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AllQuestion and Answers: Page 1932

Question Number 13725 Answers: 0 Comments: 2

(1/7)=.142857^(−) (1/7) is a recurring decimal of period 6. What will be the period of (1/7^(20) )?

$$\frac{\mathrm{1}}{\mathrm{7}}=.\overline {\mathrm{142857}} \\ $$$$\frac{\mathrm{1}}{\mathrm{7}}\:\mathrm{is}\:\mathrm{a}\:\mathrm{recurring}\:\mathrm{decimal}\:\mathrm{of}\:\mathrm{period}\:\mathrm{6}. \\ $$$$ \\ $$$$\mathrm{What}\:\mathrm{will}\:\mathrm{be}\:\mathrm{the}\:\mathrm{period}\:\mathrm{of}\:\frac{\mathrm{1}}{\mathrm{7}^{\mathrm{20}} }? \\ $$

Question Number 13724 Answers: 2 Comments: 3

Expansion of 1000! has 249, 0′s at the end Find the first non−zero digit from right. 1000!=......d000...00 What is the value of d?

$$\mathrm{Expansion}\:\mathrm{of}\:\mathrm{1000}!\:\mathrm{has}\:\mathrm{249},\:\mathrm{0}'{s}\:\mathrm{at}\:\mathrm{the}\:\mathrm{end} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{first}\:\mathrm{non}−\mathrm{zero}\:\mathrm{digit}\:\mathrm{from} \\ $$$$\mathrm{right}. \\ $$$$\mathrm{1000}!=......{d}\mathrm{000}...\mathrm{00} \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{d}? \\ $$

Question Number 13721 Answers: 0 Comments: 0

what is NBS?

$$\mathrm{what}\:\mathrm{is}\:\mathrm{NBS}? \\ $$

Question Number 13706 Answers: 1 Comments: 0

The volume of a right circular cone is 5 litres . Calculate the volumes of the two parts into which the cone is divided by a plane parallel to the base , One third of the way down from the vertex to the base. Give your answer to the nearest ml.

Question Number 13705 Answers: 1 Comments: 3

Assuming no air resistance, and angle of projection α=(π/4) , find the ratio of the length of trajectory L of a projectile motion (by the time it hits the ground) to its horizontal range R on ground. (L/R)=?

$${Assuming}\:{no}\:{air}\:{resistance}, \\ $$$${and}\:{angle}\:{of}\:{projection}\:\alpha=\frac{\pi}{\mathrm{4}}\:, \\ $$$${find}\:{the}\:{ratio}\:{of}\:{the}\:{length}\:{of} \\ $$$${trajectory}\:\boldsymbol{{L}}\:{of}\:{a}\:{projectile}\:{motion}\: \\ $$$$\left({by}\:{the}\:{time}\:{it}\:{hits}\:{the}\:{ground}\right) \\ $$$${to}\:{its}\:{horizontal}\:{range}\:\boldsymbol{{R}}\:{on}\: \\ $$$${ground}.\:\:\:\:\:\:\frac{\boldsymbol{{L}}}{\boldsymbol{{R}}}=? \\ $$

Question Number 13695 Answers: 1 Comments: 2

Volume of a bubble is 3 times larger when it reaches the surface from the bottom of the lake. What is the depth of the lake? (A) 10 m (D) 40 m (B) 20 m (E) 50 m (C) 30 m

$$\mathrm{Volume}\:\mathrm{of}\:\mathrm{a}\:\mathrm{bubble}\:\mathrm{is}\:\mathrm{3}\:\mathrm{times}\:\mathrm{larger} \\ $$$$\mathrm{when}\:\mathrm{it}\:\mathrm{reaches}\:\mathrm{the}\:\mathrm{surface}\:\mathrm{from}\:\mathrm{the}\:\mathrm{bottom} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{lake}. \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{depth}\:\mathrm{of}\:\mathrm{the}\:\mathrm{lake}? \\ $$$$ \\ $$$$\left(\mathrm{A}\right)\:\mathrm{10}\:\mathrm{m}\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{D}\right)\:\mathrm{40}\:\mathrm{m} \\ $$$$\left(\mathrm{B}\right)\:\mathrm{20}\:\mathrm{m}\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{E}\right)\:\mathrm{50}\:\mathrm{m} \\ $$$$\left(\mathrm{C}\right)\:\mathrm{30}\:\mathrm{m} \\ $$

