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Question Number 17150    Answers: 0   Comments: 3

If the equation cos x + 3 cos (2Kx) = 4 has exactly one solution, then (1) K is a rational number of the form (P/(P + 1)), P ≠ −1 (2) K is irrational number whose rational approximation does not exceed 2 (3) K is irrational number (4) K is a rational number of the form (P/(P − 1)), P ≠ 1

$$\mathrm{If}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{cos}\:{x}\:+\:\mathrm{3}\:\mathrm{cos}\:\left(\mathrm{2}{Kx}\right)\:=\:\mathrm{4} \\ $$$$\mathrm{has}\:\mathrm{exactly}\:\mathrm{one}\:\mathrm{solution},\:\mathrm{then} \\ $$$$\left(\mathrm{1}\right)\:{K}\:\mathrm{is}\:\mathrm{a}\:\mathrm{rational}\:\mathrm{number}\:\mathrm{of}\:\mathrm{the}\:\mathrm{form} \\ $$$$\frac{{P}}{{P}\:+\:\mathrm{1}},\:{P}\:\neq\:−\mathrm{1} \\ $$$$\left(\mathrm{2}\right)\:{K}\:\mathrm{is}\:\mathrm{irrational}\:\mathrm{number}\:\mathrm{whose} \\ $$$$\mathrm{rational}\:\mathrm{approximation}\:\mathrm{does}\:\mathrm{not} \\ $$$$\mathrm{exceed}\:\mathrm{2} \\ $$$$\left(\mathrm{3}\right)\:{K}\:\mathrm{is}\:\mathrm{irrational}\:\mathrm{number} \\ $$$$\left(\mathrm{4}\right)\:{K}\:\mathrm{is}\:\mathrm{a}\:\mathrm{rational}\:\mathrm{number}\:\mathrm{of}\:\mathrm{the}\:\mathrm{form} \\ $$$$\frac{{P}}{{P}\:−\:\mathrm{1}},\:{P}\:\neq\:\mathrm{1} \\ $$

Question Number 17137    Answers: 0   Comments: 0

Solve the differential equation 2x[ye^x − 1]dx + e^y dy = 0

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{differential}\:\mathrm{equation}\: \\ $$$$\mathrm{2x}\left[\mathrm{ye}^{\mathrm{x}} \:−\:\mathrm{1}\right]\mathrm{dx}\:+\:\mathrm{e}^{\mathrm{y}} \:\mathrm{dy}\:=\:\mathrm{0} \\ $$

Question Number 17129    Answers: 1   Comments: 0

Question Number 17142    Answers: 0   Comments: 2

Find two primes a and b such that a−b=995

$$\mathrm{Find}\:\mathrm{two}\:\mathrm{primes}\:{a}\:\mathrm{and}\:{b}\:\mathrm{such} \\ $$$$\mathrm{that}\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:{a}−{b}=\mathrm{995} \\ $$

Question Number 17119    Answers: 0   Comments: 2

Question Number 17117    Answers: 1   Comments: 0

A clock has a pendulum made of iron rod of length 2.5m, if the clock keeps accurate time at 0°C. By how much time will it be late running at a temperature 30°C for 1 day. coefficient of linear expansivity of iron is 1.2 × 10^(−5) per k.

$$\mathrm{A}\:\mathrm{clock}\:\mathrm{has}\:\mathrm{a}\:\mathrm{pendulum}\:\mathrm{made}\:\mathrm{of}\:\mathrm{iron}\:\mathrm{rod}\:\mathrm{of}\:\mathrm{length}\:\mathrm{2}.\mathrm{5m}, \\ $$$$\mathrm{if}\:\mathrm{the}\:\mathrm{clock}\:\mathrm{keeps}\:\mathrm{accurate}\:\mathrm{time}\:\mathrm{at}\:\mathrm{0}°\mathrm{C}.\:\mathrm{By}\:\mathrm{how}\:\mathrm{much}\:\mathrm{time}\:\mathrm{will}\:\mathrm{it}\:\mathrm{be}\:\mathrm{late} \\ $$$$\mathrm{running}\:\mathrm{at}\:\mathrm{a}\:\mathrm{temperature}\:\mathrm{30}°\mathrm{C}\:\mathrm{for}\:\mathrm{1}\:\mathrm{day}.\:\mathrm{coefficient}\:\mathrm{of}\:\mathrm{linear}\:\mathrm{expansivity}\:\mathrm{of} \\ $$$$\mathrm{iron}\:\mathrm{is}\:\:\mathrm{1}.\mathrm{2}\:×\:\mathrm{10}^{−\mathrm{5}} \mathrm{per}\:\mathrm{k}. \\ $$

