ELEVENTH YEAR MATHEMATICS

Transcription

1 The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ELEVENTH YEAR MATHEMATICS Wednesday, August 14, :30 to 3 :30 p.m., only The last page of the booklet is the answer sheet. Fold the last page along the perforations and, slowly and carefully, tear off the answer sheet. Then fill in the heading of your answer sheet. The "Reference Tables for Mathematics" which you may need to answer some questions in this examination are stapled in the center of this booklet. When directed to do so, open the booklet and carefully remove the reference tables. Note that the remainder of Reference Table C is located on a separate page. Fold this page along the perforations and, slowly and carefully, tear off this page. Then close the booklet and leave it face up. DO NOT OPEN THIS EXAMINATION BOOKLET UNTIL THE' SIGNAL IS GIVEN

2 Part I Answer all questions in this part. Questions la, Ib, a, and b count 1 credit each. Questions 3 through 30 count credits each. No partial credit will be allowed. Unless otherwise specified, answers may be left in terms of 11' or in radical form. Directions (1-): Write in the space provided on the separate answer sheet the numeral preceding the expression that best completes each statement. 1 Questions a and b refer to the straight line graph of the equation.x + y ==. a The slope of this line is (1) 1 () (3) -1 (4) - b The graph of this line does not pass through quadrant (1) I (3) III () II (4) IV Questions a and b refer to the equation.x + $ - 3 == 0 whose roots are m and n. a A possible value of m is (1) 1 () b The value of mn is (1) - () (3) -1 (4) - (3) 3 (4) -3 Directions (3-9): Write your answers in the spaces provided on the separate answer sheet. 3 The point (0,3) satisfies this system of equations: X + y = 9 %+y=3 Find the other coordinate pair in the solution set of this system. 4 Express V \1 18 as a monomial in terms of i. 5 The axis of symmetry of a parabola is the line whose equation is X' = 4. One of the points at which the parabola crosses the.x-axis is (3,0). What are the coordinates of the other point at which this parabola crosses the x-axis? 6 What is the numerical value of cos ~? 7 Find, in degrees, the value of () in the interval 0 0 < () < which satisfies the equation sin" () - sin (J = O. 8 Using principal values, express in terms of 7(', the value of the expression arc cos \/3. 9 In triangle ABC) a = 5, b = 8, and mlb = 70. Find sin A to the nearest hundredth. Directions (10-30) : Write in the space provided on the separate answer sheet the numeral preceding the expression that best completes each statement or answers each question. 10 If a parabola has its turning point at the origin and an ellipse is centered at the origin, how many points must the parabola and the ellipse have in common? (1) 1 (3) 0 () (4) 4 11 If 4 :C- l = 64, then.x is equal to (1) 1 (3) 3 () (4) ~ 1 Which is not a multiplicative inverse for the rational number -!!:!:...? v (1) () 1 m v v m (3) (4) -v m v -m. 4 + \/ The expression _ IS equiva ent to 4 - (1) \/3 7\/3 (3) \/ () V3 13 (4) 80 sin f) cos The expression IS equivalent to cos 9 SIn fj (1) 1 () sec (J (3) 1 esc (J (4) 1 sin () cos (J Math. II-Aug. '74 []

