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Important Questions for Class 11 Maths Chapter 13 Limits and Derivatives

Important Questions for Class 11 Maths Chapter 13 has been provided here. Students can refer to these questions before their examinations for better preparation.
authorImageNeha Tanna20 Nov, 2024
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Important Questions for Class 11 Maths Chapter 13

Important Questions for Class 11 Maths Chapter 13: Chapter 13 of Class 11 Mathematics, Limits and Derivatives, introduces foundational calculus concepts. It covers the idea of a limit, including evaluating limits of algebraic, trigonometric, and rational functions, and basic rules for calculating them.

The derivative is introduced as the rate of change or slope of a function, along with methods for finding derivatives of polynomials and simple functions. Important questions include solving problems on limit evaluation using formulas, derivatives of basic functions using first principles, and applying derivatives to real-world scenarios. Mastery of this chapter is crucial for understanding advanced calculus in higher classes.

Important Questions for Class 11 Maths Chapter 13 Overview

Chapter 13, Limits and Derivatives is a foundational topic in Class 11 Mathematics. It introduces the fundamental concepts of calculus, including understanding the behavior of functions as inputs approach a certain value (limits) and the rate of change of functions (derivatives). These concepts are crucial for advanced mathematics, physics, and engineering. They form the basis for solving real-world problems like motion, optimization, and growth models. By mastering this chapter, students gain insights into graphical interpretations, continuity, and instantaneous rates of change, which are essential for succeeding in higher studies and competitive exams.

Important Questions for Class 11 Maths Chapter 13 PDF Download

Below, we have provided a comprehensive PDF containing important questions for Class 11 Maths Chapter 13: Limits and Derivatives. These questions are designed to help students understand key concepts and practice effectively for exams. Download the PDF now to access detailed solutions and enhance your preparation for better performance in this crucial chapter.

Important Questions for Class 11 Maths Chapter 13 PDF

Important questions for class 11 maths chapter 13 with solutions

Below we have provided Important Questions for Class 11 Maths Chapter 13 Limits and Derivatives -

Question 1:

Find the derivative of the function x 2 cos x.

Solution:

Given function is x 2 cos x Let y = x 2 cos x Differentiate with respect to x on both sides. Then, we get: dy/dx = (d/dx)x 2 cos x Now, using the formula, we can write the above form as: dy/dx = x 2 (d/dx) cos x + cos x (d/dx)x 2 Now, differentiate the function: dy/dx = x 2 (-sin x) + cos x (2x) Now, rearrange the terms, we will get: dy/dx = 2x cos x – x 2 sin x

Question 2:

Find the positive integer “n” so that lim x → 3 [(x n – 3 n )/(x – 3)] = 108.

Solution:

Given limit: lim x → 3 [(x n – 3 n )/(x – 3)] = 108 Now, we have: lim x → 3 [(x n – 3 n )/(x-3)] = n(3) n-1 n(3) n-1 = 108 Now, this can be written as: n(3) n-1 = 4 (27) = 4(3) 4-1 Therefore, by comparing the exponents in the above equation, we get: n = 4 Therefore, the value of positive integer “n” is 4.

Question 3:

Find the derivative of f(x) = x 3 using the first principle.

Solution:

By definition, f’(x) = lim h→ 0 [f(x+h)-f(x)]/h Now, substitute f(x)=x 3 in the above equation: f’(x) = lim h→ 0 [(x+h) 3 -x 3 ]/h f’(x) = lim h→ 0 (x 3 +h 3 +3xh(x+h)-x 3 )/h f’(x) = lim h→ 0 (h 2 +3x(x+h)) Substitute h = 0, we get: f’(x) = 3x 2 Therefore, the derivative of the function f’(x) = x 3 is 3x 2 .

Question 4:

Determine the derivative of cosx/(1+sin x).

Solution:

Given function: cosx/(1+sin x) Let y = cosx/(1+sin x) Now, differentiate the function with respect to “x”, we get dy/dx = (d/dx) (cos x/(1+sin x)) Now, use the u/v formula in the above form, we get dy/dx = [(1+sin x)(-sin x) – (cos x)(cos x)]/(1+sin x) 2 dy/dx = (-sin x – sin 2 x-cos 2 x)/(1+sin x) 2 Now, take (-) outside from the numerator, we get: dy/dx = -(sin x + sin 2. x + cos 2 x)/(1+sin x) 2 We know that sin 2. x + cos 2 x = 1 By substituting this, we can get: dy/dx = -(1+sin x)/(1+sin x) 2 Cancel out (1+sin x) from both numerator and denominator, we get: dy/dx = -1/(1+sin x) Therefore, the derivative of cosx/(1+sin x) is -1/(1+sin x).

Question 5:

lim x→ 0 |x|/x is equal to: (a)1 (b)-1 (c)0 (d)does not exists

Solution:

A correct answer is an option (d)

Explanation:

The limit mentioned here is x→0 It has two possibilities: Case 1: x→0 + Now, substitute the limit in the given function: lim x→ 0+ |x|/x = x/x = 1 Case 2: x→0 – Now, substitute the limit in the given function: lim x→ 0- |x|/x = -x/x = -1 Hence, the result for both cases varies, the solution is an option (D)

Question 6:

Evaluate the derivative of f(x) = sin 2 x using Leibnitz product rule.

Solution:

Given function: f(x) = sin 2 x Let y= sin 2 x Now, by using Leibnitz product rule, we can write it as: dy/dx = (d/dx) sin 2 x Sin 2 x can be written as (sin x)(sin x) Now, it becomes: dy/dx = (d/dx) (sin x)(sin x) dy/dx = (sin x)’(sin x) + (sin x)(sin x)’ dy/dx = cos x sin x + sin x cos x dy/dx = 2 sin x cos x dy/dx = sin 2x Therefore, the derivative of the function sin 2 x is sin 2x.

Benefits of Important Questions for Class 11 Maths Chapter 13

Solving important questions for Class 11 Maths Chapter 13 Limits and Derivatives offers several benefits, especially when preparing for exams and building a solid foundation for advanced mathematics. Here are some key benefits:

Strong Conceptual Understanding

Helps in grasping the foundational concepts of limits and derivatives, which are essential for calculus in higher classes.

Improved Problem-Solving Skills

Enhances the ability to solve problems methodically and apply formulas correctly.

Better Exam Preparation

Familiarizes students with the types of questions likely to appear in exams, boosting confidence.

Clarity on Applications

Demonstrates real-life applications of limits and derivatives, making the subject more relatable.

Time Management

Regular practice improves speed and accuracy, which is crucial during exams.

Foundation for Higher Studies

Builds a strong base for advanced topics in Class 12 and competitive exams.

Important Questions for Class 11 Maths Chapter 13 FAQs

What is the purpose of limits and derivatives?

Limit refers to the value that a sequence or function approaches when the input approaches a certain value. This is because the derivative assesses the steepness of a function's steepness on a graph at a point on the graph. The value of a function when the input approaches a specific value can be defined as a Limit.

What is the first principle of limits and derivatives?

y = f(x) with respect to its variable x. If this limit exists and is finite, then we say that: Wherever the limit exists is defined to be the derivative of f at x. This definition is also called the first principle of derivative.

What are limits and derivatives used for in real life?

Limits and derivatives are used in many applications, including optimisation problems, curve sketching and finding the area under a curve. In optimisation problems, limits and derivatives are used to find the maximum or minimum values of a function.

How do derivatives connect to limits?

Derivatives be used to help us evaluate indeterminate limits of the form 0 0 through L'Hopital's Rule, which is developed by replacing the functions in the numerator and denominator with their tangent line approximations.
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