Question Number 17148    Answers: 1   Comments: 0

The number of solutions of the equation cos (π(√(x − 4))) cos (π(√x)) = 1 is

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{solutions}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\mathrm{cos}\:\left(\pi\sqrt{{x}\:−\:\mathrm{4}}\right)\:\mathrm{cos}\:\left(\pi\sqrt{{x}}\right)\:=\:\mathrm{1}\:\mathrm{is} \\ $$

Question Number 17102    Answers: 0   Comments: 3

compute: Σ_(k = 0) ^∞ ((2k + 1)/2^(2(k + 1)) )

$$\mathrm{compute}:\:\:\:\underset{\mathrm{k}\:=\:\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\mathrm{2k}\:+\:\mathrm{1}}{\mathrm{2}^{\mathrm{2}\left(\mathrm{k}\:+\:\mathrm{1}\right)} } \\ $$

Question Number 17100    Answers: 1   Comments: 0

sec xcos 5x+1=0 find number of solution

$$\mathrm{sec}\:{x}\mathrm{cos}\:\mathrm{5}{x}+\mathrm{1}=\mathrm{0} \\ $$$${find}\:{number}\:{of}\:{solution} \\ $$

Question Number 17086    Answers: 1   Comments: 6

∫_(−1) ^2 (1/x^2 ) dx

$$\underset{−\mathrm{1}} {\overset{\mathrm{2}} {\int}}\:\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\:{dx} \\ $$

Question Number 17080    Answers: 2   Comments: 0

sin^4 θ/2+cos^4 θ/2≥1/2

$$\mathrm{sin}^{\mathrm{4}} \theta/\mathrm{2}+\mathrm{cos}\:^{\mathrm{4}} \theta/\mathrm{2}\geqslant\mathrm{1}/\mathrm{2} \\ $$

Question Number 17093    Answers: 0   Comments: 16

If f(x) is a polynomial function satisfying f(x).f((1/x)) = f(x) + f((1/x)) ; x ∈ R − {0} and f(3) = 28, then f(4) is equal to

$$\mathrm{If}\:{f}\left({x}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{polynomial}\:\mathrm{function} \\ $$$$\mathrm{satisfying}\:{f}\left({x}\right).{f}\left(\frac{\mathrm{1}}{{x}}\right)\:=\:{f}\left({x}\right)\:+\:{f}\left(\frac{\mathrm{1}}{{x}}\right)\:; \\ $$$${x}\:\in\:{R}\:−\:\left\{\mathrm{0}\right\}\:\mathrm{and}\:{f}\left(\mathrm{3}\right)\:=\:\mathrm{28},\:\mathrm{then}\:{f}\left(\mathrm{4}\right)\:\mathrm{is} \\ $$$$\mathrm{equal}\:\mathrm{to} \\ $$

Question Number 17075    Answers: 2   Comments: 1

Given that: log((x/(y − z))) = log((y/(z − x))) = log((z/(x − y))) Show that : x^x × y^y × z^z = 1

$$\mathrm{Given}\:\mathrm{that}:\:\:\mathrm{log}\left(\frac{\mathrm{x}}{\mathrm{y}\:−\:\mathrm{z}}\right)\:=\:\mathrm{log}\left(\frac{\mathrm{y}}{\mathrm{z}\:−\:\mathrm{x}}\right)\:=\:\mathrm{log}\left(\frac{\mathrm{z}}{\mathrm{x}\:−\:\mathrm{y}}\right) \\ $$$$\mathrm{Show}\:\mathrm{that}\::\:\:\:\mathrm{x}^{\mathrm{x}} \:×\:\mathrm{y}^{\mathrm{y}} \:×\:\mathrm{z}^{\mathrm{z}} \:=\:\mathrm{1} \\ $$

Question Number 17073    Answers: 0   Comments: 4

Question Number 17065    Answers: 0   Comments: 4

if y=x^x what is the range of this function?

