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Question Number 860 Answers: 1 Comments: 1

f(x^2 )=[f(x)]^2 f(1)=1

$${f}\left({x}^{\mathrm{2}} \right)=\left[{f}\left({x}\right)\right]^{\mathrm{2}} \\ $$$${f}\left(\mathrm{1}\right)=\mathrm{1} \\ $$

Question Number 808 Answers: 1 Comments: 1

if the equations of the sides of the triangle are 7x+y−10=0,x−2y+5=0 and x+y+2=0, find the orhocentre of the triangle

$${if}\:{the}\:{equations}\:{of}\:{the}\:{sides}\:{of}\:{the}\:{triangle}\:{are}\:\mathrm{7}{x}+{y}−\mathrm{10}=\mathrm{0},{x}−\mathrm{2}{y}+\mathrm{5}=\mathrm{0}\:{and}\:{x}+{y}+\mathrm{2}=\mathrm{0},\:{find}\:{the}\:{orhocentre}\:{of}\:{the}\:{triangle} \\ $$

Question Number 758 Answers: 1 Comments: 1

∫xtan^(−1) xdx

$$\int{x}\mathrm{tan}^{−\mathrm{1}} {xdx} \\ $$$$ \\ $$

Question Number 737 Answers: 1 Comments: 1

(1/T)∫_t_1 ^t_2 Vsin ωt−V_γ dt=? t_1 and t_2 are solution to Vsin ωt=V_γ V≥V_γ V_γ ≥0 and t_1 <t_2

$$\frac{\mathrm{1}}{{T}}\underset{{t}_{\mathrm{1}} } {\overset{{t}_{\mathrm{2}} } {\int}}{V}\mathrm{sin}\:\omega{t}−{V}_{\gamma} \:{dt}=? \\ $$$${t}_{\mathrm{1}} \:{and}\:{t}_{\mathrm{2}} \:{are}\:{solution}\:{to} \\ $$$${V}\mathrm{sin}\:\omega{t}={V}_{\gamma} \\ $$$${V}\geqslant{V}_{\gamma} \\ $$$${V}_{\gamma} \geqslant\mathrm{0} \\ $$$${and}\:{t}_{\mathrm{1}} <{t}_{\mathrm{2}} \\ $$

Question Number 652 Answers: 1 Comments: 0

if f:R→R is continuous and f(x+y)=f(x)+y 1. find f(x) 2. proof or disproof that f′(x)=1 3. if f(0)=0 proof or disproof that f(x)=x

$${if}\:{f}:\mathbb{R}\rightarrow\mathbb{R}\:{is}\:{continuous}\:{and} \\ $$$${f}\left({x}+{y}\right)={f}\left({x}\right)+{y} \\ $$$$\mathrm{1}.\:{find}\:{f}\left({x}\right) \\ $$$$\mathrm{2}.\:{proof}\:{or}\:{disproof}\:{that}\:{f}'\left({x}\right)=\mathrm{1} \\ $$$$\mathrm{3}.\:{if}\:{f}\left(\mathrm{0}\right)=\mathrm{0}\:{proof}\:{or}\:{disproof}\:{that}\:{f}\left({x}\right)={x} \\ $$

Question Number 650 Answers: 1 Comments: 0

f:R→R g:R→R f(x+y)=f(x)+f(y)g(y) g(x+y)=f(x)g(x)+g(y)

$${f}:\mathbb{R}\rightarrow\mathbb{R} \\ $$$${g}:\mathbb{R}\rightarrow\mathbb{R} \\ $$$${f}\left({x}+{y}\right)={f}\left({x}\right)+{f}\left({y}\right){g}\left({y}\right) \\ $$$${g}\left({x}+{y}\right)={f}\left({x}\right){g}\left({x}\right)+{g}\left({y}\right) \\ $$

