NIMCET 2009 Mathematics PYQ — Let where denotes the greatest integer function. Then the number … | Mathem Solvex | Mathem Solvex
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NIMCET 2009 — Mathematics PYQ
NIMCET | Mathematics | 2009
Let f(x)=[x2−3] where [⋅] denotes the greatest integer function. Then the number of points in the interval (1,2) where the function is discontinuous is:
Choose the correct answer:
A.
4
B.
2
(Correct Answer)
C.
6
D.
None
Correct Answer:
2
Explanation
1. Property of the Greatest Integer Function
The function f(x)=[g(x)] is discontinuous at points where g(x) takes on an integer value, provided that g(x) is not constant or at a local extremum at those points.
2. Identify the Inner Function
In the given problem, the inner function is g(x)=x2−3. We need to find how many integers this function passes through as x moves from 1 to 2.
3. Evaluate the Range of g(x)
Let's find the values of g(x) at the boundaries of the interval (1,2):
As x→1+, g(x)=(1)2−3=1−3=−2.
As x→2−, g(x)=(2)2−3=4−3=1.
Since g(x)=x2−3 is a strictly increasing function for x > 0, it will take all values in the range (−2,1).
4. Find Integer Values in the Range
The integers strictly between −2 and 1 are:
−1,0
5. Locate the Points of Discontinuity
The function f(x) will be discontinuous when x2−3 equals these integers:
Case 1:x2−3=−1
x2=2⟹x=2 (Since 1 < \sqrt{2} < 2)
Case 2:x2−3=0
x2=3⟹x=3 (Since 1 < \sqrt{3} < 2)
The points x=2 and x=3 both lie within the open interval (1,2).
Conclusion:
There are exactly 2 points in the interval (1,2) where the function is discontinuous.
Final Answer:
The correct option is (b) 2.
Explanation
1. Property of the Greatest Integer Function
The function f(x)=[g(x)] is discontinuous at points where g(x) takes on an integer value, provided that g(x) is not constant or at a local extremum at those points.
2. Identify the Inner Function
In the given problem, the inner function is g(x)=x2−3. We need to find how many integers this function passes through as x moves from 1 to 2.
3. Evaluate the Range of g(x)
Let's find the values of g(x) at the boundaries of the interval (1,2):
As x→1+, g(x)=(1)2−3=1−3=−2.
As x→2−, g(x)=(2)2−3=4−3=1.
Since g(x)=x2−3 is a strictly increasing function for x > 0, it will take all values in the range (−2,1).
4. Find Integer Values in the Range
The integers strictly between −2 and 1 are:
−1,0
5. Locate the Points of Discontinuity
The function f(x) will be discontinuous when x2−3 equals these integers:
Case 1:x2−3=−1
x2=2⟹x=2 (Since 1 < \sqrt{2} < 2)
Case 2:x2−3=0
x2=3⟹x=3 (Since 1 < \sqrt{3} < 2)
The points x=2 and x=3 both lie within the open interval (1,2).
Conclusion:
There are exactly 2 points in the interval (1,2) where the function is discontinuous.
