Mathematies Part -2 SECTION -A Single Correct Choice Type Thls section contains.gm{1i4" choice questions. Each question has 4 choices (A), (B), (c) and (D) for its answer, out which ONLY ONE is correct. 1. lf lul=1,lvl=1,uv*-1,and z=Jl+v f + *, then (u, v e C and have non-zero imaginary parts) (A) lz l= 1 (B) Re(z) = s (c) tm(z) = e (DiRetzi= tm(z) 2. The set of reai values of a for which the equation zai + x2 2x 'l ,t -*T -;;7;7 * * -u = 0 has a unique sotulton ts (A) (-o,1) (B) (_1,o) (c) (-1,1) (D) none of these 3. Let an = 111.......1. The remainder when ?12a iS divided by 27,1 is n times (A) 23 (B) 25 (c) 2r (o)zs 4. Let n = 10050 -43, the sum of the digits in the decimal notation of n is (A) 8e0 (B) 8e3 (c) 8e4 (o)esz Rough Work5 Thevaiues=L,,uf+rl' equars in{Lr'-r)' ( c'-/(A)* (c) 1f r+-1+...*l) ' n+1\ 2 n.l et 3-(rn1* *f ) n+1\ 2 n) (D) none of these 6' The system of homogeneous equations tx+(t+1)y+(t -1)z=0,.(t+1)x+ty +(l+2)z=0 and (t -1)x + (t + 2)y + tz = 0 has non-trivial solutions for (t + c2;xl (1+ c2 )xl then f(x) is a potynomiat of degree 1+ c'x I T. lf a2 + b2 +c2 = -2 , and n-, = l,;i::; l1r + a2;x (A) exactly three real values of t (C) exactly one real value of t (A) 0 (c)2 The least positive value of x satisfying (A) nt3 ' $) 2nt3 (B) exactly two real values of t (D) infinite number of values of t (t+b2)x 1 +b'x (t+b2)x (B) 1 (D) 3 8. v. A letter is taken at random from the letters of the word 'sTATtsTlcs' and another letter is taken at random from the letters of the word 'ASSISTAN_T'. The probability that tnev aie same, is ltl ii13, [B]ll13' sin2 2x+ 4sina x -4sin2 xcos2 x 1 --;;}r;fG;t-= e (" + (2n + 1)nt2) is (B) n/o (D) srcl6 Rough WorkReasoning TYPo This section contains 4 reasoning type questions. Each question has 4 choices (A), (B)' (c) and (D)' out of which ONLY ONE is correct. 10. 5TATEMENT 1 : The minimum value of l2x-'ll+ l3x -21+l4x -3 | is ? and STATEMENT 2 : lfa < b < c, then minimum value of A lx-al + B lx-bl + c lx-cl (A, B, C > O)is attained at x = b' (A) Statement -1 is True, Statement -2 is True; Statement-2 is a correct explanation Statement-1 (B) Statement -1 is True, Statement -2 is True; Statement-2 is NOT a correct explanation Statement-1 (C)Statement -1 is True, Statement -2 is False iOi St"t"ment -1 is False, Statement -2 is True 1,t. 5TATEMENT 1: lf In = J-/2sinn xdx where n is a positive integer then ln is rational if n is odd. and STATEMENT 2: tn= Tt , (A) Statement -1 is True, Statement -2 is True; statement-2 is a Statement-1 idi it"t"*"nt -1 is True, Statement -2 is True; Statement-2 is NOT a Statement-1 (C) Statement -1 is True, Statement -2 is False iOi St"t"t"nt -1 is False, Statemg$,.-2 !s fgg for for correct explanation for correct explanation for Rough Work12. STATEMENT 1 : Let pn be the probability that 2 balls drawn from a bag containing n white and n black balls will be of the same colour. Then lim p" =: . n-+o z and STATEMENT -N+1 z: P" = fr+71 (A) Statement -1 is True, Statement -2 is True; Statement-2 is a correct explanation for Statement-1 (B) Statement -1 is True, Statement -2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement -1 is True, Statement -2 is False (D) Statement -1 is False, Statement -2 is True 13. STATEMENT 1 :lf y(1)=t, y'(1)=1and xyy" +xy'2 =3yy' then y'(Z)=e1' z and STATEMENT2:Thesotutiontothedifferentialequationxyy"+xy'2=3yy'isY=CrX3+Cz (A) Statement -1 is True, Statement -2 is True; Statement-2 is a correct explanation for Statement-1 (B) Statement -1 is True, Statement -2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement -1 is True, Statement -2 is False (D) Statement -1 is False, Statement -2 is True Rough WorkLinked Comprehension Type This section contains 2 paragraphs. Based upon each paragraph, 3 nrultiple choice questions have to be 9!9y€reqj_199! i99st1q! has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct. Paragraph for Question Nos. 14 to t6 Using differentiability and continuity of a function f which satisfies certain functional equation, determine in some cases the function explicity. E.g. if f satisfies f(x + y; = f(x) f(y) for ail x, y 'e f(x)*0 forany x eR andf'(0) = 1 thenf(x) = e'. we can R and 14. for real x and y and f'(2) = 3, then f(x) is equal to (B)24log (3x + 2) (D) lx2 +z '4 15. lf f is differentiable function on R and f'(0) = 2 sarisfoing f(x+y)=fr# then f(n/8) is lf afunctionf satisfy ,( x+Y ) 2+f(x)+f(y) '[ 3 J= 3 (A) -1x3 +x2 12 (C)3x+2 equalto (A) 1t2 (c)3t2 (B) 1 (D) tann/8 16. lf f'(x) = f(x) for all x and f'(0; = 4 then f(x) is equal to (A) 2?'. _ (B) "o" (C) x" + 4x' + 4x (D) 4e'Paragraph for euestion Nos. {7 to .t9 l-et k be the length of any edge of a regutar tetrahedron. (A tetrahedron whose edges are all equal in length is called a regular ietrahedron). Trie angle betwe"n " tin" and a pra,ne is equar to the comprement of the angle between the line and the normal tJ the plane *n"i"u" the angle oetwJen two pranes is equal to the angle between the normals' Let o be the origin or rereience and A, B and c vertices with position vectors d, b and 6 respectively of the regulartetrahedron. 't7 18. 19. The angle between any edge and a face not containing the edge is (R) cos-1(vz) (c)cos'ttJs (B)cos-'(1/4) (D) z/3 The angle between any two faces is (n) cos-11./5 (C) ru/3 I he value of [a b c]' is (A) k" (c) (1/3)k6 (B) cos-11/+ (D) cos-11,,3 (B) (1/2)k6 (D) (2/3)k6 Rough WorkSECTION -B Matrix -Match TYPo rhis section contains 3 questions' El"lt qyTli:i.:?:3::ift:T,:*"^n'"^?: "'iff Hffi i.l, **'Jn i l" " j"-n "'*t cn ed r h e stSl:'-T :ll""l'*:l l3 :' :': iH;,ffiX:';:'A #;,';nile-inl "t"tements in Golumn ll are labelled p, q' r, r ---L^.,^ ^^.r^,+ -ofnhinn with 5H'l"l?"?il;;t#;;J;-t'.i;r1;1"1-""Lhi"-"^":::.*^T1t:n'l"n*H:.: ; ft [" b {"ilt 3'#' J,X"' ""ii Jl i n-c"r"" ^, I --rlL . i?t?t'':::,:*l':: il;#Tdffii" *,"1""*"t" io these questions have to be darkened as lllustrited in the following example: ill;:ffi;;;"",;;:';i"!' -[ " "n9 ]'. B -I ?11-'i,9 ;l 3:g3,#d D -s #;,ile;il';;;;;; Jutr"ni'ij of bubbtes will look like the followins: @@@cc -t_. @centreof atrianOle'J the affixes of whose veftices u'" f" *'' 7 1w neing imaginary root of ] unity) is lgl--'t'"=9,{i?'''"!fEi,Ll= ll?lil"rr" rt' ' -r m r Column -ll (p) 6 (q) 3 (r) 0 (C) Qn [u, zl rne lrtaxtttr'ttt "o'"= "' " ' | (D) lf f(x)= f"f .'tjA;gU (where [.] denotesthe greatest inteser I tunction) then (s) = (s) 7 Rough Work2. Consider the expression t(x) = x-+ mX + m-+ 6m, where m ls a parameter, match the Column -| Column -ll (A) f(x) > 0 for all x (p) (-o. -8) u,10. o) (B) f(x)Oforallx>0 (r) (--o,-8)t,(0,0o) (D) f(x) < o for all x (s) tt# 4+2Jr) xt*mx*mt 3. Let ABCDEF be a convex Hexagon in 2 dimensional plane where A is origin AB ll DE, BC ll EF and CD ll FA. The y-coordinates of vertices B, C, D, E, F are the five distinct elements of the set 4 6.8 10). Match the x-co-ordrnates ot these veftlces Column -| Column -ll (A) B (p) 6.F (B) c (o) 2 (c) D (r) # (D) E is) 0 Rough WorkMg,t;fr,,ernafrcs Paft-2 SECTION -A Single Correct Ghoice Type This section contains 9 multiple choice questions. Each question has 4 choices (A), (B), (c) and (D) for its answer, out which ONLY ONE is correct. 