Question

Assertion :If a quadratic curve touches the line $y=x$ at the point $(1,1)$, then the values of $x$ for which the curve has a negative gradient are $x<\frac{1}{2}$ Reason: The equation of the curve is $y=x^2-x+1$

• A. Assertion is true and Reason is true; Reason is a correct explanation for Assertion.
• B. Assertion is True, Reason is true; Reason is not a correct explanation for Assertion.
• C. Assertion is true, Reason is false
• D. Assertion is false, Reason is true

Answer 1

The correct option is A Assertion is true and Reason is true; Reason is a correct explanation for Assertion.

We have $y=x^2+bx+c\Rightarrow \frac{dy}{dx}=2x+b$

Since the curve touches the line $y=x$ at the point $(1,1)$

${[2x+b]}_{(1,1)}=1\Rightarrow 2+b=1\Rightarrow b=-1$

Also, the curve passes through the point $(1,1)$

$\therefore 1=1+b+c\Rightarrow c=-b=1$

$\therefore y=x^2-x+1\Rightarrow \frac{dy}{dx}=2x-1$

Now, $\frac{dy}{dx}<0\Rightarrow 2x-1<0\Rightarrow x<\frac{1}{2}$