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 = 7 7 0 8 8. The smallest ostve agle whch satfes the euato s θ + cos θ + = 0 5 9. 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) 0.8 0.9 0. 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
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 7 + + = 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 + 5 + + + 7 + = 0 ta m, the m = 5 7 5 7 0. 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 + + 5 = 0 + 5 = 0 + + + 5 = 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
+ + a + + b + = 0 + a b = 0 + + 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 + + 9 = 0 + + 9 = 0 ( + ) + = 0. The umber of commo tagets to the crcles + + 9 = 0 ad + 8 + = 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 0 0 0 5. 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
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 - - - - - - 55. The euato of the taget to the curve = 9- at the ot where the ordate ad the abscsa are eual, + = 0 + + = 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 > 0 58. 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
8 5. z d eual to 0 7 7 7 7 8 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 985 + 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 0 8 50 5 78. 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 0 8 8. 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
+ + l = 0 + l + = + + = l 8. The euato 5 + 5 =.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 85. 5 7 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, 8 7 87. If there are ersos a art, ad f each of them shakes hads wth each other, the umber of hadshakes hae the art 8 7 88. 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, 809 809 9 879 90. The greatest term (umercall) the easo of ( 5) whe = 5 55 9 9 55 9. The value of the eresso ( + 0 ), f the thrd term the easo 0,00,000, 0 0 0 5/ 0 5/ 9. If 7 0 dvded b 5, the the remader 0 8 5 9. The coeffcet of the easo of a b e 9. 95. ( ) ( ) ( 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 0 98. 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 005 0 005 /, A = 0 ad Q = PAP T, the / 005 0 / 0 005
Aswer Kes.... 5.. 7. 8. 9. 0..... (c,d) 5.. 7. 8. 9. 0..... 5.. 7. 8. 9. 0..... 5.. 7. 8. 9. 0..... 5.. 7. 8. 9. 50. 5. 5. 5. 5. 55. 5. 57. 58. 59. 0..... 5.. 7. 8. 9. 70. 7. 7. 7. 7. 75. 7. 77. 78. 79. 80. 8. 8. 8. 8. 85. 8. 87. 88. 89. 90. (b, c) 9. 9. 9. 9. 95. 9. 97. (a, b) 98. 99. 00.