LUZERNE COUNTY MATHEMATICS CONTEST. Luzerne County Council of Teachers of Mathematics Wilkes University Junior Examination (Section I)

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1 LUZERNE COUNTY MATHEMATICS CONTEST Luzerne County Council of Teachers of Mathematics Wilkes University - 0 Junior Examination (Section I) NAME: SCHOOL: Address: City/ZIP: Telephone: Directions: For each problem, write your answer in the space provided. Do not use approximations. Simplify all fractions and radicals. Your answer must be complete to receive credit for a problem. ) Find all real solutions to the equation 6x + = 0. ), ) Write + i + i in the form a + bi. ) + i ) What is the greatest common divisor of 80 and 806? ) 8 ) What is the remainder when 5x 0 x + 9x is ) divided by x -? 5) What is the principal square root of 0-0 (b) ± 0 0 (d) ± 0 ( )? 5) 6) Find all values of k such that x kx + is a complete square. 6) k = ± ln 7) Find all real solutions to x x + = 0. 7) ln 8) If f (x) = x +, x x, x >, then what is the value of f f () (d) ( )? 8) 5 (b) (d) 9 9) Find all real solutions to the inequality log x 5. 9) 9 x 8 0) If a point A :(a + b, ) is symmetric to another point 0) 5 B :(6, a + b) about the x-axis, then what is a + b? (OVER)

2 ) If (, y ) and (, y ) are on the straight line y = x +, ) then which of the following is true? y > y y < y (b) y = y (d) y and y are not comparable ) Suppose f (x) = x +. Find f (x). ) f (x) = (x + ) log ) Find all real values of x such that 8 = 7. ) 6 ) How many distinct ordered pairs, (x, y), where x and y are ) 8 integers, are solutions to the inequality x + y < 0? 5) Find the maximum value of y = sin x cos x sin x. 5) 6) The solutions to x 0 lie in which interval below? x + 6) (, ] (, ) [, + ) (b) [, ) (d) (, ] [, + ) 7) Find tanα if sinα + cosα sinα cosα =. 7) 8) If < x <, what is the value of (x ) + x? 8) 9) If a and b are two real solutions to x + x 009 = 0 9) 008 then what is the value of a + a + b? 0) In the figure below, m AOP = m BOP = 5, 0) PC OA and PD OA. If m PC ( )? what is m PD ( ) =, then B C P O D A

3 LUZERNE COUNTY MATHEMATICS CONTEST Luzerne County Council of Teachers of Mathematics Wilkes University - 0 Junior Examination (Section II) NAME: SCHOOL: Address: City/ZIP: Telephone: Directions: For each problem, write your answer in the space provided. Do not use approximations. Simplify all fractions and radicals. Your answer must be complete to receive credit for a problem. ) For what values of A does the slope of the line passing through the ) A = 7 points ( A, ) and ( 9, ) equal 8? ) If f (x) = x + and g(x) = x what is ( f g f )()? ) 57 ) George is 8 years older than Curtis, and Curtis is five times as old ) 5 as Sue. The sum of their ages is 85. How old is Curtis? ) Suppose f (x) = x + C. Find all values of C such that ) C =, 5 [ ]. f () = f () 5) Find the value of x in the triangle shown x 0 5) 8 at the right. 6 6) If two sides of a triangle are 8 in. and 6 in. long, and the third 6) side has a length which is a real solution of x x + 0 = 0, then the perimeter of the triangle is (b) 6 and 6 6 (d) 6 5 7) Express 0 as a positive integer. 7) 8) If x + + y = 0, then what is the value of x + y? 8) 9) If f (x) = log x, and f (ab) =, then what is the value 9) of f (a ) + f (b )? 0) If in a right triangle ABC, m CBD = m ABD, DE BC, 0) m ABC = 60 and m BC of DEC? ( ) = 0, then what is the perimeter A D (OVER) B E C

4 ) In the sequence where a = 5, a n = a n, n, what is the value of a 0? ) (b) 5 (d) 5 ) If f (x) = sin x + ϕ π, ϕ [ 0, π) is an even function, then what ) ϕ = is the value of ϕ? ) If sinα cosα = and α ( 0, π), then sinα =? ) - (b) (d) ) A point P is randomly selected on a line segment AB of length 0. ) 5 A square is built upon AP. What is the probability that the square has an area between 5 cm and 9 cm? A P B 5) What is the inverse function of f (x) = x + if x? 5) y = x, x 0 y = x, x (b) y = x +, x 0 (d) y = x +, x 6) What is the value of a + a + 00 if a + a 5 =? 6) 0 7) The rectangular coordinate representation of the polar equation 7) (d) r = is given by which of the following? cosθ + sinθ y = x y = + x (b) y = + x (d) y = x 8) What is the value of x + y if x + ( y + = 0? 8) (b) 0 (b) (d) 9) What is the coefficient of the x y term in the expansion of 9) 988 ( x + y )? 0) Find all real solutions to x x > x +. 0) (, ) (, ) (5, )

