Official Quant thread for CAT 2013

MBA CAT 2024
9981
@karan20 said:
Option 3 : 347Since the nos. leaves the same remainder theri diff should be divisible by the required 3 digit no.so 67588-63424/ ( required no ) = 0from the options 347 satisfies the condition.PS: sum1 plz post how to solve in the absence of options...
Well, in absence of options, we would factorize (67588-63424) = 4164 and find the least three digit number we can form using the the factors, so in this case, why would we do that when options are already there :)
9982
If "3 + 2i" is one of the root of x^2 + bx + 10 = 2i ; Then find the another root and value of "b"
9983
1/x + 1/y = 1/z

if all x,y,z are integers

find no. of solutions of this eqn, for a given value of x


9984
If R=[(0.15)^55+(0.25)^ 55]/[(0.15)^54+(0.25)^54]
1.R>0.20
2.R>0.1
3.R=4
4.R
9985
@YouMadFellow

value of b is -5 and another root will be 3-2i ??? is it correct
9986
@bullseyes said:
1/x + 1/y = 1/z if all x,y,z are integers find no. of solutions of this eqn, for a given value of x
Depends on the number of factors of x ?

Basically, If number of factors of x are N, then number of solutions = 4N - 2 ?

Let x = n , y = -(n + a) => z = n*(n+ a)/a => z will be an integer when n or (n + a) is divisible by a -> 2N - 1 values of a (both positive and negative)-> 2N - 1 values for z

Similarly, we can fix z = (n+a) and get 2N - 1 values for y

So, total = 4N - 2 , where N = factors of x
9987
@shattereddream said:
@YouMadFellowvalue of b is -5 and another root will be 3-2i ??? is it correct
Value of b = -5, but the other root is not (3-2i) 😃
9988

A square playground is surrounded by a runway 1.5 Mts wide . The area of the runway is 1/4 times that of the playground . Find the area of the playground ?

9989
@YouMadFellow said:
If "3 + 2i" is one of the root of x^2 + bx + 10 = 2i ; Then find the another root and value of "b"
2 - 2i and b = -5 ?

x^2 + bx + (10 - 2i) = 0

pq = (10 - 2i)
q = (10 - 2i)/(3+2i) = (10 - 2i)(3 - 2i)/(9 + 4) = (26 - 26i)/13 = 2 - 2i

b = -(p + q) = -(3 + 2i + 2 - 2i) = -5
9990
@YouMadFellow said:
If "3 + 2i" is one of the root of x^2 + bx + 10 = 2i ; Then find the another root and value of "b"

putin x=3 + 2i in
x^2 + bx + 10 - 2i = 0

9 -4 + 12i + 3b + 2bi + 10 - 2i = 0
15 + 10i + b(3+2i) = 0
b = -15-10i / 3+2i


9991
@bullseyes said:
|x| + 2|y| = 100 no. of integral solutions
y varies from 0 to 50 only. both x and y can take + and - values in any order. so total 4*50= 200 integral pairs.
9992
@meenu05 said:
A square playground is surrounded by a runway 1.5 Mts wide . The area of the runway is 1/4 times that of the playground . Find the area of the playground ?
Side of square = x

Area of inner square/ Area of square = 3/4 => (x - 3)^2 / x^2 = 3/4

=> (x-3)/x = root(3)/2 => x = 6*(root(3) + 2)

Area of playground => (x-3)^2 = (3*(2root(3) + 3))^2 = 9*3*(4 + 3 + 4root(3)) = 27*(7 + 4root(3))

there may be calculation mistakes 😐
9993
@meenu05 said:
A square playground is surrounded by a runway 1.5 Mts wide . The area of the runway is 1/4 times that of the playground . Find the area of the playground ?
Let the length of playground side be x.
(x+3)^2 - x^2 = x^2/4.
Solve for x

9994
@YouMadFellow said:
If "3 + 2i" is one of the root of x^2 + bx + 10 = 2i ; Then find the another root and value of "b"
Product of roots = 10 - 2i
x(3 + 2i) = 10 - 2i
x = (10 - 2i)(3 - 2i)/13 = 2 - 2i

Sum of roots = 3 + 2i + 2 - 2i = 5

b = -5
@shattereddream said:
@YouMadFellowvalue of b is -5 and another root will be 3-2i ??? is it correct
The concept you have used is only applicable when all the coefficients in a quadratic eq are rational.
@bullseyes said:
1/x + 1/y = 1/z if all x,y,z are integers find no. of solutions of this eqn, for a given value of x
1/z - 1/y = 1/x
zy - xy + xz = 0
(y + x)(z - x) = -x^2
So, number of integral solutions will be given by:-
a = {2(no of factors of x^2) - 1}

If no of factors of x^2 is 'n', then
no of integral solutions are (2n - 1)
9995
Let a, b and c be the three positive integers such that sum of the reciprocals of any two integer among them is and integer multiple of the reciprocal of the third number. Find the minimum value of a + b + c.
OPTIONS

1) 5
2) 7
3) 6
4) 11
5) 12
9996
@sujamait said:
Let a, b and c be the three positive integers such that sum of the reciprocals of any two integer among them is and integer multiple of the reciprocal of the third number. Find the minimum value of a + b + c.OPTIONS1) 5 2) 7 3) 6 4) 11 5) 12
11?
9997
@sujamait said:
Let a, b and c be the three positive integers such that sum of the reciprocals of any two integer among them is and integer multiple of the reciprocal of the third number. Find the minimum value of a + b + c.OPTIONS1) 5 2) 7 3) 6 4) 11 5) 12
4) 11.

The numbers are 2,3 and 6.
9998
@chillfactor hemant sir __/ \__ and yes i have used that concept thats y taken as complex conjugate .
9999
@chillfactor said:
Product of roots = 10 - 2ix(3 + 2i) = 10 - 2ix = (10 - 2i)(3 - 2i)/13 = 2 - 2iSum of roots = 3 + 2i + 2 - 2i = 5b = -5The concept you have used is only applicable when all the coefficients in a quadratic eq are rational.1/z - 1/y = 1/xzy - xy + xz = 0(y + x)(z - x) = -x^2So, number of integral solutions will be given by:-a = {2(no of factors of x^2) - 1}If no of factors of x^2 is 'n', thenno of integral solutions are (2n - 1)
lets just say x = 12
x^2 = 144
no. of factors = 15 and according to formula the solutions are 29 rit?
unfortunately this aint the answer
10000
@sujamait said:
Let a, b and c be the three positive integers such that sum of the reciprocals of any two integer among them is and integer multiple of the reciprocal of the third number. Find the minimum value of a + b + c.OPTIONS1) 5 2) 7 3) 6 4) 11 5) 12
11?
2,3,6