official quant thread for cat08

[ikruz]

Originally Posted by ahaprashant

Not exactly trial n error,
Look at the terms in sum form: (3x + 4y)
Look at the terms in prod form: (x^2, y^3)
Looking at these, can we say we need to think of some X-term, two of which when multiplied offer an x^2 term with certain coefficient, and when summed up offer 3x as the sum, which suggests it shud be 3x/2. Similarly for Y case.

Hope that helps. (Btw its prashant )


Pagal Padha.

Thanks a lot prashant.....It definitely helped.Its very clear for me now...

I have posted my doubt to similar question above. to maximise 2x^2 + 5y^2 + 8z^2 given 2x+3y+4z=100.

how will you approach that?.

[prakharc]

Originally Posted by ikruz

The question is to find out maximum value of 2x^2 + 5y^2 + 8z^2 ,if 2x+3y+4z=100.




Answer is indeed 4534.

Is trail and error the only possible way for handling this question.

The answer given by TIME is as follows -

For maximum value of the expression is when y is maximised because the ratio of the squares of the coeffs of x,y,z in 2x+3y+4z=100 is 4:9:16 whereas the ratio of coeffs of x^2,y^2, z^2 in 2x^2 + 5y^2 + 8z^2 is 4:10:16. Therefore y^2 has relatively more weightage.

But i could nt understand this logic as well..
Can any one throw some light.

Thanks..

They have done what I did......calculated the weightage to find out which one to maximize.....

What I did wasn't hit n trial.....I calculated that we shd maximize y and then did that.......I didn't try several values of x,y,z to get results

[ikruz]

Originally Posted by prakharc

They have done what I did......calculated the weightage to find out which one to maximize.....

What I did wasn't hit n trial.....I calculated that we shd maximize y and then did that.......I didn't try several values of x,y,z to get results

Hi prakhar.

Sorry. i am not able to follow your logic.Could you explain the approach a bit more.

my doubt is -
1)what relation does the ratio of squares of the coeff of
x,y,z in 2x+3y+4z=100 have with the ratio of coeffs of x,y,z in 2x^2 + 5y^2 + 8z^2.?..

2) from this relation how do we conclude that y has to maximised?.

In your approach you have tried to tried 3 possible cases
a) x,y=0 z maximum
b) x,z=0 y maximum
c) y,z =0 x maximum.

and then concluded b) is the best option..am i right?.

Thanks for your help friend.

[prakharc]

Originally Posted by ikruz

Hi prakhar.

Sorry. i am not able to follow your logic.Could you explain the approach a bit more.

my doubt is -
1)what relation does the ratio of squares of the coeff of
x,y,z in 2x+3y+4z=100 have with the ratio of coeffs of x,y,z in 2x^2 + 5y^2 + 8z^2.?..

2) from this relation how do we conclude that y has to maximised?.

In your approach you have tried to tried 3 possible cases
a) x,y=0 z maximum
b) x,z=0 y maximum
c) y,z =0 x maximum.

and then concluded b) is the best option..am i right?.

Thanks for your help friend.

See you have to find max where we have square of x,y,z ...... so we have to maximize either x,y,z to get max value.

Now sum that we have to maximize is not symmetric in x,y,z......so we have to find relative importance of each term so as to find out which is most significant

So let's first maximize x.... x=50,y=0,z=0...then y=33.33,x=0,z=0...then z=25,x=0,y=0....this way we can see maximizing which term gives us highest sum => which term has the highest weightage in sum

We found that to be y...so we maximize y


what time people did was sot of like this only
2x+3y+4z=100...now if we take y,z=0 weightage of x^2=4..similarly for y and z...9,16 respectively

So in condition we have weightage of x^2,y^2,z^2 as 4:9:16

In sum we have 4:10:16.....so x and z remain same...but 9 of y in condition is giving 10 of y in sum....so if we maximize y we will get higher sum

[ikruz]

question)The equation 2ax^2 + 3bx + 8 = 0 has equal roots when a and b are real numbers. Find the maximum value of a+ 2b.

a) -36/5 b) -22/3 c) -7 d) -64/9

How do we go about solving this problem

my first thought was that i i shud try to arrive at (a)(b^2) = k so that i can put a=b=b ..to get the minimum value of a+2b.

But here i can see 9b^2 = 64a.
from this i cant arrive at (a)(b^2) = k .So i am stuck here.

How to proceed?.

[naiquevin]

find out maximum value of 2x^2 + 5y^2 + 8z^2 ,if 2x+3y+4z=100

consider 2x +3y + 4z = 100

had the question been to find maximum value of 4x^2 + 9y^2 + 16z^2 which is same as (2x)^2 + (3y)^2 + (4z)^2........(I)
then we would have maximised any one of 2x, 3y or 4z and proceeded
to find the maximum value

But we are asked to maximise 2x^2 + 5y^2 + 8z^2.......(II)
The ratio of coefficients is 4:10:16

whereas the ratio of coeffcients in exprn I is 4:9:16

if x and z are maximised then the increase in value of II over I will not be as much as when y is maximised

[ikruz]

thanks naiquevin and prakhar..you are people are so helpful, i got the concept now!!...

thanks for ur time.

[masoom]

a graph is given as attcahed...

using this sketch the graph of y vs x...

[Vishal2704]

Originally Posted by masoom

a graph is given as attcahed...

using this sketch the graph of y vs x...

y-x = k(y+x) k>1

y/x = (k+1)/1-k

hence slope of y to x is -ve as 1-k <0

[MaskedMenace]

Originally Posted by masoom

a graph is given as attcahed...

using this sketch the graph of y vs x...

v get y=[(1+k)/(1-k)]x line for k>1