3 15 If log sin () = a and log cos () = b, the expression log(sin (J cos 8) is equivalent to (1) a ~ b (3) alb () a -- b (4) ab 3 The accompanying diagram represents the graph of which function? 16 The solution set of 1x + 1 I == 3x is (1) {I} (3) { 1_} 5 1 () {I, --.S-} (4) { } 17 The formula for the area of a rectangular solid is S = Hw + hi ~ hw. The value of h, expressed in terms of S, g) and tu, is S -- Hw (1) ~ + w S - ZHw -- hi () w 18 When written in simplest form) the expression 1+~ b IS equlva ent to ~-1 b (1) 1 () -1 (3) w + 8 S - ZUw (4) S - 8w g + w (3) b + a a-b ( 1) )' == cos z () y == cos x 1 (3) y = cos-~ 1 (4) y == cos -,X 4 The product (1. X 10 8 ) X (1. X 10-4 ) equals (1) 1. X 10 4 (3) 1.44 X () 1.44 X 10 4 (4) 1.44 X What is the slope of a line perpendicular to the line whose equation is y = _I_x +? (1) - () (3) 1_ (4) 1 19 The value of (J in the interval 90 0 < () < 70 which satisfies the equation 4 sin = 3 is (1) 10 0 (3) 10 () 150 (4) 40 0 What is the domain of the function defined by the % - 1 equation f(z) = z? (1) {z I % = I} () {z Ix = } (3) {x I z = -I} (4) {zlz =O} 1 The expression tan 0 is equivalent to (1) tan 40 (3) --tan 40 () cot 40 (4) --eat 40 The measure, in degrees, of an angle of one radian is (1) 180 () 180?r (3) 360 (4) 360 7r 6 The expression sin 35 cos + cos 35 sin is equal to (1) sin 13 (3) cos 13 () sin 57 (4) cos 57 7 If the total cost of n articles is 50 cents, the number of articles that can be bought at the same unit price for 40 cents is 5 (1) 4n () 4n 5 (3) 5n 4 (4) 0 n 8 In triangle ABC, if a = 7, b = 6, and c = 5, what is the value of cos A? (1) *9 (3) % () %0 (4) % Math. II-Aug. '74 [3] [OVER]

4 9 If sin A cos A? (1) % () -% VI! =--, 6 what is the numerical value of (3) -%8 (4) ~8 30 If m LB == 30, b == 8, and c = 16, then ( 1) a right triangle can be constructed () no triangle can be constructed (3) only an obtuse triangle can be constructed (4) two distinct triangles can be constructed }.I ath. tt-aug. '14 [4]

5 Part II Answer twenty questions from this part, Each correct answer will receive credits. No partial credit will be allowed. Questions marked * are based upon optional topics in the syllabus. Write your answers in the spaces provided on the separate answer sheet. 31 By what natural number must a fraction be multiplied in order to produce an equivalent fraction? 3 Find a value of % which will make the following system of equations true: Z - 4z + 3 = Y y=.1:+9 33 Express, in radical form, the roots of the equation.,1; - Z - 1 = O If x is a positive acute angle and tan x =----=, find V7 the numerical value of sin x. 35 Factor: 4 sin" z - 4 sin.,1; If x = 7, find the numerical value of zo --:- z-t. 37 A circular arc of length S subtends an angle of 8 radians at the center of its circle. Express the radius of the circle in terms of Sand In triangle RST, s = 54, sin S =.7, and sin T =.8. Find the value of t. 39 If f(x) = z + 3% + 1, find the value of f(-i). 40 Find the logarithm of If log tan () = , find the positive acute angle to the nearest minute. 44 If sin % = -% and tan % = %, then angle z lies in quadrant (1) I () II (1) cos A - () cos A (3) cos A (4) cos A (8)(13) (3) III (4) IV 45 In triangle ABC, a = 15, b = 13, and c = 8. Which equation can be used to find angle A? (8)(13) (13) (15) (15) (8) 46 The solution set of 3z + > SoX' - 4 is (1) {.,1; I z > -3} (3) {z I x < -3} () {x I x < 3} (4) {z I z > 3} 47 Which is not an identity? (1) sin" A + cos" A = 1 () sec" A = 1 + tan" A (3) sin (A + B) = sin A sin B + cos A cos B (4) sin (A - B) = sin A cos B - sin B cos A 48 The units digit of a two-digit number is three times the tens digit (t). The number, expressed in terms of t, is (1) 13t (3) 3t () 6t (4) 4t 4 Express 1 form. ---as x-i a SIng. lee fracti raction In. s'tmp 1est x + 1 Directions (43-60): Write in the space provided on the separate answer sheet the numeral preceding the expression that best completes each statement or answers each question. 43 A man can do a job in 4 days working alone. His helper can do the same job in 8 days working alone. What part of the job can they do in one day if they work together at their respective rates? (1) % (3) ~ () % (4) % 1 _ 4 9 The expression 1 og % + - 1og y 15 equrv a1 ent to log (1) zy () xy"y (3) yx y (4) :c y y 50 One interval in which sin z increases throughout the entire interval is (1) 0 ~ x L. 7r () -1r L. x L. 0 (3) (4) ]4~. I1-Jlug. '74 [5] [OVER]

