### GRAPH OF FUNCTIONS

To draw the graph for any function, we need to take some positive and negative values of \(\displaystyle x,\) then find the corresponding values of \(\displaystyle y.\) And make the line or curve by joining the points obtained. (1) Straight Line \(\displaystyle \left( Ax+By+C \right)=0\). (2) Circle \(\displaystyle \left[ {{\left( x-h \right)}^{2}}+{{\left( y-k \right)}^{2}}={{r}^{2}} \right]\) […]

### Integrate root cos2x divided by sinx with respect to x

\(\displaystyle \int{\frac{\sqrt{\cos 2x}}{\sin x}dx}\) \(\displaystyle =\int{\frac{\sqrt{\frac{1-{{\tan }^{2}}x}{1+{{\tan }^{2}}x}}}{\sin x}}\,\,dx\) \(\displaystyle =\int{\frac{\sqrt{1-{{\tan }^{2}}x}}{\sec x.\sin x}\,dx}\) \(\displaystyle =\int{\frac{\sqrt{1-{{\tan }^{2}}x}}{\tan x}\,\,dx}\) \(\displaystyle =\int{\frac{\sqrt{1-{{\tan }^{2}}x}}{\tan x\left( 1+{{\tan }^{2}}x \right)}\,{{\sec }^{2}}x\,\,dx}\) \(\displaystyle =\int{\frac{\sqrt{1-{{\tan }^{2}}x}}{{{\tan }^{2}}x\left( 1+{{\tan }^{2}}x \right)}}\,\,\tan x{{\sec }^{2}}x\,\,dx\) Let \(\displaystyle \sqrt{1-{{\tan }^{2}}x}\,\,=t\) \(\displaystyle 1-{{\tan }^{2}}x={{t}^{2}}\) Differentiate both sides w.r.t x \(\displaystyle -2\tan x{{\sec }^{2}}x\,\,dx\,\,=2\,tdt\) \(\displaystyle \Rightarrow I=-\int{\frac{t}{\left( 1-{{t}^{2}} \right)\left( […]

### PROPERTIES OF LOGARITHM

Logarithms are very useful for lengthy calculation. There are two types of logs: (1) log (2) ln (Natural log) (1) The base is 10 in log (2) The base is e in natural log Here, we will discuss the properties of log. Some important properties with example are given below: \(\displaystyle {{\log }_{a}}m+{{\log }_{a}}n={{\log }_{a}}\left( […]

### WHY sin x AND cosec x ARE POSITIVE IN SECOND QUADRANT?

जब भी मै बच्चो से पूछता हूँ कि sin x and cosec x, IInd quadrant में positive क्यों होता है? cos x and sec x, IVth में और tan x and cot x IIIrd में? हमेशा answer आता हैं, Sir, ALL SCHOOL TO COLLEGE OR ADD SUGAR TO COFFEE, etc. कुछ ही बच्चों को इसका […]

### TRIGONOMETRY FORMULAE

Trigonometry Formulae: Don’t try to remember just know the concept of derivation With the help of sin (A + B), we can derive all the remaining trigonometric formulae. Let’s see, how ….. Compound Angle (1) \(\sin \left( A+B \right)=\sin A\cos B+\cos A\sin B\) (2) \(\sin \left( A-B \right)=\sin A\cos B-\cos A\sin B\) (3) \(\cos \left( […]

### MULTIPLICATION TECHNIQUES

Let us consider two digit numbers ab and cd. For example: 23 × 45 Now, Now, we can try in one step.

### OPERATIONS ON SETS

(1) Union of Sets: The union of two sets A and B is the set of all the elements which are either in A or in B or in both A and B. It is denoted by \(A\cup B\). Let \(\text{A}=\left\{ 1,\,\,2,\,\,\,3 \right\}\) \(\text{B}=\left\{ 2,\,\,\,4,\,\,6 \right\}\) \(\displaystyle \therefore \text{A}\cup \text{B}=\left\{ 1,\,\,2,\,\,3,\,\,4,\,\,6 \right\}\) (2) Intersection of […]

### LAWS OF SET OPERATIONS

(1) Idempotent laws : \(\text{A}\cap \text{A}=\text{A}\) \(\text{A}\cup \text{A}=\text{A}\) (2) Identity laws: \(\displaystyle \text{A}\cap U\text{=A}\) \(\text{A}\cup \phi =\text{A}\) (3) Commutative laws: \(\text{A}\cup \text{B}\,\text{=}\,\text{B}\cup \text{A}\) \(\text{A}\cap \text{B}\,\text{=}\,\text{B}\cap \text{A}\) (4) Associative laws: \(\left( \text{A}\cup \text{B} \right)\cup \text{C}\,=\text{A}\cup \left( \text{B}\cup \text{C} \right)\) \(\text{A}\cap \left( \text{B}\cap \text{C} \right)=\left( \text{A}\cap \text{B} \right)\cap \text{C}\) (5) Distributive Laws: \(\text{A}\cup \left( \text{B}\cap \text{C} \right)=\left( […]

### CARDINAL NUMBER OF SETS

Some important results derived from Venn diagrams are as follows: A, B and C are three sets. \(\displaystyle n\left( \text{A}\cup \text{B} \right)=n\left( \text{A} \right)+n\left( \text{B} \right)-n\left( \text{A}\cap \text{B} \right)\) \(\displaystyle n\left( \text{A}\cup \text{B} \right)=n\left( \text{A} \right)+n\left( \text{B} \right)\Leftrightarrow \,\) A and B are disjoint sets. \(n\left( \text{A}-\text{B} \right)=n\left( \text{A} \right)-n\left( \text{A}\cap \text{B} \right)\) \(\displaystyle n\left( […]

### VENN DIAGRAM

The pictorial representation of sets is called Venn diagram. The universal set is represented by a rectangular region and its subsets by circles inside the rectangle. In the above given Venn diagram, \(\text{B}\subseteq \text{A}\,\)and \(\text{A}\subseteq U\) and \(\text{B}\,\subseteq \,U\). Here,\(\text{A}\subset U\)and \(\text{B}\subseteq U\). If, there is no common elements in A and B and […]