Official Quant thread for CAT 2013

MBA CAT 2024
6461
@maddy2807 said:
How many values of n satisfy the equation, if n is an integer and x is a natural no.n+n^2+5n = n^x1)12)23)04)More than 3
4??
2, 3, -2, -3, -5
6462
@maddy2807 said:
How many values of n satisfy the equation, if n is an integer and x is a natural no.n+n^2+5n = n^x1)12)23)04)More than 3
4)??????
6463
If log(4^x + 1) + 2x = log 15 + x log4, then nx is a prime number.What is n?(All logs are to the base 10)
6464
@19rsb said:
If log(4^x + 1) + 2x = log 15 + x log4, then nx is a prime number.What is n?(All logs are to the base 10)
n=4??
6465
@19rsb said:
If log(4^x + 1) + 2x = log 15 + x log4, then nx is a prime number.What is n?(All logs are to the base 10)
was solved yesterday
10^2x=15*4^x/(4^x+1)
x=1/2

=>4x=2=prime

n=4
6466
@gautam22 said:
sir 6x=3....ye bhi to prime hai aise to kayi sari values ho sakti hain....main kya galat kar raha hoon?
bhai aap kabhi galat nai hote, kal options the, usme bas 4 possible tha isliye 4 likha
6467
@krum said:
bhai aap kabhi galat nai hote, kal options the, usme bas 4 possible tha isliye 4 likha
actually it is my laziness in writing d ques.....last lyn was this.......
least possible value of n where n is a positive integer....
APOLOGIES
6468
@catter2011 said:
A number N when divided by 25 gives a result, with the fractional part of the form 0.ab. What is b if N=(3^10-6^24+7^50-2^99) ?
6?
6469
@19rsb said:
If log(4^x + 1) + 2x = log 15 + x log4, then nx is a prime number.What is n?(All logs are to the base 10)
4
6470
Consider points A(1, €“3) and B(5, 2). Let P is a point on the line y = x. Find co-ordinates of P for which |AP + PB| is minimum.
6471
@pankaj1988 said:
Consider points A(1, €“3) and B(5, 2). Let P is a point on the line y = x. Find co-ordinates of P for which |AP + PB| is minimum.
((1-c)^2+(c+3)^2)+((5-c)^2+(2-c)^2)=16+25
=>1+c^2-2c+c^2+9+6c+25+c^2-10c+4+c^2-4c=41
=>4c^2-10c-2=0

1/4(5+/-_/33)

:splat:
6472
@krum said:
((1-c)^2+(c+3)^2)+((5-c)^2+(2-c)^2)=16+25=>1+c^2-2c+c^2+9+6c+25+c^2-10c+4+c^2-4c=41=>4c^2-10c-2=01/4(5+/-_/33)
..Actually this is the test funda ques of the day for today.
Options are
(10/9,10/9)
(11/7,11/7)
(11/9,11/9)
13/7,13/7)
Can solve it with options but then too looking for a proper method..
6473
@pankaj1988 said:
..Actually this is the test funda ques of the day for today.Options are (10/9,10/9) (11/7,11/7)(11/9,11/9)13/7,13/7)Can solve it with options but then too looking for a proper method..
what are u getting by solving from the options?
6474
@krum said:
what are u getting by solving from the options?
Bhai options se bhi lag jaegi solve karne me.
Anyway I will post the official solution from testfunda tomorrow.
6475
@pankaj1988 said:
Consider points A(1, €“3) and B(5, 2). Let P is a point on the line y = x. Find co-ordinates of P for which |AP + PB| is minimum.
crux of the problem: smallest distance between two points is given by st. line joining these two points.
if the two points in the given question were on different side of the line x=y then we would have joined the points and point of intersection with line y=x would give the cordinates of P. But in this case both the points are on same side of x=y. So we take reflection of point A about line x=y. Say reflection is A'
Now join BA' and say P is the point where BA' interesects x=y. Now PA' = PA (by symmetry).
We know BA' = (A'P+PB)is least possible distance hence AP+PB also must be least possible value.
Calculations
reflection of point (1,-3) about y=x is (-3,1)
eqn of line passing through (-3,1) and (5,2) is given by
(x+3)/(-3-5) = (y-1)/(1-2) =>(x+3) = 8(y-1)
=>x-8y +11 = 0
intersection of x-8y+11=0 and x=y is (11/7, 11/7)
ATDH.
6476
@anytomdickandhary said:
crux of the problem: smallest distance between two points is given by st. line joining these two points. if the two points in the given question were on different side of the line x=y then we would have joined the points and point of intersection with line y=x would give the cordinates of P. But in this case both the points are on same side of x=y. So we take reflection of point A about line x=y. Say reflection is A'Now join BA' and say P is the point where BA' interesects x=y. Now PA' = PA (by symmetry). We know BA' = (A'P+PB)is least possible distance hence AP+PB also must be least possible value.Calculationsreflection of point (1,-3) about y=x is (-3,1)eqn of line passing through (-3,1) and (5,2) is given by (x+3)/(-3-5) = (y-1)/(1-2) =>(x+3) = 8(y-1)=>x-8y +11 = 0intersection of x-8y+11=0 and x=y is (11/7, 11/7)ATDH.
You made it so simple sir...Thanks a lot...wasted a lot of time on it
6477
@pankaj1988 said:
Bhai options se bhi lag jaegi solve karne me.Anyway I will post the official solution from testfunda tomorrow.
damn it
D=[(1-c)^2+(c+3)^2]^1/2+[(5-c)^2+(2-c)^2]^1/2

