Official Quant thread for CAT 2013

MBA CAT 2024
24708

@bodhi_vriksha said:
Guys here is first one for today:If the equation (y^2 - 2my -4(m^2 + 1))(y^2 - 4y -2m(m^2 + 1)) = 0 has exactly 3 different roots, then how many values can "m" take?(a) 4 (b) 3 (c) 2 (d) 1 Team BV - Vineet
For exactly three roots, one of the 2 quadratic equations will be having exactly one root; i.e. D= b^2-4*a*c=0

Now first equation y^2 - 2my -4(m^2 + 1)=0 cannot have exact one root as constant term

For the equation y^2 - 4y -2m(m^2 + 1) =0,
D= 16+8m(m^2+1) =0
or, m^3+m+2=0
=> m^3+m^2-m^2-m+2m+2 =0
=> m^2(m+1)-m(m+1)+2(m+1)=0
=> (m^2-m+2)*(m+1)=0

first equation m^2-m+2 =0 donot have any real roots.

Hence, for only one value m=-1 the equation has three roots (-2, 2, 4)
24709
@bodhi_vriksha said:
Another one:For all real k, f(k) satisfies 2f(k) + f(1-k) = 2*k^2 + 1. Then which among the following istrue for all k?(a) f(k) = -1 (c) f(k) >= 1 (d) f(k) Team BV - Vineet
2f(k) + f(1-k) = 2*k^2 + 1
By replacing k with (1-k),
2f(1-k) + f(k) = 2*(1-k)^2 + 1=3+2*k^2-4*k

By solving thse two equations,
3*f(k)= 2*k^2+4*k-1 = 2(k+1)^2-3
=> f(k) = 2/3(k+1)^2-1

2/3(k+1)^2=> 0

=> f(k)>=-1

Hence, option (b)
24710
@Dexian said:
There are 2 brothers among a group of 20 persons. In how many ways can the group be arranged around a circle so that there is exactly one person between the two brothers?(1) 2 * 17! (2) 18! * 18 (3) 19! * 18 (4) 2 * 18! (5) 2 * 17! * 17!
(4) 2 * 18!
24711
@bodhi_vriksha said:
Another one:For all real k, f(k) satisfies 2f(k) + f(1-k) = 2*k^2 + 1. Then which among the following istrue for all k?(a) f(k) = -1 (c) f(k) >= 1 (d) f(k) Team BV - Vineet
On replacing k with (1-k),
2f(1-k) + f(k) = 2*(1-k)^2 + 1=3+2*k^2-4*k

solving we get :
3*f(k)= 2*k^2+4*k-1

min value of the quadratic expression is -3 at k=-1

so, 3*f(k)>=-3
f(k>)=-1

option B

24712
@vijay_chandola said:
For exactly three roots, one of the 2 quadratic equations will be having exactly one root; i.e. D= b^2-4*a*c=0Now first equation y^2 - 2my -4(m^2 + 1)=0 cannot have exact one root as constant term For the equation y^2 - 4y -2m(m^2 + 1) =0, D= 16+8m(m^2+1) =0or, m^3+m+2=0=> m^3+m^2-m^2-m+2m+2 =0=> m^2(m+1)-m(m+1)+2(m+1)=0=> (m^2-m+2)*(m+1)=0first equation m^2-m+2 =0 donot have any real roots.Hence, for only one value m=-1 the equation has three roots (-2, 2, 4)
what if the two equations have a common root , then also they can have 3 diff roots
24713
@vijay_chandola said:
2f(k) + f(1-k) = 2*k^2 + 1
By replacing k with (1-k),
2f(1-k) + f(k) = 2*(1-k)^2 + 1=3+2*k^2-4*k

By solving thse two equations,
3*f(k)= 2*k^2+4*k-1 = 2(k+1)^2-3
=> f(k) = 2/3(k+1)^2-1

