Question and Answers Forum

AllQuestion and Answers: Page 1964

Question Number 9474 Answers: 0 Comments: 0

Question Number 9471 Answers: 1 Comments: 0

prove by mathematical induction a + (a + d) + (a + 2d) + ... + [a + (n − 1)d] = (1/2)n[2a + (n − 1)d]

$$\mathrm{prove}\:\mathrm{by}\:\mathrm{mathematical}\:\mathrm{induction} \\ $$$$\mathrm{a}\:+\:\left(\mathrm{a}\:+\:\mathrm{d}\right)\:+\:\left(\mathrm{a}\:+\:\mathrm{2d}\right)\:+\:...\:+\:\left[\mathrm{a}\:+\:\left(\mathrm{n}\:−\:\mathrm{1}\right)\mathrm{d}\right]\:=\:\frac{\mathrm{1}}{\mathrm{2}}\mathrm{n}\left[\mathrm{2a}\:+\:\left(\mathrm{n}\:−\:\mathrm{1}\right)\mathrm{d}\right]\: \\ $$

Question Number 9470 Answers: 1 Comments: 2

prove by mathematical induction. (1/(1.3)) + (1/(2.5)) + (1/(3.7)) + ... + (1/((2n − 1)(2n + 1))) = (n/(2n + 1))

$$\mathrm{prove}\:\mathrm{by}\:\mathrm{mathematical}\:\mathrm{induction}. \\ $$$$\frac{\mathrm{1}}{\mathrm{1}.\mathrm{3}}\:+\:\frac{\mathrm{1}}{\mathrm{2}.\mathrm{5}}\:+\:\frac{\mathrm{1}}{\mathrm{3}.\mathrm{7}}\:+\:...\:+\:\frac{\mathrm{1}}{\left(\mathrm{2n}\:−\:\mathrm{1}\right)\left(\mathrm{2n}\:+\:\mathrm{1}\right)}\:=\:\frac{\mathrm{n}}{\mathrm{2n}\:+\:\mathrm{1}} \\ $$

Question Number 9467 Answers: 1 Comments: 0

∫x^2 ((√(1 − x^2 ))) dx

$$\int\mathrm{x}^{\mathrm{2}} \left(\sqrt{\mathrm{1}\:−\:\mathrm{x}^{\mathrm{2}} }\right)\:\mathrm{dx} \\ $$

Question Number 9458 Answers: 0 Comments: 1

If the zeta function of 2 is 𝛇(2) = Σ_(n=1) ^∞ (1/n^2 ) 𝛇(2) = (𝛑^2 /6) the sum of infinite rational numbers, why converges for (𝛑^2 /6), an irrational number?

$$\mathrm{If}\:\mathrm{the}\:\mathrm{zeta}\:\mathrm{function}\:\mathrm{of}\:\mathrm{2}\:\mathrm{is} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\zeta}\left(\mathrm{2}\right)\:=\:\underset{\boldsymbol{{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\boldsymbol{{n}}^{\mathrm{2}} } \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\zeta}\left(\mathrm{2}\right)\:=\:\frac{\boldsymbol{\pi}^{\mathrm{2}} }{\mathrm{6}} \\ $$$$\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{infinite}\:\boldsymbol{\mathrm{rational}}\:\mathrm{numbers}, \\ $$$$\mathrm{why}\:\mathrm{converges}\:\mathrm{for}\:\frac{\boldsymbol{\pi}^{\mathrm{2}} }{\mathrm{6}},\:\mathrm{an}\:\boldsymbol{\mathrm{irrational}} \\ $$$$\mathrm{number}? \\ $$

Question Number 9455 Answers: 1 Comments: 0

Solve the following system of equations 3x−8z=−11 2y−3x=−3 y−4z=−7 See the comment of Q#9439.

