MATHEMATICAL WORLD

Relation between Differentiation and Integration Table Of Contents Look at the information given below. \[\mathbf{ y=f(x)}\] \[ \mathbf{ f'(x)\rightarrow \text{Derivatives of f(x)}}\] \[ \mathbf{\displaystyle \int_a^b f’(x) = ?}\] Can you tell me the value of above integral? Yes, it will be equal to f(b)-f(a) . We have already known this result. It tells us that integration is just the reverse of differentiation, integral of the derivative of the function f(x) is just equal to the difference in the function f(x) evaluated at the limits of integration. Indefinite integration- Notes and Formula Part 1 Now with this topic, we will understand how to apply this result to find the integral of a function.  Consider this function \(\mathbf{g(x)=x^2}\) Let's find the integral of this function from a to b i.e \(\mathbf{\displaystyle \int_a^b g(x) \, dx}\) .  Can you think how we can apply this result to find ...

TRIGONOMETRIC RATIOS & IDENTITIES Table Of Contents 1. RELATION BETWEEN SYSTEM OF MEASUREMENT OF ANGLES : \(\dfrac{D}{90}=\dfrac{G}{100}=\dfrac{2 C}{\pi}\) 1 Radian \(=\dfrac{180}{\pi}\) degree \(\approx 57^{\circ} 17^{\prime} 15^{\prime \prime}\) (approximately) 1 degree \(=\dfrac{\pi}{180}\) radian \(\approx 0.0175\) radian All Chapter Notes, Concept and Important Formula 2. BASIC TRIGONOMETRIC IDENTITIES : (a) \(\sin ^{2} \theta+\cos ^{2} \theta=1\) or \(\sin ^{2} \theta=1-\cos ^{2} \theta\) or \(\cos ^{2} \theta=1-\sin ^{2} \theta\) (b) \(\sec ^{2} \theta-\tan ^{2} \theta=1\) or \(\sec ^{2} \theta=1+\tan ^{2} \theta\) or \(\tan ^{2} \theta=\sec ^{2} \theta-1\) (c) If \(\sec \theta+\tan \theta\) \(=\mathrm{k} \Rightarrow \sec \theta-\tan \theta\) \(=\dfrac{1}{\mathrm{k}} \Rightarrow 2 \sec \theta\) \(=\mathrm{k}+\dfrac{1}{\mathrm{k}}\) (d) \(\operatorname{cosec}^{2} \theta-\cot ^{2} \theta=1\) or \(\operatorname{cosec}^{2} \theta=1+\cot ^{2} \th...

TRIGONOMETRIC EQUATION 1. TRIGONOMETRIC EQUATION : An equation involving one or more trigonometrical ratios of unknown angles is called a trigonometric equation. All Chapter Notes, Concept and Important Formula 2. SOLUTION OF TRIGONOMETRIC EQUATION : A value of the unknown angle which satisfies the given equations is called a solution of the trigonometric equation. (a) Principal solution :- The solution of the trigonometric equation lying in the interval \([0,2 \pi]\) . (b) General solution :- Since all the trigonometric functions are many one & periodic, hence there are infinite values of \(\theta\) for which trigonometric functions have the same value. All such possible values of \(\theta\) for which the given trigonometric function is satisfied is given by a general formula. Such a general formula is called general solutions of trigonometric equation. 3. GENERAL SOLUTIONS OF SOME TRIGONOMETRICE EQUATIONS (TO BE REMEMBERED) :   (a) If \(\sin \theta=0\) , then \(\theta=...

SEQUENCE & SERIES 1. ARITHMETIC PROGRESSION (AP) : AP is sequence whose terms increase or decrease by a fixed number. This fixed number is called the common difference . If ‘a’ is the first term & ‘d’ is the common difference, then AP can be written as a, a + d, a + 2d, ..., a + (n – 1) d, ... (a) \(n^{\text {th }}\) term of this AP \(\boxed{T_{n}=a+(n-1) d}\) , where \(d=T_{n}-T_{n-1}\) (b) The sum of the first \(n\) terms : \(\boxed{S_{n}=\frac{n}{2}[2 a+(n-1) d]=\frac{n}{2}[a+\ell]}\) ,  where \(\ell\) is the last term. (c) Also \(n ^{\text {th }}\) term \(\boxed{T _{ n }= S _{ n }- S _{ n -1}}\) Note: (i) Sum of first n terms of an A.P. is of the form \(A n^{2}+B n\) i.e. a quadratic expression in n, in such case the common difference is twice the coefficient of \(n ^{2}\) . i.e. 2A (ii) \(n ^{\text {th }}\) term of an A.P. is of the form \(An + B\) i.e. a linear expression in \(n\) , in such case the coefficient of \(n\) is the common difference of the ...