Question Number 13688 Answers: 1 Comments: 0

Question Number 13687 Answers: 0 Comments: 4

Question Number 13681 Answers: 1 Comments: 3

Question Number 13658 Answers: 1 Comments: 0

Prove that cos^2 x + cos^2 3x + cos^2 5x + ... to n terms = (1/2)[n + ((sin4nx)/(2sin2x))]

$$\mathrm{Prove}\:\mathrm{that} \\ $$$$\mathrm{cos}^{\mathrm{2}} {x}\:+\:\mathrm{cos}^{\mathrm{2}} \mathrm{3}{x}\:+\:\mathrm{cos}^{\mathrm{2}} \mathrm{5}{x}\:+\:...\:\mathrm{to}\:{n}\:\mathrm{terms} \\ $$$$=\:\frac{\mathrm{1}}{\mathrm{2}}\left[{n}\:+\:\frac{\mathrm{sin4}{nx}}{\mathrm{2sin2}{x}}\right] \\ $$

Question Number 13654 Answers: 1 Comments: 0

The sum of the series sinθ + sin(((n − 4)/(n − 2)))θ + sin(((n − 6)/(n − 2)))θ + ... n terms is equal to (1) sin(((nθ)/(2 − n))) (2) cos(((2nθ)/(2 − n))) (3) tannθ (4) cotnθ

$$\mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{series} \\ $$$$\mathrm{sin}\theta\:+\:\mathrm{sin}\left(\frac{{n}\:−\:\mathrm{4}}{{n}\:−\:\mathrm{2}}\right)\theta\:+\:\mathrm{sin}\left(\frac{{n}\:−\:\mathrm{6}}{{n}\:−\:\mathrm{2}}\right)\theta\:+\:...\:{n}\:\mathrm{terms} \\ $$$$\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{sin}\left(\frac{{n}\theta}{\mathrm{2}\:−\:{n}}\right) \\ $$$$\left(\mathrm{2}\right)\:\mathrm{cos}\left(\frac{\mathrm{2}{n}\theta}{\mathrm{2}\:−\:{n}}\right) \\ $$$$\left(\mathrm{3}\right)\:\mathrm{tan}{n}\theta \\ $$$$\left(\mathrm{4}\right)\:\mathrm{cot}{n}\theta \\ $$

Question Number 13649 Answers: 1 Comments: 0

2xyy′+(x−1)y^2 =x^2 e^x

$$\mathrm{2}{xyy}'+\left({x}−\mathrm{1}\right){y}^{\mathrm{2}} ={x}^{\mathrm{2}} {e}^{{x}} \\ $$

Question Number 13647 Answers: 0 Comments: 6

x^2 +y^2 =5.....(1) 3x^2 +xy+y^2 =1.....(2) please help find x and y

$$\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} =\mathrm{5}.....\left(\mathrm{1}\right) \\ $$$$\mathrm{3x}^{\mathrm{2}} +\mathrm{xy}+\mathrm{y}^{\mathrm{2}} =\mathrm{1}.....\left(\mathrm{2}\right) \\ $$$$ \\ $$$$ \\ $$$$\mathrm{please}\:\mathrm{help}\:\mathrm{find}\:\mathrm{x}\:\mathrm{and}\:\mathrm{y} \\ $$

Question Number 13645 Answers: 0 Comments: 0

please is factor theorem and error and trial the same? please help cause i think theres a difference but i cant explain it. Thankz.

