Cylinder

Let Us Learn About Cylinder

A cylinder is one of the most basic curvilinear geometric shapes: the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder. The surface area and the volume of a cylinder have been known since deep antiquity.

In differential geometry, a cylinder is defined more broadly as any ruled surface spanned by a one-parameter family of parallel lines. A cylinder whose cross section is an ellipse, parabola, or hyperbola is called an elliptic cylinder, parabolic cylinder, or hyperbolic cylinder. A prism is a cylinder whose cross-section is a polygon.

A cylinder is one of the most curvilinear basic geometric shapes:It has two faces, zero vertices, and zero edges. The surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder. The surface area and the volume of a cylinder have been known since deep antiquity.

Solid line measuring jars, circular pillars, circular pencils, Circular pipes, road rollers and gas cylinders are said to have a cylindrical shape.

There is no edge for the cylinder.

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Pascal’s triangle

Let Us Learn about Pascal's triangle.

The coefficients of the expansions are arranged in an array. This array is
called Pascal’s triangle.

One of the most interesting Number Patterns is Pascal's Triangle (named after Blaise Pascal, a famous French Mathematician and Philosopher).

Patterns Within the Triangle

Binomial Coefficient

Let Us Understand What Is Binomial Coefficient.

the binomial coefficient is the coefficient of the x ^k term in the polynomial expansion of the binomial power (1 + x) ⁿ.

A binomial coefficient equals the number of combination of r items that can be selected from a set of n items. It also represents an entry in Pascal's triangle. These numbers are called binomial coefficients because they are coefficients in the binomial theorem.

Example:

Binomial Theorem

Let Us Learn What Is Binomial.


The sum of two monomial is called Binomial.

Binomial Theorem for Positive Integral Indices

Let us have a look at the following identities done earlier:

(a+ b)0 = 1 a + b ≠ 0
(a+ b)1 = a + b
(a+ b)2 = a2 + 2ab + b2
(a+ b)3 = a3 + 3a2b + 3ab2 + b3
(a+ b)4 = (a + b)3 (a + b) = a4 + 4a3b + 6a2b2 + 4ab3 + b4

In these expansions, we observe that

(i) The total number of terms in the expansion is one more than the index. For
example, in the expansion of (a + b)2 , number of terms is 3 whereas the index of
(a + b)2 is 2.

(ii) Powers of the first quantity ‘a’ go on decreasing by 1 whereas the powers of the
second quantity ‘b’ increase by 1, in the successive terms.

(iii) In each term of the expansion, the sum of the indices of a and b is the same and
is equal to the index of a + b.

Examples Using Binomial Theorem

Factorial Notation

Let Us Learn About Factorial Notation.


Question:What is Factorial Notation?
Answer: Let n be a positive integer. The continued product of first n natural numbers is called factorial n and is denoted as n

The notation n! represents the product of first n natural
numbers, i.e., the product 1 × 2 × 3 × . . . × (n – 1) × n is denoted as n!. We read this
symbol as ‘n factorial’. Thus, 1 × 2 × 3 × 4 . . . × (n – 1) × n = n !
1 = 1 !
1 × 2 = 2 !
1× 2 × 3 = 3 !
1 × 2 × 3 × 4 = 4 ! and so on.

We define 0 ! = 1
We can write 5 ! = 5 × 4 ! = 5 × 4 × 3 ! = 5 × 4 × 3 × 2 !
= 5 × 4 × 3 × 2 × 1!

Clearly, for a natural number n
n ! = n (n – 1) !
= n (n – 1) (n – 2) ! [provided (n ≥ 2)]
= n (n – 1) (n – 2) (n – 3) ! [provided (n ≥ 3)]
and so on.

Permutations when all the objects are distinct

Topic: Permutations are solved when all the objects are distinct

Question: How to Permutations are solved when all the objects are distinct?
Answer: The number of permutations of n different objects taken r at a time,
where 0 < r ≤ n and the objects do not repeat is n ( n – 1) ( n – 2). . .( n – r + 1), which is denoted by nPr. Proof There will be as many permutations as there are ways of filling in r vacant

places . . . by ← r vacant places →

the n objects. The first place can be filled in n ways; following which, the second place
can be filled in (n – 1) ways, following which the third place can be filled in (n – 2)
ways,..., the rth place can be filled in (n – (r – 1)) ways. Therefore, the number of
ways of filling in r vacant places in succession is n(n – 1) (n – 2) . . . (n – (r – 1)) or
n ( n – 1) (n – 2) ... (n – r + 1)

This expression for nPr is cumbersome and we need a notation which will help to
reduce the size of this expression. The symbol n! (read as factorial n or n factorial )
comes to our rescue. In the following text we will learn what actually n! means.

How to Solve Simultaneous Linear Equation Graphically

The graph of a linear equation ax+ by = c is a straight line equation.

Two distinct lines always intersect at exactly one point unless they are parallel (have the same slope).

The coordinates of the intersection point of the lines is the solution to the simultaneous linear equations describing the lines. So we would normally expect a pair of simultaneous equations to have just one solution.

Let's look at an example graphically:
2x + 3y = 7
4x + y = 9



From the graph we see that the point of intersection of the two lines is (2, 1)
Hence, the solution of the simultaneous equations is x = 2, y =1.