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AllQuestion and Answers: Page 1966

Question Number 10854 Answers: 1 Comments: 0

f : R → R f(x . f(x) + f(y)) = (f(x))^2 + y x,y ∈ R f(x) = ??

$${f}\::\:\mathbb{R}\:\rightarrow\:\mathbb{R} \\ $$$${f}\left({x}\:.\:{f}\left({x}\right)\:+\:{f}\left({y}\right)\right)\:=\:\left({f}\left({x}\right)\right)^{\mathrm{2}} \:+\:{y}\:\:\:\:\:\:\:\:\:\:{x},{y}\:\in\:\mathbb{R} \\ $$$$ \\ $$$${f}\left({x}\right)\:=\:?? \\ $$

Question Number 10853 Answers: 1 Comments: 0

(x + y)^n = Σ_(k=0) ^n ((n),(k) )x^k y^(n−k) (x − y)^n = ???????

$$\left({x}\:+\:{y}\right)^{{n}} \:=\:\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\begin{pmatrix}{{n}}\\{{k}}\end{pmatrix}{x}^{{k}} {y}^{{n}−{k}} \\ $$$$\left({x}\:−\:{y}\right)^{{n}} \:=\:??????? \\ $$

Question Number 10849 Answers: 0 Comments: 0

Given that f(x) = f(x + 1000) for every x ∈ R If ∫_0 ^3 f(x) = 30,what is the value of ∫_3 ^5 f(x + 2016) dx ?

$$\mathrm{Given}\:\mathrm{that}\:{f}\left({x}\right)\:=\:{f}\left({x}\:+\:\mathrm{1000}\right)\:\mathrm{for}\:\mathrm{every}\:{x}\:\in\:\mathbb{R} \\ $$$$\mathrm{If}\:\underset{\mathrm{0}} {\overset{\mathrm{3}} {\int}}\:{f}\left({x}\right)\:=\:\mathrm{30},\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\underset{\mathrm{3}} {\overset{\mathrm{5}} {\int}}\:{f}\left({x}\:+\:\mathrm{2016}\right)\:{dx}\:? \\ $$

Question Number 10846 Answers: 1 Comments: 0

Two parallel chords of length 24 cm and 10 cm which lies on opposite sides of a circle are 17 cm apart. Calculate the radius of the circle to the nearest whole number.

$$\mathrm{Two}\:\mathrm{parallel}\:\mathrm{chords}\:\mathrm{of}\:\mathrm{length}\:\mathrm{24}\:\mathrm{cm}\:\mathrm{and}\:\mathrm{10}\:\mathrm{cm}\:\mathrm{which}\:\mathrm{lies}\:\mathrm{on}\:\mathrm{opposite} \\ $$$$\mathrm{sides}\:\mathrm{of}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{are}\:\mathrm{17}\:\mathrm{cm}\:\mathrm{apart}.\:\mathrm{Calculate}\:\mathrm{the}\:\mathrm{radius}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circle} \\ $$$$\mathrm{to}\:\mathrm{the}\:\mathrm{nearest}\:\mathrm{whole}\:\mathrm{number}. \\ $$

Question Number 10837 Answers: 1 Comments: 0

given that sinA=((12)/(13))and sinB=(4/5), where A and B are acute angles, find cos(A−B) and sin(A+B)

$${given}\:{that}\:{sinA}=\frac{\mathrm{12}}{\mathrm{13}}{and}\:{sinB}=\frac{\mathrm{4}}{\mathrm{5}}, \\ $$$${where}\:{A}\:{and}\:{B}\:{are}\:{acute}\:{angles}, \\ $$$${find}\:{cos}\left({A}−{B}\right)\:{and}\:{sin}\left({A}+{B}\right) \\ $$$$ \\ $$

Question Number 10830 Answers: 2 Comments: 0

lim_(x→∞) ((3^x − 3^(−x) )/(3^x + 3^(−x) ))

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\:\frac{\mathrm{3}^{\mathrm{x}} \:−\:\mathrm{3}^{−\mathrm{x}} }{\mathrm{3}^{\mathrm{x}} \:+\:\mathrm{3}^{−\mathrm{x}} } \\ $$

Question Number 10825 Answers: 1 Comments: 0

Given that sin(x) − sin(y) = sin(θ) cos(x) + cos(y) = cos(θ) Show that cos(x + y) = −(1/2)

$$\mathrm{Given}\:\mathrm{that}\: \\ $$$$\mathrm{sin}\left(\mathrm{x}\right)\:−\:\mathrm{sin}\left(\mathrm{y}\right)\:=\:\mathrm{sin}\left(\theta\right) \\ $$$$\mathrm{cos}\left(\mathrm{x}\right)\:+\:\mathrm{cos}\left(\mathrm{y}\right)\:=\:\mathrm{cos}\left(\theta\right) \\ $$$$\mathrm{Show}\:\mathrm{that}\: \\ $$$$\mathrm{cos}\left(\mathrm{x}\:+\:\mathrm{y}\right)\:=\:−\frac{\mathrm{1}}{\mathrm{2}} \\ $$

