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PART-A Select the correct alternative : (Only one is correct) Q.1 Let f be a real valued function such that f (x) + 2 f 2002 = 3x x for all x > 0. Find f (2). (A) 1000 (B) 2000 (C) 3000 (D) 4000 Q.2 Point Alies on the line y = 2x and the sum of its abscissa and ordinate is 12. Point B lies on the x-axis and the line AB is perpendicular to the line y = 2x. Let 'O' be the origin. The area of the triangle AOB is (A) 20 (B) 40 (C) 60 (D) 80 Q.3 Minimum period of the function f (x) = | sin32x | + | cos32x | is (A) (B) 2 (C) 4 3 (D) 4 Q.4 The number k is such that tan{arc tan(2) + arc tan(20k)} = k. The sum of all possible values of k is 19 (A) – 40 21 (B) – 40 1 (C) 0 (D) 5 Q.5 Let A = the number of ways of selecting a committee of 5 from a group of 9 persons. B = the number of permutations of the word PRELEPCO taken all at a time. C = the number of ways in which 8 people can be arranged in a line if two particular people must stand next to each other. AB The value of C equals (A) 63 (B) 126 (C) 360 (D) none Q.6 Which one of the following depicts the graph of an odd function? (A) (B) (C) (D) 3 3 i 100 Q.7 If 349(x + iy) = 2 + 2 and x = ky then the value of k equals (x, y R) (A) (B) – (C) (D) – Q.8 If z1 is purely imaginary then 2 is equal to : (A) 1 (B) 2 (C) 3 (D) 0 Q.9 If sin = 12 , cos = – 13 5 , 0 < < 2. Consider the following statements. 13 –1 5 –1 12 I. = cos 13 II. = sin 13 –1 12 12 III. = – sin 13 IV. = tan 5 12 V. = – tan–1 5 then which of the following statements are true? (A) I, II and IV only (B) III and V only (C) I and III only (D) I, III and V only Q.10 Domain of definition of the function f (x) = log 10·3x2 9x1 1 + cos1(1 x) is (A) [0, 1] (B) [1, 2] (C) (0, 2) (D) (0, 1) Q.11 Two points A(x1, y1) and B(x2, y2) are chosen on the graph of f (x) = ln x with 0 < x1 < x2. The points C and D trisect line segment AB with AC < CB. Through C a horizontal line is drawn to cut the curve at E(x3, y3). If x1 = 1 and x2 = 1000 then the value of x3 equals (A) 10 (B) (C) (10)2/3 (D) (10)1/3 x x Q.12 The period of the function f(x) = sin 2x + sin 3 + sin 5 is (A) 2 (B) 6 (C) 15 (D) 30 Q.13 Triangle ABC has BC = 1 and AC = 2. The maximum possible value of the angle A is (A) 6 1 (B) 4 3 (C) 3 (D) 2 Q.14 The sum tan n 1 (A) 3 + cot1 2 4 n2 + n 1 is equal to (B) + cot1 3 2 (C) (D) + tan1 2 2 Q.15 If f (x) = 2x3 + 7x – 5 then f–1(4) is (A) equal to 1 (B) equal to 2 (C) equal to 1/3 (D) non existent Q.16 Let z be a complex number, then the region represented by the inequality z + 2< z + 4 is given by (A) Re (z) > 3 (B) Im (z) < 3 (C) Re (z) < 3 & Im (z) > 3 (D) Re (z) < 4 & Im (z) > 4 Q.17 A square OABC is

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- MATHEMATICS
