Summary

Summary In this chapter, we acquired the knowledge of • Factorial of a natural number n is the product of the first n natural numbers. • n ! = n ( n − )! , for any integer n ≥ .

• The number of ways of arranging n unlike objects is n ! . • The number of distinct permutations of r objects which can be made from n distinct objects is n P r = n ! ( n − r )!

= n ( n − )( n − ) · · · ( n − r + ) . • The number of permutations of n objects taken all at a time where P objects one of first kind, P objects one of second kind, · · · P k objects one of the k th kind and the rest, if any are all different and is given by n ! P ! P !

· · · P k ! . • Order matters for a permutation where as order does not matter for a combination. • The number of combinations of n different objects taken r at a time denoted by n C r is given by n C r = n !

r !( n − r )! = n ( n − )( n − ) · · · ( n − r + ) r ! Summary In this chapter we have acquired the knowledge of • Binomial theorem for any n ∈ N , ( a + b ) n = n C a n b + n C a n − b + · · · + n C n a b n . • n C + n C + · · · + n C n = n .

• n C + n C + + n C · · · + = n C + n C + + n C · · · + = n − • AM ≥ GM ≥ HM • The n th term of an AP is given by T n = a + ( n − ) d . • The n th term of an GP is given by T n = ar n − . • The n th term of an AGP is given by T n = ( a + ( n − ) d ) r n − . • For any positive numbers a and b , we have AM = a + b , GM = ab, HM = ab a + b .

• The sum of first n terms of an AP is given by S n = n ( a + ( n − ) d ) . • The sum of first n terms of an GP is given by S n = a ( − r n ) − r for r = . • The sum of first n terms of an AGP is given by S n = a − ( a + ( n − ) d ) r n − r + dr − r n − ( − r ) for r = . • n Σ k = k = + + + · · · + n = n ( n + ) .

+ x − x + · · · for all values of x satisfying | x | < . • log( − x ) = − x − x − x − x −· · · for all values of x satisfying | x | < . • log + x − x = h x + x + x + · · · i for all values of x satisfying | x | < . Summary The types of straight lines related to the information.

S.No Information given Equation of the lines Slope( m ) and y-intercept ( b ) y = mx + b Slope ( m ) and point ( x , y ) y − y = m ( x − x ) Two points ( x , y ) and ( x , y ) y − y y − y = x − x x − x x -intercept ( a ) and y -intercept ( b ) a + y b = Normal length ( p ) , angle ( α ) x cos α + y sin α = p Parametric form: parameter- r x − x cos θ = y − y sin θ = r The general equation ax + by + c = Form of lines Condition for Condition for parallel perpendicular y = m x + c and y = m x + c m = m m m = − a x + b y + c = and a x + b y + c = a b = a b a a + b b = A point P ( x , y ) is on the origin side or non origin side of the line ax + by + c = ( c = ) , according as ax + by + c and c are of the same sign or opposite sign. The distance between two points ( x , y ) and ( x , y ) is given by the formula D = q ( x − x ) + ( y − y ) The distance from a point P ( x , y ) to a line ax + by + c = is ax + by + c a + b The distance between two parallel lines a x + b y + c = and a x + b y + c = is | c − c | a + b The line parallel to ax + by + c = through a point ( x , y ), is ax + by = ax + by and the perpendicular line is bx − ay = bx − ay Summary The coordinates of the image of the point ( x , y ) with respect to the line ax + by + c = can be obtained by the line x − x a = y − y b = − ( ax + by + c ) a + b Pair of straight lines Condition for parallel Condition for perpendicular ax + hxy + by = h − ab = , a + b = The equation of the bisectors of the angle between the lines ax + hxy + by = is x − y a − b = xy h . The condition that the general second degree equation ax + hxy + by + gx + fy + c = should represent a pair of straight lines is abc + fgh − af − bg − ch = (i) The angle between them is θ = tan - h − ab a + b If a + b = , then the lines are perpendicular. (ii) The point of intersection P hf − bg ab − h , gh − af ab − h (iii) Two straight lines represented by the equation ax + hxy + by + gx + fy + c = are parallel if a h = h b = g f or bg = af (iv) If ax + hxy + by + gx + fy + c = represents a pair of parallel straight lines, then the distance between them is s ( g − ac ) a ( a + b ) or s ( f − bc ) b ( a + b ) Two Dimensional Analytical Geometry