6 51 The lengths of two sides of a triangle are 8 and 7 and the included angle is 60. What is the area of the triangle? (1) 14 () 14V3 (3) 8 (4) 8\/3 5 When sketched on the same set of axes in the interval, L. '1T', how many common solutions will the o L.. % graphs of y == cos ~x and y == tan x have? (1) 1 (3) 3 () (4) 4 53 Which statement about the graph defined by y = x - z + is true? ( 1) It is tangent to the x-axis. () It intersects the.x-axis at two points, only. (3) It intersects the x-axis at three points. (4) It does not intersect the x-axis. 54 Given: y varies directly as the square of z. The graph of this relation is a ( 1) straight line (3) parabola () circle (4) hyperbola 55 If cos z == -l, the positive value of tan ~x is (1) v1 (3) V3 () V (4) vi 57 In 6.ABC~ AB == 10, Be = 9, and m LA = 3. Then, angle C can be ( 1) an acute angle, only () an obtuse angle, only (3) a right angle ( 4) either an acute or an obtuse angle 58 If A and B are complementary angles, then which is always true? ( 1) sin A == cos B () sin A + cos B = 1 ( 3) sin A - cos B-1 ( 4) sin A cos B == 1 *59 The sum of -3 - i and - 3i is a complex number which, when represented graphically, lies in quadrant (1) I () II (3) III (4) IV *60 The sum of the ages of three children in a family is 33 years. If the ages of the children form an arithmetic progression and the youngest of these children is 8, what is the age of the oldest child? (1) 13 (3) 15 () 14 (4) 16 * This question is based on an optional topic in the syllabus. S6 For which value of k will the following three lines intersect at a point? y =- z - 3y = 8 kz + Sy = 1 (1) -5 () (3) 8 (4) 11 Math. II-Aug. '74 [6]

7 Common Logarithms of Numbers* NI I li I f5 5j : S N Common Logarithms of Numbers* N S M gg N " ~~... o C'D o... CD 3 :3 o n T- CD r-t ~ -Q o CT., =;;'CD ::Ten 3 en... o 0... ~3: 3 0 CT.. CD =r ~ ~ -3 Q.. _. n en This table gives the mantissas of numbers with the decimal point omitted in each case. Characteristics are determined from the numbers by inspection.

8 Angle Values of Trigonometric Functions I Sin Cos Tan Cot 0 00' ' SO ' go 00' ' SO ' ' ' ' SO ' ' SO ' ' ' ' ' j4 8 00' go 00' ' Cos Sin Cot Tan I AnKle Values of Trigonometric Functions Angle Sin Cos Tan Cot 'iii go 00' ' ' ' ' ' ' ' ' ' ' SO ' ' ' ' SO ' ' ' ' Cos Sin Cot Tan

9 Values of Trigonometric Functions Angle I Sin I Cos I Tan Cot 18 00' ' '.356.' ' ' ' SO ' ' ' ' ' ' ' ' ' ' ' ' I.5095 I I 63 00', Cos Sin I Cot l Tan I Angle Angle Values of Trigonometric Functions I Sin Cos Tan Cot 7 00' ' SO ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' I ' Cos Sin t Cot t Tan < C

10 @ Values of Trigonometric Functions Angle Sin Cos Tan 36 00' !~ 00' CJ ' ' ' ' ' ' ' SO 00' ' ' ' ' DO' ' ' Cos Sin Cot Tan Angle I Cot < Q c: - CD en ọ... -t., _. to.. o:j o 3 CO., ne Logarithms of Trigonometric Functions's Angle L Sin L Cos L Tan L Cot 0 00' I" 00' ' ' ' ' ' ' ' ' ' ' ' ' ' ' go 00' ' L Co L Sin L Cot L Tan Angle * These tables give the logarithms increased by 10. Hence in each case 10 should be subtracted.