- PART-A Select the correct alternative : (Only one is correct) Q.1 Let f be a real valued function suchthat f (x) + x 2002 2 f = 3x for all x > 0. Find f (2). (A) 1000 (B) 2000 (C) 3000 (D) 4000 Q.2 PointAliesontheliney=2x andthesumofitsabscissaandordinateis12.PointBlies onthex-axisand the lineAB is perpendicular to the line y= 2x. Let 'O' be the origin.The areaof thetriangleAOB is (A) 20 (B) 40 (C) 60 (D) 80 Q.3 Minimumperiodofthefunction f (x) = | sin32x | + | cos32x | is (A) (B) 2 (C) 4 (D) 4 3 Q.4 The number k is such that ) k 20 tan( arc ) 2 tan( arc tan =k. The sum ofall possible values of k is (A) – 40 19 (B) – 40 21 (C) 0 (D) 5 1 Q.5 Let A = the number of ways of selecting a committee of 5 from a group of 9 persons. B = the number of permutations of the word PRELEPCO taken all ata time. C=thenumberofwaysinwhich8peoplecanbearrangedinalineiftwoparticularpeoplemust stand next to each other. The value of C AB equals (A) 63 (B) 126 (C) 360 (D) none Q.6 Whichoneof thefollowingdepictsthegraph ofanodd function? (A) (B) (C) (D)
- Q.7 If 349(x + iy) = 100 3 2 3 2 i and x = ky then the value of k equals (x, y R) (A) 3 (B) – 3 (C) 3 1 (D) – 3 1 Q.8 If 2 1 z z is purelyimaginarythen 2 1 2 1 z z z z is equal to : (A) 1 (B) 2 (C) 3 (D) 0 Q.9 If sin = 13 12 , cos = – 13 5 , 0 < < 2. Consider the following statements. I. = cos–1 13 5 II. = sin–1 13 12 III. = – sin–1 13 12 IV. = tan–1 5 12 V. = – tan–1 5 12 thenwhichofthefollowingstatementsaretrue? (A) I, IIand IV only (B) IIIand V only (C) I and III only (D) I, IIIand V only Q.10 Domain ofdefinitionofthefunction f(x) =log 1 9 3 · 10 1 x 2 x + ) x 1 ( cos 1 is (A) [0, 1] (B) [1, 2] (C) (0, 2) (D) (0, 1) Q.11 Two pointsA(x1, y1) and B(x2, y2) are chosen on the graph of f (x) = ln x with 0 < x1 < x2.The points C andDtrisect linesegmentABwithAC <CB.ThroughC ahorizontal lineis drawn to cutthecurveat E(x3, y3). If x1 = 1 and x2 = 1000 then the value of x3 equals (A) 10 (B) 10 (C) (10)2/3 (D) (10)1/3 Q.12 The period of the function f(x) = sin 2x + sin x 3 + sin x 5 is (A) 2 (B) 6 (C) 15 (D) 30 Q.13 TriangleABC has BC = 1 andAC = 2.The maximum possible value of theangleAis (A) 6 (B) 4 (C) 3 (D) 2 Q.14 The sum n 1 tan1 1 n n 3 2 is equal to (A) 2 cot 4 3 1 (B) 3 cot 2 1 (C) (D) 2 tan 2 1
- Q.15 If f (x) = 2x3 + 7x – 5 then f–1(4) is (A) equal to 1 (B) equal to 2 (C) equal to 1/3 (D)nonexistent Q.16 Let z be a complex number, then the region represented by the inequality z + 2< z + 4is given by (A) Re (z) > 3 (B) Im(z) < 3 (C) Re (z) < 3 & Im (z) > 3 (D) Re (z) < 4 & Im (z) > 4 Q.17 Asquare OABC is formed byline pairs xy= 0 and xy + 1 = x + ywhere 'O' is the origin.Acircle with centre C1 inside thesquare is drawn to touch the line pair xy= 0 and another circle with centre C2 and radius twice that of C1, is drawn to touch the circle C1 and the other line pair.The radius of the circle with centre C1 is (A) 1 2 3 2 (B) 1 2 3 2 2 (C) 1 2 3 2 (D) 2 3 1 2 Q.18 Given f (x) = x 1 8 x 1 8 and g (x) = ) x (cos f 4 ) x (sin f 4 then g(x) is (A) periodic with period /2 (B) periodicwith period (C) periodic with period 2 (D) aperiodic Q.19 LetABC be a