Maths-
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Question

Statement-I : If fraction numerator x squared plus 3 x plus 1 over denominator x squared plus 2 x plus 1 end fraction equals A plus fraction numerator B over denominator x plus 1 end fraction plus fraction numerator C over denominator left parenthesis x plus 1 right parenthesis squared end fraction text  then  end text bold italic A plus bold italic B plus bold italic C equals bold 0
Statement-II :If fraction numerator x squared plus 2 x plus 3 over denominator x cubed end fraction equals A over x plus B over x squared plus C over x cubed text  then  end text A plus B minus C equals 0

Which of the above statements is true

  1. only I is true

  2. only II is true

  3. both I  II are true

  4. neither I nor II true

Hint:

In this question we will solve both the questions to know whether they are true or not. In both statements we will first solve the fractions in RHS. Then we will find the coefficients of x squared, x and constant term after that we will compare both sides to get the value of A,B,C.

The correct answer is:

only II is true


    Statement-I: If fraction numerator x squared plus 3 x plus 1 over denominator x squared plus 2 x plus 1 end fraction equals A plus fraction numerator B over denominator x plus 1 end fraction plus fraction numerator C over denominator left parenthesis x plus 1 right parenthesis squared end fraction text  then  end text bold italic A plus bold italic B plus bold italic C equals bold 0
    equals fraction numerator A left parenthesis x plus 1 right parenthesis squared plus B left parenthesis x plus 1 right parenthesis plus C over denominator left parenthesis x plus 1 right parenthesis squared end fraction
equals fraction numerator A left parenthesis x squared plus 2 x plus 1 right parenthesis plus B left parenthesis x plus 1 right parenthesis plus C over denominator left parenthesis x plus 1 right parenthesis squared end fraction
equals fraction numerator A x squared plus A 2 x plus A plus B x plus B plus C over denominator left parenthesis x plus 1 right parenthesis squared end fraction
equals fraction numerator A x squared plus x open parentheses 2 A plus B close parentheses plus A plus B plus C over denominator left parenthesis x plus 1 right parenthesis squared end fraction
    Now, A=1         2A+B=3              A+B+C=1
    rightwards double arrow 2 plus B equals 3
rightwards double arrow B equals 1         rightwards double arrow 1 plus 1 plus C equals 1
rightwards double arrow C equals 1 minus 2
rightwards double arrow C equals negative 1
    A+B+C=1+1-1=-1
    Statement 1 is not true.
    Statement-II :If
    fraction numerator x squared plus 2 x plus 3 over denominator x cubed end fraction equals A over x plus B over x squared plus C over x cubed text  then  end text A plus B minus C equals 0
space space space space space space space space space space space space space space space space space space space space equals fraction numerator A x squared plus B x plus C over denominator x cubed end fraction
    Now, A=1, B=2, C=3
    A+B-C=0
    1+2-3=0
    So, statement 2 is true.

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