Teachoo MCQ Test

Assertion Reasoning Quiz

Chapter 12 Class 11 Limits and Derivatives | 7 questions

Timed test. Answer every question before time runs out. Correct answers and solutions appear after submission.
Time remaining 672
Question 1 of 7
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Question 1 of 7
Assertion (A): If \( \lim_{x \to a} f(x) \) and \( \lim_{x \to a} g(x) \) both exist, then \( \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} \) for all functions \( g(x) \).
Reason (R): The quotient law of limits holds provided the limit of the denominator in the limit process is non-zero.
Select one option. Answers are shown after the test.
Question 2 of 7
Assertion (A): \( \lim_{x \to 0} \frac{\tan x}{x} = 0 \).
Reason (R): \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \) and \( \lim_{x \to 0} \cos x = 1 \).
Select one option. Answers are shown after the test.
Question 3 of 7
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 \( x = 100 \).
Select one option. Answers are shown after the test.
Question 4 of 7
Assertion (A): If the tangent line to \( y = f(x) \) at point \( P(a, f(a)) \) makes an angle \( \psi \) with the positive x-axis, then \( f'(a) = \tan \psi \).
Reason (R): As point \( Q(a+h, f(a+h)) \) approaches point \( P(a, f(a)) \) along the curve, the secant chord slope \( \frac{f(a+h) - f(a)}{h} \) approaches the tangent slope \( \tan \psi \).
Select one option. Answers are shown after the test.
Question 5 of 7
Assertion (A): The derivative of a constant function \( f(x) = c \) at any point \( x = a \) is equal to \( c \).
Reason (R): The derivative measures rate of change, and since a constant function does not change, its rate of change is zero everywhere.
Select one option. Answers are shown after the test.
Question 6 of 7
Assertion (A): If the left-hand limit of a function at \( x = a \) is 2 and the right-hand limit at \( x = a \) is 3, then the limit \( \lim_{x \to a} f(x) = 2.5 \).
Reason (R): The limit of a function \( f(x) \) as \( x \to a \) exists if and only if both left-hand limit and right-hand limit exist and are equal.
Select one option. Answers are shown after the test.
Question 7 of 7
Assertion (A): The derivative of \( h(x) = \frac{x^5 - \cos x}{\sin x} \) is \( h'(x) = \frac{-x^5 \cos x + 5x^4 \sin x + 1}{\sin^2 x} \).
Reason (R): The Pythagorean identity states that \( \sin^2 x + \cos^2 x = 1 \).
Select one option. Answers are shown after the test.