triangle right angled at C. The value of a log · a log a log a log b c c b b c c b (b + c 1, c– b 1) equals (A) 1 (B) 2 (C) 3 (D) 1/2 Q.20 Thesumoftheinfinitetermsoftheseries cot 1 1 3 4 2 + cot 1 2 3 4 2 + cot 1 3 3 4 2 + ..... is equal to : (A) tan–1 (1) (B) tan–1 (2) (C) tan–1 (3) (D) tan–1 (4) Q.21 The period of thefunction f (x) = | x cos x sin | | x cos | | x sin | is (A) /2 (B) /4 (C) (D) 2 Q.22 Number ofeven numbers between 1000and 9999 havingalldistinct digits, is (A) 1848 (B) 2296 (C) 2298 (D) 2300 Q.23 Suppose, f (x, n) = n 1 k x x k log , then the value of x satisfying the equation f (x, 10) = f (x, 11), is (A) 9 (B) 10 (C) 11 (D) none
- Q.24 If be a complex cube root of unity, then the value of 1 1 2 1 1 1 2 is : (A) 0 (B) 1 (C) 1 (D) none Q.25 The value of tan1 1 2 2 tan A +tan 1(cotA)+tan1(cot3A) for 0 <A< (/4) is (A) 4 tan1 (1) (B) 2 tan1 (2) (C) 0 (D) none Q.26 The maximum value of function f (x) = – (ex/2 + e–x/2),is (A) 0 (B) – 1 (C) – 2 (D) – e Q.27 Let f(x)=lnx& g(x)= x x x x x x 4 3 2 2 2 3 2 2 2 2 1 .Thedomainofthecompositefunctionfog(x) is: (A) (, ) (B) [0, ) (C) (0, ) (D) [1, ) Q.28 In a ABC, a b c. If C sin B sin A sin c b a 3 3 3 3 3 3 = 27, then the maximum value of 'a' (A) 3/2 (B) 2 (C) 3 (D) cannot be determined Q.29 The value of sec sin sin cos cos 1 1 50 9 31 9 is equal to (A) sec 10 9 (B) sec 9 (C) 1 (D) –1 Q.30 Let f beafunctionsatisfying f (xy)= y ) x ( f forall positive real numbersx and y. If f (30)=20,then the value of f (40) is (A) 15 (B) 20 (C) 40 (D) 60 Q.31 If thex intercept of the liney= mx +2 is greaterthan 1/2 then the gradient ofthe linelies in the interval (A) (–1, 0) (B) (–1/4, 0) (C) (– , – 4) (D) (– 4, 0) Q.32 If be a complex cube root of unity, then the number; (1 2)3 + ( 1 2)3 + (2 1)3 is : (A) divisible by3 but not by8 (B) divisible by8 but not by3 (C) divisible byboth 3 & 8 (D) none of these Q.33 Let f (x) = sin2x + cos4x + 2 and g(x) = cos (cos x) + cos (sin x).Also let period of f (x) and g (x) be T1 and T2 respectivelythen (A) T1 = 2T2 (B) 2T1 = T2 (C) T1 = T2 (D) T1 = 4T2 Q.34 Given i = 1 and (x + yi)2 = 45 + ai, where x and yare positive integers, then the smallest possible valueforawhichsatisfiestheinequalities? (A) 10 < a < 20 (B) 20 < a < 30 (C) 30 < a < 40 (D) 40 < a < 50
- Q.35 If x = tan1 1 cos1 1 2 + sin1 1 2 ; y = cos 1 2 1 8 1 cos then : (A) x = y (B) y = x (C) tan x = (4/3) y (D) tanx = (4/3)y Q.36 The periodof the function f(x)=sin cos x 2 +cos(sinx) equal (A) 2 (B) 2 (C) (D) 4 Q.37 Let the function f : D R, f (x)= ) 1 x 2 ( log log log 8 3 1 5 where Dis themaximumdomain of f(x). If S represents the sum ofthe absolute values ofall integers from D.Then the value ofS, is (A) 15 (B) 10 (C) 6 (D) 3 Q.38 A function f (x) = x 2 1 + x is defined from D1 D2 and is onto. If the set D1 is its complete domain then the set D2 is (A) 2 1 , (B) (– , 2) (C) (– , 1) (D) (– , 1] Q.39 A triangle hassides of length 13, 30 and 37. The radius of