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Question

The number of points of discontinuity of f(x)={[cosπx],0x1|2x3|[x2],1<x2 is/are where [.] denotes the greatest integral function

A
Two
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B
Three
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C
Four
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D
Zero
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Solution

The correct option is C Three
Consider x[0,1]
From the graph given, it is clear that [cosπx] is discontinuous at x=0,1/2 (1)
Now, consider x(1,2].
f(x)=[x2]|2x3|
For x(1,2),[x2]=1, and for x=2,[x2]=0.
Also, |2x3|=0 or x=3/2.
Therefore, x=3/2 and 2 may be the points at which f(x) is discontinuous.
f(x)=⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪1,x=0x,0<x121,12<x1(32x),1<x3/2(2x3),3/2<x20,x=2
Thus, f(x) is continuous when x[0,2]{0,1/2,2}.

option B is correct

421167_119539_ans_197db890f66f4076892e2d51629a1604.png

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