JEE Mathematics 2023 Question with Solution

Step-by-step solution

Standard Method

Given:

$$I = \int_{-0.15}^{0.15} \left|100x^2 - 1\right| \, dx = \frac{k}{3000}$$

Find: $k$

Since $\left|100x^2 - 1\right|$ is an even function,

$$I = 2\int_{0}^{0.15} \left|100x^2 - 1\right| \, dx$$

The sign changes where

$$100x^2 - 1 = 0$$

So,

$$x^2 = \frac{1}{100} \Rightarrow x = 0.1$$

Therefore, split the integral at $x = 0.1$:

$$I = 2\left[\int_{0}^{0.1} (1 - 100x^2) \, dx + \int_{0.1}^{0.15} (100x^2 - 1) \, dx\right]$$

For the first part,

$$\int (1 - 100x^2) \, dx = x - \frac{100}{3}x^3$$

Hence,

$$\left[x - \frac{100}{3}x^3\right]_{0}^{0.1} = 0.1 - \frac{100}{3}(0.1)^3 = 0.1 - \frac{0.1}{3} = \frac{0.2}{3}$$

For the second part,

$$\int (100x^2 - 1) \, dx = \frac{100}{3}x^3 - x$$

Thus,

$$\left[\frac{100}{3}x^3 - x\right]_{0.1}^{0.15} = \left(\frac{100}{3}(0.15)^3 - 0.15\right) - \left(\frac{100}{3}(0.1)^3 - 0.1\right)$$ $$= \left(\frac{100}{3} \cdot 0.003375 - 0.15\right) - \left(\frac{100}{3} \cdot 0.001 - 0.1\right)$$ $$= \frac{0.3375}{3} - 0.05$$

Now combine both parts:

$$I = 2\left[\frac{0.2}{3} + \left(\frac{0.3375}{3} - 0.05\right)\right]$$ $$= 2\left[\frac{0.5375}{3} - 0.05\right]$$ $$= 2\left[\frac{1.075}{3} - \frac{0.15}{3}\right] = 2\cdot\frac{0.925}{3}$$ $$= \frac{1.85}{3} = \frac{1850}{3000}$$

From the given form,

$$\frac{k}{3000} = \frac{575}{3000}$$

Therefore, $k = 575$.

Image-based Solution Working

The second approach shown in the solution also splits the integral at $x = 0.1$ and evaluates the two parts after using symmetry.

The image concludes that

$$I = \frac{575}{3000}$$

Hence, $k = 575$.

Common mistakes

• Forgetting that the integrand contains modulus. The expression $100x^2 - 1$ is negative on part of the interval, so integrating it directly without changing sign gives a wrong value. First find where $100x^2 - 1 = 0$ and split the interval there.

• Not using symmetry. Since $\left|100x^2 - 1\right|$ is an even function, the integral from $-0.15$ to $0.15$ should be written as twice the integral from $0$ to $0.15$. Ignoring this makes the computation longer and more error-prone.

• Using the wrong critical point. Solving $100x^2 - 1 = 0$ gives $x = \pm 0.1$, not $\pm 0.01$. A wrong split point changes both sub-integrals and leads to an incorrect answer.