the inscribed circle is (A) 7 + 2 (B) 2 9 (C) 7 – 2 (D) 2 7 Q.40 cos cos cos tan tan 1 1 7 8 7 has the value equal to (A) 1 (B) –1 (C) cos 7 (D) 0 Q.41 If Pistheaffix ofzintheArganddiagram&Pmovesso that z i z 1 isalwayspurelyimaginary, then the locus of z is : (A) circle centre 1 2 1 2 , , radius 1 2 (B) circle centre 1 2 1 2 , , radius 1 2 (C) circle centre (2,2) and radius 1/2 (D) none of these Q.42 Let f (x) = } x sgn e { | x | e and g (x) = ] x sgn e [ | x | e , x R where { x } and [ ] denotes the fractional part and integral part functions respectively.Also h (x) = ln ) x ( f + ln ) x ( g then forall real x, h (x) is (A) anodd function (B)anevenfunction (C) neitheran odd nor aneven function (D) both odd as well as even function
- Q.43 Number ofways in which4married couples can beseated round a table such that men andwomen are alternate and not all women adjacent to her husband, is (A) 144 (B) 132 (C) 120 (D) 36 Q.44 Let f(x) = (x + 1)² 1, (x 1) . Then the set S = {x: f(x) = f1 (x)} is (A) 0 1 3 3 2 3 3 2 , , , i i (B) {0, 1, 1} (C) {0, 1} (D)empty Q.45 If | x – 2 | = p where x < 2 then (x – p) equals (A) – 2 (B) 2 (C) 2 – 2p (C) 2p – 2 Q.46 If x = cos–1 (cos 4) ; y= sin–1 (sin 3) then which of the following holds ? (A) x – y = 1 (B) x + y + 1 = 0 (C) x + 2y = 2 (D) tan (x + y) = – tan7 Q.47 Range of the function f (x) = tan–1 [ ] [ ] | | x x x x 2 1 2 is where[*] is thegreatest integerfunction. (A) 1 4 , L N M I K J (B) 1 4 2 R S T U V W , g (C) 1 4 2 , R S T U V W (D) 1 4 2 , L N M O Q P Q.48 The vertices ofa triangleA, B and C have the coordinates as (b, c), (–a, 0) and (a, 0) respectively.The coordinates of thepoint of concurrencyof its perpendicularbisectors are (A) c 2 a c b , 0 2 2 2 (B) a 2 b a c , 0 2 2 2 (C) b 2 c b a , 0 2 2 2 (D) none Q.49 The sum i + 2i2 + 3i3 + ........ + 2002i2002, where i = 1 is equal to (A) – 999 + 1002i (B) – 1002 + 999i (C) – 1001 + 1000i (D) – 1002 + 1001i Q.50 What percent of the domain ofthe function f (x) = 4 2 | 5 x 2 | 9 x 9 consists ofpositive numbers. (A) 40% (B) 50% (C) 30% (D) 65% Q.51 The number of solutions of the equation tan–1 3 x + tan–1 2 x = tan–1 x is (A) 3 (B) 2 (C) 1 (D) 0
- Q.52 Write the correct order sequence in respect of followingfourstatements, where 'T' stands for true and 'F' stands for False. I If a and b be positive real number different from 1, then logba + logab 2. II If 89 log19 n = (89)2 then n = 361. III Two different real numbers y and z are the roots of the quadratic equation ax2 + c = 0 with a, c 0 then y3 + z3 = 0. IV The quadratic equation x | x | + px + q = 0 where p, q R, q 0 has at least one real root. (A) F T T F (B) F T T T (C) T F T T (D) T T F F Q.53 Let f (x) = irrational is x if x rational is x if 0 and g (x) = rational is x if x irrational is x if 0 Thenthefunction(f– g)x is (A) odd (B)even (C) neither odd nor even (D) odd as well as even Q.54 Let f(x)=sin [ ] a x (where [ ]denotesthegreatestintegerfunction) . If