Modern Gyani

Dear students, In this chapter we will learn about Probability. Below ⬇️ are the tutorial notes and videos. Probability means possibility. It is a branch of mathematics that deals with the occurrence of a random event. The value is expressed from zero to one. Probability has been introduced in Maths to predict how likely events are to happen.   Probability Definition in Math Probability is a measure of the likelihood of an event to occur. Many events cannot be predicted with total certainty. We can predict only the chance of an event to occur i.e. how likely they are to happen, using it. Probability can range in from 0 to 1, where 0 means the event to be an impossible one and 1 indicates a certain event.. The probability of all the events in a sample space adds up to 1. For example, when we toss a coin, either we get Head OR Tail, only two possible outcomes are possible (H, T). But if we toss two coins in the air, there could be three possibilities of events to occur, such as both th

Dear students, In this chapter we will learn about Arithmetic Progression. Below ⬇️ are the tutorial notes and videos. Sequence and series                       A sequence is defined as an arrangement of numbers in a particular order.  On the other hand, a series is defined as the sum of the elements of a sequence.   A sequence is an arrangement of any objects or a set of numbers in a particular order followed by some rule.   If  ₐ₁,ₐ₂,ₐ₃,ₐ₄,......... etc. denote the terms of a sequence, then 1,2,3,4,.....denotes the position of the term. A sequence can be defined based upon the number of terms i.e. either finite sequence or infinite sequence.   If  ₐ₁,ₐ₂,ₐ₃,ₐ₄,.........   is a sequence, then the corresponding series is given by Sₙ= ₐ₁+  ₐ₂+ₐ₃+ₐ₄……ₐₙ Note : The series is finite or infinite depending if the sequence is finite or infinite. PROGRESSION A progression is a series that advances in a logical and predictable pattern. In mathematics, for example, the series 2, 4, 6, 8 is a

Dear students, In this chapter we will learn about Quadratic Equations. Below ⬇️ are the tutorial notes and videos. QUADRATIC EQUATIONS The polynomial of degree two is called quadratic polynomial and equation corresponding to a quadratic polynomial P(x) is called a quadratic equation in variable x. Thus, P(x) = ax^2+ bx + c =0, a ≠ 0, a, b, c ∈ R is known as the standard form of quadratic equation. There are two types of quadratic equation. (i) Complete quadratic equation : The equation ax^2 + bx + c =0 where a ≠ 0, b ≠ 0,c ≠ 0  (ii) Pure quadratic equation : An equation in the form of ax^2= 0, a ≠ 0, b = 0, c = 0 Question: Find the roots of the equation 2x^2 – 5x + 3 = 0, by factorisation.  Solution : Let us first split the middle term – 5x as –2x –3x [because (–2x) × (–3x) = 6x^2= (2x^2) × 3]. So, 2x^2–5x+3=2x^2 –2x–3x+3=2x(x–1)–3(x–1)=(2x–3)(x–1) Now,2x^2 –5x+3=0 can we written  as (2x–3)(x–1)=0. So, the values of x for which 2x^2 –5x+3=0 are the same for which (2x–3)(x–1)=0, i.e.,