| Assertion [A]: The tangent to curve \[y=4{{x}^{3}}-3x+5\] at \[x\text{ }=\text{ }1\] is perpendicular to the line \[x+9y+3=0.\] |
| Reason [R]: Slope of line \[ax+by+c=0\] is |
| \[-\frac{Coeff\,.\,of\,x}{Coeff\,.\,of\,y}\] |
A) Both A and R are individually true and R is the correct explanation of A.
B) Both A and R are individually true and R is not the correct explanation of A.
C) 'A' is true but 'R' is false
D) 'A' is false but 'R' is true
E) Both A and R are false.
Correct Answer: B
Solution :
Given \[y=4{{x}^{3}}-3x+5\] \[\frac{dy}{dx}=12{{x}^{2}}-3=3(4{{x}^{2}}-1)\] At x = 1, slope of tangent, \[{{m}_{1}}=3(4\times {{(1)}^{2}}-1)\] \[=3\times 3=9\] Slope of line \[x+9y+3=0\] is, \[{{m}_{2}}=-\frac{1}{9}\] Clearly \[{{m}_{1}}{{m}_{2}}=9\times \left( -\frac{1}{9} \right)=-1\] \[\Rightarrow \] Assertion [A] is true. We know that slope of line \[ax+by+c=0\] is \[-\frac{a}{b}\] \[\therefore \] Reason (R) is true. But It is not correct explanation of Assertion [A]. Hence option [B] is the correct Answer.
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