Ultimate Quant Marathon Blog For IIM CAT

I could not post due to some engagements. Here are a bunch of problems to compensate

Question 1)

A + B + C + D = D + E + F + G = G + H + I = 17 where each letter represent a number from 1 to 9. Find out number of ordered pairs (D,G) if letter A = 4.
a) 0                       b) 1                 c)2                        d) 3                 e) none of these

Question 2)

The sequence 1, 3, 4, 9, 10, 12….. includes all numbers that are a sum of one or more distinct powers of 3. Then the 50th term of the sequence is
a. 252                    b. 283                     c. 327                      d. 360                  e) none of these

Question 3)

Given that g(h(x)) = 2x² + 3x and h(g(x)) = x² + 4x − 4 for all
real x. WHich of the following could be the value of g(-4)?
a)1                     b) -1                          c) 2                 d) -2                   e) -3

Question 4)

If a, x, b and y are real numbers and ax+by = 4 and ax² +by² = 2 and
ax³ + by³= −3 then find (2x − 1)(2y − 1)
a)4                      b) 3                    c) 5                 d) -3          e) cannot be determined.

Question 5)

K1,K2,K3…K30 are thirty toffees. A child places these toffees on a circle, such that there are exactly n ( n is a positive integer) toffees placed between Ki and Ki+1 and no two toffees overlap each other. Find n
a)4                        b) 5                     c) 9                 d) 12                       e) 13

Question 6)
For the n found in previous  question, which of the two toffees are adjacently
placed on the circle? ( All other conditions remaining same)
a) K11 and K13                    b) K6 and K23                   c) K2 and K10              d) K11 and K18
e) K20 and K28

Let x,y,z be distinct positive integers such that x+y+z=11. Find the maximum value of (xyz+xy+yz+zx)?

Suppose we have 3 real numbers x,y and z such that x^3=3y-2, y^3=3z-2 and z^3=3x-2 . Find the number of triplets of (x,y,z) which exist?

A) 0     B) 1                         C)  2          D) 3                E) None of these

Let a,b,c be real numbers such that for all real numbers x, such that |x|<=1, we have |ax^2+bx+c|<=100. Determine the maximum value of |a|+|b|+|c|

1)Two real non negative numbers satisfy that ab>=a^3+b^3, find the maximum value of a+b

a) 1/2  b)  1 c) 3/2 d) 2 e) none of these

2) Let x(n) be a sequence of real numbers such that x(1)=2 and x(n+1)=2x(n)/3+1/(3x(n))

then  for all n>1 which is always true

a) x(n) >1  b) x(n) <2 c) 1<x(n) <3/2 d) 1<x(n)<2  e) 3/2<x(n) <2

3) if p  and q are real positive numbers such that p+q=1 then fidn the minimum value of (p+1/p)^2+(q+1/q)^2

a) 5/2         b) 25/2           c) 15/2          d) 6 e) none of these

Concept 2 Inequalities

Lets move on to our next concept, i.e Inequalities. Inequalities are generally present in cat and similar MBA papers, the question can be direct or indirect.

Concept 2.1 AM-GM Inequality

It means that AM( arithemetic mean) of  a set of positive numbers is always greater than or equal to the GM( geometric mean). The equality holds when the numbers are equal

(a+b+c)/3 >=(a+b+c)^(1/3)……….( 2.1)

Example 2.1 If a,b,c are positive numbers prove that (a+b)(b+c)(c+a)>=8abc

what we will do is  use AM-GM multiple times

(a+b)/2 >=sqrt(ab)

=>(a+b)>=2sqrt(ab)

similarly for others

(b+c)>=2sqrt(bc)

(c+a)>=2sqrt(ac)

then multiplying these three inequalities we get the desired result!

Practice Problem 2.1show that (n^n)[(n+1)/2]^(2n)>(n!)^3

Practice Problem 2.2 if x,y,z be the lengths of the sides of a triangle then prove that (x+y+z)^3>=27(x+y-z)(y+z-x)(z+x-y)

Practice Problem 2.3 show that for any natural number n, (n+1)^n>2.4.6….2n

Example 2.2 Show that for any natural number n 2^n>=1 +n.2^[(n-1)/2]

Lets see how we do this

2^n>=1+n.2^[(n-1)/2]

2^n-1>=n.2^[(n-1)/2] ( can you recognise the form?)

its the sum of a GP

we need to use AM-GM on the sum of GP

[1+2+2^2...+2^(n-1)]/n>(1.2.2^2…2^(n-1))^(1/n)

(2^n-1)/n> ( 2^(1+2+3..+n-1))^(1/n)=(2^[n(n-1)/2])^(1/n)=2^((n-1)/2)

so

2^n-1>2^((n-1)/2)

so we are done !!

Concept 2.2 Cauchy- Schwartz Inequality

If a,b,c and x,y,z be real numbers ( positive, negative or zero) then

(ax+by+cz)^2<=(a^2+b^2+c^2)(x^2+y^2+z^2)

Equality holds iff  a:b:c::x:y:z

Example 2.3 if x^4+y^4+z^4 =27 find min value of x^6+y^6+z^6

use cauchy on x^3,y^3,z^3 and  x,y,z

then (x^6+y^6+z^6)(x^2+y^2+z^2)>=(x^4+y^4+z^4)^2….(1)

use cauchy on the numbers x^2,y^2,z^2 and 1,1,1

then (x^4+y^4+z^4)(1+1+1)>=(x^2+y^2+z^2)^2

3(x^4+y^4+z^4)>=(x^2+y^2+z^2)^2…(2)

squaring both sides of 1 and using 2 we get

(x^4+y^4+z^4)^4<=3[(x^6+y^6+z^6)^2](x^4+y^4+z^4)

putting x^4+y^4+z^4=27 and taking positive square root we get

x^6+y^6+z^6>=81

Practice Problem2.4  if a,b,c be positive numbers such that a+b+c=4 find minimum value of a^3+b^3+c^3

Practice Problem 2.5 Find the min value of 2x+y if xy=8 and x,y are positive numbers

For any queries, post your doubts here itself !