Question and Answers Forum

AllQuestion and Answers: Page 1805

Question Number 21738 Answers: 1 Comments: 0

What is the last digit from the sum of 1 . 2^1 + 2 . 2^2 + 3 . 2^3 + ... + 50 . 2^(50) ?

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{last}\:\mathrm{digit}\:\mathrm{from}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of} \\ $$$$\mathrm{1}\:.\:\mathrm{2}^{\mathrm{1}} \:+\:\mathrm{2}\:.\:\mathrm{2}^{\mathrm{2}} \:+\:\mathrm{3}\:.\:\mathrm{2}^{\mathrm{3}} \:+\:...\:+\:\mathrm{50}\:.\:\mathrm{2}^{\mathrm{50}} \:? \\ $$

Question Number 21784 Answers: 0 Comments: 0

Call a positive integer n good if there are n integers, positive or negative, and not necessarily distinct, such that their sum and product are both equal to n (e.g. 8 is good since 8=4∙2∙1∙1∙1∙1(−1)(−1)=4+2+1+1+1 +1+(−1)+(−1)). Show that integers of the form 4k + 1 (k ≥ 0) and 4l (l ≥ 2) are good.

$$\mathrm{Call}\:\mathrm{a}\:\mathrm{positive}\:\mathrm{integer}\:{n}\:\boldsymbol{\mathrm{good}}\:\mathrm{if}\:\mathrm{there} \\ $$$$\mathrm{are}\:{n}\:\mathrm{integers},\:\mathrm{positive}\:\mathrm{or}\:\mathrm{negative},\:\mathrm{and} \\ $$$$\mathrm{not}\:\mathrm{necessarily}\:\mathrm{distinct},\:\mathrm{such}\:\mathrm{that}\:\mathrm{their} \\ $$$$\mathrm{sum}\:\mathrm{and}\:\mathrm{product}\:\mathrm{are}\:\mathrm{both}\:\mathrm{equal}\:\mathrm{to}\:{n} \\ $$$$\left(\mathrm{e}.\mathrm{g}.\:\mathrm{8}\:\mathrm{is}\:\boldsymbol{\mathrm{good}}\:\mathrm{since}\right. \\ $$$$\mathrm{8}=\mathrm{4}\centerdot\mathrm{2}\centerdot\mathrm{1}\centerdot\mathrm{1}\centerdot\mathrm{1}\centerdot\mathrm{1}\left(−\mathrm{1}\right)\left(−\mathrm{1}\right)=\mathrm{4}+\mathrm{2}+\mathrm{1}+\mathrm{1}+\mathrm{1} \\ $$$$\left.+\mathrm{1}+\left(−\mathrm{1}\right)+\left(−\mathrm{1}\right)\right). \\ $$$$\mathrm{Show}\:\mathrm{that}\:\mathrm{integers}\:\mathrm{of}\:\mathrm{the}\:\mathrm{form}\:\mathrm{4}{k}\:+\:\mathrm{1} \\ $$$$\left({k}\:\geqslant\:\mathrm{0}\right)\:\mathrm{and}\:\mathrm{4}{l}\:\left({l}\:\geqslant\:\mathrm{2}\right)\:\mathrm{are}\:\boldsymbol{\mathrm{good}}. \\ $$

Question Number 21782 Answers: 0 Comments: 0

a_1 =1, a_(n+1) =(a_n /(√(a_n +n+1))) Σ_(n=1) ^∞ a_n =?

$${a}_{\mathrm{1}} =\mathrm{1},\:{a}_{{n}+\mathrm{1}} =\frac{{a}_{{n}} }{\sqrt{{a}_{{n}} +{n}+\mathrm{1}}} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}{a}_{{n}} =? \\ $$

Question Number 21781 Answers: 2 Comments: 0

Which is greater 10^(11) or 11^(10) ?

$$\mathrm{Which}\:\mathrm{is}\:\mathrm{greater}\:\mathrm{10}^{\mathrm{11}} \:\mathrm{or}\:\mathrm{11}^{\mathrm{10}} ? \\ $$

Question Number 21733 Answers: 1 Comments: 0

If p is one of roots from x^2 − 2x + 6 = 0 then p^4 + 16p is equal to ...

$$\mathrm{If}\:{p}\:\mathrm{is}\:\mathrm{one}\:\mathrm{of}\:\:\mathrm{roots}\:\mathrm{from}\:{x}^{\mathrm{2}} \:−\:\mathrm{2}{x}\:+\:\mathrm{6}\:=\:\mathrm{0} \\ $$$$\mathrm{then}\:{p}^{\mathrm{4}} \:+\:\mathrm{16}{p}\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:... \\ $$