1' lf h denotes the arithmetic mean and.k denotes the geometric mean of the intercepts made on the coordinate axes by the lines passing through the point (1, 1), ttren tne'poi^iit, rl lies on (A) a circle (B) a parabola (C) a straight line iOia'nyperUota 2. Thenonzerovalueof aforwtrichthelines a<-4y+Tz+ 16= O=4x+ 3y_22+ 3andx_3y+ 42+ 6= 0 = x -y + z+ 1 are coplanar is (B) 37 (D) None of these 3. Let f : R -+ R then f is differentiable on R if (A) lf I f(x)-f(y)F k I x-y I for ail x,y e R and somek > 0 (B) lf(x)-f(y) F k lx-y lt,, for ail x,y e R and somek > O (C) | f(x)-f(V)ls k I x-y f tor ail x,y e R and somek >O (D) f2 is differentiable on R (A) 35 (c) 47 (x 4. Let f(x)=]Jtt+lt-tl)ot, if x>2 l0 "^--'then l5x-2, if x<2 (A) f is not continuous at x = 2 (C) f is differentiable everywhere (B) f is continuous but not differentiable at x = 2 (D) f'(2+) doesnt exist 5' The value of m for which the area of the triangfe induded behreen the axes and any tangent to the curve X'y = b' is constant, is (Al1t2 (B) 1 Rough Work't '= 1ffix then I equals (A) (1+-f''[]t,+x)2 -f tr*"1.i]"" (c) (1+*)''' [*' -*^**]." (B) (1+ "F" [ft'' * *)' -;--* *9t'r;] -g (D) (1 +'F'' [f t'' * x)2 -f ^ -i] -" (B) -l"oto cosec0 (D) -1coto seco 8. v. f 1. o < I, value of (A) -1t not."., (c) -1t"n,x cosec ct rhe sorution " [:])' (A) y+e x =const (C)y=e-^+const (A) f(x) -9(x)+ c (c) .,/(x) *.nffi + c 1n t2 sin2x I ----___=------ox J -rt2 J1+ sin2c sin x -*,.'+e *)+1=o (B)x=log(c+y; (D)y=e^+const (B) f(x) + g(x) + c (D) None of these g(x)= ffi"nlco"1x -,rlz;*J"in'" *i'f ol tn"n I equals Rough Work asinx+bcosxReasoning Type This section contains 4 reasoning type questions. Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct. 10. STATEMENT 1:lf a> b>0theneccentricityoftheellipse ax2 +by2+cx+dyre=0i".f9-! ll b and STATEMENT 2: Eccentricity e of etlipse +.+= (a > b) is given by b2 = a21t-e2y. 'azb' (A) Statement -1 is True, Statement -2 is True; Statement-2 is a correct explanation for Statement-1 (B) Statement -'l is True, Statement -2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement -1 is True, Statement -2 is False (D) Statement;1 is False, Statement -2 is True 11. STATEMENT 1: lf a tangent to the curve xt * yt = a3 at (xt, yr) passes through (a, a) then (x1, yr) lies on a circle of radius a. and STATEMENT 2: The curve x3 + y3 = a3 and x' +y' = a2 cut orthogonally. (A) Statement -1 is True, Statement -2 is True; Statement-2 is a corect explanation for Statement-1 (B) Statement -1 is True, Statement -2 is True; Statement-2 ie NOT a correct explanation for Statement-1 (C) Statement -1 is True, Statement -2 is False D) Statement -1 is False, Statement -2 is True Rough Work12. STATEMENT 1: For any large positive integer n, the integer next to r-1*1-1*.....*-l --l-i"z 234 2n-1 2n and 5TATEMEN r 2: t -7 * ! -L+..... -! = -1-* -1=*. .. * -l-* + . 