$$\mathrm{please}\:\mathrm{is}\:\mathrm{factor}\:\mathrm{theorem}\:\mathrm{and}\:\mathrm{error} \\ $$$$\mathrm{and}\:\mathrm{trial}\:\mathrm{the}\:\mathrm{same}?\:\mathrm{please}\:\mathrm{help}\: \\ $$$$\mathrm{cause}\:\mathrm{i}\:\mathrm{think}\:\mathrm{theres}\:\mathrm{a}\:\mathrm{difference} \\ $$$$\mathrm{but}\:\mathrm{i}\:\mathrm{cant}\:\mathrm{explain}\:\mathrm{it}. \\ $$$$ \\ $$$$\mathrm{Thankz}. \\ $$

Question Number 13636 Answers: 1 Comments: 0

A steam envine of efficiency 70% burns 20g of coal to produce 10kJ of energy.If it burns 200g of coal per second,calculate its output power.

$$\mathrm{A}\:\mathrm{steam}\:\mathrm{envine}\:\mathrm{of}\:\mathrm{efficiency}\:\mathrm{70\%} \\ $$$$\mathrm{burns}\:\mathrm{20g}\:\mathrm{of}\:\mathrm{coal}\:\mathrm{to}\:\mathrm{produce}\:\mathrm{10kJ} \\ $$$$\mathrm{of}\:\mathrm{energy}.\mathrm{If}\:\mathrm{it}\:\mathrm{burns}\:\mathrm{200g}\:\mathrm{of}\:\mathrm{coal}\: \\ $$$$\mathrm{per}\:\mathrm{second},\mathrm{calculate}\:\mathrm{its}\:\mathrm{output} \\ $$$$\mathrm{power}. \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 13627 Answers: 1 Comments: 0

Question Number 13626 Answers: 1 Comments: 0

Prove that cosech^(−1) (x) = sinh^(−1) ((1/x))

$$\mathrm{Prove}\:\mathrm{that} \\ $$$$\mathrm{cosech}^{−\mathrm{1}} \left(\mathrm{x}\right)\:=\:\mathrm{sinh}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{x}}\right) \\ $$

Question Number 13609 Answers: 1 Comments: 0

Question Number 13608 Answers: 0 Comments: 2

if distance is given by x(t)=2t+5 then ,the acceleration at 4s is............

$$\mathrm{if}\:\mathrm{distance}\:\mathrm{is}\:\mathrm{given}\:\mathrm{by}\:\mathrm{x}\left(\mathrm{t}\right)=\mathrm{2t}+\mathrm{5}\:\mathrm{then}\:,\mathrm{the} \\ $$$$\mathrm{acceleration}\:\mathrm{at}\:\mathrm{4s}\:\mathrm{is}............ \\ $$

Question Number 13606 Answers: 1 Comments: 0

Show that 19^(93) − 13^(99) is a positive integer divisible by 162.

$$\mathrm{Show}\:\mathrm{that}\:\mathrm{19}^{\mathrm{93}} \:−\:\mathrm{13}^{\mathrm{99}} \:\mathrm{is}\:\mathrm{a}\:\mathrm{positive} \\ $$$$\mathrm{integer}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{162}. \\ $$

Question Number 13732 Answers: 1 Comments: 3

Sum the following: tan x+2tan 2x+2^2 tan 2^2 x+...+2^n tan 2^n x

$$\mathrm{Sum}\:\mathrm{the}\:\mathrm{following}: \\ $$$$\mathrm{tan}\:{x}+\mathrm{2tan}\:\mathrm{2}{x}+\mathrm{2}^{\mathrm{2}} \mathrm{tan}\:\mathrm{2}^{\mathrm{2}} {x}+...+\mathrm{2}^{{n}} \mathrm{tan}\:\mathrm{2}^{{n}} {x} \\ $$

