Class 12 Maths Chapter 3 Notes

• Important ch 3 maths class 12 important questions for Intermediate part-II students.
• INTRODUCTION: When the derived function (or differential coefficient) of a function is known, then the aim to find the function itself can be achieved. The technique or method to find such a function whose derivative is given involves the inverse process of differentiation, called anti-derivation or integration. We use differentials of variables while applying method of substitution in integrating process. Before the further study of anti-derivation, we first discuss the differentials of variables.
• Important ch 3 maths class 12 for Intermediate part-II students.
• Since log of zero and negative numbers does not exist therefore in above formula mod assure that we are taking a log of positive quantity.
• Important class 12 maths ch 3 for Intermediate part-II students.
• Note. Instead of dy, we can write df, that is, df = f'(x) dx where f'(x) being coefficient of differential is called differential coefficient.
• Important class 12 maths ch 3 ex 3.1 for Intermediate part-II students.
• INTEGRATION AS ANTI-DERIVATIVE (INVERSE OF DERIVATIVE): In chapter 2, we have been finding the derived function (differential coefficient) of a given function. Now we consider the reverse (or inverse) process i.e., we find a function when its derivative is known. In other words we can say that if phi' * (x) = f(x) then phi(x) is called an anti-derivative or an integral of f(x) For example, an anti-derivative of f(x) = 3x ^ 2 is phi(x) = x ^ 3 because phi' * (x) = d/dx (x ^ 3) = 3x ^ 2 = f(x)
• Important class 12 maths ch 3 ex 3.2 for Intermediate part-II students.
• The inverse process of differentiation i.e., the process of finding such a function whose derivative is given is called anti-differentiation or integration. 
• Important class 12 maths ch 3 ex 3.3 for Intermediate part-II students.
• While finding the derivatives of the expressions such as x ^ 2 + x x ^ 2 + x - 3 etc., we see that the derivative of each of them is 2x + 1, that is, x ^ 2 + x + 5
• Then Phi(x) = x ^ 2 + x is not only anti-derivative of (i), but all anti-derivatives of f(x) = 2x + 1 are included in x ^ 2 + x + c where c is the arbitrary constant which can be found if further information is given.
• Important ex 3.1 class 12 maths solutions for Intermediate part-II students.
• As c is not definite, so Phi(x) + c is called the indefinite integral of f(x) , that is, integrate f(x) dx = Phi(x) + c (ii) In (ii), f(x) is called integrand and e is named as the constant of integration.
• The symbol [....dx indicates that integrand is to be integrated w.r.t.x. Note that x and J...dx are inverse operations of each other.
• Find the slope and inclination of the line joining the points (-2,4);(5,11)
• Important ex 3.2 class 12 maths solutions for Intermediate part-II students.
• Find k so that the line joining A(7,3);B(k,-6) and the line joining C(-4,5); D(-6,4) are perpendicular.
• Find an equation of the line bisecting the I and III quadrants.
• Important exercise 3.3 class 12 maths solutions for Intermediate part-II students.
• Find an equation of the line for x-intercept:-3 and y-intercept:4
• Find the distance from the point P(6,-1) to the line 6x-4y+9=0
• Find whether the given point (5,8) lies above or below the line 2x-3y+6=0
• Important class 12 maths chapter 3 exercise 3.1 for Intermediate part-II students.
• Find the angle measured from the line l_1  to the line l_2 where      l_1: Joining (2,7) and (7,10)      l_2: Joining (1,1) and (-5,5)
• Express the given system of equations in matrix form            2x+3y+4=0;x-2y-3=0;3x+y-8=0
• Find the angle from the line with slope –7/3 to the line with slope 5/2.
• Find an equation of each of the lines represented by 20x^2+17xy-24y^2=0
• Define Homogenous equation.
• Write down the joint equation.