Test Paper-II. 1. If sin θ + cos θ = m and sec θ + cosec θ = n, then (a) 2n = m (n 2 1) (b) 2m = n (m 2 1) (c) 2n = m (m 2 1) (d) none of these

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1 Test Paer-II. If s θ + cos θ = m ad sec θ + cosec θ =, the = m ( ) m = (m ) = m (m ). If a ABC, cos A = s B, the t s C a osceles tragle a eulateral tragle a rght agled tragle. If cos B = cos ( A+ C), the ta A, ta B, ta C are cos ( A C) A.P. G.P. H.P.. If =, the ta α ta α ta α... ta ( ) α α eual to 5. If s ( cos θ ) = cos ( s θ ), the the value of I cos θ + K J. If cos α + cos β = 0 = s α + s β, the cos α + cos β = s (α + β) cos (α + β ) s (α + β) cos (α + β) 7. cos 7 cos cos = The smallest ostve agle whch satfes the euato s θ + cos θ + = Let α, β be a two ostve values of for whch cos, cos ad cos are G.P. The mmum value of α β / / / 0. The geeral soluto of s θ secθ + taθ = 0 + θ = + ( ), θ =, Z θ =, Z + θ = + ( ), Z θ =, Z. The euato a s + cos = a 7 ossesses a soluto f a > a. cosec (cos ) real f [, ] R a odd multle of a tegral multe of. α, β are γ are three agles gve b α = ta ( ) ad γ = cos. The α > β α > γ. The value of cos( cos 0.8) , β = s + s β > γ > α 5. I a ABC, ac s (A B + C) = a + b c c + a b b c a c a b. abc A B C I a ABC, cos cos cos a+ b+ c = 7. Let the agles A, B, C of ABC be A.P. ad let b : c = :. The agle A 75º 5º 0º 8. If the agles A ad B of a ABC satf the euato s A + s B = (cos B cos A), the the

2 dffer b 9. A ma of heght ft. observes the to of a tower ad the foot of the tower at agles of 5º ad 0º of elevato ad deresso resectvel. The heght of the tower ( + )m ( + )m ( + )m 0. Two vertcal oles 0 m ad 80 m hgh stad aart o a horzotal lae. The heght of the ot of tersecto of the les jog the to of each ole to the foot of the other 5 m m 8 m 50 m. A vertcal tree stads at a ot A o a bak of caal. The agle of elevato of ts to from a ot B o the other bak at the caal ad drectl ooste to A 0º. The agle of elevato of the to from aother ot C 0º. If A, B ad C are o the same horzotal lae, ABC = 0º ad BC = 0 m, the heght of the tree 5 ( + ) 5 ( + ) 5 ( ) 5 ( ). The agle of elevato of a statoar cloud from a ot 500 m above a lake 5º ad the agle of deresso of ts reflecto the lake 5º. The heght of cloud above the lake level 500 metres 500 metres 500 metres. If α, β, γ are the real roots of the euato + = 0, the the cetrod of the tragle havg vertces α, I αk J I, β, ad γ β I, γ are (, ) (, ) (, ) (, ). A rectagle has two ooste vertces at the ots (, ) ad (5, 5). If the other vertces le o the le =, the the coordates of the other vertces are (, ), (, ) (, ), (, 5) (, ), (, ) (, ), (, ) 5. Wthout chagg the drecto of coordates aes, org trasferred to (α, β ) so that the lear terms the euato = 0 are elmated. The ot (α, β ) (, ) (, ) (, ) (, ). A suare costructed o the orto of the le + = 5 whch terceted betwee the aes, o the sde of the le awa from org. The euatos to the dagoals of the suare are = 5, = 5 = 5, = 5 = 5, = 5 = 5, = 5 7. The cetrod of the tragle formed b the ar of les = 0 ad the le = 0 (, ) (, ) (, 0) 8. The three les whose combed euato ( + )( + ) = 0 form a tragle whch eulateral rght agled obtuse agled 9. If the agle betwee the two les rereseted b = 0 ta m, the m = The legth of tercet made b the le l + m + =0 betwee the ar of les a + h + b = 0 ( l + m ) h ab am hlm + bl ( l + m )( h ab) ( am hlm + bl ) ( h ab)( l + m ) am hlm + bl. The euato of a crcle assg through the org ad makg tercets, 5 o the coordate aes = = = 0. The abscsae of two ots A ad B are the roots of the euato + a b = 0 ad ther ordates are the roots of the euato + = 0. The euato of the crcle wth AB as dameter

