Tiêu đề 01-Basic-Mathematics-in-Physics.pdf ✅

Physics Smart Booklet 1 1.Basic Mathematics In Physics Physics Smart Booklet Theory + NCERT MCQs + Topic Wise Practice MCQs + NEET PYQs
Physics Smart Booklet 2
Physics Smart Booklet 3 Basic Mathematics in Physics Quadratic equation An algebraic equation of second order (highest power of variable is 2) is called a quadratic equation Example: ax2 + bx + c = 0, a  0 It has solution for two values of x which are given by 2 b b 4ac x 2a − + − = and 2 b b 4ac x 2a − − − = The quantity (b 2 – 4ac) is called discriminant of the equation. Binomial theorem (i) The binomial theorem for any positive value of n is (x + a)n = xn + nC1 a x n–1 + nC2 a 2 x n–2 + ..... + nCr a r x n–r + ..... + an Where ‘a’ is a constant and n r n! C r!(n r)! = − Here, n! = n(n – 1) (n – 2) ........ 3  2  1 For example, 6! = 6  5  4  3  2  1 = 720 (ii) n 2 3 n(n 1) n(n 1)(n 2) (1 x) 1 nx x x ... 2! 3! − − − + = + + + + For |x| << 1, we can neglect higher powers of x So, (1 + x)n = 1 + nx Similarly (1 – x)n = 1 – nx (1 + x)–n = 1 – nx and (1 – x)–n = 1 + nx Here ‘n’ may have any value Arithmetic progression [A.P] A sequence like a, a + d, a + 2d, .......... is called arithmetic progression Here ‘d’ is the common difference (i) The nth term of an A.P. is given by n a a (n 1)d = + − (ii) The sum of first ‘n’ terms of an A.P. is given by st th n 1 n n n S [1 term n term] (a a ) 2 2 = + = + Here a1 = a and an = a + (n – 1)d So, n n S [2a (n 1)d] 2 = + − Geometric progression G.P The progression like a, ar, ar2 , .......... is called geometric progression. Here ‘r’ is called geometric ratio or common ratio. (i) The nth term of G.P. is given by n 1 n a a r − = (ii) The sum of the first n terms of G.P. is given by n n a(r 1) S for r 1 (r 1) − =  − and n n a(1 r ) S for r 1 (1 r) − =  − (iii) The sum of infinite terms of G.P., for | r | < 1, is given by