Question A circular cylinder is to fit inside a sphere of radius 10 cm. Calculate the maximum possible volume of the cylinder.(It is probably best to take as your independent variable the height,or half the height of the cylinder.) Answer Volume of the sphere is 4*pi*r^3 = 4*pi*10^3 =4000*pi Now, volume of a cylinder...
Question Is this continuous? How do i check if it is continuous or not? g(x) = e^(1/x) for x < 0 Sin inverse (x) for 0 =< x =< 1 Tan inverse [1 / (x-1)] for x > 1 Answer Its difficult to do “limits” and all by “typing”. I’ll give you an idea....
Question 1*2+2*3+3*4+4*5+…+n*(n+1)=1/3*n*(n+1)*(n+2) Answer 1*2+2*3+3*4+4*5+…+n*(n+1)=? Let S = 1*2+2*3+3*4+4*5+…+n*(n+1) … ( 1) Use. 1 + 2 + … + n = n*( n + 1 ) / 2 and 1^2 + 2^2 + … + n^2 = [ n*(n+1)*(2n+1)/6 ] So S = Sum [ k =1 to k =n for k *(...
Comments Off on Select the reason that proves this statement
Question Select the reason that proves this statement. Theorem: If a + c = b + c, then a = b. 1. Identity property of addition 2. Property of opposites 3. Given 4. Commutative property of addition 5. Associative property of addition Addition property of equality Answer Its actually “the additive inverse property”. A...
Comments Off on If a = b and c = d, then a + c = b + d. is this Deductive or Inductive?
Answer Its “deductive”. The idea is that “deductive” arguments means that you “deduced” your conclusion based on earlier “facts”. Inductive argument however is different. For example, whats the nth term or “pattern” for the below : 2, 4, 6 , … You “guess” that its 2(1), 2(2), 2(3) … and “inductively” guess that the...
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Question Prove that if a>0, then the absolute value of x < a iff -a<x<a? Answer a > 0. In order for | x | < a, we must have either if x is negative then its ‘value’ must be ‘less negative’ than ‘-a’ and hence ‘smaller in absolute value’ meaning x < –...
Comments Off on For how many whole numbers between 0 and 1000 does the sum of the digits equal 9 ?
Answer Let your whole number be ‘X’. Now, 0 < X < 1000 means that your numbers are either 3 digit, 2 digit and 1 digit. We consider each case seperately. Case 1. ( 1 digit ) I believe its obvious that only ‘9’ fits the criteria. Case 2 ( 2 digit number )...
Comments Off on How many 4-digit numbers are divisible by 3 and have 23 as their last two digits??
Answer Let X be your 4 digit number. X = ‘ab23’ = 1000a + 100b + 2*10 + 3 = ( a + b + 2 ) + 999a + 99b + 21 = ( a + b + 2 ) + 3 x ( 333a + 33b + 7 ) Thus, if 3...
Question The polynomial a8x8 +a7x7 + … + a0 has a8 = 1, a7 = -4, a6 = 7 and all its roots positive and real. Find the possible values for a0. Answer Let the roots be xi. We have Sum xi2 = 42 – 2·7 = 2. By Cauchy-Schwarz inequality, we have (x1·1...
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