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{following}\:\:\mathrm{system}\:\mathrm{of}\:\mathrm{equations} \\ $$$$\mathrm{3x}−\mathrm{8z}=−\mathrm{11} \\ $$$$\mathrm{2y}−\mathrm{3x}=−\mathrm{3} \\ $$$$\mathrm{y}−\mathrm{4z}=−\mathrm{7} \\ $$$${See}\:{the}\:{comment}\:{of}\:{Q}#\mathrm{9439}. \\ $$

Question Number 9453 Answers: 1 Comments: 0

find dc′s dr′s of a mormal to the plane 2x+(5/2)y+(7/8)z=23

$$\mathrm{find}\:\mathrm{dc}'\mathrm{s}\:\mathrm{dr}'\mathrm{s}\:\mathrm{of}\:\mathrm{a}\:\mathrm{mormal}\:\mathrm{to}\:\mathrm{the}\: \\ $$$$\mathrm{plane}\:\mathrm{2x}+\frac{\mathrm{5}}{\mathrm{2}}\mathrm{y}+\frac{\mathrm{7}}{\mathrm{8}}\mathrm{z}=\mathrm{23} \\ $$

Question Number 9449 Answers: 1 Comments: 0

2xy = x + y ...... (i) 6xz = 6z − 2x ..... (ii) 3yz = 3y + 4z ....... (iii) Solve for x, y and z

$$\mathrm{2xy}\:=\:\mathrm{x}\:+\:\mathrm{y}\:\:\:\:\:......\:\left(\mathrm{i}\right) \\ $$$$\mathrm{6xz}\:=\:\mathrm{6z}\:−\:\mathrm{2x}\:\:\:\:.....\:\left(\mathrm{ii}\right) \\ $$$$\mathrm{3yz}\:=\:\mathrm{3y}\:+\:\mathrm{4z}\:\:\:.......\:\left(\mathrm{iii}\right) \\ $$$$ \\ $$$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{x},\:\mathrm{y}\:\mathrm{and}\:\mathrm{z} \\ $$

Question Number 9439 Answers: 3 Comments: 1

Solve for x , y and z (x + 1)(y + 3) = 8 ....... (i) (y + 3)(z − 1) = 3 ......... (ii) (z − 1)(x + 1) = 2 ......... (iii)

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{x}\:,\:\mathrm{y}\:\mathrm{and}\:\mathrm{z} \\ $$$$\left(\mathrm{x}\:+\:\mathrm{1}\right)\left(\mathrm{y}\:+\:\mathrm{3}\right)\:=\:\mathrm{8}\:\:\:.......\:\left(\mathrm{i}\right) \\ $$$$\left(\mathrm{y}\:+\:\mathrm{3}\right)\left(\mathrm{z}\:−\:\mathrm{1}\right)\:=\:\mathrm{3}\:\:.........\:\left(\mathrm{ii}\right) \\ $$$$\left(\mathrm{z}\:−\:\mathrm{1}\right)\left(\mathrm{x}\:+\:\mathrm{1}\right)\:=\:\mathrm{2}\:\:\:.........\:\left(\mathrm{iii}\right) \\ $$

Question Number 9438 Answers: 1 Comments: 0

Solve for x, y and z 2xy = x + y ...... (i) 6xz = 6z − 2x ....... (ii) 3yz = 3y + 4z ......... (iii) That is the correct question sir.

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{x},\:\mathrm{y}\:\mathrm{and}\:\mathrm{z} \\ $$$$\mathrm{2xy}\:=\:\mathrm{x}\:+\:\mathrm{y}\:\:\:\:\:\:......\:\left(\mathrm{i}\right) \\ $$$$\mathrm{6xz}\:=\:\mathrm{6z}\:−\:\mathrm{2x}\:\:\:\:.......\:\left(\mathrm{ii}\right) \\ $$$$\mathrm{3yz}\:=\:\mathrm{3y}\:+\:\mathrm{4z}\:\:\:\:.........\:\left(\mathrm{iii}\right) \\ $$$$ \\ $$$$\mathrm{That}\:\mathrm{is}\:\mathrm{the}\:\mathrm{correct}\:\mathrm{question}\:\mathrm{sir}. \\ $$

Question Number 9436 Answers: 2 Comments: 0

If they said that Σ_(k=1) ^∞ k diverges, why 1 + 2 + 3 + 4 + ... = − (1/(12)) ?

$$\mathrm{If}\:\mathrm{they}\:\mathrm{said}\:\mathrm{that}\:\underset{{k}=\mathrm{1}} {\overset{\infty} {\sum}}\:{k}\:\mathrm{diverges},\:\mathrm{why} \\ $$$$\mathrm{1}\:+\:\mathrm{2}\:+\:\mathrm{3}\:+\:\mathrm{4}\:+\:...\:=\:−\:\frac{\mathrm{1}}{\mathrm{12}}\:? \\ $$

Question Number 9460 Answers: 1 Comments: 0

Question Number 9461 Answers: 1 Comments: 0

(dy/dx) = ((1 + y)/(2 + x)) solve the differential equation.