Question Number 21732 Answers: 0 Comments: 2

(i) Find the first three terms in the expansion of (2 − x)^6 in ascending power of x. (ii) Find the value of k for which there is no term in x^2 in the expansion (1 + kx)(2 − x)^6

$$\left(\mathrm{i}\right)\:\mathrm{Find}\:\mathrm{the}\:\mathrm{first}\:\mathrm{three}\:\mathrm{terms}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\:\left(\mathrm{2}\:−\:\mathrm{x}\right)^{\mathrm{6}} \:\mathrm{in}\:\mathrm{ascending}\:\mathrm{power} \\ $$$$\mathrm{of}\:\mathrm{x}. \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{k}\:\mathrm{for}\:\mathrm{which}\:\mathrm{there}\:\mathrm{is}\:\mathrm{no}\:\mathrm{term}\:\mathrm{in}\:\mathrm{x}^{\mathrm{2}} \:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion} \\ $$$$\left(\mathrm{1}\:+\:\mathrm{kx}\right)\left(\mathrm{2}\:−\:\mathrm{x}\right)^{\mathrm{6}} \\ $$

Question Number 21731 Answers: 0 Comments: 0

A very flexible uniform chain of mass M and length L is suspended vertically so that its lower end just touches the surface of a table. When the upper end of the chain is released, it falls with each link coming to rest the instant it strikes the table. Find the force exerted by the chain on the table at the moment when x part of chain has already rested on the table.

Question Number 21722 Answers: 1 Comments: 0

Question Number 21721 Answers: 1 Comments: 0

∫_0 ^(π/2) sin^2 xcos^3 xdx

$$\int_{\mathrm{0}} ^{\pi/\mathrm{2}} {sin}^{\mathrm{2}} {xcos}^{\mathrm{3}} {xdx} \\ $$

Question Number 21720 Answers: 1 Comments: 0

∫sin^5 θdθ

$$\int{sin}^{\mathrm{5}} \theta{d}\theta \\ $$

Question Number 21713 Answers: 0 Comments: 7

A constant force F = 20 N acts on a block of mass 2 kg which is connected to two blocks of masses m_1 = 1 kg and m_2 = 2 kg. Calculate the accelerations produced in all the three blocks. Assume pulleys are frictionless and weightless.

$$\mathrm{A}\:\mathrm{constant}\:\mathrm{force}\:{F}\:=\:\mathrm{20}\:\mathrm{N}\:\mathrm{acts}\:\mathrm{on}\:\mathrm{a} \\ $$$$\mathrm{block}\:\mathrm{of}\:\mathrm{mass}\:\mathrm{2}\:\mathrm{kg}\:\mathrm{which}\:\mathrm{is}\:\mathrm{connected}\:\mathrm{to} \\ $$$$\mathrm{two}\:\mathrm{blocks}\:\mathrm{of}\:\mathrm{masses}\:{m}_{\mathrm{1}} \:=\:\mathrm{1}\:\mathrm{kg}\:\mathrm{and} \\ $$$${m}_{\mathrm{2}} \:=\:\mathrm{2}\:\mathrm{kg}.\:\mathrm{Calculate}\:\mathrm{the}\:\mathrm{accelerations} \\ $$$$\mathrm{produced}\:\mathrm{in}\:\mathrm{all}\:\mathrm{the}\:\mathrm{three}\:\mathrm{blocks}.\:\mathrm{Assume} \\ $$$$\mathrm{pulleys}\:\mathrm{are}\:\mathrm{frictionless}\:\mathrm{and}\:\mathrm{weightless}. \\ $$

Question Number 21707 Answers: 0 Comments: 3

∫_(π/6) ^(π/3) (1/2)cot^2 2θdθ

$$\int_{\pi/\mathrm{6}} ^{\pi/\mathrm{3}} \frac{\mathrm{1}}{\mathrm{2}}{cot}^{\mathrm{2}} \mathrm{2}\theta{d}\theta \\ $$

Question Number 21702 Answers: 0 Comments: 1

∫_(π/6) ^(π/3) 1/2cot^2 2θdθ

$$\int_{\pi/\mathrm{6}} ^{\pi/\mathrm{3}} \mathrm{1}/\mathrm{2}{cot}^{\mathrm{2}} \mathrm{2}\theta{d}\theta \\ $$