PERMUTATION & COMBINATION 1. FUNDAMENTAL PRINCIPLE OF COUNTING (counting without actually counting): If an event can occur in 'm' different ways, following which another event can occur in 'n' different ways, then the total number of different ways of (a) Simultaneous occurrence of both events in a definite order is \(m\) \(× n\) . This can be extended to any number of events (known as multiplication principle). (b) Happening of exactly one of the events is \(m + n\) (known as addition principle). All Chapter Notes, Concept and Important Formula 2. FACTORIAL: A Useful Notation \(: n != n ( n -1)( n -2) \ldots \ldots\) . \(2.1\) ; \(n !=n .(n-1) !\) where \(n \in W\) \(0 !=1 !=1\) \((2 n) !=2^{n} \cdot n ![1.3 .5 .7 \ldots \ldots . .(2 n-1)]\) Note that : (i) Factorial of negative integers is not defined. (ii) Let \(p\) be a prime number and \(n\) be a positive integer, then exponent of \(p\) in \(n !\) is denoted by \(E _{ p }( n !)\) and is given by \(...

QUADRATIC EQUATION 1.  SOLUTION OF QUADRATIC EQUATION & RELATION BETWEEN ROOTS & CO-EFFICIENTS : (a) The solutions of the quadratic equation, \(a x^{2}+b x+c=0\) is given by \(\mathbf{x =\dfrac{- b \pm \sqrt{ b ^{2}-4 a c }}{ 2 a }}\) (b) The expression \(b^{2}-4 a c \equiv D\) is called the discriminant of the quadratic equation. (c) If \(\alpha\, \& \, \beta\) are the roots of the quadratic equation \(a x^{2}+b x+c=0\) , then; (i) \(\alpha+\beta=-b / a\)      (ii) \(\alpha \beta= c / a\)      (iii) \(|\alpha-\beta|=\sqrt{ D } /| a |\) (d) Quadratic equation whose roots are \(\alpha \)   &  \(\beta\) is \((x-\alpha)(x-\beta)=0\) i.e.  \(x ^{2}-(\alpha+\beta) x +\alpha \beta=0\)   i.e.  \(x ^{2}-\) (sum of roots) \(x +\) product of roots \(=0\) (e) If \(\alpha, \beta\) are roots of equation \(a x^{2}+b x+c=0\) , we have identity in \(x\) as \(a x^{2}+b x+c=a(x-\alpha)(x-\beta)\) All Chapter Notes...

BINOMIAL THEOREM \((x+y)^{n}={ }^{n} C_{0} x^{n}+{ }^{n} C_{1} x^{n-1} y+{ }^{n} C_{2} x^{n-2} y^{2}+\ldots\) \( . .+{ }^{n} C_{r} x^{n-r} y^{r}+\ldots\) \( . .+{ }^{n} C_{n} y^{n}\) \(=\displaystyle \sum_{ r =0}^{ n }{ }^{ n } C _{ r } x ^{ n - r } y ^{ r }\) , where \(n \in N\) . 1. IMPORTANT TERMS IN THE BINOMIAL EXPANSION ARE : (a) General term: The general term or the \((r+1)^{\text {th }}\) term in the expansion of \((x+y)^{n}\) is given by \(T _{ r +1}={ }^{ n } C _{ r } x ^{ n \cdot r } \cdot y ^{r}\) (b) Middle term : The middle term (s) is the expansion of \((x+y)^{n}\) is (are) : (i) If \(n\) is even, there is only one middle term which is given by \(T _{( n +2) / 2}={ }^{ n } C _{ n / 2} \cdot x ^{ n / 2} \cdot y ^{ n / 2}\) (ii) If \(n\) is odd, there are two middle terms which are \(T _{( n +1) / 2}\) & \(T _{[( n +1) / 2]+1}\) (c) Term independent of x : Term independent of \(x\) contains no \(x\) ; Hence find the value of r for which the exponent of \(x\)...