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Question Number 110525 Answers: 0 Comments: 1

Question Number 110524 Answers: 1 Comments: 0

A blind man is to place 6 letters into 6 pigeon holes, how many ways can atleast 5 letters be wrongly placed? (Note that only one letter must be in a pigeon hole).

$$\mathrm{A}\:\mathrm{blind}\:\mathrm{man}\:\mathrm{is}\:\mathrm{to}\:\mathrm{place}\:\mathrm{6}\:\mathrm{letters}\:\mathrm{into}\:\mathrm{6} \\ $$$$\mathrm{pigeon}\:\mathrm{holes},\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can} \\ $$$$\mathrm{atleast}\:\mathrm{5}\:\mathrm{letters}\:\mathrm{be}\:\mathrm{wrongly}\:\mathrm{placed}? \\ $$$$\left(\mathrm{Note}\:\mathrm{that}\:\mathrm{only}\:\mathrm{one}\:\mathrm{letter}\:\mathrm{must}\:\mathrm{be}\:\mathrm{in}\:\mathrm{a}\right. \\ $$$$\left.\mathrm{pigeon}\:\mathrm{hole}\right). \\ $$

Question Number 110523 Answers: 0 Comments: 0

In the square PQRS, K is the midpoint of PQ, L is the midpoint of QR, M is the midpoint RS, N is the midpoint of SP and O is the midpoint of KM. A line segment is drawn from each pair of points from (K,L,M,N,O,P,Q,R,S). These line segments create points of intersections not contained in (K,L,M,N,O,P,Q,R,S). How many distinct such points are there?

$$ \\ $$$$\mathrm{In}\:\mathrm{the}\:\mathrm{square}\:\mathrm{PQRS},\:\mathrm{K}\:\mathrm{is}\:\mathrm{the} \\ $$$$\mathrm{midpoint}\:\mathrm{of}\:\mathrm{PQ},\:\mathrm{L}\:\mathrm{is}\:\mathrm{the}\:\mathrm{midpoint}\:\mathrm{of} \\ $$$$\mathrm{QR},\:\mathrm{M}\:\mathrm{is}\:\mathrm{the}\:\mathrm{midpoint}\:\mathrm{RS},\:\mathrm{N}\:\mathrm{is}\:\mathrm{the} \\ $$$$\mathrm{midpoint}\:\mathrm{of}\:\mathrm{SP}\:\mathrm{and}\:\mathrm{O}\:\mathrm{is}\:\mathrm{the}\:\mathrm{midpoint} \\ $$$$\mathrm{of}\:\mathrm{KM}.\:\mathrm{A}\:\mathrm{line}\:\mathrm{segment}\:\mathrm{is}\:\mathrm{drawn}\:\mathrm{from} \\ $$$$\mathrm{each}\:\mathrm{pair}\:\mathrm{of}\:\mathrm{points}\:\mathrm{from} \\ $$$$\left(\mathrm{K},\mathrm{L},\mathrm{M},\mathrm{N},\mathrm{O},\mathrm{P},\mathrm{Q},\mathrm{R},\mathrm{S}\right).\:\mathrm{These}\:\mathrm{line} \\ $$$$\mathrm{segments}\:\mathrm{create}\:\mathrm{points}\:\mathrm{of} \\ $$$$\mathrm{intersections}\:\mathrm{not}\:\mathrm{contained}\:\mathrm{in} \\ $$$$\left(\mathrm{K},\mathrm{L},\mathrm{M},\mathrm{N},\mathrm{O},\mathrm{P},\mathrm{Q},\mathrm{R},\mathrm{S}\right).\:\mathrm{How}\:\mathrm{many} \\ $$$$\mathrm{distinct}\:\mathrm{such}\:\mathrm{points}\:\mathrm{are}\:\mathrm{there}? \\ $$

Question Number 110510 Answers: 0 Comments: 0

Question Number 110519 Answers: 1 Comments: 2

17x ≡ 3 (mod 29)

$$\:\:\:\:\:\:\mathrm{17x}\:\equiv\:\mathrm{3}\:\left(\mathrm{mod}\:\mathrm{29}\right) \\ $$

Question Number 110504 Answers: 0 Comments: 0

Given a transformation , T: C → C ; z → ω Show that if ω = ((z−i)/(z +1)) then z = ((ω + i)/(1−ω)). Hence the image of the line ∣z−i∣ = ∣z + 2∣ under the transformation T the ω−plane is a circle with center (−2,−i) and radius (√(10)) .