D'=1/2*(2c-2+2c+6)*[(1-c)^2+(c+3)^2]^-1/2 + 1/2*(2c-10+2c-4)*[(5-c)^2+(2-c)^2]^-1/2
D'=1/2*(4c+4)*[(1-c)^2+(c+3)^2]^-1/2 + 1/2*(4c-14)*[(5-c)^2+(2-c)^2]^-1/2=0

=>(2c+2)*_/(2c^2-14c+29) + (2c-7)*_/(2c^2+4c+10)=0
=>(2c+2)^2*(2c^2-14c+29)=(2c-7)^2*(2c^2+4c+10)
=>(4c^2+4+8c)*(2c^2-14c+29)=(4c^2+49-28c)*(2c^2+4c+10)
=>14c^2-260c+374=0
=>7c^2-130c-187=0
=>(c-17)(c-11/7)=0
=>c=11/7

(11/7,11/7)


6478
@anytomdickandhary said:
crux of the problem: smallest distance between two points is given by st. line joining these two points. if the two points in the given question were on different side of the line x=y then we would have joined the points and point of intersection with line y=x would give the cordinates of P. But in this case both the points are on same side of x=y. So we take reflection of point A about line x=y. Say reflection is A'Now join BA' and say P is the point where BA' interesects x=y. Now PA' = PA (by symmetry). We know BA' = (A'P+PB)is least possible distance hence AP+PB also must be least possible value.Calculationsreflection of point (1,-3) about y=x is (-3,1)eqn of line passing through (-3,1) and (5,2) is given by (x+3)/(-3-5) = (y-1)/(1-2) =>(x+3) = 8(y-1)=>x-8y +11 = 0intersection of x-8y+11=0 and x=y is (11/7, 11/7)ATDH.
sir __/\__ magic, i tried 3 diff. ways,
1) right angled triangle
2) isosceles triangle :splat:
3) differentiation
6479
@maddy2807 ..ans..6..
2,-2,3,-3,-5,-7..
6480
@gautam22 said:
A tank of ht. 3h has and inlet pipe A and two outlet pipes B and C at heights 2h and h from the base.A can fill tank in 12 hrs. The time in which B and C can empty it are in the ratio 1:2. All the pipes are opened simultaneously. The tank is filled in 16 hrs 48 mins. Find the time in which B can empty it?24273036
crux of the problem: if the total work to be done is constant then if the rate of working decreases then time taken increases.

i.e R*T=W (Work: filling of the tank)

also, if rate of work decreases by 1/x then time taken increases by 1/(x-1)

i.e R(1-1/x)*T(1 + 1/x-1) = W.

Application to the given problem: divide tank in three parts, base to h->v1, h to 2h-> V2 and 2h to 3h -> v3. if outlet pipes were not there then all the parts would be filled in time T= 4 hr each.

Say rate of A is R. Rate of C is R/x and B is 2R/x

Effective rate while v2 is getting filled = R-R/x = R(1-1/x) [rate has decreased by 1/x] hence time must increase by 1/(x-1) i.e new time = 4*(1+1/x-1) = 4x/(x-1)

Effective rate while v3 is getting filled = R-R/x -2R/x = R{1-1/(x/3) } [rate has decreased by 1/(x/3)] hence time must increase by 1/(x/3-1) i.e new time = 4*{1+1/(x/3 -1)} = 4x/(x-3 )

Calculations:(This is the point where you would start from in real exam if above concept is known, so observe how quick and simple things are from here on)

=>total time to fill = time for v1 + time v2 + time v3 =16hr48 min
=> 4 + 4x/(x-1) + 4x/(x-3) = 16 + 4/5
=> x/(x-1) + x/(x-3) = 16/5

=> x = 6

hence rate of B = 2R/6 = R/3 hence it will take 3 time the time of A i.e 12*3 = 36 hrs


The good thing about this method is that it deals in single variable so we do not get lost in multiple eqns.

ATDH.