2/3(k+1)^2=> 0

=> f(k)>=-1

Hence, option (b)
@@amresh_maverick said:
On replacing k with (1-k),
2f(1-k) + f(k) = 2*(1-k)^2 + 1=3+2*k^2-4*k

solving we get :
3*f(k)= 2*k^2+4*k-1

min value of the quadratic expression is -3 at k=-1

so, 3*f(k)>=-3
f(k>=-1

option B

Neat work guys!
Team BV - Vineet
24714
@amresh_maverick said:
what if the two equations have a common root , then also they can have 3 diff roots
Cross-check is required I guess.. Here we are getting only one value of m. And for that values of y are -2,2,4. :D
24715

Attched fig:

24716

b?

c> = 90
b> = 66;
e>= 180-24-x;
also, e> = 2x ;
3x = 156;

x = 52;
Theta = 90 - 52 = 38 (b)
@amresh_maverick said:
Attched fig:
B I U S :)

    albiesriram 09:06 PM, 05 Mar '13
    Saving ...
    24717
    @amresh_maverick said:
    Attched fig:
    38 ?
    24718
    @Dexian said:
    There are 2 brothers among a group of 20 persons. In how many ways can the group be arranged around a circle so that there is exactly one person between the two brothers?(1) 2 * 17! (2) 18! * 18 (3) 19! * 18 (4) 2 * 18! (5) 2 * 17! * 17!
    18c1*17!*2 =2*18!
    24719
    @amresh_maverick is that b ?
    24720

    THE ENTRANCE TICKET AT MINERVA THEATRE IS RS250. WHEN PRICE OF TICKET LOWERED , SALES OF TICKETS INCREASED BY 50% WHILE THE COLLECTIONS FELL BY 17.5%. FIND THE DEDUCTION IN TICKET PRICE ?

    150
    112.5
    105
    120

    24721
    @raopradeep said:
    THE ENTRANCE TICKET AT MINERVA THEATRE IS RS250. WHEN PRICE OF TICKET LOWERED , SALES OF TICKETS INCREASED BY 50% WHILE THE COLLECTIONS FELL BY 17.5%. FIND THE DEDUCTION IN TICKET PRICE ?150112.5105120
    112.5 it will be
    24722
    @abhishek.2011 POST APPROACH

    24723

    @raopradeep said:
    THE ENTRANCE TICKET AT MINERVA THEATRE IS RS250. WHEN PRICE OF TICKET LOWERED , SALES OF TICKETS INCREASED BY 50% WHILE THE COLLECTIONS FELL BY 17.5%. FIND THE DEDUCTION IN TICKET PRICE ?150112.5105120
    .825 n* 250 = 1.5 n*x
    x = 137.5 => reduction = 112.5 (b)
    24724
    @raopradeep said:
    THE ENTRANCE TICKET AT MINERVA THEATRE IS RS250. WHEN PRICE OF TICKET LOWERED , SALES OF TICKETS INCREASED BY 50% WHILE THE COLLECTIONS FELL BY 17.5%. FIND THE DEDUCTION IN TICKET PRICE ?150112.5105120
    It is 9/20 of 250 = 112.5

    Team BV - Kamal Lohia
    24725
    @raopradeep said:
    THE ENTRANCE TICKET AT MINERVA THEATRE IS RS250. WHEN PRICE OF TICKET LOWERED , SALES OF TICKETS INCREASED BY 50% WHILE THE COLLECTIONS FELL BY 17.5%. FIND THE DEDUCTION IN TICKET PRICE ?150112.5105120

    Price * Sales = Collection

    250 *y = C
    (250*x)*1.5y = 0.825C

    y = Sales C = Collection x = =%change

    Solving,

    x = 0.55

    New Price = 137.5

    Decrease = 112.5





    24726

    112.5

    THE ENTRANCE TICKET AT MINERVA THEATRE IS RS250. WHEN PRICE OF TICKET LOWERED , SALES OF TICKETS INCREASED BY 50% WHILE THE COLLECTIONS FELL BY 17.5%. FIND THE DEDUCTION IN TICKET PRICE ?150112.5105120

    24727
    @Nihilist1002 said:
    @amresh_maverick is that b ?
    OA: is B 38