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Mathematics - Combinations

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Published in: Mathematics
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Revision Notes on Combinations.

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  1. Revision Notes on Combinations In order to solve a quadratic equation of the form ax2 + bx + c, we first need to calculate the discriminant with the help of the formula D = b2— 4ac. The solution of the quadratic equation ax2 + bx + 0 is given by x = [-b ± b2— 4ac] / 2a If a and ß are the roots of the quadratic equation ax2 + bx + c = 0, then we have the following results for the sum and product of roots: cx + c/a It is not possible for a quadratic equation to have three different roots and if in any case it happens, then the equation becomes an identity. Nature of Roots: Consider an equation ax2 + bx + c 0, then we have the following cases: 0, where a, b and c e R and a # 1. 2. 3. 4. D > 0 iff the roots are real and distinct i.e. the roots are unequal D = 0 iff the roots are real and coincident i.e. equal D < 0 iffthe roots are imaginary The imaginary roots always occur in pairs i.e. if a+ib is one root of a quadratic equation, then the other root must be the conjugate i.e. a-ib, where a, b e R and i = 4-1. Consider an equation ax2 + bx + c 0, then 0, where a, b and c EQ and a # If D > 0 and is also a perfect square then the roots are rational and unequal. 2. If a = p + qq is a root of the equation, where 'p' is rational and 'Iq is a surd, then the other root must be the conjugate of it i.e. ß = p - qq and vice versa. If the roots of the quadratic equation are known, then the equation may be constructed with the help of the formula ( Sum So if a equation For the quadratic of roots) x + (Product of roots) and 13 are the roots of equation then is the quadratic quadratic ß)x+aß expressiony 0 = ax2 + bx + c, where a, b, c e R and a 0, then the graph between x and y is always a parabola.
  2. 2. If a > 0, then the shape of the parabola is concave upwards If a < 0, then the shape of the parabola is concave upwards Inequalities of the form P(x)/ Q(x) > 0 can be easily solved by the method of intervals of number line rule. The maximum and minimum values of the expression y = ax2 + bx + c occur at the point x = -b/2a depending on whether a > 0 or a< 0. y / 4a, 00] if a > 0 2. If a < 0, then y e [-00, (4ac-b2) / 4a] The quadratic function of the form f(x, y) = ax2+by2 + 2hxy + 2gx + 2fy + c = 0 can be resolved into two linear factors provided it satisfies the following condition: abc + 2fgh —af2 — bg2 — ch2 = 0 In general, if al,a2, ag, are the roots of the equation aoxn -kalxn-l + a2xn-2 + ......an— + + an then 2.2 3.2 Every equation of nth degree has exactly n roots (n 21) and if it has more than n roots then the equation becomes an identity. If there are two real numbers 'a' and 'b' such that f(a) and f(b) are of opposite signs, then f(x) = 0 must have at least one real root between 'a' and 'b'. Every equation f(x) = 0 of odd degree has at least one real root of a sign opposite to that of its last term.

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