Question Number 21701 Answers: 2 Comments: 1

∫2cot^2 2t

$$\int\mathrm{2}{cot}^{\mathrm{2}} \mathrm{2}{t} \\ $$

Question Number 21686 Answers: 0 Comments: 0

I have 6 friends and during a vacation I met them during several dinners. I found that I dined with all the 6 exactly on 1 day; with every 5 of them on 2 days; with every 4 of them on 3 days; with every 3 of them on 4 days; with every 2 of them on 5 days. Further every friend was present at 7 dinners and every friend was absent at 7 dinners. How many dinners did I have alone?

$$\mathrm{I}\:\mathrm{have}\:\mathrm{6}\:\mathrm{friends}\:\mathrm{and}\:\mathrm{during}\:\mathrm{a}\:\mathrm{vacation} \\ $$$$\mathrm{I}\:\mathrm{met}\:\mathrm{them}\:\mathrm{during}\:\mathrm{several}\:\mathrm{dinners}.\:\mathrm{I} \\ $$$$\mathrm{found}\:\mathrm{that}\:\mathrm{I}\:\mathrm{dined}\:\mathrm{with}\:\mathrm{all}\:\mathrm{the}\:\mathrm{6}\:\mathrm{exactly} \\ $$$$\mathrm{on}\:\mathrm{1}\:\mathrm{day};\:\mathrm{with}\:\mathrm{every}\:\mathrm{5}\:\mathrm{of}\:\mathrm{them}\:\mathrm{on}\:\mathrm{2}\:\mathrm{days}; \\ $$$$\mathrm{with}\:\mathrm{every}\:\mathrm{4}\:\mathrm{of}\:\mathrm{them}\:\mathrm{on}\:\mathrm{3}\:\mathrm{days};\:\mathrm{with} \\ $$$$\mathrm{every}\:\mathrm{3}\:\mathrm{of}\:\mathrm{them}\:\mathrm{on}\:\mathrm{4}\:\mathrm{days};\:\mathrm{with}\:\mathrm{every}\:\mathrm{2} \\ $$$$\mathrm{of}\:\mathrm{them}\:\mathrm{on}\:\mathrm{5}\:\mathrm{days}.\:\mathrm{Further}\:\mathrm{every}\:\mathrm{friend} \\ $$$$\mathrm{was}\:\mathrm{present}\:\mathrm{at}\:\mathrm{7}\:\mathrm{dinners}\:\mathrm{and}\:\mathrm{every} \\ $$$$\mathrm{friend}\:\mathrm{was}\:\mathrm{absent}\:\mathrm{at}\:\mathrm{7}\:\mathrm{dinners}.\:\mathrm{How} \\ $$$$\mathrm{many}\:\mathrm{dinners}\:\mathrm{did}\:\mathrm{I}\:\mathrm{have}\:\mathrm{alone}? \\ $$

Question Number 21685 Answers: 0 Comments: 0

In a group of ten persons, each person is asked to write the sum of the ages of all the other 9 persons. If all the ten sums form the 9-element set {82, 83, 84, 85, 87, 90, 91, 92} find the individual ages of the persons (assuming them to be whole numbers of years).

$$\mathrm{In}\:\mathrm{a}\:\mathrm{group}\:\mathrm{of}\:\mathrm{ten}\:\mathrm{persons},\:\mathrm{each}\:\mathrm{person} \\ $$$$\mathrm{is}\:\mathrm{asked}\:\mathrm{to}\:\mathrm{write}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{ages}\:\mathrm{of} \\ $$$$\mathrm{all}\:\mathrm{the}\:\mathrm{other}\:\mathrm{9}\:\mathrm{persons}.\:\mathrm{If}\:\mathrm{all}\:\mathrm{the}\:\mathrm{ten} \\ $$$$\mathrm{sums}\:\mathrm{form}\:\mathrm{the}\:\mathrm{9}-\mathrm{element}\:\mathrm{set}\:\left\{\mathrm{82},\:\mathrm{83},\:\mathrm{84},\right. \\ $$$$\left.\mathrm{85},\:\mathrm{87},\:\mathrm{90},\:\mathrm{91},\:\mathrm{92}\right\}\:\mathrm{find}\:\mathrm{the}\:\mathrm{individual} \\ $$$$\mathrm{ages}\:\mathrm{of}\:\mathrm{the}\:\mathrm{persons}\:\left(\mathrm{assuming}\:\mathrm{them}\:\mathrm{to}\right. \\ $$$$\left.\mathrm{be}\:\mathrm{whole}\:\mathrm{numbers}\:\mathrm{of}\:\mathrm{years}\right). \\ $$

