FORM GR9367 67 | THE GRADUATE RECORD EXAMINATIONS® MATHEMATICS TEST Do not break the seal until you are told to so The contents of this test are confidential Disclosure or reproduction of any portion of it is prohibited M THIS TEST BOOK MUST NOT BE TAKEN FROM THE ROO TESTING SERVICE and the ETS logo RECORD EXAMINATIONS, ETS, EDUCATIONAL Service Testing are registered trademarks of Educational 29 All rights reserved Service Testing ional Copyright © 1990, 1991, 1993 by Educat Princeton, N.J 08541 GRE, GRADUATE HS ME ETEer th 02 MATHEMATICS TEST Time—170 minutes 66 Questions Hị Directions: or incomplete statements below is followed by five suggested answers or compleEach of the qiéstiéns tions In each case, select the one that is the best of the choices offered and then mark the corresponding space on the answer sheet Computation and scratchwork may be done in this examination book Note: In this examination: (1) All logarithms are to the base e unless otherwise specified (2) The set of all x such that a S$ x < b is denoted by [a, 4] Hf f(e(x)) = and f(x) = x +3 (A) x — li tan x x, then g(x) = (B) -x xp COSX (Cc) TTI (D) (E) = (D) I (C) (B) -1 (A) ~œ forall real (BE) +0 [08 eae = (B) a) Let — denote (A) oz (py EHS _ I {xe A: x¢ B} If (A — B)U B = A, which of the following must be true? B is empty (BR) As B (CC) BAA (D) (8B -~A)U4 (E) =B None of the above GO ON TO THE NEXT PAGE 30 (E) log4 ~ In f(x) = (x| + 3x? forall real x, then /’(—~1) (A) -7 ý (B =5 For what value of & is the value of 5, (A) (B) ~! is () bth, (x? + x) dx (C) (D) (E) nonexistent ¬ a minimum? -2 (D) =3 In how many of the eight standard octants of xyz-space does the graph of z = e**) (A) One (B) Two (C) Three (D) @Œ) appear? Four Suppose that the function / is defined on an interval by the formula f(x) = /tan?x — which of the following intervals could be its domain? GO ON TO THE NEXT PAGE 32 -4 (E) Eight If / is continuous, @) ~j 10 If ƒ”({x) = f(x) (A) I~ e* Độ for all real x, © )= and if f (0) = and /'(0) = (B) e* ~ wo) 837 —I, then f(x) (C) e"* ~) = (D) e~* il If d(x, y, 2) = x? + 2xy + xz?, which of the following partial derivatives are identically zero? 1, 28 * ay? ag H- gậy HE ag (A) (B) TIE only [and IT only (D) If and II! only (C) (E) iand III only 111, and Ul GO ON TO THE NEXT PAGE 34 we (E) -e* 13 g on sin 2x (l +x)log(1 + x) (a) -2 = @®) -} (G0 (py (E) Em [hát = tan (A) (B) I (C)e (D) x (E) +00 14 Ata 15 percent annual inflation rate, the value of the dollar would decrease by approximately one-half every years At this inflation rate, in approximately how many years would the dollar be worth i 1ò ũõ of its present value? (A) 25 (B) 50 (C) 75 GO ON TO THE NEXT PAGE 36 @) 100 (E) 125 15, Let f(o= : ƒ —tl+r dt for all real x (A) y - = E(x -2) eee (B) y ~ Arctan = d(x ~ 2) ape (D) y — Arctan + = F(x — 2) 16 Let f(x) = e#)h(x) An equation of the line tangent to the graph of / at the point (2, /(2)) {C) y ~ | = (Aretan 2)(x — 2) (Œ)y =5 = (Arelan 2)(x — 2) and h'(x) = ~g' (x)A(x) forall real x Which of the following must be true? (A) f is a constant function ie sin l7 (A) linear nonconstant function constant function, linear nonconstant function of the above Arcoos 5) ~ cos % a 24 = (B) — cos % —— 1+ cos (@./ _ GO ON TO THE NEXT PAGE, 38 24 in (B) / is a (C) g is a (D) g is a (E) None ĐT is i 18 HS) bị = œ ¬ » (~1?x*:forall (8)Mộc (a) sin x 19 Tế, n=0 = x e(0, Ð), then ƒ') = ' (C) : T1 dầy t 35dy —y 33a ==0? n2 (A) cụết + ctef + c;2et (B) cục + ce”! + cy2e~f (C) cel ~ ew + cyte” (D) ce! + ce?! + c;e? (E) cổ? + cạế—? GO ON TO THE NEXT PAGE 40 LX We (D) x» Which of the following is the general solution of the differential equation dầy m _~ me a9 (BE) dễ 20 (Which of the following double integrals represents the volume of the solid bounded above by the graph of = ~ x? ~ 2y" and bounded below by the graph of z = —2 + x? + 2y2? ị m2 () (B) po ema © yf? pe JE Sag ~ 2x 2)— Ay) dy de proven ee Wye SGT RST (8 —— 2v2 2x — 4y*)dy? dy dx x= /Gn aye Se Verge OY (Ð) prfone pema? Seong @ > 2x? — Ay?) dx dy Œ 2| EMR og — ant — ay de dy =0 21, Let a bea number in the interval (0, 1) and let f be a function defined on [0, 1] by _ ja? if OS x for all xe [0, 1) {A) 24 If A T only and (B) II only 8B “= and (D) 48 Land II only (D) are events in a probability space such that < P(A) following CANNOT (A) (C) be true? B are independent = AUB (B) A I] and TH only = P(B) = P(AQB) isa proper subset of 2B (E) P(A)P(B) < P(AMB) GO ON TO THE NEXT PAGE (E) 1, 0, and HI < 1, which of the (C) A #8 25 Let / bea real-valued function with domain [0, 1] If there is some for all x and y in [0, 1], which of the following must be true? > such that I(x) -fQ) Ss K|x -y| :(A) f is discontinuous at each point of (0, 1) ¡(B) f is not continuous on (0, 1), but is discontinuous at only countably many points of (0, 1) (C) f is continuous on (0, 1), but is differentiable at only countably many points of (0, 1) “(D) f is continuous on (0,1); bit may not be differentiable on (0, 1) '(E) / ïs differentiable on (0, 1) 26 Let i = (1, 0,0), j = (0, 1,0), and vy = (A)i+i—k and k = (0,0, 1) The vectors (B)i~j+k 27 If the curve in the yz-plane with equation surface of revolution is vị and (€Œ)i+k z = f(y) x? +27 = f(y) x? +272 =1/(y)| yp? +2? =1/(y)| yt 2=[/ (xP GO ON TO THE NEXT PAGE 46 (@D)j-k (E)i+j is rotated around the y-axis, an equation of the resulting (A) x? +2? =[f()P (B) (C) (D) () v; are orthogonal if vị = i +j-k t n be an integer greater than |, Which of the following conditions guarantee that the equation awl TT se > ajx! has at least one root in the interval (0, 1)? ¿=0 ay > and nÌ > đị < ] ¡=0 HH aạ > and na] > aj >t ¿=0 HI ao < and ni] > a> ¡=0 (A) None (C) (D) (E) Il only HI only Tand Tf (B) Ionly 44 If x isareal number and P is a polynomial function, then (A) (B) 6P'(x) lim P(x (C) 3P"(x) GO ON TO THE NEXT PAGE 60 + 3h) + P(x ~ 3h) — 2P(x) (D) 9P"(x) = Ỉ h # 45 ‘Consider the system of equations ax? + by) = ee i dx? + ey =f real solutions bd The maximum possible number of # ae and s tant cons real are f and where a, b, c, d, e, (x, y) of the system is (A) 46 If x —x (C) two (B) one none +1 = ay tax — 2) + ax (1, ~1,0, D (C) (7, 6, 10, 1) (D) (7, II, 12, 6) (E) (7, 11,6, 1) GO ON TO fHE NEXT PAGE 62 three (E) five is — 2) + a(x ~ 2) for all real numbers x, then (ap, a, 42, a) (A) (-š 0, -8) (B) (D) et C be the ellipse with center(0, 0), major axis of length 2a, and minor axis of length 2b @ x dy — pdx The value is bị XA) mV25 + bề — (B) 2zVa? + bỲ (C) 2nab (D) nab (E) b 48 Let Gy denote the cyclic group of order n Which of the following direct products is NOT cyclic? (A) (B) (C) (D) Œ) Gir Gy Gy Gy Gà x x x x X Gi Gy x Gs G33 G33 Gi GO ON TO THE NEXT PAGE 64 _ i 49, Let X be arandom variable with probability density function fed gl =“) 1s ash, otherwise What is the standard deviatiéi 6f “X (A) 30 (â) v I (B) (D) Đ @) is (A) (B) (C) (D) (E) ~— iol 10 Com oO Pox OOn 50 The set of all points (x, y, z) in Euclidean 3-space such that a plane containing the points (1, 0, 0), (0, 1, 0), and (0, 0, 1) a sphere with center at the origin and radius a surface containing the point (1, 1, 1) a vector space with basis {(1, 0, 0), (0, 1, 0), (0, 0, 1)} none of the above GO ON TO THE NEXT PAGE 66 ki ...HS ME ETEer th 02 MATHEMATICS TEST Time—170 minutes 66 Questions Hị Directions: or incomplete statements below is followed... arbitrary} a is arbitrary and a = (E ~—b and = of b isarbitrary} the empty set 42 What is the greatest value of for which any real-valued function f that satisfies the following properties must... ”@)1< b for all x in 0, 1) (A) I (B) (C) GO ON TO THE NEXT PAGE 38 (D) 12 (EB) 24 t n be an integer greater than |, Which of the following conditions guarantee that the equation awl TT se > ajx! has