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SOLVED, PAPER, , 2017, , SECTION -A, , Multiple Choice Questions (MCQ), Q.1-Q.10 carry one mark each., 1. Consider the function f(r,y) = 5 — dsine + y* for, , 0 < 2 < 28 and y € BR. The set of critical points of, f(x,y) consists of, , (a) A point of local maximum and a point of local, mini, (b)_ A paint of local maximum and a saddle point, , (c) A point of local maximum, a point of local, minimum and a saddle point, , (d) A point of local minimum and a saddle point, [Functions of two Variables], Sol. (d) The given function is defined by, , S(x,y) =5 —dsing + 7,0 < 2 < yx, , of, , = je 7 tour, of |, , => ay 7!, , ‘Therefore, set of critical points are given by, of of, , = =0, —=0, , ox dy, , TN, , , , eosrel, yed, , z=(2n-l)§=n=0, => tr=§,8,y=0, y=0, Hence, critical points are ($0) , (4,0), Now,, , r= ef @ dsinz, , O24, , and, , wrt=s? =Bsing 2 Ofor0 <a < 2r, So, at (F,0) f has minimam and at (2,0) has a saddle, point,, , 2. Leto: R = RB be a differentiable function such that, is strictly increasing with 4'(1) = 0. Let « and (7 denote, the minimum and maximum valucs of ¢(:) on the, interval [2,3], respectively. Then which one of the, , following is TRUE?, (a) 4 = 4(3) (b) a = 42.5), (c) 3 = (2.5) (d) a = 3), [Functions of one real Variable}, , Sol. (a) o is strictly increasing function such that o"(1) « O., Also: R > Risa differentiable function,

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Let a, 3 € (2,3) are a point of extrema., Suppose, a, 3 € (2,3), then, , (a) =0= f(R), -, dis strictly increasing such that o‘(1) = 0, , = ¢{z)>0, @{a),o'(3) > 0,, , So a, 3 ¢ (2,3)., , Suppose a, J € {2,3}, then, (2) =a, o(3)=8, 3. The number of generators of the additive group Zag is, equal to, (a6 (b) 12, (c) 18 (d) 6, , [Group Theory], Sol. (b) Number of generators of Zsq is equal to 6(36). But, , (36) = o(2*.3"), , 25 5, (a) 7 (b) 3, 2 Sr, {c) 5 (d) 2, [Functions of one Real Variables], Sol. (c) Here, =. 7 ork, ht +, din Yes (5 2 =), , , , HIT JAM 2017 Solved Paper, , But we know that, , cos. a + casa + d) + cos(a + ad) +..., , + con(a+ (n= 1d) see ies, , Therefore, , y«(), co (wf) sin ((n +38) _,, sin &, ont. sin (392) — sin Oe, , sin 5, im (F + GF) +8in F) — ain, , , , , , in, § (sin (3 + 5) — sin #), , in S=, st, L gin (50 on, _ asin (+3), ~ in DE, sin = 2, feos = r, , ", be, sin = on ~ 2n, , , , Therefore, as, Sak, noo, lim = iye(F), «tin Zo HE ty 2 2?, , me 5 8 Gn ain SE nao 2 5, , Alternatively: We have, (§+5 bx 5), Eye :, a= n, , i, =* [ an($ 4-2) a, lo 2 2, ‘ ix, [= (F)«, , 5. Let f : R + R be a twice differential function. If, (u,v) = f(u? — v2), then He + Fe =, {a) 4(u? — v7) f"(u? — 0?), (by 4(u? — v?) f"(a? = v?), , we A

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4 HIT JAM 2017 Solved Paper, , (c) Both L. and R exist 10. If, (d) Neither 1. nor R exists, l+z, fo xr<0, [Functions of one Real Variable] f(x) = ;, (l-2x)(pr+q). if 220, Sol. (a), fiz)= sth ts +i) aa(? i #0 satishies the assumptions of Relle's theorem in the, imerval [—1, 1], then the ordered pair (p,q) is, fa) (2,—1) (b) (—2,-1), R= tin f(z) = tim ATA +A) n(; i: h>o (©) (2,1) (4) (2.1), 20+ ho A A [Real analysis], = Him(2 +f) sin (;), 1 1 Sol. (d) Since, f(x) satisfies of Rolle’s Theorem. Therefore,, = jim 2sin (=) + fi sin = f(x) must be continuous in [-1,1] and differential in, which does not exist. (Lb., and Hence, f(r) is continuous utc = 0, b= Jim £2) > lim fz) = Jim, ffx), lim Cet nO =”) in (5) h>0 => l=q, ho —h oh i, 4 —h+ A(1 — A) 1 ad =, “A (-i), 1 Also, f(r) must be derivable at ar = 0,, = — lim(h}sin (;) =0, AO) A, which exits, => Tim, f(z) = lim, s"(e), Therefore, L exist but R does not exist, * i= Jim [-(nz +9) +p(1 -z)), je Tr 2 a 20, 9. A limyse fy ede = 3, then x p-qel, limo. [, we * dr =, m > p=2, o, Therel » (2,1, (c) van (d) 25 oe D1), , [Functions of one Real Variable] Q.H -Q.30 Carry two marks each., , Sol. (a) limy ya. ff" ede = F (Given), Now, 11. The flux of the vector field, , T oe, . 2-2 2-2" Pt aim, J, 7° de= f = Pa (are @F)i4 + (2n7-#)3, , ae + ,-t A q =, =i edt (Let 2 ot, > lads = dt) along the outword normal, across the ellipse, , _l * Aa, tat a? + 16y? = 4 is equal to, “2h, (a) dx? - 2 (b) 20? -4, five (by x? 2 (a) 27, , 11, ai feetly 7 er [Vector Calculus}

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IIT JAM Mathematics, , Sol. (b), , , , , , , , , , F-F = 1 (20? — 44 dry? + 2x72), Let ¢ = 2roosé,y = brsind, , *>rel, , i, , 2. dyda:, , .). flux = ff F Ft, L [+ (2#- 4442, , x i § eos sin? § + 2n? x 2rcost) rdrde, , -i[ [er- 4)r?, , + (2r4 sin? 6 + dr?x?) cos Adedr, , = i [on -4) BG, _ z ‘em 4) [ae, , , , st, , , , , , _ 2n an? 4, # 2, = 2x? - 4,, , 12. Let p be the set of all invertible 5 x 5, matrices with entries O and 1, For each Moe M, let, , m(M) and no(M) denote the number of 1"s and 0's, in M respectively. Then mir yyea¢|m(M) — no( M)| =, (ay {b) 3, (v5 (d) 15, {Linear Algebra], Sol. (a) Consider the matrix, 1 Oy2 Ozy Ory Os, Me O 1) Oz: Oz Cos, 0 0 Lf Oy Os, o 0 0 0 1, , 5, my(Af) | ng(Mf), 5 10, i il, 8 12, Therefore, minarene|m(M) = no(M)| = 1, 28 EI, 13. Let M = i i and x = | . Thea lim, 0 M™2, (a) does not exit (h) is |, (c) is fl (d) is 1, {Linear Algebra}, boy 3, iven that Mos |? 4 =, Sol. (c) Given that 1 ; | and x if, , Let 4 be an eigen value of matrix M. Then A= 4.1, , Consider (A =H) X = 0, 0 $) ix) _ jo, ~ 0 sly) fo, , - wd, , - [Papp, o 0, sel, »> =|?, Therefore,, s-[! 3], 0 L, => sia! ?, a 0, Hence, MS =SA, where