$$\mathrm{Given}\:\mathrm{a}\:\mathrm{transformation}\:,\:\mathcal{T}:\:\mathbb{C}\:\rightarrow\:\mathbb{C}\:;\:{z}\:\rightarrow\:\omega \\ $$$$\:\mathrm{Show}\:\mathrm{that}\:\mathrm{if}\:\omega\:=\:\frac{{z}−{i}}{{z}\:+\mathrm{1}}\:\mathrm{then}\:{z}\:=\:\frac{\omega\:+\:{i}}{\mathrm{1}−\omega}.\:\mathrm{Hence} \\ $$$$\mathrm{the}\:\mathrm{image}\:\mathrm{of}\:\mathrm{the}\:\mathrm{line}\:\mid{z}−{i}\mid\:=\:\mid{z}\:+\:\mathrm{2}\mid\:\mathrm{under}\:\mathrm{the}\:\mathrm{transformation} \\ $$$$\mathcal{T}\:\:\:\:\mathrm{the}\:\omega−\mathrm{plane}\:\mathrm{is}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{with}\:\mathrm{center}\:\left(−\mathrm{2},−{i}\right)\:\mathrm{and}\:\mathrm{radius}\:\sqrt{\mathrm{10}}\:.\: \\ $$

Question Number 110503 Answers: 4 Comments: 7

Question Number 110715 Answers: 1 Comments: 2

a,b,c,d are unit digits whose pairwise sums form an arithmetic progression. Given that a+b+c+d is even, find the common positive difference of the arithmetic progression.

$$\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\:\mathrm{are}\:\mathrm{unit}\:\mathrm{digits}\:\mathrm{whose} \\ $$$$\mathrm{pairwise}\:\mathrm{sums}\:\mathrm{form}\:\mathrm{an}\:\mathrm{arithmetic} \\ $$$$\mathrm{progression}.\:\mathrm{Given}\:\mathrm{that}\:\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}\:\mathrm{is} \\ $$$$\mathrm{even},\:\mathrm{find}\:\mathrm{the}\:\mathrm{common}\:\mathrm{positive} \\ $$$$\mathrm{difference}\:\mathrm{of}\:\mathrm{the}\:\mathrm{arithmetic} \\ $$$$\mathrm{progression}. \\ $$

Question Number 110498 Answers: 1 Comments: 3

How many ways can the letters in the word MATHEMATICS be rearranged such that the word formed either starts or ends with a vowel, and any three consecutive letters must contain a vowel?

$$\mathrm{How}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{the}\:\mathrm{letters}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{word}\:\mathrm{MATHEMATICS}\:\mathrm{be} \\ $$$$\mathrm{rearranged}\:\mathrm{such}\:\mathrm{that}\:\mathrm{the}\:\mathrm{word}\:\mathrm{formed} \\ $$$$\mathrm{either}\:\mathrm{starts}\:\mathrm{or}\:\mathrm{ends}\:\mathrm{with}\:\mathrm{a}\:\mathrm{vowel},\:\mathrm{and} \\ $$$$\mathrm{any}\:\mathrm{three}\:\mathrm{consecutive}\:\mathrm{letters}\:\mathrm{must} \\ $$$$\mathrm{contain}\:\mathrm{a}\:\mathrm{vowel}? \\ $$

Question Number 110490 Answers: 1 Comments: 0

e^(iπ) = −1

$$ \\ $$$${e}^{\mathrm{i}\pi} \:=\:−\mathrm{1} \\ $$

Question Number 110480 Answers: 0 Comments: 0

Question Number 110471 Answers: 1 Comments: 0

the sum of all x values (x^2 −6x+9)^((x−3)/(x−1)) = (x−3)^((x−4))

$$\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{all}\:\mathrm{x}\:\mathrm{values} \\ $$$$\left({x}^{\mathrm{2}} −\mathrm{6}{x}+\mathrm{9}\right)^{\frac{{x}−\mathrm{3}}{{x}−\mathrm{1}}} \:=\:\left({x}−\mathrm{3}\right)^{\left({x}−\mathrm{4}\right)} \\ $$

Question Number 110463 Answers: 0 Comments: 2

Find the gcd(n−1,n+1)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{gcd}\left({n}−\mathrm{1},{n}+\mathrm{1}\right) \\ $$

Question Number 110467 Answers: 2 Comments: 1

Question Number 110460 Answers: 0 Comments: 0

Find all continuous real function f such that f(ln(x+y))=f(sin(xy))+f(cos(y/x)) for all x,y∈R^+ .

