Pre-Calculus Individual Test 2017 February Regional

Transcription

1 The abbreviatin NOTA means Nne f the Abve answers and shuld be chsen if chices A, B, C and D are nt crrect. N calculatr is allwed n this test. Arcfunctins (such as y = Arcsin( ) ) have traditinal restricted ranges. cisθ = cs( θ) + i sin( θ), and i =. f( ) = sin( ) and g ( ) = Arcct( ). Find the value f g i f. D In RST, m S = 60, m T = 0 and ST=0. Which is an epressin fr the length f side RT? cs(0 ) cs(0 ) sin(0 ) D. sin(0 ). Line L cntains the pints (, 0) and (,). Line L is perpendicular t L and cntains the pint (,0). L and L intersect at the pint with crdinates (,) rs. Give the value f r+ s. 6 D February Reginal R 60 0 S 0 T. RST has tw sides with lengths and 6 respectively, and the included angle has measure. Use the apprimatin f sin t find the apprimate fr ( ) length f the third side f RST. D.. RST has tw sides with lengths and 6 respectively, and the included angle has measure. Use the apprimatin f fr sin ( ) t find the apprimate area f RST. 6 D The equatin cs ( ) cs( ) = has fur slutins rst,, and u ver [0, ], and r < s< t < u. Give the slutin t. 6 D The smallest real slutin fr the equatin lg ( ) = lg (7 ) is p. Give the value f p +. 8 D.

2 8. Fifty single peple attend a meeting. Of thse fifty, 0 have a jb. 0 are male, and 0 drive a car that is less than three years ld. What is the least pssible number f the meeting s attendees which are male, single, have a jb and drive a car that is less than years ld? D February Reginal. The angle between the vectrs given by, and, has a csine f a fr a and b relatively prime b whle numbers. Give a+ b. 0 9 D The equatin f a circle is May nt y ( 6) + y = 6. P be A secnd and drawn t cngruent circle is scale. drawn as shwn, tangent t the y-ais, cplanar with the first circle. Pint P has the greatest y-crdinate f the unin f the tw Q graphs, and pint Q, in Quadrant IV, has the least. If PQ=0 and the equatin f the secnd circle is ( + a) + ( y+ b) = c then a+ b=? 8 D. -. The graphs f r = 8 csθ and r = 6 intersect at pints P and Q. What is the distance frm P t Q? 6 D. 6. The fur furth rts f 6 are written r a cis θ r a cis θ r = ai cis θ = i ( ), = i ( ), ( ) and r = ai cis( θ ) fr 0 θ < θ < θ < θ <. Give the θ value f. a 0. The graph f y + + 0y 00= 0 has which linear equatin belw as an asymptte? D. y = + + y = y = + + D. y = +. f( ) = + and g ( ) = +. f( g( k )) = 90. If k is an integer, find the value f k k. 0 8 D. 9

3 . Fr f( ) =, and the functin + g, s that f( g( )) = g( f( )) = fr all values f in the dmains f bth functins. Which is an epressin fr the functin g? D The set f pints P ( y, ) in the y-plane such that the sum f the distances frm P t (,) and frm P t (,0) is 0. What is the equatin that represents the set f pints? ( + ) ( y 7) + = 6 ( + ) ( y 7) + = 6 ( + ) ( y 7) + = 9 ( + ) ( y 7) D. = 9 7. In btuse RST, RS=0, ST=6 and m R =0. What is the length f the third side f the triangle? D February Reginal Q The fur blades f a wind turbine each has length 00 ft. That is, a pint Q n the tip f ne blade is 00 ft frm the center f rtatin. As Q travels in a circle when the turbine s blades rtate, its least distance abve grund is 0 feet. The functin which gives the distance in feet abve grund f Q at time t minutes has equatin dt () = asin( b t) + c, given that Q makes a cmplete rtatin in minute. ( abc,, ) =? (00,,0) (00,, 0) (0,,0) D. (00,, 0) 9. Which f the fllwing is NOT equal t fr all values f in the dmain f the epressin? + sin ( ) cs ( ) sin( ) sin( )cs( ) sec ( ) tan ( ) D. cs( ) cs ( ) 0. The real slutins t r and s fr r D. 7 = are > s. Give the value f rs.

4 . The functin y = f( ) is btained by slving ln y = + c, given that f () =. Which is an epressin fr f( )? 6 e e+ 6 e 6 e + D. 6 e. The graphs f 07 February Reginal ( ) = f and g ( ) = + intersect n the -ais at =. Cnsider the set f the five -intercepts f the tw graphs. Give the psitive difference f the greatest and least elements f the set... D... The circle with equatin + y = 0 is graphed n the same crdinate aes as a parabla with equatin f( ) = a + b+ c fr a > 0. If the circle and parabla share the same minimum pint and bth -intercepts, then find the value f f (). 9 6 D The graph f + a + b f( ) = + c has eactly ne -intercept which is at =. It has tw remvable discntinuities (hles), ne f which is at the pint,. Give the value f a c 7 D.. Given the parametric equatins y = t t 6 find the average = t (arithmetic mean) f the -intercepts f the graph n the y-plane, when graphed ver all real values f t. D. 6. The system f three linear equatins yz,, is slved by using with slutin ( ) Cramer s Rule. If = then what is the value f the z-crdinate f the slutin t the system? D. 9

5 7. The directri f the graph f + y = and the directri f the graph f 8y = 8 intersect at the pint (,) rs. Give the prduct 8rs. 8. D. 6 6 y 07 February Reginal y=f() n 9. f( ) = fr!!! n! n nn-zer integers and a real f() f() = f() i a+ b number. If ( ) and a > b then which statement is NOT true? b < 0 ln a = b is a ratinal number D. f( b) = f () 6 0. y = f( ) is a plynmial functin with rts at =, = 9, and = nly. f is nt necessarily a cubic plynmial. Which must be a rt f y = f( )? The functin y = f( ), shwn, is a plynmial functin f degree. The graph f f is tangent t the -ais at =, and has rts 0,, and D. Which statement(s) must be true? I. The graph f y = f( ) has the same rts as y = f( ). II. The graph f y = f ( ) has five distinct rts. III. Fr g ( ) f( ) IV. Fr h ( ) f( ) =, ( 0) =, ( ) g < g. g 0 < g. all must be true I, IV nly I, II, III nly D. I, II, IV nly