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Math Team Skills for March Rounds Round 1 – Alg 2: Simultaneous Equations and Determinants The determinant of a 2x2 matrix is determined using a formula a b The determinant of matrix c d a b Denoted: ad bc c d Alternatively, the negatively sloped diagonal product minus the positively sloped diagonal product a b c d bc ad To find the determinant of a 3x3 or larger square matrix… The sum of the negatively sloped diagonal products minus the sum of the positively sloped diagonal products. Negatively sloped diagonal products a b d e g h c f i bfg cdh aei a b d e g h Positively sloped diagonal products a b d e g h ceg afh c f i bdi c f ( aei bfg cdh ) (afh bdi ceg ) i Solving Simultaneous Equations – Reduce the number of variables then backfill. Example: (2 equations, 2 unknowns) 2x y 5 x 3y 6 Multiply both sides of the top equation by 3. Add the two equations and the y-terms will cancel. Solve for x. 3(2 x y ) (5)3 6 x 3 y 15 x 3y 6 Substitute the known value into either of the first equation to find the other unknown. 7 x 21 x3 2x y 5 2(3) y 5 6 y 5 y 1 Example: (3 equations, 3 unknowns) x 2y z 8 2x y z 7 x 2 y z 6 1( x 2 y z ) (8)( 1) 2x y z 7 x 2 y z 8 2x y z 7 x y 1 2x y z 7 x 2 y z 6 3x y 1 x y 1 3x y 1 x y 1 3x y 1 2x 2 x 1 Backfill into previous equations to find y and z. Round 2 – Alg 1: Exponents and Radicals Rules of Exponents x m x n x mn xm m n x xn x n n 1 xn m y x y m x n x0 1 x m n x mn xy x y n x y n n n x a b xn n y If n is a positive integer and x is a nonnegative real number, then x1/ n n x. Therefore, rules for radicals are the same as the rules for exponents. Power Root Ex.) Simplify a) 3 b) 5 52 5x 2 5(12 x 2) 5x 5 c) 3x 2 y 6 z 0 3x5 y 6 y 2 1 x 3 y 8 1 3 8 x y 5 2 2 9x y 9x 3 3 d) x 2y 4 3 2 27 2 3 32 xy 1 3 27 2 3 2 4 3 1 27 2 3 1 1 10 8 2 1 3 9 9 Matching Bases Solving equations involving exponents often requires matching the bases of the terms. If the bases are matched, the exponents must be equal. Ex.) Solve 27 x 1 3 9 x 2 3 3 3 3 3 3 3 x 1 3 x 1 1 1 2 x 2 2 x 2 33 x 3 31 32 x 4 33 x 3 32 x 5 3x 3 2 x 5 x 8 All the terms can be expressed using base 3. Power to a power - Multiply (distribute) Product of powers - Add (comb. Like terms) Bases are matched, exponents are equal Simplifying and Manipulating Radicals To simplify a radical, we basically take the perfect squares out. (Or perfect cubes for cube roots) Ex) 18 x 9 9 2 x 8 x 9 x 8 2 x 3x 4 2 x But remember, if it is convenient, we can put numbers or variables back into the radical. Ex) Solve for a and b. 8 3 6 a 24 b 4 2 3 6 a 24 b 4 4 3 6 a 12 b 4 12 6 a 12 b a 4, b 6 Round 3 – Trigonometry: Anything • See Trigonometry Rounds and info from previous months Round 4 – Alg 1: Anything • • • • • • Solving equations Proportions Percent and mixture problems The coordinate plane Factoring Solving rational equations … just to name a few topics Round 5 – Plane Geometry: Anything • • • • • Regular polygons and their properties Triangles Transversals Interior and exterior angle theorems Circles … just to name a few topics Round 6 – Probability and the Binomial Theorem Using the Binomial Theorem for a binomial expansion. To expand ( x y )n , we use the nth row of Pascal’s triangle to find the coefficients. The powers of x start with n in the first term and decrease by one for each successive term. The powers of y start with zero and increase by one for each successive term. Ex) ( x y ) 2 Row 0 Row 1 1x 2 2 xy 1 y 2 Row 2 x 2 2 xy y 2 Row 3 Row 4 Row 5 Ex) Expand ( x y ) Row 0 4 Row 1 1x 4 x y 6 x y 4 xy 1 y 4 3 2 2 3 Row 2 4 Row 3 x 4 4 x 3 y 6 x 2 y 2 4 xy 3 y 4 Row 4 Row 5 2x y 1 2 x 3 2 x y 3 2 x ( y ) 8 x 3 4 x y 3 2 x y y 2 3 Ex) Expand 3 2 3 2 2 2 4 8 x 12 x y 6 xy y 3 2 2 2 2 4 6 6 1 y 2 3 Probability • • • • Independent events Dependent events Compound probability Combinatorics • Counting principles • Combinations • Permutations