathrm{is} \\ $$$$\left({a}\right)\:\mathrm{frictional}\:\mathrm{force}\:\mathrm{along}\:\mathrm{westward} \\ $$$$\left({b}\right)\:\mathrm{muscle}\:\mathrm{force}\:\mathrm{along}\:\mathrm{southward} \\ $$$$\left({c}\right)\:\mathrm{frictional}\:\mathrm{force}\:\mathrm{along}\:\mathrm{south}-\mathrm{west} \\ $$$$\left({d}\right)\:\mathrm{muscle}\:\mathrm{force}\:\mathrm{along}\:\mathrm{south}-\mathrm{west} \\ $$

Question Number 22047 Answers: 2 Comments: 1

If x > 0 and the 4^(th) term in the expansion of (2 + (3/8)x)^(10) has maximum value then find the range of x.

$$\mathrm{If}\:{x}\:>\:\mathrm{0}\:\mathrm{and}\:\mathrm{the}\:\mathrm{4}^{\mathrm{th}} \:\mathrm{term}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion} \\ $$$$\mathrm{of}\:\left(\mathrm{2}\:+\:\frac{\mathrm{3}}{\mathrm{8}}{x}\right)^{\mathrm{10}} \:\mathrm{has}\:\mathrm{maximum}\:\mathrm{value} \\ $$$$\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of}\:{x}. \\ $$

Question Number 22050 Answers: 0 Comments: 0

Calculate the energy emitted when electrons of 1 g atom of hydrogen undergo transition giving the spectral line of lowest energy in the visible region of its atomic spectrum (R_H = 1.1 × 10^7 m^(−1) , c = 3 × 10^8 ms^(−1) , h = 6.62 × 10^(−34) Js)

$$\mathrm{Calculate}\:\mathrm{the}\:\mathrm{energy}\:\mathrm{emitted}\:\mathrm{when} \\ $$$$\mathrm{electrons}\:\mathrm{of}\:\mathrm{1}\:\mathrm{g}\:\mathrm{atom}\:\mathrm{of}\:\mathrm{hydrogen} \\ $$$$\mathrm{undergo}\:\mathrm{transition}\:\mathrm{giving}\:\mathrm{the}\:\mathrm{spectral} \\ $$$$\mathrm{line}\:\mathrm{of}\:\mathrm{lowest}\:\mathrm{energy}\:\mathrm{in}\:\mathrm{the}\:\mathrm{visible} \\ $$$$\mathrm{region}\:\mathrm{of}\:\mathrm{its}\:\mathrm{atomic}\:\mathrm{spectrum} \\ $$$$\left(\mathrm{R}_{\mathrm{H}} \:=\:\mathrm{1}.\mathrm{1}\:×\:\mathrm{10}^{\mathrm{7}} \:\mathrm{m}^{−\mathrm{1}} ,\:{c}\:=\:\mathrm{3}\:×\:\mathrm{10}^{\mathrm{8}} \:{ms}^{−\mathrm{1}} ,\right. \\ $$$$\left.{h}\:=\:\mathrm{6}.\mathrm{62}\:×\:\mathrm{10}^{−\mathrm{34}} \:\mathrm{Js}\right) \\ $$

Question Number 22044 Answers: 1 Comments: 0

Let A = {1, 2, 3, ....., n}, if a_i is the minimum element of the set A; (where A; denotes the subset of A containing exactly three elements) and X denotes the set of A_i ′s, then evaluate Σ_(A_i ∈X) a.

$$\mathrm{Let}\:{A}\:=\:\left\{\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:.....,\:{n}\right\},\:\mathrm{if}\:{a}_{{i}} \:\mathrm{is}\:\mathrm{the} \\ $$$$\mathrm{minimum}\:\mathrm{element}\:\mathrm{of}\:\mathrm{the}\:\mathrm{set}\:{A};\:\left(\mathrm{where}\right. \\ $$$${A};\:\mathrm{denotes}\:\mathrm{the}\:\mathrm{subset}\:\mathrm{of}\:{A}\:\mathrm{containing} \\ $$$$\left.\mathrm{exactly}\:\mathrm{three}\:\mathrm{elements}\right)\:\mathrm{and}\:{X}\:\mathrm{denotes} \\ $$$$\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:{A}_{{i}} '\mathrm{s},\:\mathrm{then}\:\mathrm{evaluate}\:\underset{{A}_{{i}} \in{X}} {\sum}{a}. \\ $$

Question Number 22043 Answers: 1 Comments: 0

In how many ways we can choose 3 squares on a chess board such that one of the squares has its two sides common to other two squares?

$$\mathrm{In}\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{we}\:\mathrm{can}\:\mathrm{choose}\:\mathrm{3} \\ $$$$\mathrm{squares}\:\mathrm{on}\:\mathrm{a}\:\mathrm{chess}\:\mathrm{board}\:\mathrm{such}\:\mathrm{that}\:\mathrm{one} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{squares}\:\mathrm{has}\:\mathrm{its}\:\mathrm{two}\:\mathrm{sides}\:\mathrm{common} \\ $$$$\mathrm{to}\:\mathrm{other}\:\mathrm{two}\:\mathrm{squares}? \\ $$