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{continuous}\:\mathrm{real}\:\mathrm{function}\:{f}\: \\ $$$$\mathrm{such}\:\mathrm{that}\: \\ $$$${f}\left(\mathrm{ln}\left({x}+{y}\right)\right)={f}\left(\mathrm{sin}\left({xy}\right)\right)+{f}\left(\mathrm{cos}\left({y}/{x}\right)\right) \\ $$$$\mathrm{for}\:\mathrm{all}\:{x},{y}\in\mathbb{R}^{+} . \\ $$

Question Number 110456 Answers: 1 Comments: 0

If (f(x).g(x))′ = f(x)′ . g(x)′ find the function of f(x) .

$$\:\:\:{If}\:\left({f}\left({x}\right).{g}\left({x}\right)\right)'\:=\:{f}\left({x}\right)'\:.\:{g}\left({x}\right)'\: \\ $$$$\:{find}\:{the}\:{function}\:{of}\:{f}\left({x}\right)\:. \\ $$

Question Number 110451 Answers: 1 Comments: 0

calculate U_n =∫_([(1/n),n[^2 ) (x^2 −y^2 )e^(−x^2 −y^2 ) dxdy and lim_(n→+∞) U_n

$$\mathrm{calculate}\:\mathrm{U}_{\mathrm{n}} =\int_{\left[\frac{\mathrm{1}}{\mathrm{n}},\mathrm{n}\left[^{\mathrm{2}} \right.\right.} \:\:\:\:\left(\mathrm{x}^{\mathrm{2}} −\mathrm{y}^{\mathrm{2}} \right)\mathrm{e}^{−\mathrm{x}^{\mathrm{2}} −\mathrm{y}^{\mathrm{2}} } \mathrm{dxdy} \\ $$$$\mathrm{and}\:\mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \:\mathrm{U}_{\mathrm{n}} \\ $$

Question Number 110450 Answers: 1 Comments: 0

find ∫∫_([0,1]^2 ) ln(x^2 +3y^2 ) e^(−x^2 −3y^2 ) dxdy

$$\mathrm{find}\:\int\int_{\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{2}} } \:\:\:\mathrm{ln}\left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{3y}^{\mathrm{2}} \right)\:\mathrm{e}^{−\mathrm{x}^{\mathrm{2}} −\mathrm{3y}^{\mathrm{2}} } \:\mathrm{dxdy} \\ $$

Question Number 110449 Answers: 2 Comments: 0

find lim_(x→0) ((arctan(x−sinx)−arctan(1−cosx))/x^2 )

$$\mathrm{find}\:\mathrm{lim}_{\mathrm{x}\rightarrow\mathrm{0}} \:\frac{\mathrm{arctan}\left(\mathrm{x}−\mathrm{sinx}\right)−\mathrm{arctan}\left(\mathrm{1}−\mathrm{cosx}\right)}{\mathrm{x}^{\mathrm{2}} } \\ $$

Question Number 110448 Answers: 1 Comments: 0

calculate ∫_(−∞) ^(+∞) ((cos(2x))/((x^2 −4i)^3 ))dx (i=(√(−1)))

$$\mathrm{calculate}\:\:\int_{−\infty} ^{+\infty} \:\frac{\mathrm{cos}\left(\mathrm{2x}\right)}{\left(\mathrm{x}^{\mathrm{2}} −\mathrm{4i}\right)^{\mathrm{3}} }\mathrm{dx}\:\:\:\:\:\left(\mathrm{i}=\sqrt{−\mathrm{1}}\right) \\ $$

Question Number 110447 Answers: 1 Comments: 0

calculate ∫_(−∞) ^(+∞) (dx/((x^2 −ix +1)^2 )) (i=(√(−1)))

$$\mathrm{calculate}\:\int_{−\infty} ^{+\infty} \:\frac{\mathrm{dx}}{\left(\mathrm{x}^{\mathrm{2}} −\mathrm{ix}\:+\mathrm{1}\right)^{\mathrm{2}} }\:\:\left(\mathrm{i}=\sqrt{−\mathrm{1}}\right) \\ $$

Question Number 110440 Answers: 1 Comments: 0

Given lim_(x→0) (f(x)+(1/(f(x)))) = 2 , find the value of lim_(x→0) f(x).

$$\mathrm{Given}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\mathrm{f}\left(\mathrm{x}\right)+\frac{\mathrm{1}}{\mathrm{f}\left(\mathrm{x}\right)}\right)\:=\:\mathrm{2}\:,\:\mathrm{find}\:\mathrm{the} \\ $$$$\mathrm{value}\:\mathrm{of}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\mathrm{f}\left(\mathrm{x}\right). \\ $$

