Inequalities - Two more questions


1. (|x| - 2) ( x + 5) < 0. What is the range of values x can take?

2. a and b are roots of the equation x^2 - px + 12 = 0. If the difference between the roots is at least 12, what is the range of values p can take?

Batches starting in Chennai for CAT 2013. Weekend batches @ Anna Nagar from 27th October, @ Mylapore from 10th November, @ Velachery from 28th October

Solutions to the above questions

1. (|x| - 2) ( x + 5) < 0 - 

This can be true in two scenarios

Scenario I - (|x| - 2)  < 0 and ( x + 5) > 0
 Or |x| < 2 and x > -5.This gives us the range (-2,2)

Scenario II - (|x| - 2)  > 0 and ( x + 5) < 0
 Or |x| > 2 and x < -5. This gives us the range (-Infinity, -2)

So, the overall range is (-infinity, -2) or (-2,2)

2. a and b are roots of the equation x^2 - px + 12 = 0. If the difference between the roots is at least 12, what is the range of values p can take?

The roots are a and b
a + b = p ab = 12
(a + b )^2 = p^2
(a -b)^2 = (a + b ) ^2 - 4ab
=> (a-b) ^2 = p^2 - 12*4 = p^2 - 48

If |a-b| > 12 { Difference between the roots is at least 12}
then, (a-b)^2 > 144

p^2 - 48 > 144
p^2 > 192
P > 8sqrt(3) or P < -8 sqrt(3)


IIM CAT Preparation Tips: Inequalities - Two more questions

Aug 4, 2012

Inequalities - Two more questions


1. (|x| - 2) ( x + 5) < 0. What is the range of values x can take?

2. a and b are roots of the equation x^2 - px + 12 = 0. If the difference between the roots is at least 12, what is the range of values p can take?

Batches starting in Chennai for CAT 2013. Weekend batches @ Anna Nagar from 27th October, @ Mylapore from 10th November, @ Velachery from 28th October

Solutions to the above questions

1. (|x| - 2) ( x + 5) < 0 - 

This can be true in two scenarios

Scenario I - (|x| - 2)  < 0 and ( x + 5) > 0
 Or |x| < 2 and x > -5.This gives us the range (-2,2)

Scenario II - (|x| - 2)  > 0 and ( x + 5) < 0
 Or |x| > 2 and x < -5. This gives us the range (-Infinity, -2)

So, the overall range is (-infinity, -2) or (-2,2)

2. a and b are roots of the equation x^2 - px + 12 = 0. If the difference between the roots is at least 12, what is the range of values p can take?

The roots are a and b
a + b = p ab = 12
(a + b )^2 = p^2
(a -b)^2 = (a + b ) ^2 - 4ab
=> (a-b) ^2 = p^2 - 12*4 = p^2 - 48

If |a-b| > 12 { Difference between the roots is at least 12}
then, (a-b)^2 > 144

p^2 - 48 > 144
p^2 > 192
P > 8sqrt(3) or P < -8 sqrt(3)


2 Comments:

At 1:40 PM , Blogger rajput said...

I am here to discuss about trapezium,A trapezium has no parallel sides and any quadrilateral drawn randomly would probably be a trapezium and if the quadrilateral had one pair of parallel sides then it is a trapezoid and if both pairs of sides are parallel its a parallelogram.
How to Construct a Trapezium

 
At 2:37 PM , Blogger Bee said...

Hi Rajput,

The distinction between trapezium and trapezoid depends largely on whether one follows American nomenclature or nomenclature outside of North America. The Americans call the quadrilateral with one pair of parallel sides a trapezoid, the British call it Trapezium.

 

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