Suppose a, b, c, d are four real numbers andΔ(x)=x+ax+bx+a−cx+bx+cx−1x+cx+dx−b+d,Statement-1: If a, b, c, d are in A.P. and ∫02 Δ(x)dx=−4 then common difference of the A.P. is ±1Statement-2: If a, b, c, d are in A.P., then ∆(x) is independent of x.

Suppose a, b, c, d are four real numbers and

Δ(x)=x+ax+bx+acx+bx+cx1x+cx+dxb+d,
Statement-1: If a, b, c, d are in A.P. and 02Δ(x)dx=4 then common difference of the A.P. is ±1

Statement-2: If a, b, c, d are in A.P., then (x) is independent of x.

  1. A

    STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1 

  2. B

     STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1 

  3. C

    STATEMENT-1 is True, STATEMENT-2 is False 

  4. D

    STATEMENT-1 is False, STATEMENT-2 is True

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    Solution:

    Let D be the common difference of the A.P. |
    Using R3 R3-R2 and R2 R2-R1, we get
    Δ(x)=x+ax+bx+acDD2D1DD2D+1

    Again using C1 C1-C2, we get
    Δ(x)=Dx+bx+ad0D2D10D2D+1=2D2
    Now, 022D2dx=44D2=4D=±1
    Also, (x) is independent of x.

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