In this article, you will find Class 11 Maths NCERT solutions in PDF format. These are notes thoroughly organized by HashLearn teachers making it ideal for exams like **JEE & other state engineering exams**, **CBSE and ICSE Board exams**.…

# Class 12 Maths NCERT solutions

George Koshy March 4, 2019Students preparing for various competitive exam often dive directly into the subject without having a look at the syllabus first. A deep understanding of the syllabus can help with familiarizing the subject, understanding its depth, difficulty level and so on. Hence, the first step in your preparation process is to have a deep understanding of the syllabus. …

Read More# Descartes’ Rule of Signs explained by an IITian

Natarajan C February 23, 2017This explanation of Descartes’ Rule of Signs was submitted by Ayush, a HashLearn tutor from IIT, Dhanbad.

Consider a polynomial equation p(x)=0 (descending powers of x)

i) Number of positive roots of the equation p(x)=0 cannot exceed the number of changes of sign in p(x).…

# Study Tips: Maxima and Minima

Natarajan C February 18, 2017This tip on calculus was contributed by one of our top tutors, Darshan Maheshwari.

The last edition of study tips, we posted an interesting trick to differentiate an infinite series. This time we’re following it up with helpful formulae to quickly find maxima or minima for functions involving two variables.…

Read More# Study Tips: Differentiation of an Infinite Series

Natarajan C February 16, 2017This tip on calculus was contributed by one of our top tutors, Shashwat.

What you will learn from this tip:

The trick to differentiate an infinite series.

Consider the following question:

A function y is defined as

Find at x = 0.…

Read More# Study Tips: Finding the Roots of Unity

Natarajan C February 14, 2017This tip on roots of an equation, and complex numbers was contributed by one of our top tutors, Sudheer Naidu.

- What you will learn from this tip:
- An understanding of the relation between complex numbers and the roots of unity.
- The trick to calculate the n
^{th}roots of unity.

# Study Tips: Sum of Infinite Telescoping Series

Natarajan C February 11, 2017This tip on sequences and series was contributed by one of our top tutors, Rushil Shah.

- What you will learn from this tip:
- An understanding of a telescoping series
- The trick to calculate the sum of a telescoping series

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# Telescoping series

A telescoping series is any series where nearly every term cancels with a preceding or following term.…

Read More# Trick to solve seating arrangement problems

Natarajan C January 7, 2017This tip to solve permutation and combinations problems was submitted by Ayush, a HashLearn tutor from IIT, Dhanbad.

In this tip, you will learn how to quickly solve one kind of seating arrangement problems.

**Question**

*n* people are sitting on a round table.…

# Study Tips: Finding Values For Trigonometric Identities

Natarajan C December 6, 2016With this tip, you will learn:

- The 4 types of trigonometric identities
- The formulae to find maximum & minimum values for trigonometric identities for each type
- An example problem to solve

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**What are the different types of trigonometric identities?**

…Read More # Study Tips: Properties of Triangles That You Need To Know

Natarajan C November 30, 2016This Coordinate Geometry tip was contributed by Kshitij of BITS Pilani, Goa Campus.

What you will learn from this tip:

— Formulae to calculate trigonometric ratios of half-angles

— The properties of every triangle

### Trigonometric ratios of half-angles

Here are some important formulae that you should note down to calculate the trigonometric ratios of half-angles.…

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