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Question Number 110888 Answers: 3 Comments: 0

....calculus.... please solve : Ω_1 =∫_0 ^( (π/4)) ((√(tan(x))) +(√(cot(x))) )dx=?? Ω_2 =∫_0 ^(π/4) tan(x)ln((1+tan^2 (x)))dx =?? ...M.N.july 1970#... Good luck

$$\:\:\:\:\:\:\:\:\:\:....{calculus}.... \\ $$$${please}\:{solve}\:: \\ $$$$ \\ $$$$\Omega_{\mathrm{1}} =\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{4}}} \left(\sqrt{{tan}\left({x}\right)}\:+\sqrt{{cot}\left({x}\right)}\:\right){dx}=?? \\ $$$$\:\Omega_{\mathrm{2}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} {tan}\left({x}\right){ln}\left(\left(\mathrm{1}+{tan}^{\mathrm{2}} \left({x}\right)\right)\right){dx}\:=?? \\ $$$$\:\: \\ $$$$\:\:\:\:\:\:\:\:...\mathscr{M}.\mathscr{N}.{july}\:\mathrm{1970}#... \\ $$$$\:\mathscr{G}{ood}\:{luck} \\ $$$$ \\ $$$$ \\ $$

Question Number 110887 Answers: 1 Comments: 2

Question Number 110528 Answers: 1 Comments: 0

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