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

[implex]

Originally Posted by m2shines

If you go through above 2 solution u can easily prove 3 to be wrong

if cat moves in horizontally. Then distance will be 60/(13-7) = 10

and if Cat moves at 30degree to north then distance will be 15/2 = 7.5

so its sure that the catching point doesnt lies on a circle as distance is nt same.

Correct if i am wrong somewhere.

thats not the distance bro. its time
distance will be 7*10=70!!
and 7*3=21!!

[m2shines]

Originally Posted by implex

thats not the distance bro. its time
distance will be 7*10=70!!
and 7*3=21!!

Its un intentional error. Thanks for telling me Implex.

Its been corrected now

[sanchit_states]

I did'nt get today's question.
Replying yesterday question:

How many (a, b, c) satisfy log(2ab) = loga*logb, log(bc) = logb*logc, log(2ac) = loga*logc ?

(1) none (2) 1 (3) 2 (4) 4 (5) none of these


Let log a=A log b=B log c=C
So equ:
log2+A+B=AB-----1
B+C=BC----2
log2+A+C=AC----3
By equ2
B=C/(C-1)
By equ (1-3)
B-C=A(B-C) thus either B=C OR A=1
By B=C in equ 2
we have either C=2 or B=0
By C=2---->c=100
Since C=B=2---->b=100
Putting the values in equ 1:
log2+2=A=log2+log100----->a=200
thus (a,b,c)=(200,100,100)
ANOTHER Possiblity:B=0----b=1
Since B=C----c=1
Thus a=1/2 by equ 1.
Second pair (a,b,c)--(1/2,1,1)
Another condition A=1---a=10
Put in equ1
log2+1+B=B-----not popssible,discarded.
Thus 2 pairs is ans.
Option (3)

[slam]

People,
Here's my take:

(1) can be proved by solving with the values given (I am not repeating this part of the solution as it has already been posted)

(3) as already stated by many, the distance will be minimum in case the cat running perpendicular to the line between pagal and the cat. (again not repeating this part of the solution)

(2) The max distance covered by the cat will be when it runs towards east, that is, directly away from pagal.
In this case, time = 60/6 = 10 seconds.
Assuming cat's original position as origin, pt of contact will be (70,0) in this case.
The min distance covered by the cat will be when it runs towards west, that is, directly towards pagal.
In this case, the time will be 60/20 = 3 seconds.
Pt of contact will be (-21,0)

If at all these two points lie on a circle, then the line joining these two points has to be a diameter(Since they denote the max and min distance from an external point).
In such a case, centre would be (49/2,0) and radius will be (91/2).
Taking X as the angle at which Pagal runs, and Y as the angle at which cat runs, we get
13T cosX = 60 + 7T cosY --(1)
13T sinX = 7T sinY--(2)

Now, the general form of the point of contact would be (7T cosY, 7T sinY)
Distance b/w this point and centre = D
D^2
= (on solving)
= (49/2)^2 + 49(T^2 - 7TcosY)

Square (1) and (2) and add them, to get
T^2 - 7T cosY = 30
so, D^2 = (substituting)
= 49*169/4
= (91/2)^2, which is the same as the radius.

Hence, option (2) is also correct.

=> option(4) all of the above are correct.

----------
slam.

[pavanpadekal]

Originally Posted by slam

People,
Here's my take:

(1) can be proved by solving with the values given (I am not repeating this part of the solution as it has already been posted)

(3) as already stated by many, the distance will be minimum in case the cat running perpendicular to the line between pagal and the cat. (again not repeating this part of the solution)

(2) The max distance covered by the cat will be when it runs towards east, that is, directly away from pagal.
In this case, time = 60/6 = 10 seconds.
Assuming cat's original position as origin, pt of contact will be (70,0) in this case.
The min distance covered by the cat will be when it runs towards west, that is, directly towards pagal.
In this case, the time will be 60/20 = 3 seconds.
Pt of contact will be (-21,0)

If at all these two points lie on a circle, then the line joining these two points has to be a diameter(Since they denote the max and min distance from an external point).
In such a case, centre would be (49/2,0) and radius will be (91/2).
Taking X as the angle at which Pagal runs, and Y as the angle at which cat runs, we get
13T cosX = 60 + 7T cosY --(1)
13T sinX = 7T sinY--(2)

Now, the general form of the point of contact would be (7T cosY, 7T sinY)
Distance b/w this point and centre = D
D^2
= (on solving)
= (49/2)^2 + 49(T^2 - 7TcosY)

Square (1) and (2) and add them, to get
T^2 - 7T cosY = 30
so, D^2 = (substituting)
= 49*169/4
= (91/2)^2, which is the same as the radius.

Hence, option (2) is also correct.

=> option(4) all of the above are correct.

----------
slam.

I think this should be (60+7T cosY, 7T sinY) as the origin is the position of the PaGal and not the CAT
Just look into it !

[sudeepdeb]

------------------------------------------------------
Quantitative Question # 016
------------------------------------------------------
A PaGal is stationed at C, 60 meters directly west of a CAT located at M. The CAT is trying to escape running at 7m/s in a direction 30 degrees east of north. The PaGal an expert in geometry runs at 13m/s in a suitable straight line path that will intercept the CAT as quickly as possible.
Then which among is following is true?

