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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