Question Number 21683 Answers: 1 Comments: 2

Suppose A_1 A_2 ...A_(20) is a 20-sided regular polygon. How many non-isosceles (scalene) triangles can be formed whose vertices are among the vertices of the polygon but whose sides are not the sides of the polygon?

$$\mathrm{Suppose}\:{A}_{\mathrm{1}} {A}_{\mathrm{2}} ...{A}_{\mathrm{20}} \:\mathrm{is}\:\mathrm{a}\:\mathrm{20}-\mathrm{sided}\:\mathrm{regular} \\ $$$$\mathrm{polygon}.\:\mathrm{How}\:\mathrm{many}\:\mathrm{non}-\mathrm{isosceles} \\ $$$$\left(\mathrm{scalene}\right)\:\mathrm{triangles}\:\mathrm{can}\:\mathrm{be}\:\mathrm{formed}\:\mathrm{whose} \\ $$$$\mathrm{vertices}\:\mathrm{are}\:\mathrm{among}\:\mathrm{the}\:\mathrm{vertices}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{polygon}\:\mathrm{but}\:\mathrm{whose}\:\mathrm{sides}\:\mathrm{are}\:\mathrm{not}\:\mathrm{the} \\ $$$$\mathrm{sides}\:\mathrm{of}\:\mathrm{the}\:\mathrm{polygon}? \\ $$

Question Number 21682 Answers: 1 Comments: 0

Prove that the ten′s digit of any power of 3 is even. [e.g. the ten′s digit of 3^6 = 729 is 2].

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{ten}'\mathrm{s}\:\mathrm{digit}\:\mathrm{of}\:\mathrm{any}\:\mathrm{power} \\ $$$$\mathrm{of}\:\mathrm{3}\:\mathrm{is}\:\mathrm{even}.\:\left[\mathrm{e}.\mathrm{g}.\:\mathrm{the}\:\mathrm{ten}'\mathrm{s}\:\mathrm{digit}\:\mathrm{of}\:\mathrm{3}^{\mathrm{6}} \:=\right. \\ $$$$\left.\mathrm{729}\:\mathrm{is}\:\mathrm{2}\right]. \\ $$

Question Number 21670 Answers: 1 Comments: 1

The block of mass 2 kg and 3 kg are placed one over the other. The contact surfaces are rough with coefficient of friction μ_1 = 0.2, μ_2 = 0.06. A force F = (1/2)t N (where t is in second) is applied on upper block in the direction. (Given that g = 10 m/s^2 ) 1. The relative slipping between the blocks occurs at t = 2. Friction force acting between the two blocks at t = 8 s 3. The acceleration time graph for 3 kg block is

$$\mathrm{The}\:\mathrm{block}\:\mathrm{of}\:\mathrm{mass}\:\mathrm{2}\:\mathrm{kg}\:\mathrm{and}\:\mathrm{3}\:\mathrm{kg}\:\mathrm{are} \\ $$$$\mathrm{placed}\:\mathrm{one}\:\mathrm{over}\:\mathrm{the}\:\mathrm{other}.\:\mathrm{The}\:\mathrm{contact} \\ $$$$\mathrm{surfaces}\:\mathrm{are}\:\mathrm{rough}\:\mathrm{with}\:\mathrm{coefficient}\:\mathrm{of} \\ $$$$\mathrm{friction}\:\mu_{\mathrm{1}} \:=\:\mathrm{0}.\mathrm{2},\:\mu_{\mathrm{2}} \:=\:\mathrm{0}.\mathrm{06}.\:\mathrm{A}\:\mathrm{force}\:{F}\:= \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}{t}\:\mathrm{N}\:\left(\mathrm{where}\:{t}\:\mathrm{is}\:\mathrm{in}\:\mathrm{second}\right)\:\mathrm{is}\:\mathrm{applied} \\ $$$$\mathrm{on}\:\mathrm{upper}\:\mathrm{block}\:\mathrm{in}\:\mathrm{the}\:\mathrm{direction}.\:\left(\mathrm{Given}\right. \\ $$$$\left.\mathrm{that}\:{g}\:=\:\mathrm{10}\:\mathrm{m}/\mathrm{s}^{\mathrm{2}} \right) \\ $$$$\mathrm{1}.\:\mathrm{The}\:\mathrm{relative}\:\mathrm{slipping}\:\mathrm{between}\:\mathrm{the} \\ $$$$\mathrm{blocks}\:\mathrm{occurs}\:\mathrm{at}\:{t}\:= \\ $$$$\mathrm{2}.\:\mathrm{Friction}\:\mathrm{force}\:\mathrm{acting}\:\mathrm{between}\:\mathrm{the}\:\mathrm{two} \\ $$$$\mathrm{blocks}\:\mathrm{at}\:{t}\:=\:\mathrm{8}\:\mathrm{s} \\ $$$$\mathrm{3}.\:\mathrm{The}\:\mathrm{acceleration}\:\mathrm{time}\:\mathrm{graph}\:\mathrm{for}\:\mathrm{3}\:\mathrm{kg} \\ $$$$\mathrm{block}\:\mathrm{is} \\ $$

