Sujit Prasad Kushwaha

Conservation of Linear Momentum The total quantity of motion contained in a body is called linear momentum . It is denoted by $\vec{P}$ and defined as the product of mass and velocity of the body. For a body of mass $m$ and velocity $v$, its linear momentum $\vec{P}$ in vector form is given by: $$\vec{P} = m\vec{v}$$ In terms of magnitude only, $$P = mv$$ SI unit of linear momentum is $\text{kg m s}^{-1}$ or Ns, and dimensional formula is $[MLT^{-1}]$. Law of Conservation of Linear Momentum It states that, "the total linear momentum of a system of particles remains constant, provided that no external force acts on the system" . Proof: From Newton's second law of motion , $$F = \frac{dP}{dt}$$ In absence of force, $$\frac{dP}{dt} = 0 \;\Rightarrow\; P = \text{constant}$$ Proof (for a system of particles): Let us consider a system of $n$ particles of total mass $M$. Then, $$M = m_1 + m_2 + \cdots + m_n$$ So, $$P = P_1 + P_2 + \cdots + P_n$$ Differe...

 New Syllabus Course Title: Fundamental of Statistics Course Code: STAT.101 Nature of Course: Theory Group A 1. Introduction to Statistics [5] Meaning of Statistics as a Science, Importance of Statistics; Scope of Statistics in the field of physical Sciences, Social Sciences, Management Sciences, Information Technology, Agriculture, Insurance, Education and Psychology. 2. Population and Sample [8] Types of Characteristics, Scales of measurement, qualitative, quantitative, discrete and continuous variables, entities, types of data, (i)primary data, secondary data and their sources (ii)cross sectional data, time series data, failure time data, panel data, Notion of a statistical population, finite population, infinite population, homogeneous population and heterogeneous population, Notion of sample, random sample and non-random sample, methods of sampling (description only); simple random sampling with replacement (SRSWR) and without replacement (SRSWOR). 3. Presentation of Data [6] ...

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Have you ever thought what are the most commonly used Mathematics Formulae? Yes, many of you have. Let me convert your imagination into reality. Here are some of the most commonly used mathematics formulas: 1. Quadratic Formula:  $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$ where a, b, and c are coefficients of quadratic equation. 2. Pythagoras Theorem: $$a^2+b^2=c^2$$ where a, b, and c are the length of the sides of a right-angle triangle. 3. Distance Formula: $$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$ where $(x_1,y_1)$ and $(x_2,y_2)$ are the coordinate of two points in a plane. 4. Area of a Circle: $$A=\pi{r^2}$$ where r is the radius of the circle. 5. Volume of a Sphere: $$V=\frac{4}{3}\pi{r^3}$$ where r is the radius of the sphere. 6. Volume of a Cylinder: $$V=\pi{r^2}h$$ where r is the radius and h is the height of the cylinder. 7. Area of a Triangle: $$A=\frac{1}{2}bh$$ where b is the base of the triangle and h is the height. 8. Slope-intercept formula: $$y=mx+c$$ where m is the slope o...

This is the mathematics formulae book for class 12 students. Here, we have covered all the formulae needed to solve the questions of Mathematics. Permutation The total arrangement of total objects taken some or all at a time is called permutation. The permutation of 'n' objects taken 'r' at a time is denoted by $^np_{r}$ and is defined by; 1. $^np_{r}=\frac{n!}{(n-r)!}$ 2. Permutation of an object, all different;$$=\frac{n!}{p!\,q!\,r!}$$ where,  n=total objects p=repetition of first kind  q=repetition of second kind r=repetition of third kind 3. Circular permutation= $(n-1)!$ 4. If clockwise and anticlockwise arrangement are not distinguishable [necklace of beads and beads into bracelet]; Total number of permutation=$\frac{(n-1)!}{2}$ Combination The selection of total objects taken some or all at a time is called combination. The combination of 'n objects taken 'r' at a time is denoted by $^nc_{r} \, or \, c(n,r)$ and defined by; $$^nc_{r}=\frac{n!}{(n-r)!...