11th Class

  Life on the Earth   By now you might have realised that all units of this book have acquainted you with the three major realms of the environment, that is, the lithosphere, the atmosphere and the hydrosphere. You know that living organisms of the earth, constituting the biosphere, interact with other environmental realms. The biosphere includes all the living components of the earth. It consists of all plants and animals, including all the micro-organisms that live on the planet earth and their interactions with the surrounding environment. Most of the organisms exist on the lithosphere and/or the hydrosphere as well as in the atmosphere. There are also many organisms that move freely from one realm to the other.   Life on the earth is found almost everywhere. Living organisms are found from the poles to the equator, from the bottom of the sea more...

  Biodiversity and Conservation   You have already learnt about the geomorphic processes particularly weathering and depth of weathering mantle in different climatic zones. See the Figure 6.2 in Chapter 6 in order to recapitulate. You should know that this weathering mantle is the basis for the diversity of vegetation and hence, the biodiversity. The basic cause for such weathering variations and resultant biodiversity is the input of solar energy and water. No wonder that the areas that are rich in these inputs are the areas of wide spectrum of biodiversity.   Biodiversity as we have today is the result of 2.5-3.5 billion years of evolution. Before the advent of humans, our earth supported more biodiversity than in any other period. Since, the emergence of humans, however, biodiversity has begun a rapid decline, with one species after another bearing the brunt of extinction due to overuse. more...

  PERMUTATION & COMBINATION   Learning Objectives  

• Factorial
• Permutation
• Combination

  Factorial The factorial, symbolized by an exclamation mark (!), is a quantity defined for all integers greater than or equal to 0. Mathematically, the formula for the factorial is as follows.   If n is an integer greater than or equal to 1, then n! = n (n - 1) (n - 2) (n - 3) …. (3)(2)(10).   Example: \[1!=1,\,\,2!=2,\,\,3!=6,\,\,4!=4.3.2.1=24,\,\,5!=5\times 4\times 3\times 2\times 1=120\] \[61=6\times 5\times 4\times 3\times 2\times 1=720,\text{ }7!=\text{ }5040\text{ }and\text{ }8!=40320\text{ }etc.\] The special case 0! is defined to have value 0! = 1.   Permutation   The different arrangements which can be made by taking some or all of the given things or objects at a time is called Permutation. All permutations (arrangements] made with the letters a, b, c by taking two more...

  Sequence and Series   A particular order in which related things follow each other. Called sequence. The sequence having specified patterns is called progression. The real sequence is that sequence whose range is a subset of the real numbers. A series is defined as the expression denoting the sum of the terms of the sequence. The sum is obtained after adding the terms of the sequence. If \[{{a}_{1}},\,\,{{a}_{2}}\,\,{{a}_{3}},----,\,\,{{a}_{n}}\,\]is a sequence having n terms, then the sum of the series is given by: \[\sum\limits_{K=1}^{n}{{{a}_{k}}={{a}_{1}}+{{a}_{2}}+{{a}_{3}}+----+{{a}_{n}}}\]   Arithmetic Progression (A.P.) A sequence is said to be in arithmetic progression if the difference between its consecutive terms is a constant. The difference between the consecutive terms of an A.P. is called common difference and nth term of the sequence is called general term. If \[{{a}_{1}},\,\,{{a}_{2}}\,\,{{a}_{3}},----,\,\,{{a}_{n}}\,\]be n terms of the sequence in A.P., then nth term of the sequence is more...

  Set Theory                                                           A Set A set is the collection of things which is well-defined. Here well-defined means that group or collection of things which is defined distinguishable and distinct e.g. Let A is the collection of the group M = {cow, ox, book, pen, man}. Is this group collection is a set or not? Actually this is not a set. Because it is collection of things but it cannot be defined in a single definition. For example, A = {1, 2, 3, 4, 5,... n} Here, collection A is a set because A is group or collection of natural numbers. e.g. A = {a, e, i, o, u}= {x/x : vowel of English alphabet}   Distinguish Between a Set and a Member e.g.       A = {2, 5, 8, more...

