Question

# Assertion :If a>b>0 & ac<−1<bc, then tan−1(a−b1+ab)+tan−1(b−c1+bc)+tan−1(c−a1+ac)=π Reason: tan−1x−tan−1y=tan−1(x−y1+xy)∀x,y

A
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
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B
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
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C
Assertion is correct but Reason is incorrect
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D
Assertion false but Reason is true
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Solution

## The correct option is C Assertion is correct but Reason is incorrecttan−1x−tan−1y⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩tan−1(x−y1+xy) if xy>−1π+tan−1(x−y1+xy) if x>0,y<0,xy<−1−π+tan−1(x−y1+xy) if x<0,y>0,xy<−1 Now a>b>0 so ab>0∴tan−1(a−b1+ab)=tan−1a−tan−1b .....( i )And bc>−1∴tan−1(b−c1+bc)=tan−1b−tan−1c......( ii )Also ac<−1 & a>0,c<0∴tan−1(c−a1+ca)=π+tan−1c−tan−1a.....( iii )On adding (i), (ii) & (iii) we gettan−1(a−b1+ab)+tan−1(b−c1+bc)+tan−1(c−a1+ca)∴ Assertion (A) is correct∴ By fact Reason is false

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