$$\mathrm{if}\:\mathrm{y}=\mathrm{x}^{\mathrm{x}} \\ $$$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of}\:\mathrm{this}\:\mathrm{function}? \\ $$

Question Number 17095    Answers: 1   Comments: 0

The total number of solutions of the equation tan x + sec x = 2 which lie in the interval [0, 2π] is

$$\mathrm{The}\:\mathrm{total}\:\mathrm{number}\:\mathrm{of}\:\mathrm{solutions}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equation}\:\mathrm{tan}\:{x}\:+\:\mathrm{sec}\:{x}\:=\:\mathrm{2}\:\mathrm{which}\:\mathrm{lie}\:\mathrm{in} \\ $$$$\mathrm{the}\:\mathrm{interval}\:\left[\mathrm{0},\:\mathrm{2}\pi\right]\:\mathrm{is} \\ $$

Question Number 17068    Answers: 2   Comments: 0

prove that 4tan^(−1) ((1/5))−tan^(−1) ((1/(239))) =π/4

$$\mathrm{prove}\:\mathrm{that}\: \\ $$$$ \\ $$$$\mathrm{4tan}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{5}}\right)−\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{239}}\right)\:=\pi/\mathrm{4} \\ $$

Question Number 17869    Answers: 1   Comments: 1

Question Number 17034    Answers: 1   Comments: 0

∫_0 ^( Π) (dx/(3+2sinx+cosx))

$$\int_{\mathrm{0}} ^{\:\Pi} \frac{\mathrm{dx}}{\mathrm{3}+\mathrm{2sinx}+\mathrm{cosx}} \\ $$

Question Number 17033    Answers: 1   Comments: 0

∫_((Π )/2) ^( 0) ((sinx cosx dx)/(2cos^2 x+3sin^2 x))

$$\int_{\frac{\Pi\:}{\mathrm{2}}} ^{\:\mathrm{0}} \:\frac{\mathrm{sinx}\:\mathrm{cosx}\:\mathrm{dx}}{\mathrm{2cos}^{\mathrm{2}} \mathrm{x}+\mathrm{3sin}^{\mathrm{2}} \mathrm{x}} \\ $$

Question Number 17030    Answers: 0   Comments: 1

∫ cot^2 (x^3 ) dx

$$\int\:\mathrm{cot}^{\mathrm{2}} \left(\mathrm{x}^{\mathrm{3}} \right)\:\mathrm{dx} \\ $$

Question Number 17027    Answers: 0   Comments: 0

Question Number 17019    Answers: 1   Comments: 0

Σ_(r = 1) ^3 2r − 1 = ?

$$\underset{\mathrm{r}\:=\:\mathrm{1}} {\overset{\mathrm{3}} {\sum}}\:\mathrm{2r}\:−\:\mathrm{1}\:\:=\:\:? \\ $$

Question Number 17018    Answers: 0   Comments: 8

Find the cube root of: 55 + 63 (√2)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{cube}\:\mathrm{root}\:\mathrm{of}:\:\:\:\mathrm{55}\:+\:\mathrm{63}\:\sqrt{\mathrm{2}} \\ $$

Question Number 17017    Answers: 1   Comments: 0

solve the simultaenous equation x+y=3 (2^x /x)=(2^y /y) find xand y.show ur workings....

$${solve}\:{the}\:{simultaenous}\:{equation} \\ $$$$\:{x}+{y}=\mathrm{3} \\ $$$$\:\frac{\mathrm{2}^{{x}} }{{x}}=\frac{\mathrm{2}^{{y}} }{{y}} \\ $$$$\:{find}\:{xand}\:{y}.{show}\:{ur}\:{workings}.... \\ $$

Question Number 17011    Answers: 1   Comments: 0

∫_0 ^( a) x(√((a^2 −x^2 )/(a^2 +x^2 )))dx

$$\int_{\mathrm{0}} ^{\:\mathrm{a}} \mathrm{x}\sqrt{\frac{\mathrm{a}^{\mathrm{2}} −\mathrm{x}^{\mathrm{2}} }{\mathrm{a}^{\mathrm{2}} +\mathrm{x}^{\mathrm{2}} }}\mathrm{dx} \\ $$

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