fis periodicwith fundamental period , then a belongs to : (A) [2, 3) (B) {4, 5} (C) [4, 5] (D) [4, 5) Q.55 If x = 2 2 10 and y= 2 2 10 then the value of log2(x2 + xy + y2), is (A) 0 (B) 2 (C) 3 (D) 4 Q.56 For x R, the function f (x) satisfies 2 f (x) + f (1 – x) = x2 then the value of f (4) is equal to (A) 3 13 (B) 3 43 (C) 3 23 (D) none Q.57 Number of solution(s) of the equation cos–1(1 – x) – 2cos–1x = 2 is (A) 3 (B) 2 (C) 1 (D) 0 Q.58 Let , 2 be complex cube roots of unity.Then the determinant x x x 1 1 1 2 2 2 reduces to: (A) x3 (B) 2x3 (C) 3 x3 (D) none Q.59 The function f(x) = cot1 ( ) x x 3 + cos1 x x 2 3 1 is defined on the set S, where S = (A) {0, 3} (B) (0, 3) (C) {0, 3} (D) [ 3, 0]
- Q.60 The domainof thefunction f(x) = arc x x x cot 2 2 , where [x] denotesthe greatest integernotgreater than x,is: (A) R (B) R {0} (C) R n n I : { } 0 (D) R {n : n I} Q.61 The sum of the infinite series 3 2 log + 9 4 log + 27 8 log + 81 16 log + ...... , is (A) log 8 (B) 2 2 log (C) 4 2 log 3 (D) 2 2 log 5 Q.62 There exists a positive real number x satisfying cos(tan–1x) = x. Thevalue of cos–1 2 x2 is (A) 10 (B) 5 (C) 5 2 (D) 5 4 Q.63 The range of thefunction, f(x) = cot–1 log . 0 5 4 2 2 3 x x is: (A) (0, ) (B) 0 3 4 , (C) 3 4 , (D) 2 3 4 , Q.64 Number ofways in which 5 boys and 5 girls can be seated alternativelyon a round table if a particular boys and a particular girl are neveradjacent to each other in anyarrangement, is (A) 288 (B) 552 (C) 1584 (D) 1728 Q.65 If f (x + ay, x ay) = axy then f (x, y) is equal to : (A) x y 2 2 4 (B) x y 2 2 4 (C) 4 xy (D) none Q.66 Let[x]denotethegreatestintegerinx.Thenintheinterval[0,3]thenumberofsolutionsoftheequation, x2 3x + [x] = 0 is : (A) 6 (B) 4 (C) 2 (D) 0 Q.67 Let z z c 1 2 2 then z z z z 1 2 2 1 2 2 = (A) c2 (B) c2/2 (C) 2 c2 (D) none Q.68 Number of value of x satisfyingthe equation sin–1 x 5 + sin–1 x 12 = 2 is (A) 0 (B) 1 (C) 2 (D) more than 2
- Q.69 Given the graphs of the two functions, y = f(x) & y = g(x). In the adjacent figure from point A on the graph of the function y = f(x) correspondingtothegivenvalueoftheindependent variable(say x0),a straight line is drawn parallel to the X-axis to intersect the bisector of the first and the third quadrants at point B . From the point Ba straight lineparallel totheY-axis is drawnto intersect thegraphofthefunction y = g(x)at C.Again astraight lineis drawnfrom thepoint C parallel to the X-axis, to intersect the line NN at D . If the straight line NN is parallel toY-axis,then the co-ordinates of the point D are (A)f(x0),g(f(x0)) (B) x0, g(x0) (C) x0, g(f(x0)) (D) f(x0), f(g(x0)) Q.70 If 2 f(x2) + 3 f(1/x2) = x2 1 (x 0) then f(x2) is : (A) 1 5 4 2 x x (B) 1 5 2 x x (C) 5 1 2 4 x x (D) 2 3 5 4 2 2 x x x Q.71 Range of the function f (x) = } x { 1 } x { where {x} denotes the fractional part function is (A) [0 , 1) (B) 2 1 , 0 (C) 2 1 , 0 (D) 2 1 , 0 Q.72 The domainof the function, f(x) = 3 x 4 2 x 4 3 x 2 2 x x 5 . 