Basic Maths Section [A] Algebra

University:
Santa Ana College
Course:
MATH 170 | Pre-Calculus Mathematics
Academic year:

2023
Views:

307

Pages:

2
Author:

Judsonpofq

BASIC MATHS & VECTOR SECTION [A] : ALGEBRA QUADRATIC EQUATION An algebraic equation of second order (highest power of the variable is equal to 2) is called a quadratic equation. The equation ax2 + bx + c = 0....(i) is the general form of quadratic equation where a. The general solution of above equation is x b  b 2  4ac 2a b  b 2  4ac b  b 2  4ac and x2  2a 2a Here x1 and x2 are called roots of equation (i). We can easily see that b c sum of roots = x1 + x2 = – and product of roots = x1 x2  a a Find roots of equation 2x2 – x – 3 = 0. If values of x be x1 and x2 then x1  Ex. Sol. Compare this equation with standard quadratic equation ax2 + bx + c = 0, we have a=2, b=–1, c=–3. b  b 2  4ac x   1  Now from x = ; 2a x Ex. 2 66  44 33 1  1  24 1  5   xx    xx , xx   or x = –1 44 44 22 4 4 In quadratic equation ax2 + bx + c =0, if discriminant D = b2 – 4ac, then roots of quadratic equation are: (A) real and distinct, if D > 0 (B) real and equal (repeated roots), if D = 0 (C) non-real (imaginary), if D < 0 Sol.  1  4  2   3 22 (D) None of the above (ABC) BINOMIAL EXPRESSION An algebraic expression containing two terms is called a binomial expression.  1 For example (a+b), (a+b)3, (2x––3y)–1,  x   etc. are binomial expressions. y  Binomial Theorem a  b n  a n  na n 1b1  n  n  1 n  2 2 n  n  1 2 a b  .... (1  x ) n  1  nx  x  .... 2 1 2 1 Binomial Approximation If x is very small, then terms containing higher powers of x can be neglected so (1+x)n =1 + nx Ex. The mass m of a body moving with a velocity v is given by m  m0 v2 1 2 c c = speed of light = 3 × 108 m/s. Find the value of m at v = 3 × 107 m/s. where m0 = rest mass of body = 10 kg and Sol.  v2  m  m0  1  2   c  1/2   3 107  2   10 1    8    3  10   1/2 1    10 1   100  1/2   1  1   10  10 1      10.05 kg    10  2 100 200      LOGARITHM Common formulae (i) log mn = log m+log n (ii) log m =logm-log n (iii) log mn = n log m (iv) loge m = 2.303 log10m n COMPONENDO AND DIVIDENDO RULE If p a pq ab   then q b p q ab ARITHMETIC PROGRESSION (AP) General form : a, a + d, a +2d, ..., a + (n–1)d Here a = first term, d = common difference Ex. n n [a+a+(n–1)d] = [Ist term + nth term] 2 2 Find sum of first n natural numbers. Sol. Let sum be Sn then Sum of n terms Sn = Sn = 1 + 2 + 3 +....+n; S n   n n  1   n 1  n     2 2   GEOMETRICAL PROGRESSION (GP) General form : a, ar, ar2,..., arn–1 Sum of n terms S n   a 1 r n  1r Here a = first term, r = common ratio Sum of  term S   1 1 1    ....upto  . 2 4 8 Ex. Find 1  Sol. Here, a = 1, r = 1 a So, S    2 1r 1 1 1 2 2 a 1r

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