Question Number 21662 Answers: 1 Comments: 0

Question Number 21661 Answers: 0 Comments: 2

Question Number 21679 Answers: 1 Comments: 0

∫((secθ dθ)/(1−secθ))

$$\int\frac{\mathrm{sec}\theta\:\mathrm{d}\theta}{\mathrm{1}−\mathrm{sec}\theta} \\ $$

Question Number 21656 Answers: 0 Comments: 4

Let A(x) is a cubic polynomial and B(x) = (x −1)(x − 2)(x − 3) Find how many C(x) so that B(C(x)) = B(x) . A(x)

$$\mathrm{Let}\:{A}\left({x}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{cubic}\:\mathrm{polynomial}\:\mathrm{and}\:{B}\left({x}\right)\:=\:\left({x}\:−\mathrm{1}\right)\left({x}\:−\:\mathrm{2}\right)\left({x}\:−\:\mathrm{3}\right) \\ $$$$\mathrm{Find}\:\mathrm{how}\:\mathrm{many}\:{C}\left({x}\right)\:\mathrm{so}\:\mathrm{that} \\ $$$${B}\left({C}\left({x}\right)\right)\:=\:{B}\left({x}\right)\:.\:{A}\left({x}\right) \\ $$

Question Number 21655 Answers: 1 Comments: 0

(((2017)),(( 0)) ) + (((2017)),(( 2)) ) + (((2017)),(( 4)) ) + (((2017)),(( 6)) ) + ... + (((2017)),((2016)) ) is equal to ...

$$\begin{pmatrix}{\mathrm{2017}}\\{\:\:\:\:\mathrm{0}}\end{pmatrix}\:+\:\begin{pmatrix}{\mathrm{2017}}\\{\:\:\:\:\mathrm{2}}\end{pmatrix}\:+\:\begin{pmatrix}{\mathrm{2017}}\\{\:\:\:\:\mathrm{4}}\end{pmatrix}\:+\:\begin{pmatrix}{\mathrm{2017}}\\{\:\:\:\:\mathrm{6}}\end{pmatrix}\:+\:...\:+\:\begin{pmatrix}{\mathrm{2017}}\\{\mathrm{2016}}\end{pmatrix} \\ $$$$\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:... \\ $$

Question Number 21654 Answers: 0 Comments: 0

Find the minimum value of Q that satisfy: ∣xy(x^2 − y^2 ) + yz(y^2 − z^2 ) + zx(z^2 − x^2 )∣ ≤ Q(x^2 + y^2 + z^2 )^2

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:{Q}\:\mathrm{that}\:\mathrm{satisfy}: \\ $$$$\mid{xy}\left({x}^{\mathrm{2}} \:−\:{y}^{\mathrm{2}} \right)\:+\:{yz}\left({y}^{\mathrm{2}} \:−\:{z}^{\mathrm{2}} \right)\:+\:{zx}\left({z}^{\mathrm{2}} \:−\:{x}^{\mathrm{2}} \right)\mid\:\leqslant\:{Q}\left({x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:+\:{z}^{\mathrm{2}} \right)^{\mathrm{2}} \\ $$

Question Number 21653 Answers: 0 Comments: 0

Find all pair of solutions (x,y) that satisfy the equation: ((7x^2 − 13xy + 7y^2 ))^(1/3) = ∣x − y∣ + 1

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{pair}\:\mathrm{of}\:\mathrm{solutions}\:\left({x},{y}\right)\:\mathrm{that}\:\mathrm{satisfy}\:\mathrm{the}\:\mathrm{equation}: \\ $$$$\sqrt[{\mathrm{3}}]{\mathrm{7}{x}^{\mathrm{2}} \:−\:\mathrm{13}{xy}\:+\:\mathrm{7}{y}^{\mathrm{2}} }\:=\:\mid{x}\:−\:{y}\mid\:+\:\mathrm{1} \\ $$

Pg 1800 Pg 1801 Pg 1802 Pg 1803 Pg 1804 Pg 1805 Pg 1806 Pg 1807 Pg 1808 Pg 1809

Contact: info@tinkutara.com