Question Number 10820 Answers: 1 Comments: 0

Question Number 10836 Answers: 1 Comments: 0

solve cos2θ−3cosθ=1 for o≤θ≤2π

$${solve}\:{cos}\mathrm{2}\theta−\mathrm{3}{cos}\theta=\mathrm{1} \\ $$$${for}\:{o}\leqslant\theta\leqslant\mathrm{2}\pi \\ $$

Question Number 10815 Answers: 0 Comments: 0

Evalute ∫(x^(2 ) + 9)^9 dx .

$${Evalute}\:\:\int\left({x}^{\mathrm{2}\:} \:\:+\:\mathrm{9}\right)^{\mathrm{9}} \:{dx}\:. \\ $$

Question Number 10807 Answers: 2 Comments: 2

Question Number 10795 Answers: 1 Comments: 1

by use sketching determine the range or(ranges) of the value x can take for each of the following inqualities (i) 3x^2 −19x−6≤0 (ii)2x^2 −5x−3≥0

$$\mathrm{by}\:\mathrm{use}\:\mathrm{sketching}\:\mathrm{determine}\:\mathrm{the}\:\mathrm{range} \\ $$$$\mathrm{or}\left(\mathrm{ranges}\right)\:\mathrm{of}\:\mathrm{the}\:\mathrm{value}\:\mathrm{x}\:\mathrm{can}\:\mathrm{take}\:\mathrm{for} \\ $$$$\mathrm{each}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{inqualities} \\ $$$$\left(\mathrm{i}\right)\:\mathrm{3x}^{\mathrm{2}} −\mathrm{19x}−\mathrm{6}\leqslant\mathrm{0} \\ $$$$\left(\mathrm{ii}\right)\mathrm{2x}^{\mathrm{2}} −\mathrm{5x}−\mathrm{3}\geqslant\mathrm{0} \\ $$

Question Number 10794 Answers: 0 Comments: 1

find the range or(ranges) of the value x can take for x+6>∣2x+3∣

$$\mathrm{find}\:\mathrm{the}\:\mathrm{range}\:\mathrm{or}\left(\mathrm{ranges}\right)\:\mathrm{of}\:\mathrm{the}\:\mathrm{value} \\ $$$$\mathrm{x}\:\mathrm{can}\:\mathrm{take}\:\mathrm{for}\:\mathrm{x}+\mathrm{6}>\mid\mathrm{2x}+\mathrm{3}\mid \\ $$

Question Number 10793 Answers: 1 Comments: 0

let A = determinant ((4,(4k),k),(0,k,(4k)),(0,0,4)) if det(A^2 )=16 then ∣k∣ is?

$$\mathrm{let}\:\:\mathrm{A}\:=\begin{vmatrix}{\mathrm{4}}&{\mathrm{4k}}&{\mathrm{k}}\\{\mathrm{0}}&{\mathrm{k}}&{\mathrm{4k}}\\{\mathrm{0}}&{\mathrm{0}}&{\mathrm{4}}\end{vmatrix}\:\mathrm{if}\:\mathrm{det}\left(\mathrm{A}^{\mathrm{2}} \right)=\mathrm{16} \\ $$$$\mathrm{then}\:\mid\mathrm{k}\mid\:\mathrm{is}? \\ $$

Question Number 10790 Answers: 1 Comments: 0

∫_0 ^(π/2) (√(sin x)) dx

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \sqrt{\mathrm{sin}\:{x}}\:{dx} \\ $$

Question Number 10789 Answers: 1 Comments: 0

(2+3i)x^2 −(3−2i)y=2x−3y+5i

$$\left(\mathrm{2}+\mathrm{3}{i}\right){x}^{\mathrm{2}} −\left(\mathrm{3}−\mathrm{2}{i}\right){y}=\mathrm{2}{x}−\mathrm{3}{y}+\mathrm{5}{i} \\ $$

Question Number 10788 Answers: 1 Comments: 0

factorise the expression sin4x−sinx

$${factorise}\:{the}\:{expression}\:{sin}\mathrm{4}{x}−{sinx} \\ $$

Question Number 10787 Answers: 0 Comments: 2

Question Number 10829 Answers: 2 Comments: 0

lim_(x→1) ∫_( 1) ^( x) ((e^t^2 (dt))/(x^2 − 1)) (a) 1 (b) 0 (c) e/2 (d) e

$$\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\int_{\:\mathrm{1}} ^{\:\mathrm{x}} \:\:\:\frac{\mathrm{e}^{\mathrm{t}^{\mathrm{2}} } \:\left(\mathrm{dt}\right)}{\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{1}}\: \\ $$$$\left(\mathrm{a}\right)\:\mathrm{1}\:\left(\mathrm{b}\right)\:\mathrm{0}\:\left(\mathrm{c}\right)\:\mathrm{e}/\mathrm{2}\:\left(\mathrm{d}\right)\:\mathrm{e} \\ $$

