PRECALCULUS SUPPLEMENTARY V

PRECALCULUS SUPPLEMENTARY V - AP CALCULUS PRECALCULUS...

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AP CALCULUS PRECALCULUS SUPPLEMENTARY V Problems : 1) Solve the following linear system: 713 637 4839 637 713 4611 xy += Hint : You can solve the system using the traditional methods from grade 10 (i.e. elimination, substitution, graphing) however you can find a new system of equations by adding and subtracting the equations to which its solution is the same as the original system of equations. 2) Using the quadratic formula, find the roots of ax cx b 2 0 + = and then show that their sum is equal to the product of the roots of . ax bx c 2 0 ++= O A C B 3) Let Δ ABC be inscribed in a semi-circle with center O (see the diagram to the right). Prove that Δ ABC is right-angled. Hint : What can be said about line segments OA , OB and OC ? 4) Find the coordinates that generate the minimum value for the following functions: a) yx x =− + 26 2 1 b) yx x c) y xx = −− 1 54 2 d) y = 1 2 2 5) Prove the following trigonometric identities: a) () ( ) ( ) cos cos cos sin sin xxx x 24 1 8 8 = x b) sin sin sin x ++ = 2 3 4 3 0 ππ c) 1 10 3 10 4 sin cos oo −= d) ( )( ) cos cos 75 15 1 4 = e) sin cos sin cos cos 2 36 6 2 3 x x + + = π x f) tan tan sec 44 22 = x 6) A polynomial is divided by fx ( ) xaxb . Show that the remainder, ( ) Rx is fa ab xb fb ba xa = −+ . Hint : The degree of divisor, is two. Therefore, the remainder must be a polynomial of degree at most one. Let where p and q are constants to be determined. ( ) px q =+
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Solutions : 1) If we add the equations, we get the following: which simplifies to . If we subtract the original equations, we get the following: which reduces to . The solution to the new linear system is the same as the solution to the original linear system, even though the individual lines are different. In effect, all these lines go through a common point (i.e. the solution to the linear systems; math aficionados call this a point of concurrency
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PRECALCULUS SUPPLEMENTARY V - AP CALCULUS PRECALCULUS...

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