GULF MATHEMATICS OLYMPIAD 04 CLASS : XII Date of Eamiatio: Maimum Marks : 50 Time : 0:30 a.m. to :30 p.m. Duratio: Hours Istructios to cadidates. This questio paper cosists of 50 questios. All questios are compulsor 3. Each questio carries mark 4. Each questio offers four suggested aswers 5. Choose the correct aswer ad idicate our choice o the Aswer Sheet b darkeig the circle represetig our choice usig blue or black ball poit pe.. There is o egative markig for wrog aswers 7. Write our roll umber i the aswer sheet 8. Had over the Aswer Sheet to the ivigilator at the ed of the eamiatio 9. Use of calculator is ot permitted 0. Log tables ca be used for calculatio GULF MATHEMATICS OLYMPIAD - 04 Page
GULF MATHEMATICS OLYMPIAD 04. Let R be a relatio defied o the set of real umbers b arbiff ab, the R is Refleive ad smmetric Smmetric Trasitive ol Ati smmetric ol. 3 If f: RR give b f() = 3 3, the fof() is 3 3 3 3 3. If ta - ad ta - 3 are the two agles of a triagle, the the third agle is 75 0 45 35 4. If ta ta ta... ta..3 3.4 of is + = ta -, the the value 5. If A ad B are two matrices such that AB = B ad BA = A, the A + B is AB BA A + B AB. If A = 0 3 4 ad ka = 0 3 4, the the values of k, ad are respectivel, 7. If A.-,-8 -,4, 9 -, -4, -9 -., 8 cos si si cos A A A, the A. A is A A A 8. If A = 3 5, the A - is give b 5 5 3 3 5 3 5 3 GULF MATHEMATICS OLYMPIAD - 04 Page
9. If i ABC, 9 4 c a 0, the the value of Si A+Si B+Si C is a b b c 9 4 3 3 3 3 0. If 0, the the value of a b c a b c abc 0 abc is. If k is a scalar ad A is a square matri of order, the ka is ka k A k A ka. Let A ad B are two square matrices such that A+B = AB, the which of the followig is true? AB = BA AB = -BA AB + BA = 0 AB + BA = 0 3. si, The values of p ad q such that f() = psi q, is cotiuous everwhere. cos, P=0, q= p=, q= p=, q = p =, q = 0 k, k k 4. Let g ( ), the 3 k, k g is discotiuous at = k g is cotiuous at = k but is ot differetiable at = k g is differetiable but ot cotiuous at = k g is differetiable everwhere 5. If a( ), the d d is. The secod derivative of = a Cos 3 t with respect to = asi 3 t at t = is 4 4 3a a 0 GULF MATHEMATICS OLYMPIAD - 04 Page 3
, the d d d d is equal to 0 7. If = 8. If = Sec Si 0, the d d is 9. The distace S of a particle i time t is give b S = t 3 t 4t -8. Its acceleratio vaishes at t =. 3 4 0. The radius of a spherical balloo is chagig at the rate of whe its volume chages as same as its surface area is 4m ½ m m 4 m 4 m /s. The radius of the balloo. The iterval i which ta si cos is icreasig 0,, 4 0, 4 5, 4 4. The fuctio f() = 3 3 4 is alwas Icreasig strictl icreasig decreasig strictl decreasig 3. The equatio of the ormal to the curve = + at the poit (,0) is = + = 0 + = 4 + = 4 4. If the taget to the curve + k + 3 = 0 at (, ) is parallel to the lie + = 4, the the value of k is 5. The radius of a circle is chagig from 5cm to 5.cm. The the approimate chage i area is 4. The maimum value of the the fuctio f() = 3 3 i the iterval [0,] is 0 GULF MATHEMATICS OLYMPIAD - 04 Page 4
7 The total reveue i rupees received from the sale of uits of a product is give b R() = 3 + 3 + 5. What is the margial reveue whe = 5? 9 90 8. It is give that at =, the fuctio f() = 4 4 + + 9 attais maimum value i the iterval 0. The the value of 8 3 4 9. 3 e d is equal to e C e C e C e C 30. The itegral of a Si b Cos with respect to is ata ta C ab b a a ta ta C b b b b ta ta C a a b a ta ta C a b 3 The itegral of Si. Cos with respect to is Ta + cot + C Ta - cot + C Tacot + C cot - ta + C d is 0 3. The value of ( )( ) 33. 0 si cos 4 si d 4 8 GULF MATHEMATICS OLYMPIAD - 04 Page 5
34. If f (si ) d A f (si ) d, the the value of A is 0 0 0 35. The area eclosed betwee the curve = 4 ad the lie = is 8 3 4 3 3 3. The parabolas = 4, = 4 divide the square regio bouded b the lies = 4, = 4 ad the co ordiate aes. If S, S ad S 3 are respectivel the areas of the parts, the S : S : S 3 is ::3 :: :: :: 3 d d d 37. The degree of the differetial equatio si 0 is d d d 3 ot defied 38. d A homogeeous differetial equatio of the form h(, ) d substitutio, ca be solved b makig the = v v = = v = v 39. Y = Acos + Bsi is the geeral solutio of the differetial equatio d 0 d d 0 d d 0 d d d 0 40. If is the agle betwee the vectors a ad b such that ab. = a b, the is equal to 0 4 4. A uit vector perpedicular to the plae determied b the poits (,-,), (,0,-) ad (0,,) ˆ i ˆj kˆ iˆ ˆj kˆ iˆ ˆj kˆ ˆ i ˆj kˆ GULF MATHEMATICS OLYMPIAD - 04 Page
4. If is the agle betwee the vectors e ad e, the e e e e Si Cos Ta Ta 43. The lie passig through poits (5,,a) ad (3,b,) crosses the YZ plae at the poit 7 3 0,,, the a = 8, b = a =, b = 8 a = 4, b = a =, b = 4 44. A plae makes itercepts 4 ad 3 o the ad z aes. If it is parallel to the - ais, the its equatio is, 3 +4z = 3z + 4 = 3 +4z = 3z + 4 = is 45. The lies 3 z 4 ad k 4 z 5 are coplaar if k 4. K = 3 or -3 k = 0 or - k = or - k = 0 or -3 The distace betwee the plaes is r. i ˆ ˆj 3kˆ 4 ad r iˆ ˆj kˆ. 3 9 3 0 is 5 0 0 5 3 4 3 4 3 4 3 4 47. The graph of the iequatio + > 5 is Half plae that cotais origi Ope half plae ot cotaiig the origi Whole plae ecept the poits lig o the lie + = 5 Half plae cotaiig the positive ais 48.. A card is draw from a well shuffled pack of 5 cards. The probabilit that the card is black or club is 3 4 49. If the stadard deviatio of the biomial distributio is give b the the mea is P(X ) C p q, is, 4 8 50. A ad B are two idepedet evets such that P A B ad P A B 5, the P( is or 4 5 or or 3 or 3 3 4 4 4 GULF MATHEMATICS OLYMPIAD - 04 Page 7