Question Number 10777 Answers: 2 Comments: 2

find the range or (ranges) of value x can take for x+6>[2x+3]

$$\:\mathrm{find}\:\mathrm{the}\:\mathrm{range}\:\mathrm{or}\:\left(\mathrm{ranges}\right)\:\mathrm{of}\:\mathrm{value}\:\mathrm{x} \\ $$$$\mathrm{can}\:\mathrm{take}\:\mathrm{for}\:\:\mathrm{x}+\mathrm{6}>\left[\mathrm{2x}+\mathrm{3}\right] \\ $$

Question Number 10776 Answers: 1 Comments: 0

if D = determinant ((1,(3sinθ),1),((sinθ),1,(3cosθ)),(1,(sinθ),1)) the maxi mum value of D is?

$$\mathrm{if}\:\:\:\mathrm{D}\:=\begin{vmatrix}{\mathrm{1}}&{\mathrm{3sin}\theta}&{\mathrm{1}}\\{\mathrm{sin}\theta}&{\mathrm{1}}&{\mathrm{3cos}\theta}\\{\mathrm{1}}&{\mathrm{sin}\theta}&{\mathrm{1}}\end{vmatrix}\:\mathrm{the}\:\mathrm{maxi} \\ $$$$\mathrm{mum}\:\mathrm{value}\:\mathrm{of}\:\mathrm{D}\:\mathrm{is}? \\ $$

Question Number 10774 Answers: 1 Comments: 0

Question Number 10773 Answers: 1 Comments: 0

Question Number 10768 Answers: 0 Comments: 2

Solve the following differential equations by power series. y^(//) −2y = 4x^2 e^x^2

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{following}\:\mathrm{differential}\:\mathrm{equations}\: \\ $$$$\mathrm{by}\:\mathrm{power}\:\mathrm{series}. \\ $$$${y}^{//} −\mathrm{2}{y}\:=\:\mathrm{4}{x}^{\mathrm{2}} {e}^{{x}^{\mathrm{2}} } \\ $$

Question Number 10762 Answers: 0 Comments: 0

in eac of the following problems you are given a function on the interval −π<x<π. Sketch several periods of the corresponding periodic function of period 2π. Expand the periodic function in a sine−consine Fourier series. F(x) = { _0 ^(x+π) _(0<x<π) ^(−π<x<0)

$$\mathrm{in}\:\mathrm{eac}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{problems}\:\mathrm{you}\:{are} \\ $$$$\mathrm{given}\:\mathrm{a}\:\mathrm{function}\:\mathrm{on}\:\mathrm{the}\:\mathrm{interval}\:\:−\pi<{x}<\pi. \\ $$$$\mathrm{Sketch}\:\mathrm{several}\:\mathrm{periods}\:\mathrm{of}\:\mathrm{the}\:\mathrm{corresponding} \\ $$$$\mathrm{periodic}\:\mathrm{function}\:\mathrm{of}\:\mathrm{period}\:\mathrm{2}\pi.\:\mathrm{Expand}\:\mathrm{the} \\ $$$$\mathrm{periodic}\:\mathrm{function}\:\mathrm{in}\:\mathrm{a}\:\mathrm{sine}−\mathrm{consine}\:\mathrm{Fourier} \\ $$$$\mathrm{series}.\:\: \\ $$$$ \\ $$$${F}\left({x}\right)\:=\:\left\{\overset{{x}+\pi} {\:}_{\mathrm{0}} \:\:\:\overset{−\pi<{x}<\mathrm{0}} {\:}_{\mathrm{0}<{x}<\pi} \right. \\ $$

Question Number 10761 Answers: 0 Comments: 0

Express the following Γ function using tables and evaluate them using a table or Γ functions. ∫_0 ^∞ x^(−1/3) e^(−8x ) dx

$$\mathrm{Express}\:\mathrm{the}\:\mathrm{following}\:\Gamma\:\mathrm{function}\:\mathrm{using}\:\mathrm{tables} \\ $$$$\mathrm{and}\:\mathrm{evaluate}\:\mathrm{them}\:\mathrm{using}\:\mathrm{a}\:\mathrm{table}\:\mathrm{or}\:\Gamma\:\mathrm{functions}. \\ $$$$ \\ $$$$\int_{\mathrm{0}} ^{\infty} {x}^{−\mathrm{1}/\mathrm{3}} {e}^{−\mathrm{8}{x}\:} {dx} \\ $$

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