3 + + a + + b + = 0 + a b = a + b = 0. The euato of the crcle whch touches both the aes ad the straght le + = the frst uadrat ad les below t + + = = = 0 ( + ) + = 0. The umber of commo tagets to the crcles = 0 ad = 0 5. If QQ' a double ordate of a arabola = 9, the the locus of ts ot of trecto = = =. The curve descrbed arametrcall b = t + t +, = t t + reresets a ar of straght les a ellse a arabola a herbola 7. The orto of a taget to a arabola = a cut off betwee the drectr ad the curve subteds a agle θ at the focus, where θ = 8. If + = m ( + ) ad + = m ( + ) are two tagets to the arabola = 8, the m + m = 0 m m = m m = 9. The doma of the fucto R f () = s S I T K JU V W [, ) [, ] (, ] [, ] [, ] [, ] (, ) (, ) 0. The doma of the fucto L M f () = cos NM QP (, ) (, ) (, ) (, ) IO P. The doma of the fucto f () = (, ) (, ) [0, ] [, ]. The doma of the fucto f () =... tmes (, ) [, ) (, ) -. e + e + cos- lm Æ0 eual to 0. Let f () be a twce dfferetable fucto ad f " f (0) = 5, the lm ( ) - f ( ) + f ( 9 ) Æ0 eual to If α ad β be the roots of a + b + c = 0, the ( α) lm ( + a + b + c) α a (α β ) e a (α β ) a (β α) e oe of these θ θ θ. lm taθ + ta + ta ta θ θ cotθ cot θ R = +, < 0 7. Let f () = S. If f () +, 0 T cotuous the terval [, ], the euals 8. If f () = ad g () = f [ f ()], the g '() for > 0 9. If f () =, the at = 0, f () dfferetable as well as cotuous f () dfferetable but ot cotuous

4 f () cotuous but ot dfferetable f () ether cotuous ot dfferetable 50. The set of ots of dcotut of the fucto f () = s s {0} { : I} φ 5. If = ( + / ) ( + / ) ( / ), the d d = 5. If = ( ), d d = ( ) ( + ) + ( ) ( + ) + ( ) ( + ) 5. If = cos + ( + ) + + +, the d d = - ( + ) 5. If = a ( ), the d d eual to The euato of the taget to the curve = 9- at the ot where the ordate ad the abscsa are eual, + = = 0 = 0 5. If f () = a ( ) 89, the greatest value of f () a o mamum value 57. f () = + a + b + 5 s a creasg fucto the set of real umbers f a ad b satf the codto a b 5 > 0 a b + 5 > 0 a b + 5 < 0 a > 0 b > The euato of the ormal to the curve = / at the ot of tersecto wth the -a = 0 + = 0 + = 0 b + g 59. z + d eual to b g L + I NM K J O QP + C C [{ ( + )} ( ) ] L NM + I K J + + I K J O QP + C C 0. z e d eual to e + e e + e j sec (e ) + C e e + e j + sec (e ) + C e e e j sec (e ) + C. Let f () = d + z d / ad f (0) = 0, the f () =. z s dbcosg eual to s + C s s + I + C H K z0 cos I K J + C. The value of the teger e cos ( + ) d, teger, 0. If f ( ) d = ad zb f ( ) g d = 7, the the z b g z value of f d 0 9

