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If F ( X ) = ⎧ ⎪ ⎨ ⎪ ⎩ X 2 + B , 0 ≤ X < 1 4 , X = 1 X + 3 , 1 < X ≤ 2 , Then the Value of (A, B) for Which F (X) Cannot Be Continuous at X = 1, is (A) (2, 2) (B) (3, 1) (C) (4, 0) (D) (5, 2) - Mathematics

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Question

If  \[f\left( x \right) = \left\{ \begin{array}a x^2 + b , & 0 \leq x < 1 \\ 4 , & x = 1 \\ x + 3 , & 1 < x \leq 2\end{array}, \right.\] then the value of (ab) for which f (x) cannot be continuous at x = 1, is

 

Options

  • (2, 2)

  • (3, 1)

  • (4, 0)

  • (5, 2)

MCQ
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Solution

(5, 2) 

If f(x) is continuous at x = 1, then

\[\lim_{x \to 1^-} f\left( x \right) = f\left( 1 \right)\]

\[\Rightarrow \lim_{h \to 0} f\left( 1 - h \right) = 4 \left[ \because f\left( 1 \right) = 4 \right]\]
\[ \Rightarrow \lim_{h \to 0} a \left( 1 - h \right)^2 + b = 4 \]
\[ \Rightarrow \left( a + b \right) = 4\]

Thus, the possible values of (ab) can be  

\[\left( 2, 2 \right), \left( 3, 1 \right), \left( 4, 0 \right)\] . But
\[\left( a, b \right) \neq \left( 5, 2 \right)\].

Hence, for  

\[\left( a, b \right) = \left( 5, 2 \right)\],
\[f\left( x \right)\]  cannot be continuous at = 1.

 

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Notes

Disclaimer: The question in the book has some error. The solution here is created according to the question given in the book.

  Is there an error in this question or solution?
Chapter 9: Continuity - Exercise 9.4 [Page 46]

APPEARS IN

RD Sharma Mathematics [English] Class 12
Chapter 9 Continuity
Exercise 9.4 | Q 32 | Page 46

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