Top Mathematics Questions from IIT JEE Mains 2021: A Comprehensive Guide

Introduction to IIT JEE Mains 2021 Mathematics Questions

As thousands of aspirants gear up for the prestigious Indian Institutes of Technology Joint Entrance Examination (JEE) Main, understanding the types of questions asked in the previous years becomes crucial. In this article, we will explore some of the notable mathematics questions from the IIT-JEE Mains in 2021. These questions are designed to test the depth of mathematical knowledge and problem-solving abilities of the candidates.

Algebra: A Foundation for Success

One of the predominant areas in the mathematics section of JEE Mains, algebra questions often appear in a broad spectrum, ranging from simple quadratic equations to complex inequalities. In one of the notable questions from the 2021 exam, students were required to solve the following equation:

Example 1: Advanced Algebra Problem in IIT JEE Mains 2021

Given the function f(x) x^3 - 6x^2 11x - 6. Find the roots of the function and determine the intervals where the function is increasing and decreasing.

Practical Steps to Solve:

Identify the roots of the equation, given that one root is x 1. Use synthetic division or polynomial division to find the other roots. Differentiate the function to find the critical points. Analyze the intervals defined by these critical points to determine where the function is increasing or decreasing.

Calculus: A Critical Component of the Exam

Calculus questions in the IIT JEE Mains often require a deep understanding of concepts like differentiation, integration, and their applications. Here’s a challenging calculus problem that appeared in the 2021 exam:

Example 2: Advanced Calculus Problem in IIT JEE Mains 2021

Consider the function F(x) ∫ e^(-t^2) dt. Find the value of F(1).

Practical Steps to Solve:

Recognize the function as a definite integral and understand the properties of the exponential function. Utilize the given bounds of integration to evaluate the function. Note that the integral does not have an elementary antiderivative, but the value of the integral at specific points can be computed using numerical methods or known series expansion.

Geometry: Shapes and Structures

Geometry questions in the IIT JEE Mains can range from basic coordinate geometry to more intricate problems involving vectors and planar geometry. Here is an example from 2021:

Example 3: Advanced Geometry Problem in IIT JEE Mains 2021

Given a triangle ABC with vertices A(0,0), B(2,0), and C(1,3). Find the equation of the circumcircle of the triangle.

Practical Steps to Solve:

Calculate the lengths of the sides AB, BC, and AC using the distance formula. Determine the coordinates of the circumcenter using the perpendicular bisectors of any two sides. Use the circumcenter and any of the triangle’s vertices to derive the equation of the circumcircle.

Combinatorics: Count and Compute

Lastly, combinatorics questions are not uncommon in the IIT JEE Mains. Here is a question that appeared in 2021:

Example 4: Advanced Combinatorics Problem in IIT JEE Mains 2021

A bag contains 5 indistinguishable red balls and 5 indistinguishable blue balls. In how many ways can 7 balls be selected such that there is at least 1 ball of each color?

Practical Steps to Solve:

Apply the principle of inclusion-exclusion to count the total number of ways to select 7 balls from 10. Subtract the cases where only red or only blue balls are selected. Ensure that at least one ball of each color is included in the selection.

Conclusion and Further Preparation Tips

The questions described above are just a snapshot of the variety and depth of the mathematics questions asked in the IIT JEE Mains 2021. Candidates aspiring to excel in this exam should allocate significant time to practice such problems and sharpen their problem-solving skills. Regular practice, along with a thorough understanding of core concepts, is key to help you further in your preparation, here are some recommended resources:

Practice past year question papers (IIT JEE Main Advanced) Join online tutorial programs and mentorship Utilize video lectures on subject-specific topics Participate in mock tests to enhance speed and accuracy

By studying these examples and adopting a structured approach to preparation, aspiring JEE Mains candidates can increase their chances of excelling in the examination.