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FEBRUARY REGIONAL A
1. Compute where: B
A = the sum of all values of θ ∈( 0, 2π ] such that PRECALCULUS TEAM CONDENSED 2sin 2 (θ ) − sin (θ ) −1
= 0 3sin 2 (θ ) − 2sin (θ ) −1
⎛ 1⎞
⎛ 1⎞
B = the value of Arctan ⎜ ⎟ + Arctan ⎜ ⎟ in radians. ⎝ 2⎠
⎝ 3⎠
2. Compute A B where: A = the height of the building (rectangle), in meters, if the measurements of α = 60 and tan ( β ) = 3 3 are taken 4 meters apart. (see picture above) B = the value of ( x + y + z ) given that x +11 = y + 33 = z + 22 = 10x +10y +10z − 7 3. In order to solve many Precalculus and Calculus questions, you’ll need a strong foundation in Algebra. Consider the following Algebra 2 concepts and compute AB. A = xy where x + y = 2 and x 3 + y 3 = 12 B = log r ( pq ) where log p ( q ) = 2 and log q ( pr ) = 3 4. Perform the partial fraction decomposition for the following fraction: 5s
A
B
C
=
+
+
3
2
s + 5s + 8s + 4 s +1 s + 2 ( s + 2 )2
Evaluate 2A + 3B − C ( )
(
)
(
)
5. Compute ABCD where A = sin 15 , B = sin 22.5 , C = sin 67.5 , and ( )
D = sin 75 6. Consider f : C → C where f ( x ) = x 3 + ax 2 + bx + c for {a, b, c} ⊂ R and b > 0. There exists at least one pure imaginary number z ≠ 0 such that f ( z ) ∈R. Find f ( z ) in terms of a, b, and c. 7. Compute A + B where: A = the fundamental period of f (θ ) = sin ( 2θ ) + 3 cos (θ ). Note: The fundamental period is the least p > 0 such that f (θ ) = f (θ + p). B = z0 where the parabola in the complex plane that passes through all complex numbers z such that 2 ( Re ( z )) + 4 Re ( z ) − Im ( z ) = 3 has its vertex occurring at 2
z = z0 . Note: Re ( z ) and Im ( z ) denote the real and imaginary parts of z , respectively. Aπ
8. Compute where: 2013B
A = the remainder when f ( x ) = 2013x 2013 + 2012x 2012 + ... + 2x 2 + x is divided by ( x −1). 16
B = the area bounded by the graph of sin 2 (θ ) + 4 cos2 (θ ) = 2 r
9. Compute A + B + C where: 5
5
A = the number of terms in the simplified expansion of ( x + y ) ( x − y ) 8
B = the coefficient on the x 5 y 3 term in the simplified expansion of ( 2y − x ) 9
C = the sum of the coefficients in the simplified expansion of (3x − 4y ) 10. Find all possible values of A + B where: A = the area of the region bounded in the complex plane by the convex hexagon with 3
⎛ B⎞
vertices that are solutions to z 6 = ⎜ ⎟ ⎝ 3⎠
i
B = the distance in the complex plane between 1 +
and a + 0i 5
11. Find the sum of all solutions to the following equation over the domain [0,2π] 1
1
1
1
sin θ cosθ tan θ + cosθ tan θ − sin θ cosθ − cosθ − sin θ tan θ − tan θ + sin θ + = 0
2
2
2
2
3x3 + 5 x 2 − 11x + 3
12. Consider the rational function f ( x ) =
x2 − 9
• There is a vertical asymptote at x=A. Find A
• There is a hole at the point (B, C). Calculate B & C
• There is an oblique asymptote at y=Dx+E. Calculate D & E
• There are two roots of f(x) at x=F and x=G, F<G. Calculate F and G
Evaluate: A-­‐B+C-­‐D+E-­‐F+G 13. A nice factorization for x 3 + y3 + z 3 is ( x + y + z ) ( x 2 + y 2 + z 2 − xy − yz − zx ) + 3xyz. Using this identity, find the number of positive integral factors of (1213 − 633 − 583 ). 14. Find the number of right triangles that exist such that each side length is a positive perfect square. 15. Consider the equation of the ellipse given by 2 x 2 + y 2 − 16 x + 2 y + 32 = 0 . A=area of the ellipse B=eccentricity of the ellipse C=maximum y value of the ellipse D=the distance from one foci to any point on the ellipse, plus the distance from that same point to the other foci of the ellipse. Evaluate AC+BD