Question Number 630 Answers: 0 Comments: 1

∫_(−(π/2)) ^(+(π/2)) ((sin x)/(cos x))dx ∫_(−(π/2)) ^(+(π/2)) ((sin x)/(cos x))cos(2nx)dx ∫_(−(π/2)) ^(+(π/2)) ((sin x)/(cos x))sin(2nx)dx n∈N^∗

$$\underset{−\frac{\pi}{\mathrm{2}}} {\overset{+\frac{\pi}{\mathrm{2}}} {\int}}\frac{\mathrm{sin}\:{x}}{\mathrm{cos}\:{x}}{dx} \\ $$$$\underset{−\frac{\pi}{\mathrm{2}}} {\overset{+\frac{\pi}{\mathrm{2}}} {\int}}\frac{\mathrm{sin}\:{x}}{\mathrm{cos}\:{x}}\mathrm{cos}\left(\mathrm{2}{nx}\right){dx} \\ $$$$\underset{−\frac{\pi}{\mathrm{2}}} {\overset{+\frac{\pi}{\mathrm{2}}} {\int}}\frac{\mathrm{sin}\:{x}}{\mathrm{cos}\:{x}}\mathrm{sin}\left(\mathrm{2}{nx}\right){dx} \\ $$$${n}\in\mathbb{N}^{\ast} \\ $$

Question Number 516 Answers: 0 Comments: 1

Find all triangles with consecutive integer sides and having an angle twice another angle.

$${Find}\:{all}\:{triangles}\:{with}\:{consecutive} \\ $$$${integer}\:{sides}\:{and}\:{having}\:{an}\:{angle}\:{twice} \\ $$$${another}\:{angle}. \\ $$

Question Number 510 Answers: 0 Comments: 1

proof or given a counter example: if p,q are prines with p>q, and ∃s prime such s∈(q,p) then p−q≤Σ_(r∈(q,p),r is prime) r

$${proof}\:{or}\:{given}\:{a}\:{counter}\:{example}: \\ $$$${if}\:{p},{q}\:{are}\:{prines}\:{with}\:{p}>{q},\:{and}\:\exists{s}\:{prime} \\ $$$${such}\:{s}\in\left({q},{p}\right)\:{then} \\ $$$${p}−{q}\leqslant\underset{{r}\in\left({q},{p}\right),{r}\:{is}\:{prime}} {\sum}{r} \\ $$$$ \\ $$

Question Number 497 Answers: 1 Comments: 0

∫e^x sin 2x dx

$$\int{e}^{{x}} \mathrm{sin}\:\mathrm{2}{x}\:{dx} \\ $$

Question Number 495 Answers: 1 Comments: 0

∫sec^3 xdx

$$\int\mathrm{sec}^{\mathrm{3}} {xdx} \\ $$

Question Number 487 Answers: 1 Comments: 0

∫(1/(1+(√(2x)))) dx

$$\int\frac{\mathrm{1}}{\mathrm{1}+\sqrt{\mathrm{2}{x}}}\:\:{dx} \\ $$

Question Number 483 Answers: 1 Comments: 0

∫(1/(√(x−1)))dx

$$\int\frac{\mathrm{1}}{\sqrt{{x}−\mathrm{1}}}{dx} \\ $$$$ \\ $$

Question Number 480 Answers: 0 Comments: 1

proof or given a counter example: for s∈{2,3,4,5} Σ_(i=1) ^n [(1/s^i )−(((−1)^i )/i^s )]≤Σ_(i=1) ^n ((s+1)/(si^s ))

$${proof}\:{or}\:{given}\:{a}\:{counter}\:{example}: \\ $$$${for}\:{s}\in\left\{\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{5}\right\} \\ $$$$\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\left[\frac{\mathrm{1}}{{s}^{{i}} }−\frac{\left(−\mathrm{1}\right)^{{i}} }{{i}^{{s}} }\right]\leqslant\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{{s}+\mathrm{1}}{{si}^{{s}} } \\ $$

Question Number 472 Answers: 1 Comments: 0

proof or give a counter−example: if nm is prime then mdc(n^2 ,m^2 )=1

$${proof}\:{or}\:{give}\:{a}\:{counter}−{example}: \\ $$$${if}\:{nm}\:{is}\:{prime}\:{then}\:\mathrm{mdc}\left({n}^{\mathrm{2}} ,{m}^{\mathrm{2}} \right)=\mathrm{1} \\ $$

Question Number 465 Answers: 0 Comments: 1

A particle moves with a central acceration varies as the cube of the distance. if it be projected from an apse at distance a from the origin with a velocity which is (√(2 )) times the velocity for a circle of radius a. show that the equation of path is its rcosθ/(√2) = a

$${A}\:{particle}\:{moves}\:{with}\:{a}\:{central}\:{acceration}\:{varies}\:{as}\:{the}\:{cube}\:{of}\:{the}\:{distance}.\:{if}\:{it}\:{be}\:{projected}\:{from}\:{an}\:{apse}\:{at}\:{distance}\:{a}\:{from}\:{the}\:{origin}\:{with}\:{a}\:{velocity}\:{which}\:{is}\:\sqrt{\mathrm{2}\:}\:{times}\:{the}\:{velocity}\:{for}\:{a}\:{circle}\:{of}\:{radius}\:{a}.\:{show}\:{that}\:{the}\:{equation}\:{of}\:{path}\:{is}\:{its}\:{r}\mathrm{cos}\theta/\sqrt{\mathrm{2}}\:=\:{a} \\ $$