5 8 5. z d eual to cos. z d eual to 0 7. The dfferetal euato of faml of arabolas wth foc at the org ad a alog the -a d d di d K J + I d d K J + d I d d K J + + d d = 0 d = 0 = 0 d d 8. Soluto of the euato d + d + + = 0 = ta = ta = ta c+ + c+ + c I I I 9. A soluto of the dfferetal euato di d d K J + = 0 d = = = = 70. The order of the dfferetal euato whose geeral soluto gve b = (c + c ) cos ( + c ) c e + c 5 where c, c, c, c, c 5 are arbtrar costats, 5 7. The smallest teger for whch + I K J =, 8 7. The locus rereseted b z = z + a crcle of radus a ellse wth foc at ad a le through the org a crcle o the jo of ad as dameter 0 k ki 7. The value of s cos k K J = 7. The commo roots of the euatos z + z + z + = 0 ad z z 00 + = 0 are, ω, ω ω, ω 75. The smallest teger for whch + =, 8 7. The locus rereseted b z = z + a crcle of radus a ellse wth foc at ad a le through the org a crcle o the jo of ad as dameter 77. The umber of odd umbers betwee 0 ad The sum to terms of the seuece a, ar, ar,... a r a r a r 79. If the frst, secod ad last terms of a A.P. are a, b ad a resectvel, the ts sum ab ab ( b a) b a ab ( b a) 80. Betwee two umbers whose sum, a eve umber of arthmetc meas are serted. If the sum of these meas eceeds ther umber b ut, the the umber of meas are The set of values of for whch the roots of the euato + + ( ) = 0 are of ooste sg (, 0) (0, ) (, ) (0, ) 8. If the rato of the roots of l + + = 0 :, the + l + = 0 I K J

6 + + l = 0 + l + = + + = l 8. The euato =.7 has o soluto oe soluto two solutos more tha two solutos 8. If s θ ad cos θ are the roots of the euato a + b + c = 0, the (a c) = b c (a c) = b + c (a + c) = b c (a + c) = b + c C + 5 j = j C = 5 C 5 C 5 C 8. A ma has got seve freds. The umber of was whch he ca vte oe or more of h freds to der, If there are ersos a art, ad f each of them shakes hads wth each other, the umber of hadshakes hae the art I a eamato there are three multle choce uestos ad each uesto has choces. Number of was whch a studet ca fal to get all aswers correct 7 9 I K J, whe eaded 89. The 8th term of + ascedg ower of, The greatest term (umercall) the easo of ( 5) whe = The value of the eresso ( + 0 ), f the thrd term the easo 0,00,000, / 0 5/ 9. If 7 0 dvded b 5, the the remader The coeffcet of the easo of a b e ( ) ( ) ( a b ) ( a + b) ( ) ( b + a ) C(, 0) + C(, ) C(, ) eual to P(, ) = e e + e wll be eual to! 5! 7! e e e e 9. The sum of these seres = 0 + e e e e 97. If AB = A ad BA = B, the B eual to B A Let A be a vertble matr, whch of the followg ot true? (A' ) = (A )' A = A (A ) = (A ) 99. The matr A = Nlotet Idemotet Orthogoal Ivolutar a tegral multe of 00. If P = P(Q 005 )P T eual to /, A = 0 ad Q = PAP T, the / / 0 005

7 Aswer Kes (c,d) (b, c) (a, b)

INTERNATIONAL BACCALAUREATE ORGANIZATION GROUP 5 MATHEMATICS FORMULAE AND STATISTICAL TABLES

INTERNATIONAL BACCALAUREATE ORGANIZATION GROUP 5 MATHEMATICS ORMULAE AND STATISTICAL TABLES To be used the teachg ad eamato of: Mathematcs HL Mathematcal Methods SL Mathematcal Studes SL urther Mathematcs