2' 3 4 "-2n n+1 n+2 2n-1 2n (A) Statement -1 is True, Statement -2 is True; Statement-2 is a correct explanation for Statement-1 (B) Statement ,l is True, Statement -2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement -1 is True, Statement -2 is False (D) Statement -1 is False, Statement -2 is True 13. STATEMENTl:lf nisoddthentheproductP=(1-i1X2-izX3-b).....(n-in)whereir,iz,ig,.",in are distinct integ'ers taken from the set {1,2,3,.....,n}, is certainly even' and STATEMENT 2'. P can be zero for some choice of i1, i2, ""., in (A) Statement -1 is True, Statement -2 is True; Statement2 is a correct explanation for Statement-1 (B) Statement -1 is True, Statement -2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement -1 is True, Statement -2 is False (D) Statement -1 is False, Statement -2 is True Rough WorkLinked GomPrehension TYPo This section contains 2 paragraphs. Based upon each parlgraph' 3 multiple choice questions have to be "n"*"r"0 r""n ou"iiJi'r.'u1ai"noi""" (n), ia), (c) .ro (o), trt tt *hi.n oHtY oue it cotttt' Paragraph for Question Nos' {4to 16 cr is a circle on the major axis and c2 is a circle on the minor axis of the elipse p,*n*='t u" 25 16 diameters 14. lf I is the length of the tangent from any point on the circle cr to the ellipse P, $hen (A\l<213 (B)l=3 (D)l= 3/a (c) l< 3/4 15. lf e is a point on the circle C2 and R is the point of the ellipse such that Q lies on the ordinate of the point R, theri +-.9+ is equalto absclssa or K (A) 5/4 (B)4/5 (c)v+ (D) 3/5 16. Length of the chord of contact of the tangents from any point on the circle c1 to the circle cz is 6)72t5 F) 24t5 ic) zals (D) 3215 Rough WorkParagraph for Question Nos. iZto i9 A (3,^7) and B(6, 5) are two points C : x' + y' -4x-6y-3 = 0 is a circle. 17. The chords in which the concurrent at (A) (2, 3) (c) (3,23t2l' circle c cuts the members of the family S of circles through A and B are (B) (2,23t3) (D) (3, 2) (B) x'+y2 -5x+Oy*1=0 (D) x'+y2+5x-6y-1=0 18. Equation of the member of the family S which bisects the circumference of C is (A) x'+y2-5x*1=o (C) xt +y2 _5x-6y-1=o 19' lf o is the origin and P is the centre of C, then difference of the square of the lengths of the tangents^from A and B to the circle C is equal to (A) (ABr-(B)(oP)' Rough WorkSECTION -B Matrix -Match TYPe Tl.lis section contains 3 questions. Each question contains statements given in two cotumns, which have to be matched. The statements in column I are taoetteo A, B, c and D, whi|e the statements in Co|umn || are |abe||ed p, q, r,sandt.AnygivenstatementinCo|umnIcanhavecorrectmatchingwith tiUE On frlbne statement(s) in Column ll. The appropriate bubbles correspondingtotheanswerstothesequestionshavetobedarkenedas illustrated in the following e,xample: iiin" "orr"a matches ur" n-p, s ancl t; B -q and r; C -p alg q; and D -s and t: then the correct darkening of bubbles will look iike the following: t\B(l D @@c@@o@@oc @@ooc @@o@@1. Match th Column -ll (p) 1i24 (q) nlZ (r) Log4 (s) 3/8 Rough Work2. Letf(x)=Pe2*+Qe*+Rxsatisficsf(0)=-1,1'11og2)=sr,fiotrtx)-Rx)dx=19.5,thenmatchthe Column -| Column -ll (A) P aI -6 (B) a (q) 5 (c) R ]IL2 (D) P+Q+R (s) 3 3. Match the Column -| Column -ll (A) The period of the function 3x -[3xl (p) 180 (B) The distance between the points on the ciiCit-+-tz+ 2x * 19 = 0 which are points of contact of tangents from ( 1. 6) (q) 2008 (c) The value of the integr"t J'"XXtr-Ti: (r) J40 (D) The shortqst distance between origin and theifrve 1z+lr +xv=60 (s) J24 Rough Work