Question Number 433 Answers: 1 Comments: 0

find tagent plane of surface x^2 +y^3 =z^4 at the point (28,8,6)

$$\mathrm{find}\:\mathrm{tagent}\:\mathrm{plane}\:\mathrm{of}\:\mathrm{surface} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{3}} ={z}^{\mathrm{4}} \\ $$$$\mathrm{at}\:\mathrm{the}\:\mathrm{point}\:\left(\mathrm{28},\mathrm{8},\mathrm{6}\right) \\ $$

Question Number 435 Answers: 0 Comments: 2

∫(√(cos x)) dx =....

$$\int\sqrt{{cos}\:{x}}\:{dx}\:=.... \\ $$$$ \\ $$

Question Number 405 Answers: 1 Comments: 0

}}4×66

$$\left.\right\}\left.\right\}\mathrm{4}×\mathrm{66} \\ $$

Question Number 382 Answers: 1 Comments: 3

Evaluate lim_(n⇒∞) ∫_0 ^1 (x^n /(cos x)) dx

$$\mathrm{Evaluate}\: \\ $$$$\mathrm{li}\underset{\mathrm{n}\Rightarrow\infty} {\mathrm{m}}\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\frac{{x}^{{n}} }{{cos}\:{x}}\:{dx} \\ $$

Question Number 374 Answers: 1 Comments: 0

Show that the function f(x)=(√x) continuous on [0,∞)

$${S}\mathrm{how}\:\mathrm{that} \\ $$$$\mathrm{the}\:\mathrm{function}\:{f}\left({x}\right)=\sqrt{{x}}\:\mathrm{continuous}\:\mathrm{on}\:\left[\mathrm{0},\infty\right) \\ $$

Question Number 368 Answers: 1 Comments: 1

lim_(x→0) [f((a/x)+b)−(a/x)f ′((a/x)+b)]=α. Then lim_(x→∞) f(x)=....

$${li}\underset{{x}\rightarrow\mathrm{0}} {{m}}\left[{f}\left(\frac{{a}}{{x}}+{b}\right)−\frac{{a}}{{x}}{f}\:'\left(\frac{{a}}{{x}}+{b}\right)\right]=\alpha.\:{Then} \\ $$$$\:{li}\underset{{x}\rightarrow\infty} {{m}f}\left({x}\right)=.... \\ $$$$ \\ $$$$ \\ $$

Question Number 364 Answers: 2 Comments: 0

lim_(n→∞) [((f(a+1/n))/(f(a)))]^n =....

$${li}\underset{{n}\rightarrow\infty} {{m}}\left[\frac{{f}\left({a}+\mathrm{1}/{n}\right)}{{f}\left({a}\right)}\right]^{{n}} =.... \\ $$

Question Number 264 Answers: 0 Comments: 3

If f(1)=1 and f ′(x)=(1/(x^2 +(f(x))^2 )) then lim_(x→∞) f(x)= ....

$${If}\:{f}\left(\mathrm{1}\right)=\mathrm{1}\:{and}\:{f}\:'\left({x}\right)=\frac{\mathrm{1}}{{x}^{\mathrm{2}} +\left({f}\left({x}\right)\right)^{\mathrm{2}} }\:{then}\: \\ $$$${li}\underset{{x}\rightarrow\infty} {{m}}\:{f}\left({x}\right)=\:.... \\ $$

Question Number 147 Answers: 1 Comments: 0

∫_0 ^3 ∫_0 ^3 (√(9−y^2 )) dydx = ....

$$\underset{\mathrm{0}} {\overset{\mathrm{3}} {\int}}\underset{\mathrm{0}} {\overset{\mathrm{3}} {\int}}\sqrt{\mathrm{9}−{y}^{\mathrm{2}} \:}\:{dydx}\:=\:.... \\ $$

Question Number 143 Answers: 2 Comments: 0

tan (90−θ)=?

$$\mathrm{tan}\:\left(\mathrm{90}−\theta\right)=? \\ $$

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