Question Stem for Question 15 and 16:

Consider the curve $C_1$ given by $y=e^{-x}$ for $x\in [0,10\pi ]$, and the curve $C_2$ given by $y=e^{-x(\sin x+\cos x)}$ for $x\in [0,10\pi ].\; Let$ n$be\; the\; total\; number\; of\; points\; of\; intersection\; of\; the\; curves$ C_1$and$ C_2$.$

Suppose that ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n\in [0,10\pi ]$ are the $x$ -coordinates of the points of intersection of the curves $C_1$ and $C_2$ such that ${\alpha }_1<{\alpha }_2<\dots <{\alpha }_n$.

Then the value of n is .

The correct answer is 11.

To find the total number of points of intersection, $n$, of the two curves $C_1$ and $C_2$ in the interval $x\in [0,10\pi ]$, we set their $y$-coordinates equal to each other:

$e^{-x}=e^{-x(\sin x+\cos x)}$

Since the exponential function is one-to-one, we can equate the exponents:

$-x=-x(\sin x+\cos x)$

Rearranging all terms to one side:

$x(1-(\sin x+\cos x\left)\right)=0$

This gives us two cases to consider:

Case 1: $x=0$

Since $0\in [0,10\pi ]$, $x=0$ is a valid solution.

Case 2: $\sin x+\cos x=1$

To solve this trigonometric equation, divide both sides by $\sqrt{2}$:

$\frac{1}{\sqrt{2}}\sin x+\frac{1}{\sqrt{2}}\cos x=\frac{1}{\sqrt{2}}$

Using the sine addition formula $\sin (A+B)=\sin A\cos B+\cos A\sin B$ with $\cos \left(\frac{\pi }{4}\right)=\sin \left(\frac{\pi }{4}\right)=\frac{1}{\sqrt{2}}$:

$\sin (x+\frac{\pi }{4})=\frac{1}{\sqrt{2}}$

The general solutions for this equation are:

$x+\frac{\pi }{4}=2k\pi +\frac{\pi }{4}$ or $x+\frac{\pi }{4}=2k\pi +\frac{3\pi }{4}$ for any integer $k$.

Subtracting $\frac{\pi }{4}$ from both sides gives two sets of solutions:

1) $x=2k\pi$

2) $x=2k\pi +\frac{\pi }{2}$

Now, let us find all distinct solutions within the given interval $x\in [0,10\pi ]$:

For $x=2k\pi$:

$k=0,1,2,3,4,5\Rightarrow x=0,2\pi ,4\pi ,6\pi ,8\pi ,10\pi$

(Total of 6 solutions, including $x=0$ from Case 1).

For $x=2k\pi +\frac{\pi }{2}$:

$k=0,1,2,3,4\Rightarrow x=\frac{\pi }{2},2\pi +\frac{\pi }{2},4\pi +\frac{\pi }{2},6\pi +\frac{\pi }{2},8\pi +\frac{\pi }{2}$

(Total of 5 distinct solutions).

Combining all distinct intersection points gives:

$n=6+5=11$

Thus, the total number of points of intersection is 11.