11 Logarithms of Trigonometric Functions'" Angle L Sin L Cos L Tan L Cot go 00' ' SO ' ' ' ' , ' ' ' ' ' SO ' ' ' ' ' ' ' ' ' ' L Cos L Sin L Cot L Tan Angle Logarithms of Trigonometric Functiona" Angle ILSin I LCOB IL Tan L Cot 18 00' ' ' ' SO ' ' SO ' ' ' SO ' ' ' ' ' ' ' ' ' ' ~ ~ ' ' I L Cos LSin L Cot L Tan * These tables give the logarithms increased by 10. Hence in each case 10 should be subtracted.

12 Logarithms of Trigonometric Functions'" Angle L Sin L Cos L Tan L Cot 7 00' ' ' ' ' ' ' ' ' SO ' ' SO ' ' SO ' ' SO ' ' SO ' ' SO SO ' L Coa L Sin L Cot L Tan Anal. * These tables give the logarithms increased by 10. Hence in each case 10 should be subtracted. Logarithms of Trigonometric Functions* Angle L Sin L Cos L Tan L Cot ' ' SO ' ' ' ' , ' ' ' ' ~ ' ' ' ' ' ' ' ' ' ' L Cos L Sin I L Cot I L Tan r o ca ọ....,_. zr 3 en o -h ~ cc ot --.. O 3., CD _. n." c: :::J ṉ -- + o ::J en.

13 FOR TEACH 11 ELEVENTH YEAR MATHEMATICS RS ONLY Wednesday, August 14, :30 to 3':30 p.m., only Just before the start of the examination period, distribute one examination booklet, face up, to each pupil. Instruct the pupils to read the directions on the cover of the examination booklet, detach the answer sheet, and fill in its heading. When each pupil has received a booklet and finished filling in the heading of the answer sheet, instruct the pupils to open their examination booklets and carefully remove the Reference Tables which are stapled in the center of the booklet. Instruct the pupils to also carefully remove the remainder of Reference Table C which is located on a separate perforated page. After these tables have been removed from the booklets, direct the pupils to open their examination booklets and begin work. Use only red ink or red pencil in rating Regents papers. Do not attempt to correct the pupil's work by making insertions or changes of any kind. Use checkmarks to indicate pupil errors. Unless otherwise specified, mathematically correct variations in the answers will be allowed. In problems involving logarithms, answers should be left correct to four significant digits unless directions say otherwise. Units need not be given when the wording of the questions allows such omissions. SCORING KEY Part I Allow 1 credit for each correct answer for le, b, a) and b ; allow no partial credit. Allow credits for each correct answer for 3-30; allow no partial credit. For questions 10-30, allow credit if the pupil has written the correct answer instead of the number, 1,, 3, or 4. (la) 4 (3) (3,0) (13) (3) 4 (b) 3 (4) 14iy (14) 4 (4) (a) 1 (5) (5,0) (15) 1 (5) 1 (b) 4 (6) -% (16) 1 (6) (7) 90 (17) 4 (7) (8) '1r -- 6 (18) 3 (8) 3 (9) 0.59 (19) (9) 4 (10) (0) 4 (30) 1 (11) (1) 1 (1) 3 () [OVER]

14 ELEVENTH YEAR MATHEMATICS - concluded Part II Allow credits for each of 0 of the following. For questions 43-60, allow credit if the pupil has written the correct answer instead of the number I,., 3, or 4. If a student has answered more than 0 questions on Part II, do not allow credit on those questions beyond the first twenty answered. (31) 1 (41) 4 0 ' (51) (3) 6 or -1 (4) (33) 1 -I- \/S.:r + 1 (5) (43) 4 (53) 4 (34) ~~ (44) 3 (54) 3 (35) ( sin x - 3) ( sin x + 1) (45) 1 (55) 1 (36) 18 (46) (56) 4 (37) S - (J (47) 3 (57) 4 (38) 1 (48) 1 (58) 1 (39) 0 (49) 4 (59) 3 (40) (50) 3 (60)