NIMCET Previous Year Questions (PYQs) – Quadratic Equations – Page 1 of 3

Number of Roots

Given:

$e^x\sin x=1$ has two real roots → say $x_1$ and $x_2$

Apply Rolle’s Theorem:

Since $f(x)=e^x\sin x$ is continuous and differentiable, and $f(x_1)=f(x_2)$, ⇒ There exists $c\in (x_1,x_2)$ such that $f^{\prime }(c)=0$

Compute:

$$f^{\prime }(x)=e^x(\sin x+\cos x)=0\Rightarrow \tan x=-1$$ At this point, $$e^x\cos x=-1$$

$$\boxed{{\displaystyle \text{At least one root}}}$$