PRECALCULUS SUPPLEMENTARY VII

PRECALCULUS SUPPLEMENTARY VII - AP CALCULUS PRECALCULUS...

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AP CALCULUS PRECALCULUS SUPPLEMENTARY VII Problems : 1) If 41 2 2 x x k −+ is a perfect square, find the value of k . 2) Show that () ( ) log log log log aba b a b 22 −= . 3) If the coefficient of x 2 in the expansion ( ) xa x x −− + 32 2 is 2 , then what is the value of a ? 4) Let sin θ = p q with being an acute angle. Find the value of cos sin cos sin θθ + in terms of p and q . 5) Prove the following identity: cos sin tan csc sec cos sin 21 2 x x x xx x x + += + + . 6) Consider Δ ABC such that the side lengths are related by the equation bc a + = 2 . Prove the following identities: a) 2sin sin A B C =+ b) 211 hhh abc where h i represents the altitude from the respective vertex, i . 7) ( 2005 LSU Math Contest: Team Problem ) Nine labelled squares are arranged as shown to the right (figure is NOT drawn to scale). If square A has area 1 2 cm and square B has area 81 2 cm , then what is the area, in square centimetres, of square I ? 8) ( 2004 Nova Scotia Math League Game ) a) The parabola yx x =+− 2 45 has vertex at point A . The horizontal line y = 7 meets the parabola at the point B and C . Find the area of Δ ABC . b) A parabola meets the y -axis at point A and the x -axis at points B and C . Let O be the origin. There are two parabolas for which OA BC = and the area of Δ ABC is 32. Find the equations of these two parabolas. 9) ( 2004 Nova Scotia Math League Game ) ABCD is a square with D at 10 , and C at 20 , (see diagram to the right). A line l passing through the origin divides the square into two regions of equal area. Determine the equation of line l . 10) ( 2006 Eton Press Contest ) Prove that for all natural numbers n , ( ) ( ) nn + + 11 2 1 is a perfect square.
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Solutions : 1) METHOD 1 : If 41 2 2 x x k −+ is a perfect square then we can say that () 2 2 2 2 xx k
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PRECALCULUS SUPPLEMENTARY VII - AP CALCULUS PRECALCULUS...

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