Question Number 13601 Answers: 1 Comments: 0

Let f : R − {(3/5)} → R be defined by f(x) = ((3x + 2)/(5x − 3)) . Then, (a) f^(−1) (x) = x (b) f^(−1) (x) = −f(x) (c) fof(x) = −x (d) f^(−1) (x) = (1/(19))f(x)

$$\mathrm{Let}\:{f}\::\:\mathbb{R}\:−\:\left\{\frac{\mathrm{3}}{\mathrm{5}}\right\}\:\rightarrow\:\mathbb{R}\:\mathrm{be}\:\mathrm{defined}\:\mathrm{by} \\ $$$${f}\left({x}\right)\:=\:\frac{\mathrm{3}{x}\:+\:\mathrm{2}}{\mathrm{5}{x}\:−\:\mathrm{3}}\:.\:\mathrm{Then}, \\ $$$$\left(\mathrm{a}\right)\:{f}^{−\mathrm{1}} \left({x}\right)\:=\:{x} \\ $$$$\left(\mathrm{b}\right)\:{f}^{−\mathrm{1}} \left({x}\right)\:=\:−{f}\left({x}\right) \\ $$$$\left(\mathrm{c}\right)\:{fof}\left({x}\right)\:=\:−{x} \\ $$$$\left(\mathrm{d}\right)\:{f}^{−\mathrm{1}} \left({x}\right)\:=\:\frac{\mathrm{1}}{\mathrm{19}}{f}\left({x}\right) \\ $$

Question Number 13622 Answers: 1 Comments: 0

Evaluate: ∫_0 ^(2π) e^(x/2) sin ((x/2) + (π/4))dx

$$\mathrm{Evaluate}:\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} {e}^{\frac{{x}}{\mathrm{2}}} \mathrm{sin}\:\left(\frac{{x}}{\mathrm{2}}\:+\:\frac{\pi}{\mathrm{4}}\right){dx} \\ $$

Question Number 13623 Answers: 1 Comments: 0

Evaluate: ∫_0 ^(2π) e^x cos ((π/4) + (x/2))dx

$$\mathrm{Evaluate}:\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} {e}^{{x}} \:\mathrm{cos}\:\left(\frac{\pi}{\mathrm{4}}\:+\:\frac{{x}}{\mathrm{2}}\right){dx} \\ $$

Question Number 13598 Answers: 0 Comments: 1

If g(x) = x^2 + x − 2 and (1/2) gof(x) = 2x^2 − 5x + 2, then prove that f(x) = 2x − 3.

$$\mathrm{If}\:{g}\left({x}\right)\:=\:{x}^{\mathrm{2}} \:+\:{x}\:−\:\mathrm{2}\:\mathrm{and} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\:{gof}\left({x}\right)\:=\:\mathrm{2}{x}^{\mathrm{2}} \:−\:\mathrm{5}{x}\:+\:\mathrm{2},\:\mathrm{then}\:\mathrm{prove} \\ $$$$\mathrm{that}\:{f}\left({x}\right)\:=\:\mathrm{2}{x}\:−\:\mathrm{3}. \\ $$

Question Number 13595 Answers: 1 Comments: 0

If f : R → (−1, 1) is defined by f(x) = ((−x∣x∣)/(1 + x^2 )) , then prove that f^(−1) (x) = −sgn(x)(√((∣x∣)/(1 − ∣x∣)))

$$\mathrm{If}\:{f}\::\:\mathbb{R}\:\rightarrow\:\left(−\mathrm{1},\:\mathrm{1}\right)\:\mathrm{is}\:\mathrm{defined}\:\mathrm{by} \\ $$$${f}\left({x}\right)\:=\:\frac{−{x}\mid{x}\mid}{\mathrm{1}\:+\:{x}^{\mathrm{2}} }\:,\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that} \\ $$$${f}^{−\mathrm{1}} \left({x}\right)\:=\:−\mathrm{sgn}\left({x}\right)\sqrt{\frac{\mid{x}\mid}{\mathrm{1}\:−\:\mid{x}\mid}} \\ $$

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