Question Number 110436 Answers: 2 Comments: 0

If 2f(x) + f((1/x)) = 3x what is f(x)

$$\:\:\:\:\:\mathrm{If}\:\mathrm{2f}\left(\mathrm{x}\right)\:+\:\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{x}}\right)\:=\:\mathrm{3x} \\ $$$$\:\:\:\:\:\mathrm{what}\:\mathrm{is}\:\mathrm{f}\left(\mathrm{x}\right) \\ $$

Question Number 110434 Answers: 1 Comments: 1

The product of the four terms of an increasing arithmetic progression is a, their sum is b, and the sum of their reciprocal is c. Suppose that a,b,c form a geometric progression whose product is 8000, find the sum of the first and fourth term.

$$\mathrm{The}\:\mathrm{product}\:\mathrm{of}\:\mathrm{the}\:\mathrm{four}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{an} \\ $$$$\mathrm{increasing}\:\mathrm{arithmetic}\:\mathrm{progression}\:\mathrm{is}\:\mathrm{a}, \\ $$$$\mathrm{their}\:\mathrm{sum}\:\mathrm{is}\:\mathrm{b},\:\mathrm{and}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{their} \\ $$$$\mathrm{reciprocal}\:\mathrm{is}\:\mathrm{c}.\:\mathrm{Suppose}\:\mathrm{that}\:\mathrm{a},\mathrm{b},\mathrm{c}\:\mathrm{form} \\ $$$$\mathrm{a}\:\mathrm{geometric}\:\mathrm{progression}\:\mathrm{whose} \\ $$$$\mathrm{product}\:\mathrm{is}\:\mathrm{8000},\:\mathrm{find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{first}\:\mathrm{and}\:\mathrm{fourth}\:\mathrm{term}. \\ $$

Question Number 110425 Answers: 2 Comments: 3

Which of the following is not a factor of x^6 −56x+55 A. x−1 B. x^2 −x+5 C. x^3 +2x^2 −2x−11 D. x^4 +x^3 +4x^2 −9x+11 E. x^5 +x^4 +x^3 +x^2 +x−55 Please show all workings clearly. Thanks.

$$ \\ $$$$\mathrm{Which}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{is}\:\mathrm{not}\:\mathrm{a}\:\mathrm{factor} \\ $$$$\mathrm{of}\:\mathrm{x}^{\mathrm{6}} −\mathrm{56x}+\mathrm{55} \\ $$$$\mathrm{A}.\:\mathrm{x}−\mathrm{1}\:\mathrm{B}.\:\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{5}\:\mathrm{C}. \\ $$$$\mathrm{x}^{\mathrm{3}} +\mathrm{2x}^{\mathrm{2}} −\mathrm{2x}−\mathrm{11}\:\mathrm{D}. \\ $$$$\mathrm{x}^{\mathrm{4}} +\mathrm{x}^{\mathrm{3}} +\mathrm{4x}^{\mathrm{2}} −\mathrm{9x}+\mathrm{11}\:\mathrm{E}. \\ $$$$\mathrm{x}^{\mathrm{5}} +\mathrm{x}^{\mathrm{4}} +\mathrm{x}^{\mathrm{3}} +\mathrm{x}^{\mathrm{2}} +\mathrm{x}−\mathrm{55} \\ $$$$ \\ $$$$\mathrm{Please}\:\mathrm{show}\:\mathrm{all}\:\mathrm{workings}\:\mathrm{clearly}. \\ $$$$\mathrm{Thanks}. \\ $$

Question Number 110420 Answers: 1 Comments: 0

Prove that x^5 −3x^4 −17x^3 −x^2 −3x+17 cannot be factorized completely over the set of polynomials with integral coefficients.

$$\mathrm{Prove}\:\mathrm{that} \\ $$$$\mathrm{x}^{\mathrm{5}} −\mathrm{3x}^{\mathrm{4}} −\mathrm{17x}^{\mathrm{3}} −\mathrm{x}^{\mathrm{2}} −\mathrm{3x}+\mathrm{17}\:\mathrm{cannot}\:\mathrm{be} \\ $$$$\mathrm{factorized}\:\mathrm{completely}\:\mathrm{over}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of} \\ $$$$\mathrm{polynomials}\:\mathrm{with}\:\mathrm{integral}\:\mathrm{coefficients}. \\ $$

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