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KVPY2013MathematicsDefinite Integration

For x , t R let p_ t (x)=( sint ) x²-(2 cost ) x+ sint be a family of quadratic polynomials in x with variable coefficients. Let A(t)= ₀¹ p_ t (x) d x . Which of the following statements are true? (I) A ( t ) < 0 for all t . (II) A(t) has infinitely many critical points. (III) A ( t )=0 for infinitely many t . (IV) A^ (t) < 0 for all t .

Options

  1. A(I) and (II) only
  2. B(II) and (III) only
  3. C(III) and (IV) only
  4. D(IV) and (I) only

Correct answer

B. (II) and (III) only

Step-by-step solution

array l A(t)= ₀¹ ( t) x²-(2 t) x+ t d x A(t)= t 3 - t+ t= 4 3 t- t A^ (t)= 4 t 3 + t array St. I and N are False

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