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10th Class Mathematics Polynomials Question Bank Assertion and Reason MCQs - Polynomials

  • question_answer
    Directions: Each of these questions contains two statements: Assertion [A] and Reason [R]. Each of these questions also has four alternative choices, any one of which is the correct answer. You have to select one of the codes [a], [b], [c] and [d] given below.
    Assertion [A] If the zeroes of \[{{x}^{2}}+px+q\] are two consecutive integers, then \[{{p}^{2}}-1=4q\]
    Reason [R] If \[\alpha \], \[\beta \]are zeroes of
    \[\left( x-a \right)\left( x-b \right)-c\], then a, b are zeroes of
    \[\left( x-\alpha  \right)\left( x-\beta  \right)+c\]

    A) A is true, R is true; R is a correct explanation for A.

    B) A is true, R is true; R is not a correct explanation for A.

    C) A is true; R is False.

    D) A is false; R is true.

    Correct Answer: B

    Solution :

    Assertion Let \[\alpha ,\,\beta \]be the zeroes of given polynomial \[{{x}^{2}}+px+q\]
    \[\therefore \,\,\,\,\alpha -\beta =1\]                                               … (i)
    And \[\alpha +\beta =-p\]
    \[\alpha \,.\,\beta =q\]                                         … (ii)
    from Eq. (i), squaring \[{{\left( \alpha -\beta  \right)}^{2}}=1\]
    \[\Rightarrow \,\,\,{{\left( \alpha +\beta  \right)}^{2}}-4\alpha \beta =1\]
    \[\Rightarrow \,\,{{\left( -p \right)}^{2}}-4q=1\]                   (using (ii))
    \[\Rightarrow \,\,{{p}^{2}}-1=4q\]
    Reason Given, \[p\left( x \right)=\left( x-a \right)\left( x-b \right)-c\]
    \[={{x}^{2}}-\left( a+b \right)x+\left( ab-c \right)\]
    \[\therefore \,\,\,\,\alpha +\beta =-\frac{\left\{ -\left( a+b \right) \right\}}{1}=a+b\]
    \[\Rightarrow \,\,\,a+b=\alpha +\beta \]                                   … (i)
    (\[\because \,\,\alpha \] and \[\beta \]are zeroes of \[p\left( x \right)\]]
    and  \[\alpha \beta =\frac{\left( ab-c \right)}{1}=ab-c\]
    \[\Rightarrow \,\,\,ab=\alpha \beta +c\]
    Now, family of polynomials having a and b as its zeroes is given by \[k\left[ {{x}^{2}}-\left( a+b \right)x+ab \right]\]
    \[=k\left[ {{x}^{2}}-\left( \alpha +\beta  \right)x+\left( \alpha \beta +c \right) \right]\]
    [Using Eqs. (i) and (ii)]
    Now, Taking k = 1, the polynomial with a and b as its zeroes
    \[={{x}^{2}}-\left( \alpha +\beta  \right)x+\alpha \beta +c\]
    \[=\left( x-\alpha  \right)\left( x-\beta  \right)+c\]
    Assertion is the true, Reason is true but Reason is not a correct explanation for Assertion.
     


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