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GATE 2023 MA – Question 50

Numerical Analysis · Numerical Differentiation and Integration · 2 marks · Multiple select

For h>0 define D_hf(a)=[αf(a−h)+βf(a)+γf(a+2h)]/(6h). If this equals f′(a) for every real polynomial of degree at most 2 and every real a, which hold?

  1. α+2γ=−2
  2. α+2β−2γ=0
  3. α+2γ=2
  4. α+2β−2γ=1

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Show answer and explanation

Correct answer: (A) α+2γ=−2; (B) α+2β−2γ=0

Explanation

Exactness for constants gives α+β+γ=0. Matching first-order and second-order Taylor coefficients gives −α+2γ=6 and α+4γ=0. Thus γ=1,α=−4,β=3. Substitution yields α+2γ=−2 and α+2β−2γ=0. **A,B**.

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