                                                                                      Relation and Function   Let \[A=\{1,\,2,\,3,4,\}\], \[B=\{2,\,3\}\] \[A\times B=\{1,\,2,3,\,4,\}\times \{2,3\}=\{(1,2),(2,2),(3,2),(4,2),(1,3),(2,3),(3,3),(4,3)\}\]   Let we choose an arbitrary set:   \[R=[(1,2),(2,2),(1,3),(4,3)]\] Then R is said to be the relation between a set A to B.   Definition   Relation R is the subset of the Cartesian Product\[A\times B\]. It is represented as \[R=\{(x,y):x\in A\,\] and \[y\in B\}\] {the 2nd element in the ordered pair (x, y) is the image of 1st element}   Sometimes, it is said that a relation on the set A means the all members / elements of the relation R be the elements / members of \[A\text{ }\times \text{ }A\]. e.g.      Let \[A=\{1,\,2,\,3\}\] and a relation R is defined as \[R=\{(x,y):x<y\] where \[x,y\in A\}\]   Sol.     \[\because \]\[\mathbf{A=\{1,}\,\mathbf{2,}\,\mathbf{3\}}\] \[A\times A=\{(1,1),(2,2),(3,3),(2,1),(3,1),(1,2),(3,2),(1,3),(2,3)\}\]             \[\because \,\,\,\,R=\,\,\,\because x<y\]             \[\because \,\,\,\,R=\{(x,y):x<y,and\,x,y\in A\}=\{(1,2),(2,3),(1,3)\]   Note: Let a set more...

  Limits and Derivatives   Key Points to Remember   A number, (\[\ell \] is said to be the limit of the function \[\text{y=f}(x)\] at\[x=a\], then \[\exists \] a positive number, \[\in \,\,>\,\,0\] corresponding to the small positive number \[\delta \,>\,\,0\] such that \[\left| \text{f(x)-}\left. \ell  \right| \right.\,<\,\varepsilon ,\] provided \[\left| x-\left. a \right| \right.\,<\,\delta \] Evidently, \[\underset{x\to \,a}{\mathop{\lim }}\,\,\,f(x)=\ell \] We have to learn how to evaluate the limit of the function \[y=f(x)\] Only three type of function can be evaluate the limit of the function.

Algebraic function

Trigonometric function

Logarithmic and exponential function

Now, basically, there are three method to evaluate the limit of the function.  

• By Formula Method

i.e.        \[\underset{x\to a}{\mathop{\lim }}\,\frac{{{x}^{n}}-{{a}^{n}}}{x-a}=n.{{a}^{n-1}}\]  

• Factorisation Method: In this matter numerator and denominator are factorised. The common factors are cancelled and the rest is the result.
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                                                                                                 Trigonometry   Key Points to Remember Trigonometry: It is derived/contained from two greek words trigon and metron means that the measurement of three sides of the triangle.     A triangle has three vertex, three angles and three sides in above figure. vertex be A, Band C Angle be \[\angle ABC,\] \[\angle BCA,\] and \[\angle CAB,\]or \[\angle BAC\] According to sides, type of triangle be (a)        Equilateral Triangle: All sides are equal (b)        Isosceles Triangle: Two sides are equal (c)        Scalene Triangle: All sides are different. According to Angle, There are three types of triangles: (a)        Acute Angle Triangle              (b)        Obtuse Angle Triangle   more...

  Continuity and Differentiability of a Function   Introduction   The word 'continuous' means without any break or gap. If the graph a function has no break or gap or jump, then it is said to be continuous. A function which is not continuous is called a discontinuous function. While studying graphs of functions, we see that graphs of functions sinx, x, cosx, etc. are continuous on R but greatest integer functions [x] has break at every integral point, so it is not continuous. Similarly tanx, cotx, seex, 1/x etc. are also discontinuous function on R.   Continuous Function                            Discontinuous Function                                       more...

                                                                                         Complex Number   Complex Numbers: "Complex number is the combination of real and imaginary number".   Definition: A number of the form\[x+iy,\], where \[x,y\in R\] and \[i=\sqrt{-1}\] is called a complex number and (i) is called iota. A complex number is usually denoted by z and the set of complex number is denoted by C.   \[\Rightarrow C=\{x+iy:x\in R,\,Y\in R,\,i=\sqrt{-1}\}\]   For example: \[5+3i,\] \[-1+i,\] \[0+4i,\] \[4+0i\] etc. are complex numbers.   Note: Integral powers of iota (i)   since \[i=\sqrt{-1}\] hence we have \[{{i}^{2}}=-1,\] and \[{{i}^{4}}=1.\]   Conjugate of a complex number: If a complex number \[z=a+i\,b,\] \[(a,b)\in R,\] then its conjugate is defined as \[\overline{z}=a-ib\]                              Hence, we have   \[\operatorname{Re}(z)=\frac{z+\overline{z}}{2}\] and \[\operatorname{Im}(z)=\frac{z-\overline{z}}{2i}\]   \[\Rightarrow \] Geometrically, the conjugate of z more...