$$\frac{\mathrm{dy}}{\mathrm{dx}}\:=\:\frac{\mathrm{1}\:+\:\mathrm{y}}{\mathrm{2}\:+\:\mathrm{x}} \\ $$$$\mathrm{solve}\:\mathrm{the}\:\mathrm{differential}\:\mathrm{equation}. \\ $$

Question Number 9424 Answers: 1 Comments: 0

Solve for x, y : x, y ∈ R x^3 − 3xy^2 = − 46 ........ (i) 3x^2 y − y^3 = 9 ..........(ii)

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{x},\:\mathrm{y}\::\:\:\mathrm{x},\:\mathrm{y}\:\in\:\mathrm{R} \\ $$$$\mathrm{x}^{\mathrm{3}} \:−\:\mathrm{3xy}^{\mathrm{2}} \:=\:−\:\mathrm{46}\:\:\:........\:\left(\mathrm{i}\right) \\ $$$$\mathrm{3x}^{\mathrm{2}} \mathrm{y}\:−\:\mathrm{y}^{\mathrm{3}} \:=\:\mathrm{9}\:\:\:..........\left(\mathrm{ii}\right) \\ $$

Question Number 9423 Answers: 1 Comments: 3

if x and y are two sets such that n(x) =17 , n(y)=23 and n(X∪Y) =38, find n(X∪Y).

$${if}\:{x}\:{and}\:{y}\:{are}\:{two}\:{sets}\:{such}\:{that}\:{n}\left({x}\right) \\ $$$$=\mathrm{17}\:,\:{n}\left({y}\right)=\mathrm{23}\:{and}\:{n}\left({X}\cup{Y}\right)\:=\mathrm{38}, \\ $$$$\boldsymbol{{find}}\:\boldsymbol{{n}}\left({X}\cup{Y}\right). \\ $$

Question Number 9422 Answers: 0 Comments: 0

The value of ^(95) C_4 +Σ_(j=1) ^5 ^(100−j) C_3 is

$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\:^{\mathrm{95}} {C}_{\mathrm{4}} +\underset{{j}=\mathrm{1}} {\overset{\mathrm{5}} {\sum}}\:^{\mathrm{100}−{j}} {C}_{\mathrm{3}} \:\:\mathrm{is} \\ $$

Question Number 9421 Answers: 1 Comments: 0

If the sum of the coefficients in the expansion of (a+b)^n is 4096, then the greatest coefficient in the expansion is

$$\mathrm{If}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{coefficients}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{expansion}\:\mathrm{of}\:\left({a}+{b}\right)^{{n}} \:\mathrm{is}\:\mathrm{4096},\:\mathrm{then}\:\mathrm{the} \\ $$$$\mathrm{greatest}\:\mathrm{coefficient}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{is} \\ $$

Question Number 9414 Answers: 1 Comments: 0

tanA+cot=A= 2cosec2A

$${tanA}+{cot}={A}=\:\mathrm{2}{cosec}\mathrm{2}{A} \\ $$

Question Number 9420 Answers: 0 Comments: 0

x and y are two binary numbers which are in 4 − bit 2′s complement formate, where x = 00102 and y = 11012 : clearly, y is a negative number. what is the result of x + y in decimal formate.

$$\mathrm{x}\:\mathrm{and}\:\mathrm{y}\:\mathrm{are}\:\mathrm{two}\:\mathrm{binary}\:\mathrm{numbers}\:\mathrm{which}\:\mathrm{are}\: \\ $$$$\mathrm{in}\:\mathrm{4}\:−\:\mathrm{bit}\:\mathrm{2}'\mathrm{s}\:\mathrm{complement}\:\mathrm{formate},\: \\ $$$$\mathrm{where}\:\mathrm{x}\:=\:\mathrm{00102}\:\mathrm{and}\:\mathrm{y}\:=\:\mathrm{11012}\::\:\mathrm{clearly}, \\ $$$$\mathrm{y}\:\mathrm{is}\:\mathrm{a}\:\mathrm{negative}\:\mathrm{number}.\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{result} \\ $$$$\mathrm{of}\:\mathrm{x}\:+\:\mathrm{y}\:\mathrm{in}\:\mathrm{decimal}\:\mathrm{formate}. \\ $$

Question Number 9404 Answers: 1 Comments: 0