The number of points of intersection of two curves y = 2 sin x and y = 5x² + 2x + 3 is (A) 0 (B) 1 (

Q: Wait, why can't we just solve $2 \sin x = 5x^2 + 2x + 3$ algebraically?

That is a fantastic question! The equation $2 \sin x = 5x^2 + 2x + 3$ is a transcendental equation, meaning it mixes trigonometric and algebraic terms. These are notoriously difficult to solve using standard algebraic methods because there is no simple way to isolate $x$. In such cases, range analysis—comparing the maximum of one function to the minimum of the other—is often the most elegant and powerful tool in your arsenal.

Q: How do I know if the parabola opens up or down?

Look at the coefficient of the $x^2$ term. For any quadratic $y = ax^2 + bx + c$, if $a > 0$, the parabola opens upwards like a cup, and it has a minimum value at its vertex. If $a < 0$, it opens downwards like a frown, and it has a maximum value. In our case, $a = 5$, which is positive, so it definitely opens upwards!

Q: Is it possible for the parabola to dip below the sine wave at some other point?

Not in this case! We found that the absolute minimum value of the parabola is $2.8$. Since the sine function $2 \sin x$ is bounded by $2$, it can never reach $2.8$. Because the entire parabola sits at or above $2.8$ and the entire sine wave sits at or below $2$, they are separated by a permanent gap. They simply never meet.

Question

The number of points of intersection of two curves $y = 2 \sin x$ and $y = 5x^2 + 2x + 3$ is

Options:

(A) 0

(B) 1

(C) 2

(D) $\infty$

Sigma Percentile · Accepted Answer

### The Dance of Functions: Why They Never Meet

Welcome, future IITians! Today, we are going to explore a problem that might look like a standard intersection puzzle, but it is actually a beautiful lesson in the power of range analysis.

We are tasked with finding the number of points of intersection between two very different curves: a trigonometric wave, $y = 2 \sin x$, and a quadratic parabola, $y = 5x^2 + 2x + 3$.

When you see a problem like this, your first instinct might be to try and solve for $x$ by setting the equations equal to each other. But hold on! Before you dive into complex algebra, let's pause and visualize the geometry. This is where the magic happens.

### Phase 1: The Oscillating Wave

Let's first analyze our trigonometric friend, $y = 2 \sin x$. We know that the basic sine function, $\sin x$, is a well-behaved creature.

It is strictly bounded, oscillating rhythmically between $-1$ and $1$ for all real values of $x$. When we multiply this by $2$, we are simply scaling the amplitude.

The wave now stretches, reaching a peak of $2$ and a trough of $-2$. Mathematically, we can write this as:

$$-2 \le 2 \sin x \le 2$$

This means that no matter what value of $x$ you choose, the $y$-value of this curve will never, ever exceed $2$. This is our upper boundary, our ceiling.

### Phase 2: The Upward-Opening Parabola

Now, let's turn our attention to the quadratic curve, $y = 5x^2 + 2x + 3$. Because the coefficient of the $x^2$ term is $5$, which is positive, we know this parabola opens upwards.

It is like a cup, and it has a lowest point—a vertex—that we need to find. To find this minimum, we use the classic technique of completing the square.

Let's factor out the $5$ from the $x$ terms:

$$y = 5\left(x^2 + \frac{2}{5}x\right) + 3$$

To make the expression inside the parentheses a perfect square, we take half of the coefficient of $x$ (which is $\frac{1}{5}$), square it to get $\frac{1}{25}$, and then add and subtract it inside the bracket:

$$y = 5\left(x^2 + \frac{2}{5}x + \frac{1}{25} - \frac{1}{25}\right) + 3$$

Simplifying this, we get:

$$y = 5\left(x + \frac{1}{5}\right)^2 - \frac{1}{5} + 3$$

This simplifies further to:

$$y = 5\left(x + \frac{1}{5}\right)^2 + \frac{14}{5}$$

### Phase 3: The Moment of Truth

Now, look at the expression $y = 5(x + \frac{1}{5})^2 + 2.8$. Since the squared term $5(x + \frac{1}{5})^2$ is always greater than or equal to zero, the smallest value this parabola can ever take is $2.8$, which occurs when $x = -\frac{1}{5}$.

This is our floor. Now, let's compare our two findings.

The sine wave is trapped below the line $y = 2$. The parabola is trapped above the line $y = 2.8$.

Since $2.8$ is strictly greater than $2$, there is a permanent gap of $0.8$ units between these two curves. They are like two ships passing in the night, separated by a vast, unbridgeable ocean.

They will never intersect. Therefore, the number of points of intersection is exactly **0**.

This is the beauty of range analysis—it allows us to solve complex problems by understanding the fundamental nature of the functions rather than getting lost in the weeds of calculation. Keep this intuition sharp, and you will conquer any problem the JEE throws your way!

Animated Solution for Mathematics - Quadratic Equations: The number of points of intersection of two curves $y = 2 sin x$ and $y = 5 x^{2} + 2 x + 3$ is

Select Answer:

Visualized Solution

Visualizing the Problem

Plotting $y = 2 sin x$

Range of $y = 2 sin x$

Analyzing the Quadratic Curve

Completing the Square: Part 1

Completing the Square: Part 2

Simplifying the Equation

Minimum Value of the Parabola

Comparing the Ranges

Number of Intersections

The Sigma Insight: Maximum and Minimum Values of Quadratic Expressions

The Dance of Functions

Phase 1

Phase 2

Phase 3

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Animated Solution for Mathematics - Quadratic Equations: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of points of intersection of two curves y=2sinx and y=5x2+2x+3 is