0 log 5 . 0 x is : (A) 1 2 , (B) [1, 3] (C) 1 2 1 3 2 , , (D) 1 2 1 2 1 2 1 3 2 , , , Q.73 Let k bethe valueso thatthe functiondefined byf(x)= k x 5 x will beits own inverse, thenthevalueof tan k, to the nearest tenths, is (A) – 1.6 (B) – 0.9 (C) 1.2 (D) 2.4 Q.74 The function f (x) is defined byf(x) = cos4x + K cos22x + sin4x, where K is a constant.If the function f (x) is a constant function, thevalueof k is (A) – 1 (B) – 1/2 (C) 0 (D) 1/2 (E) 1 Q.75 If w is an imaginarycube root of unity, then the value of sin w w 10 23 4 is : (A) 3 2 (B) 1 2 (C) 1 2 (D) 3 2
- Q.76 Theset ofvalues of x, satisfyingtheequation tan2(sin–1x) >1 is (A) [–1, 1] (B) 2 2 2 2 , (C) (–1, 1) – 2 2 2 2 , (D) [–1, 1] – 2 2 2 2 , Q.77 Definethefunctionf(n)wherenisanonnegativeintegersatisfyingf(0)=1andf(n)isdefinedforn > 0 as f (n) = n · 1 n 0 i ) ( f i . Let 2m be the highest power of 2 that divides f (20). The value of m is (A) 18 (B) 19 (C) 20 (D) 21 (E) 22 Q.78 Let f (x) = x 1 x and let g(x)= x 1 x r . Let S be the set of all real numbers r such that f (g(x))= g(f (x))for infinitelymanyreal number x. The numberof elements in setS is (A) 1 (B) 2 (C) 3 (D) 5 Q.79 If for all x different from both 1 and 0 we have f1(x) = 1 x x , f2(x) = x 1 1 , and for all integers n 1, we have fn + 2(x) = even is n if ) x ( f f odd is n if ) x ( f f 2 1 n 1 1 n then f4(x) equals (A)x (B) x – 1 (C) f1(x) (D) f2(x) Q.80 If | x | – x + y = 10 and x + | y | + y = 12 then x + y is equal to (A) 5 26 (B) 0 (C) – 5 26 (D) 5 42 Q.81 .. .......... 5 3 5 3 Lim n equals (A) (15)2/3 (B) (15)1/3 (C) (45)1/2 (D) (45)1/3 Q.82 For n = 1, 2, 3, ......, let Tn = 1 + 2 + ....... + n. Which of the following statements is correct (A) There is no value of n for whichTn is a positive power of 2. (A positive power of two is an integer of the form 2k where k is apositive integer) (B) There is exactlyone value of n for which Tn is a positive powerof 2. (C) There are exactlytwovalues of n for which Tn is a positive power of 2. (D) There are morethan two but finitelymanyvalues ofn for which Tn is a positive powerof 2.
- Q.83 If x2 + bx + a = 0 has roots 'm' and 'n' (m > n) then the roots of the quadratic equation (x – 2)2 + b(x – 2) + a = 0 are (A) m – 2 and n – 2 (B) m + 2 and n + 2 (C) 2 – m and 2 – n (D) m and n Q.84 If (a + ib)5 = + i where a, b, , R then (b + ia)5 equals (A) a + i) (B) + i (C) – i (D) – i Select the correct alternative : (More than one are correct) Q.85 If the function f (x) = ax + b has its own inverse then the ordered pair (a, b) can be (A) (1, 0) (B) (–1, 0) (C) (–1, 1) (D) (1, 1) Q.86 Whichofthefollowinghomogeneous functions areofdegreezero ? (A) x y ln y x + y x ln x y (B) x x y y x y ( ) ( ) (C) xy x y 2 2 (D) x sin y x y cos y x Q.87 The equation of the line making equal intercepts on the axes & touching z 2 1 2 where z = x + iy is : (A) x + y = 1 (B) x y = 1 (C) x y = 1 (D) x y = 1 Q.88 The value of sin 1 2 3 4 1 cot + cos 1 2 