Question Number 22042 Answers: 1 Comments: 0

Determine the number of ordered pairs of positive integers (a, b) such that the least common multiple of a and b is 2^3 ∙5^7 ∙11^(13) .

$$\mathrm{Determine}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ordered} \\ $$$$\mathrm{pairs}\:\mathrm{of}\:\mathrm{positive}\:\mathrm{integers}\:\left({a},\:{b}\right)\:\mathrm{such} \\ $$$$\mathrm{that}\:\mathrm{the}\:\mathrm{least}\:\mathrm{common}\:\mathrm{multiple}\:\mathrm{of}\:{a} \\ $$$$\mathrm{and}\:{b}\:\mathrm{is}\:\mathrm{2}^{\mathrm{3}} \centerdot\mathrm{5}^{\mathrm{7}} \centerdot\mathrm{11}^{\mathrm{13}} . \\ $$

Question Number 22041 Answers: 0 Comments: 0

On the modified chess board 10 × 10, Amit and Suresh two persons which start moving towards each other. Each person moving with same constant speed. Amit can move only to the right and upwards along the lines while Suresh can move only to the left or downwards along the lines of the chess boards. The total number of ways in which Amit and Suresh can meet at same point during their trip.

$$\mathrm{On}\:\mathrm{the}\:\mathrm{modified}\:\mathrm{chess}\:\mathrm{board}\:\mathrm{10}\:×\:\mathrm{10}, \\ $$$$\mathrm{Amit}\:\mathrm{and}\:\mathrm{Suresh}\:\mathrm{two}\:\mathrm{persons}\:\mathrm{which} \\ $$$$\mathrm{start}\:\mathrm{moving}\:\mathrm{towards}\:\mathrm{each}\:\mathrm{other}.\:\mathrm{Each} \\ $$$$\mathrm{person}\:\mathrm{moving}\:\mathrm{with}\:\mathrm{same}\:\mathrm{constant} \\ $$$$\mathrm{speed}.\:\mathrm{Amit}\:\mathrm{can}\:\mathrm{move}\:\mathrm{only}\:\mathrm{to}\:\mathrm{the} \\ $$$$\mathrm{right}\:\mathrm{and}\:\mathrm{upwards}\:\mathrm{along}\:\mathrm{the}\:\mathrm{lines} \\ $$$$\mathrm{while}\:\mathrm{Suresh}\:\mathrm{can}\:\mathrm{move}\:\mathrm{only}\:\mathrm{to}\:\mathrm{the}\:\mathrm{left} \\ $$$$\mathrm{or}\:\mathrm{downwards}\:\mathrm{along}\:\mathrm{the}\:\mathrm{lines}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{chess}\:\mathrm{boards}.\:\mathrm{The}\:\mathrm{total}\:\mathrm{number}\:\mathrm{of} \\ $$$$\mathrm{ways}\:\mathrm{in}\:\mathrm{which}\:\mathrm{Amit}\:\mathrm{and}\:\mathrm{Suresh}\:\mathrm{can} \\ $$$$\mathrm{meet}\:\mathrm{at}\:\mathrm{same}\:\mathrm{point}\:\mathrm{during}\:\mathrm{their}\:\mathrm{trip}. \\ $$

Question Number 22040 Answers: 0 Comments: 4

The total number of non-similar triangles which can be formed such that all the angles of the triangle are integers is

$$\mathrm{The}\:\mathrm{total}\:\mathrm{number}\:\mathrm{of}\:\mathrm{non}-\mathrm{similar} \\ $$$$\mathrm{triangles}\:\mathrm{which}\:\mathrm{can}\:\mathrm{be}\:\mathrm{formed}\:\mathrm{such} \\ $$$$\mathrm{that}\:\mathrm{all}\:\mathrm{the}\:\mathrm{angles}\:\mathrm{of}\:\mathrm{the}\:\mathrm{triangle}\:\mathrm{are} \\ $$$$\mathrm{integers}\:\mathrm{is} \\ $$

Question Number 22038 Answers: 0 Comments: 1

The symbols +, +, ×, ×, ★, •, are placed in the squares of the adjoining figure. The number of ways of placing symbols so that no row remains empty is

$$\mathrm{The}\:\mathrm{symbols}\:+,\:+,\:×,\:×,\:\bigstar,\:\bullet,\:\mathrm{are} \\ $$$$\mathrm{placed}\:\mathrm{in}\:\mathrm{the}\:\mathrm{squares}\:\mathrm{of}\:\mathrm{the}\:\mathrm{adjoining} \\ $$$$\mathrm{figure}.\:\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ways}\:\mathrm{of}\:\mathrm{placing} \\ $$$$\mathrm{symbols}\:\mathrm{so}\:\mathrm{that}\:\mathrm{no}\:\mathrm{row}\:\mathrm{remains}\:\mathrm{empty} \\ $$$$\mathrm{is} \\ $$

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