5 LUZERNE COUNTY MATHEMATICS CONTEST Luzerne County Council of Teachers of Mathematics Wilkes University - 0 Senior Examination (Section I) NAME: SCHOOL: Address: City/ZIP: Telephone: Directions: For each problem, write your answer in the space provided. Do not use approximations. Simplify all fractions and radicals. Your answer must be complete to receive credit for a problem. ) Find the domain of the function f (x) = x x. ) (, 0) (, ) ) Factor the following expression completely : x x 5x. ) (x + )(x + 7)(x + ) ) Express ( 6 )( 6 ) in the form a + bi. ) 8i x +, x 0 ) If f (x) =, then what is the value of f ( )? ) x, x < 0 (b) (d) 5 5) Find the area of the given isosceles 6 5) 6 trapezoid if its perimeter is. 0 6) If z is complex and z ( i ) =, then what is the value 6) + i of z? 7) Four consecutive even integers have a sum of 80. What is the 7) 9 smallest of the four integers? 8) The foci of the ellipse x 7 + y = are given by : 6 8) (0, ) and (0, ) (,0) and (,0) (b) (0, ) and (0, ) (d) (,0) and (,0) 9) Find all real solutions to the system x + y =. 9) (, ) x y = 8 0) If tanα and tanβ are the solutions to x x + = 0, 0) then what is the value of tan ( α + β )? - (b) - (d) (OVER)

6 ) Find the domain of f (x) = log 6 x. ) (, 6 ) ) What is the value of ab if a + b + = 0? ) ) What is the distance between the points (, 5 ) and (, 7 )? ) 9 ) How many real zeros does f (x) = x have? 0 (b) (d) x ) (b) 5) What is the center of the circle x + y x + 8y + 9 = 0? 5) (, 8 ) (b) (, 8 ) ( 7, ) (d) ( 7, ) 6) Find the area of the given triangle. 6) , π, π 7) Find all solutions to the equation sin x + = cosx in the 7) interval 0, π [ ]. 8) Find all real solutions to ( log 8) ( log 8 6) = log x. 8) 6 9) What is the volume of a pyramid which has a square base with 9) a side of length and a height of 6? 0) Find the maximum value of y = sin x cosx if 0 x π. 0)

7 LUZERNE COUNTY MATHEMATICS CONTEST Luzerne County Council of Teachers of Mathematics Wilkes University - 0 Senior Examination (Section II) NAME: SCHOOL: Address: City/ZIP: Telephone: Directions: For each problem, write your answer in the space provided. Do not use approximations. Simplify all fractions and radicals. Your answer must be complete to receive credit for a problem. ) If the diameter of a circle is doubled, then the area of the circle ) (d) increases by a factor of? (b) (d) y = ) Find the equation of the line (in slope-intercept form) that passes ) 5 x through the points ( 7, - ) and ( -, 6 ). { }and N = ( x, y ) x y = ) If M = ( x, y ) x + y = then what is M N? x =, y =, { } ) (d) { } ( ) { } (b) (, ) (d), ) Assume x > 0, y > 0 and z > 0. Eliminate negative exponents ) and reduce x y 5 z z y x ln e to lowest terms. 5) What is the nd term of the sequence 0,,,,, 0,? 5) 8 6) What is the vertex of the parabola y = x + x 9? ( 6), 8 5 ) 7) If f (x) = x + x +, find f (x). x 7) f (x) = x 8) What is the constant term in the expansion of x + (b) (d) x 8z 5 0 y? 8) (b) x 9 9) Find all values of x such that x x x 9 = x. 9) x = 0) Find the domain of f (x) = ln(x + ) + x. 0) (, ] (OVER)

8 ) How many different ways can 8 marbles be placed in a row ) 560 given that are green, are red, and are black? ) Write + i 8 + ( 8 )i ( ) 8 in the form a + bi. ) ) Find all values of k such that kx + kx + 7 = 0 has ) k = 8 exactly one real solution. ) If the sequence a n { } n is given by a = 0, a = a +, ) (b) a = a +, a = a +,, then what is the value of a 0? -005 (b) (d) -0 5) What does tanx equal if cos x = 7 and π x π? 5) 5 A 6) DB and DC are tangent to O. B 6) 67 If m D = 6, what is m A? O D C 7) What is the area of a square that has a perimeter that is equal to the 7) 6π circumference of a circle with radius 8? 8) Evaluate cos(arcsin 5 arccos 5 ). 8) ) In the given circle of radius 0, what is the area 9) 9 π of sector AOB? A 0 B O 0 0) What is the minimum value of x + y if x and y satisfy 0) x + y x y? x + 0 (b) -5 (d) -6

{ } and let N = 1, 0, 1, 2, 3

{ } and let N = 1, 0, 1, 2, 3 LUZERNE COUNTY MATHEMATICS CONTEST Luzerne County Council of Teachers of Mathematics Wilkes University - 2014 Junior Eamination (Section II) NAME: SCHOOL: Address: City/ZIP: Telephone: Directions: For