| 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 \[{{x}^{2}}+x-12\] divides \[{{x}^{2}}+x-12\] exactly, then \[a=-8\] and b = - 5 |
| Reason [R] When a polynomial \[p\left( x \right)\]is completely divided by \[\left( x-\alpha \right)\], then \[p\left( x \right)=0\]. |
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: D
Solution :
| Let \[f\left( x \right)={{x}^{2}}+x-12\] |
| \[={{x}^{2}}+4x-3x-12\] |
| \[=x\left( x+4 \right)-3\left( x+4 \right)\] |
| \[=\left( x-3 \right)\left( x+4 \right)\] |
| \[p\left( 3 \right)=0=p\left( -4 \right)\] |
| \[p\left( 3 \right)={{3}^{3}}+a{{\left( 3 \right)}^{2}}+b\left( 3 \right)-84=0\] |
| \[27+9a+3b-84=0\] |
| \[9a+3b=57\] …. (i) |
| \[p\left( -4 \right)={{\left( -4 \right)}^{3}}+a{{\left( -4 \right)}^{2}}+b\left( -4 \right)-84=0\] |
| \[\Rightarrow \,\,-64+16a-4b-84=0\] |
| \[4a-b=37\] |
| \[12a-3b=111\] .… (ii) |
| Adding Eqs. (i) and (ii) |
| \[21a=168\] |
| \[a=8\] |
| Substitute a = 8 in Eq. (i) |
| \[9\times 8+3\times b=57\] |
| \[72\text{ }+\text{ }3b\text{ }=\text{ }57\] |
| \[3b=-15\] |
| \[b=-5\] |
| Hence, Assertion is false but Reason is true. |
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