Assertion Reasoning Quiz

Assertion (A): \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \).

Reason (R): For \( 0 < |x| < \frac{\pi}{2} \), the inequality \( \cos x < \frac{\sin x}{x} < 1 \) holds, and by Sandwich Theorem, as \( x \to 0 \), both lower and upper bounds approach 1.

Assertion (A): For \( f(x) = x^2 - 2 \), the derivative at \( x = 10 \) is \( f'(10) = 20 \).

Reason (R): Power rule gives \( \frac{d}{dx}(x^2) = 2x \).

Assertion (A): For the cost function \( C(x) = 0.005 x^3 - 0.02 x^2 + 30x + 5000 \), the marginal cost when 100 units are produced is ₹ 176.

Reason (R): Marginal cost is the derivative \( C'(x) = 0.015 x^2 - 0.04 x + 30 \), which evaluates to \( 0.015(10000) - 0.04(100) + 30 = 150 - 4 + 30 = 176 \) at \( x = 100 \).

Assertion (A): For the polynomial \( f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + \dots + \frac{x^2}{2} + x + 1 \), the relation \( f'(1) = 100 f'(0) \) is satisfied.

Reason (R): Differentiating term-by-term yields \( f'(x) = x^{99} + x^{98} + \dots + x + 1 \), so \( f'(0) = 1 \) and \( f'(1) = 100 \).

Assertion (A): For the function \( f(x) = \frac{x^2 - 4}{x - 2} \) (\( x \ne 2 \)), the limit \( \lim_{x \to 2} f(x) = 4 \).

Reason (R): The value of the function at \( x = 2 \) is equal to 4, i.e., \( f(2) = 4 \).

Assertion (A): \( \lim_{x \to 5} (|x| - 5) = 0 \).

Reason (R): The absolute value function \( f(x) = |x| \) is continuous for all real numbers \( x \in \mathbb{R} \).