3 4 1 cot is equal to (A) 1 (B) 3 2 10 (C) 2 sin 1 2 3 4 1 1 1 cot cot ( ) (D) 2 sin tan ( ) tan 1 1 1 1 2 4 3 Q.89 Let f(x) = 1 x & g(x) = 1 x then : (A) f(g(x))&g(f(x))havedifferent domains (B) f(g(x)) & g(f(x)) havethe same range (C) f(g(x))is a bijectivemapping (D) g(f(x)) is neither odd nor even . Q.90 If f(x)is a polynomial functionsatisfyingthecondition f(x) .f(1/x)= f(x) +f(1/x)and f(2) =9 then (A) 2 f(4) = 3 f(6) (B) 14 f(1) = f(3) (C) 9 f(3) = 2 f(5) (D) f(10) = f(11) Q.91 Suppose that thefunctions f (x)andg (x) satisfythesystem of equation f (x) + 3g(x) = x2 + x + 6 2 f (x) + 4 g (x) = 2x2 + 4 for every x. The value of 'x' for which f (x) = g (x) can be equal to (A) – 2 (B) 2 (C) – 5 (D) 5
- Q.92 tan1x , tan1y, tan1z are in A.P. and x , y, z are also in A.P. (y 0 , 1 , 1) then (A) x , y, z are in G.P. (B) x , y, z are in H.P. (C) x = y = z (D) (x y)2 + (y z)2 + (z x)2 = 0 Q.93 D [1, 1] is the domain ofthefollowingfunctions, statewhich of them has the inverse. (A) f(x) = x2 (B) g(x) = x3 (C) h(x) = sin 2x (D) k(x)= sin (x/2) Q.94 Which ofthefollowingexpressionsarenot thetrigonometricforms ofanycomplexnumber? (A) 3 cos sin 4 4 i (B) 2 cos sin 4 4 i (C) 2 sin cos 2 3 2 3 i (D) cos 3 + i sin 2 3 Q.95 Identifythepair(s)offunctions whichare identical . (A) y = tan (cos 1 x); y = 1 2 x x (B) y = tan (cot 1 x) ; y = 1 x (C) y = sin (arc tan x); y = x x 1 2 (D) y = cos (arc tan x) ; y = sin (arc cot x) Q.96 For the equation 2x = tan(2tan–1a) + 2tan(tan–1a + tan–1a3), which of the followingis invalid? (A) a2x + 2a = x (B) a2 + 2ax + 1 = 0 (C) a 0 (D) a –1, 1 Q.97 Suppose f (x) = sin x and g (x)= 1 – x . Which of the following composite functions have the same range? (A)fog (B)gof (C)fof (D) gog Q.98 Thegraphs ofwhichofthefollowingpairs differ. (A) y = sin tan x x 1 2 + cos cot x x 1 2 ; y = sin 2x (B) y = tanx cot x ; y = sin x cosecx (C) y = cos x + sin x ; y = sec cos sec cos x ecx x ecx (D) none of these Q.99 Thefunctions whichareaperiodicare: (A) y = [x + 1] (B) y= sin x2 (C) y = sin2 x (D) y = sin1 x where [x] denotes greatest integerfunction
- Q.100 The value of cos 1 2 14 5 1 cos cos is : (A) cos 7 5 (B) sin 10 (C) cos 2 5 (D) cos 3 5 Q.101 Let x,ybereals satisfying sinx+cos y=1 and sin y+ cos x =–1. Then which ofthefollowingmust be correct? (A) sin(x + y) = 0 (B) cos(x – y) = 0 (C) cos(x + y) = 1 (D) cos 2x = cos 2y Q.102 Whichofthefollowingfunction(s)is/areperiodicwith period. (A) f(x) = sinx (B) f(x) = [x + ] (C) f(x) = cos(sinx) (D) f(x) = cos2x (where [.] denotes the greatest integer function) PART-B SUBJECTIVE Q.103 Solve thesystemofequations log10(x3) + log10(y2) = 11 log10(x2) – log10(y3) = 3. Q.104 If f (x) = x2 – x + 2; g (x) = ax + b and ) x ( g f = 9x2 – 3x + 2, determine all possible ordered pairs (a,b)whichsatisfythisrelationship. Q.105 A flight of stairs has 10 steps. A person can go up the steps one at a time, two at a time, or any combination of1's and 2's. Find the