CAT 2008: Quantitative Questions a Day 1 to 50 - The discussions

[shivendu]

Here is my take....

Let Pagal and cat meet at point A after time 't' at an angle x.
C and M are the starting points of pagal and cat respectively.

Applying sine rule in triangle CMA, we get
AC/sin 120 = AM/sin x = CM/ sin(60-x)---------(1)

We have, AC = 13t and AM = 7t --------------(2)
from (1) and (2), we get
13t/sin 120 = 7t/sin x = 60/sin(60-x)

So, sin x = 7*root(3)/26
cos x = root(1-sin^2 x) = 23/26

So, t = 60 * (sin x) / 7*sin(60-x)----(3)
After solving (3), we can get t = 15/2 s.
So, option 1 is correct.
Not sure about option 2 & 3.

[implex]

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Quantitative Question # 017
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Let R(x) be the remainder when x^16 + x^8 + x^6 + x^4 + x^2 + 1 is divided by x^3 - 1. Let D(x) be the divisor (less than degree 4) of x^6 + 4x^3 + 8. Then which among the following is true?

(1) The sum of the coefficients of R(x) and D(x) is equal
(2) The sum of the absolute value of coefficients of R(x) and D(x) is equal
(3) R(x) > D(x) for all non-positive x
(4) at least 2 of the above
(5) none of the above
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[Aarav]

Finding R(x) should be a breeze, since puys have done PHD on remainder problems in other threads. D(x) will require some thinking

[implex]

Originally Posted by implex

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Quantitative Question # 017
------------------------------------------------------

Let R(x) be the remainder when x^16 + x^8 + x^6 + x^4 + x^2 + 1 is divided by x^3 - 1. Let D(x) be the divisor (less than degree 4) of x^6 + 4x^3 + 8. Then which among the following is true?

(1) The sum of the coefficients of R(x) and D(x) is equal
(2) The sum of the absolute value of coefficients of R(x) and D(x) is equal
(3) R(x) > D(x) for all non-positive x
(4) at least 2 of the above
(5) none of the above
-------------------------------------------------------------------

we can find R(x)=2x^2+2x+2
and D(x)=x^2+(sqrt(3)-1)x+2
clearly R(x)-D(x)=x^2+(3-sqrt(3))x
now R(x)-D(X) is not positive for all non-positive x
and options 1 and 2 can be easily negated!

so option 5)

not fully sure!!

[rajeev_hts]

Originally Posted by implex

we can find R(x)=2x^2+2x+2
and D(x)=x^2+(sqrt(3)-1)x+2
clearly R(x)-D(x)=x^2+(3-sqrt(3))x
now R(x)-D(X) is not positive for all non-positive x
and options 1 and 2 can be easily negated!

so option 5)

not fully sure!!

Hi Implex how did you get D(x)=x^2+(sqrt(3)-1)x+2 ???

[sabsebadapaagal]

Originally Posted by implex

we can find R(x)=2x^2+2x+2
and D(x)=x^2+(sqrt(3)-1)x+2
clearly R(x)-D(x)=x^2+(3-sqrt(3))x
now R(x)-D(X) is not positive for all non-positive x
and options 1 and 2 can be easily negated!

so option 5)

not fully sure!!

I am not able to understand the above part which is highlighted in Bold letters.

@implex pls explain in detail

..... I tried alot but cud nt find any D(x) other than 1 so D(x)=1 this implies the option 3
I am not sure whether it is rite or wrong

[rik_12]

i kno dis method is wrong...but jus thought i can post it...

while solving i jus thought of puttin a particular value of x in2 d polynomial and testing for R(x) and D(x)...

if we put x=2 in the first expression R(x)=0

and D(x)=2....

hence i guess...answer mayb option 5...am i rit?

[selebratinglife]

Originally Posted by implex

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Quantitative Question # 017
------------------------------------------------------

Let R(x) be the remainder when x^16 + x^8 + x^6 + x^4 + x^2 + 1 is divided by x^3 - 1. Let D(x) be the divisor (less than degree 4) of x^6 + 4x^3 + 8. Then which among the following is true?

(1) The sum of the coefficients of R(x) and D(x) is equal
(2) The sum of the absolute value of coefficients of R(x) and D(x) is equal
(3) R(x) > D(x) for all non-positive x
(4) at least 2 of the above
(5) none of the above
-------------------------------------------------------------------

Is it implied that D(X) should perfectly divide the equation?
x^6 + 4x^3 + 8=0 If we consider X^3 = a the the roots would be -2+/-2i..Which is irrational. Am i missing something here?

[implex]

Originally Posted by selebratinglife

Is it implied that D(X) should perfectly divide the equation?
x^6 + 4x^3 + 8=0 If we consider X^3 = a the the roots would be -2+/-2i..Which is irrational. Am i missing something here?

non-real roots appear in pairs, so we can have factors of degree 2 which will completely divide x^6 + 4x^3 + 8 and still have real coefficients.

[implex]

Originally Posted by sabsebadapaagal

I am not able to understand the above part which is highlighted in Bold letters.

@implex pls explain in detail

..... I tried alot but cud nt find any D(x) other than 1 so D(x)=1 this implies the option 3
I am not sure whether it is rite or wrong

let x^3=t
t=-2+-2i
let us pick one of the values
x^3=-2-2i
let x=a+ib
we can easily prove that x=sqrt(2)( cos75+isin75) and other values

pick teh complex conjugate of cos75+isin75 and multiply we
x^2+(sqrt(3)-1)x+2 which is one of the divisors . so we have our D(x)