Inequalities  Nilanjan Dutta, XLRI  100 Percentile CAT 2016

This lesson is based on the event conducted by ADURO. Class was taken by Nilanjan Dutta who is an alum of XLRI, Jamshedpur. He clocked a perfect 100 percentile score in CAT 2016 and also scored 99.43 %ile in CAT 2011 and 99.35%ile in XAT 2012. Nilanjan is a Business Consultant by Profession, besides helping students on the CAT/GDPI/CRT front.
what is meant by the term Inequalities?
Normally we deal a lot in Equations in Quants, where we equate L.H.S. and R.H.S.
Similarly in Equalities we have L.H.S. and R.H.S. as well, just that the two aint necessarily equal,and could be connected in any of the following manners:
L.H.S > = R.H.S., L.H.S. > R.H.S., L.H.S. < R.H.S., L.H.S. < = R.H.S.
Some Basic Rules and Things to Remember in case of Inequalities are:
1) The concept of Arithmetic Mean > = Geometric Mean(G.M.) for any set of +ve numbers
2) If A*B > 0, then A & B are of same sign, both are +ve, or both are ve
3) If A*B < 0, then A & B are of opposite sign, If A is +ve, B is ve and viceversa
4) Concept of Modulous Function, which implies the magnitude of the variable and not the sign, 4 = 4, 4 = 4
5) MaximaMinima concepts dealing with products and sum
The easiest way to solve inequalities is by simply solving it the wrong way, That is simply EQUATE the inequalities, And for Everything else, mark the CORRECT OPTION and leave. Please DO NOT solve a Single Inequalities Question unless it becomes very mandatory to solve it,Simply Equate or Look at the Options n Mark the Correct Answer. In this note we shall be solving most questions, without solving.
What is the minimum value of ( X + Y ) / Z + ( X + Z ) / Y + ( Y + Z ) / X ?
A) 1
B ) 2
C) 3
D) 6In Any INEQUALTIES Question all you need to do is EQUATE, so all I do is EQUATE X = Y = Z, then the expression becomes 2 + 2 + 2 = 6, irrespective of the value X,Y,Z
Minimum value of ( a + b + c ) ( 1/a + 1/b + 1/c ) ?
A) 3
B ) 6
C) 8
D) 9Equate a =b = c, you get 3a * 3/a = 9
If a, b,c are distinct positive real numbers, then [a^{2} . ( b + c ) + b^{2} . ( c + a ) + c^{2} . ( a + b ) ]/abc is always
A) > 4
B ) > 6
C) > 9
D) None of TheseEquate a = b = c, we get 6a^{3} / a^{3 }= 6
Now it is given that a, b and c are distinct positive real numbers hence answers will be always > 6 ( Option B )
Solve (1 + x^{2}) / (x^{2}  5x + 6) < 0
A) x < 2
B x > 3
C) 2 < x < 3
D) Both a and bput x = 2.5 and check if LHS is +ve or not. Correct answer is Option C
If ab < = 28, bc < = 14, ac < = 8, then what is the max value of the product of a, b and c ?
A) 196
B ) 28
C) 56
D) 112multiply the three, (abc)^{2} < = 28 * 14 * 4 < = 56 * 56 < = 56^{2}, Hence abc < = 56, Max value = 56
Let p be any nonnegative integer, and 2x + p = 2y, p + y and x + y = z , For what value of p, will x + y + z be maximum ?
A) 0
B ) 1
C) 2
D) 3Valid only when p = 0
( x^{2}  3x + 24) / (x^{2}  3x + 3) < 4, Solve for range of values x can assume
A) x < 1
B ) 4 < x < 8
C) 4 < x < 6
D) None of theseUse options, Answer is D
Solve for all real values of x, if root(9x x^{2}) > 0
A) x < 0
B ) 0 < x < 9
C) x > 9
D) None of TheseOption B is correct.
x(9x) > 0
x(x9) < 0
0 < x < 9Solve the inequality: x^{3} – 5x^{2} + 8x – 4 > 0. ( Hard Question from 2iim )
A) (2, ∞)
B ) ((1, 2) ∪ (2, ∞)
C) (∞, 1) ∪ (2, ∞)
D) (∞, 1)Simply put x =0, then LHS = 4, NOT Satsfied, Eliminate Option C & D
Put x = 3, Satisfied , both A & B are possible
Then put x = 1.5, and see if it satisfies or NOT
Again DO NOT TRY SOLVING INEQUALITIESIf R = ( 30^{65}  29^{65} ) / ( 30^{64} + 29^{64 }) then find range of R
A) 00.1
B ) 0.10.5
C) 0.51
D) Greater than 1Numerator = (3029) (Denominator + some positive value) hence > 1 ( Option D )