total number ofways in which the person can go up the stairs. Q.106 TriangleABC has incentre at point I.Also, one median is lineCD, and theinscribed circle is tangent to sideABatpoint D.If thelengthofCD=3andlength ofAB=8.Findthedistanceofthevertex andfrom theincentre. Q.107 Find all triplets of natural numbers (a, b, c) such that a, b and c are in geometric progression and a + b + c = 111. Q.108 Letxbethesumofthefollowinginfiniteseries: x = 3 1 1 + 3 3 1 + 3 5 1 + 3 7 1 + Evaluatethesum 3 1 1 + 3 2 1 + 3 3 1 + 3 4 1 + intermsofx. Q.109 Findthegeneralsolutionoftheequation 2 x sin ) 2 ( log = logsin x4 sin3x
- Q.110 Find thesolution set ofinequality, 2 x 3 3. Q.111 Given that tan x 4 = 2 1 then find the value of the expression y= x 2 cos 1 x cos x 2 sin 2 . Q.112 Let p, q, r be the roots of the equation x3 – 14x2 + 26x – 4 = 0. Find the value of the expression r 1 1 q 1 1 p 1 1 . Q.113 If sin x = b a b a where 0 < a < b then find the value of x cos x cot 2 2 . Q.114 The lengths of the sides of a triangle are log1012, log1075 and log10n,where n N. Find the number of possible values of n. Q.115 Let P bea point on the line 3x + 4y= 12 which is closest to the origin. Find the sum ofthe abscissa and ordinate ofthe point P. Q.116 Find the equationof a line passingthrough(– 4, –2)having equal intercept on the coordinate axes. Q.117 Let f (x) = x x x x a a a a , where a > 0 and a 1. If f (p) = 2, then find f (2p). Q.118 Find the sum of the roots of the equation 1 2 2 2 2 x 222 1 x 111 2 x 333 . Q.119 Given tan · tan = a b , show that the expression F = 2 2 cos b sin a 1 + 2 2 cos b sin a 1 is independent of and and has the value equal to ab b a . Q.120 Let and be two different values of x satisfying the equation a cos x + b sin x = c with a2 + b2 0, prove that 4 cos2 2 cos2 2 = 2 2 2 b a ) c a ( .
- ANSWER KEY Q.1 B Q.2 D Q.3 C Q.4 A Q.5 B Q.6 D Q.7 D Q.8 A Q.9 D Q.10 C Q.11 A Q.12 D Q.13 A Q.14 A Q.15 A Q.16 A Q.17 C Q.18 A Q.19 B Q.20 B Q.21 C Q.22 B Q.23 C Q.24 A Q.25 A Q.26 C Q.27 A Q.28 C Q.29 D Q.30 A Q.31 D Q.32 C Q.33 C Q.34 B Q.35 C Q.36 D Q.37 C Q.38 D Q.39 B Q.40 B Q.41 A Q.42 A Q.43 B Q.44 C Q.45 C Q.46 D Q.47 C Q.48 A Q.49 D Q.50 A Q.51 A Q.52 B Q.53 A Q.54 D Q.55 C Q.56 C Q.57 C Q.58 A Q.59 C Q.60 C Q.61 C Q.62 C Q.63 C Q.64 D Q.65 A Q.66 C Q.67 A Q.68 B Q.69 C Q.70 D Q.71 C Q.72 D Q.73 A Q.74 B Q.75 C Q.76 C Q.77 C Q.78 B Q.79 C Q.80 A Q.81 D Q.82 A Q.83 B Q.84 B Q.85 A,B,C Q.86 A,B,C Q.87 A,D Q.88 B,C,D Q.89 B,C,D Q.90 B,C Q.91 A,D Q.92 A,B,C,D Q.93 B,D Q.94 A,B,C,D Q.95 A,B,C,D Q.96 B,C Q.97 B,D Q.98 A,B,C Q.99 A,B,D Q.100 B,C,D Q.101 A,D Q.102 A,C,D Q.103 x = 1000, y = 10 Q.104 (3, 0) and (– 3, 1) Q.105 89 Q.106 5/3 Q.107 (37, 37, 37); (1, 10, 100); (100, 10, 1) Q.108 8x/7 Q.109 x = n + (–1)n 6 Q.110 x (– , – 3] , 5 3 Q.111 – 5/6 Q.112 – 4 9 Q.113 2 2 a b ab 4 Q.114 893 Q.115 25 84 Q.116 2y = x and x + y + 6 = 0 Q.117 5/4 Q.118 111 2

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