Question

The number of solutions of \[y = {e^x}\] and \[y = \sin x\] is:
A. \[0\]
B. \[1\]
C. \[2\]
D. Infinite

Answer 1

Hint: In this question, we need to find the number of solutions of \[y = {e^x}\] and \[y = \sin x\]. For this, we need to draw the graphs of these two equations. We can determine the number of solutions based on the number of intersecting points of two graphs.

Complete step-by-step answer:
Consider first \[y = {e^x}\]
Let us plot its graph.

Image: Graph of \[y = {e^x}\]
Now, let us plot the graph of \[y = \sin x\].

Image: Graph of sinx
Now, we will combine these two graphs into one graph for finding a number of intersecting points.
Consider the following combined graph of \[y = {e^x}\] and \[y = \sin x\].

Image: Combined graphs of \[y = {e^x}\] and \[y = \sin x\]
Here, we can say that there are infinite numbers of intersecting points of two curves.
That means, there are an infinite number of solutions for the curves \[y = {e^x}\] and \[y = \sin x\].
Therefore, the correct option is (D).

Additional Information: Sine waves or sinusoidal waves are graphs of functions defined by \[y = \sin x\]. Take note of how the graph tends to repeat itself while you start moving along the x-axis. Periods are the iterations of this regular repeating. Here, the interesting part is the graph of every exponential function that passes through the point (0, 1). Also, such graphs are continuous and increasing. The domain of the graph of an exponential function is the set of all the real numbers.

Note: Here, students generally make mistakes in drawing graphs. Thus, they may get the wrong result. Here, it is necessary to combine both the graphs in one graph to visualize infinite intersecting points of two curves in a better way.