Question

Minimum value of the expression $(4 \sin^2\theta + 12 \operatorname{cosec}^2\theta - 13)$ is equal to-

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (C)

3

Answer 2

To find the minimum value of the expression $(4 \sin^2\theta + 12 \operatorname{cosec}^2\theta - 13)$, let's analyze the term $4 \sin^2\theta + 12 \operatorname{cosec}^2\theta$.

1. Define a substitution: Let $x = \sin^2\theta$. Since $\sin\theta$ is a real number, $0 \le \sin^2\theta \le 1$. Also, for $\operatorname{cosec}^2\theta$ to be defined, $\sin\theta \ne 0$, which means $\sin^2\theta \ne 0$. Therefore, the domain for $x$ is $x \in (0, 1]$.

2. Rewrite the expression in terms of x: The expression becomes $E = 4x + 12\left(\frac{1}{x}\right) - 13 = 4x + \frac{12}{x} - 13$. We need to find the minimum value of this expression for $x \in (0, 1]$.

3. Analyze the function $g(x) = 4x + \frac{12}{x}$: To find the minimum value of $g(x)$ on the interval $(0, 1]$, we can use calculus. Find the first derivative of $g(x)$ with respect to $x$: $g'(x) = \frac{d}{dx}\left(4x + \frac{12}{x}\right) = 4 - \frac{12}{x^2}$.

4. Determine the monotonicity of $g(x)$ on $(0, 1]$: For $x \in (0, 1]$, we have $x^2 \le 1$. This implies $\frac{1}{x^2} \ge 1$. Multiplying by 12, we get $\frac{12}{x^2} \ge 12$. Now, consider $g'(x) = 4 - \frac{12}{x^2}$. Since $\frac{12}{x^2} \ge 12$, it follows that $4 - \frac{12}{x^2} \le 4 - 12 = -8$. So, $g'(x) \le -8$, which means $g'(x) < 0$ for all $x \in (0, 1]$.

5. Find the minimum value: Since $g'(x) < 0$ on the interval $(0, 1]$, the function $g(x)$ is strictly decreasing on this interval. For a strictly decreasing function, the minimum value occurs at the largest possible value in its domain. In this case, the largest value of $x$ in $(0, 1]$ is $x=1$.

   Substitute $x=1$ into $g(x)$: $g(1) = 4(1) + \frac{12}{1} = 4 + 12 = 16$. This minimum value of $g(x)$ occurs when $\sin^2\theta = 1$, which is a valid value (e.g., $\theta = \frac{\pi}{2}$).

6. Calculate the minimum value of the original expression: The minimum value of the expression $(4 \sin^2\theta + 12 \operatorname{cosec}^2\theta - 13)$ is: $E_{min} = g(1) - 13 = 16 - 13 = 3$.