(1) It takes 15/2 seconds for the PaGaL to catch CAT
(2) If CAT chooses a different direction to escape, irrespective of the direction, all interception points lie on a circle.
(3) If the CAT is intercepted after running a distance of x meters in a particular direction and if the CAT had been intercepted after it had run a distance of y meters in the opposite direction then min (x+y) is 14√30
(4) All of the above
(5) Exactly 2 of the above


Please read choice (3) again from here -> the word mouse is to be replaced by CAT. That was put inadvertently, and I couldn't make last minute corrections due to issues I faced editing the NL.
-----------------------------
Ahh one of my favourites : co-ordinate geometry.
My pick on answer : (4) All the above are correct.

let (x,y) be locus of point where the CAT and PaGal meets. Point M is considered to be the origin.
(A) = angle from x-axis by which CAT runs off
(B) = angle from x-axis by which PaGal runs off.
(T) = time at which they CAT is busted.

Equations for CAT :
x=7tcos(A) , y=7tsin(A)
Equations for PaGal :
x=13tcos(B) - 60 , y=13tsin(B)
Since A,B,T are variables for locus. Eliminating them,(not writing here, as algebra solving requires lot of typoo to be done)
we get the equation
f(x,y)= (x-24.5)^2 + y^2 - 2070.25
which is equation of a circle with centre (24.5,0) and radius = sqrt(2070.25) = 45.5

So option (3) is correct.
For option (2), substitute x= 0, calculate y, and double it to get 14*root(30).
For option 1, I think its simple.

Final Answer : (4) all are correct

PS: no NL today .
PSS: do i get 4 or 5 for finding the locus ? :P

[sudeepdeb]

1 query :
Not at all related to the thread, so apologizing before.

Q: I see some users labeled as expert pagal and other such notations, how do we get that ?? By number of posts ?

[Aarav]

Originally Posted by sudeepdeb

------------------------------------------------------
Quantitative Question # 016
------------------------------------------------------
A PaGal is stationed at C, 60 meters directly west of a CAT located at M. The CAT is trying to escape running at 7m/s in a direction 30 degrees east of north. The PaGal an expert in geometry runs at 13m/s in a suitable straight line path that will intercept the CAT as quickly as possible.
Then which among is following is true?

(1) It takes 15/2 seconds for the PaGaL to catch CAT
(2) If CAT chooses a different direction to escape, irrespective of the direction, all interception points lie on a circle.
(3) If the CAT is intercepted after running a distance of x meters in a particular direction and if the CAT had been intercepted after it had run a distance of y meters in the opposite direction then min (x+y) is 14√30
(4) All of the above
(5) Exactly 2 of the above


Please read choice (3) again from here -> the word mouse is to be replaced by CAT. That was put inadvertently, and I couldn't make last minute corrections due to issues I faced editing the NL.
-----------------------------
Ahh one of my favourites : co-ordinate geometry.
My pick on answer : (4) All the above are correct.

let (x,y) be locus of point where the CAT and PaGal meets. Point M is considered to be the origin.
(A) = angle from x-axis by which CAT runs off
(B) = angle from x-axis by which PaGal runs off.
(T) = time at which they CAT is busted.

Equations for CAT :
x=7tcos(A) , y=7tsin(A)
Equations for PaGal :
x=13tcos(B) - 60 , y=13tsin(B)
Since A,B,T are variables for locus. Eliminating them,(not writing here, as algebra solving requires lot of typoo to be done)
we get the equation
f(x,y)= (x-24.5)^2 + y^2 - 2070.25
which is equation of a circle with centre (24.5,0) and radius = sqrt(2070.25) = 45.5

So option (3) is correct.
For option (2), substitute x= 0, calculate y, and double it to get 14*root(30).
For option 1, I think its simple.

Final Answer : (4) all are correct

PS: no NL today .
PSS: do i get 4 or 5 for finding the locus ? :P

You get 5 sir After many attempts, I'm able to connect -> let me reach back Delhi tonight and then I will participate in the discussions with you.

[slam]

Originally Posted by pavanpadekal

I think this should be (60+7T cosY, 7T sinY) as the origin is the position of the PaGal and not the CAT
Just look into it !

People,
Here's my take:

(1) can be proved by solving with the values given (I am not repeating this part of the solution as it has already been posted)

(3) as already stated by many, the distance will be minimum in case the cat running perpendicular to the line between pagal and the cat. (again not repeating this part of the solution)

(2) The max distance covered by the cat will be when it runs towards east, that is, directly away from pagal.
In this case, time = 60/6 = 10 seconds.
Assuming cat's original position as origin, pt of contact will be (70,0) in this case.
The min distance covered by the cat will be when it runs towards west, that is, directly towards pagal.
In this case, the time will be 60/20 = 3 seconds.
correct.

I have taken the original position of the cat as the origin. Please see text in bold.

----------
slam.

[sudeepdeb]

Originally Posted by Aarav

You get 5 sir After many attempts, I'm able to connect -> let me reach back Delhi tonight and then I will participate in the discussions with you.

yippppiii.....

Can i trade the 5 points for upgrading my status to say "trainee pagal" ?? :P

PS: got carried away, no more bakar.