Time : 3 hours 03 - Mathematics - March 2007 Marks : 100 Pg - 1 S E CT I O N - A

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Estella Blankenship
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1 Time : hours 0 - Mthemtics - Mrch 007 Mrks : 100 Pg - 1 Instructions : 1. Answer ll questions.. Write your nswers ccording to the instructions given below with the questions.. Begin ech section on new pge. S E CT I O N - A Given below re 1 to 15 multiple choice questions. Ech crries one mrk. Write the seril number ( or b or c or d ) in your nswer book of the lterntive which you feel is the correct nswer of the question For Δ ABC, A is ( 1, ), B is (, ) nd C is on X - is. If the centroid of Δ ABC is on Y - is, then find the coordintes of C. ( ) ( -, 0 ) ( b ) (, 0 ) ( c ) ( 0, - ) ( d ) none of these. Find the eqution of the perpendiculr bisector of AB, where A is (, ) nd B is (, ). ( ) y - = 0 ( b ) y + = 0 ( c ) - = 0 ( d ) - = 0. Find the rdius of circle touching X - is nd hving its centre t (, - ) ( ) ( b ) ( c ) 5 ( d ) none of these.. There is point on the prbol y =, whose - coordinte is two times the y - coordinte. Find the point. ( ) (, ) ( b ) ( 6, ) ( c ) (, 8 ) ( d ) ( 8, ) 5. Find the mesure of the ngle between the symptotes of - y = 16. ( ) ( b ) ( c ) ( d ) none of these 6. If = 5, b = nd - b =, then find. b. ( ) - 9 ( b ) 0 ( c ) 9 ( d ) none of these 7. Force i + j + k is pplied t B ( 1,, ). Find the torque round A ( - 1,, 0 ) nd its mgnitude. ( ) 1 ( b ) ( -, 1, ) ( c ) (, - 1, - ) ( d ) none of these 8. Find the mesure of the ngle between the plnes + by + d = 0 nd z = 0, ( + b 0 ). ( ) ( b ) d cos ( c ) ( d ) + b

2 Time : hours 0 - Mthemtics - Mrch 007 Mrks : 100 Pg - 9. Find sin lim. 0 ( ) - 1 ( b ) 1 ( c ) 0 ( d ) limit does not eist d sin 1 cos Find e, < 1 d sin ( ) + cos - 1 sin cos + e e ( b ) 1 - ( c ) 0 ( d ) none of these 11. Find e sec ( 1 + tn ) d. ( ) e tn + c ( b ) e sec + c ( c ) e tn + c ( d ) none of these 1. If f is n even function nd f ( ) d =, then find - ( ) 0 ( b ) ( c ) ( d ) 1 0 f ( ) d. 1. Find the re of the region bounded by the curve y = cos, X - is nd the lines = 0, =. ( ) ( b ) 1 ( c ) ( d ) none of these 1. Obtin the order of differentil eqution d y d dy =. d ( ) 1 ( b ) does not eist ( c ) ( d ) 15. A bll is projected verticlly upwrds with speed 19.6 m/s. Find the time for mimum height. ( ) seconds ( b ) seconds ( c ) 19.6 seconds ( d ) none of these S E C T I O N B Answer the following 16 to 0 questions. Ech question crries one mrk For A ( -, ) nd B (, 0 ), find the rtio in which the Y - is divides AB from A - side.

3 Time : hours 0 - Mthemtics - Mrch 007 Mrks : 100 Pg Obtin the eqution of circle given tht its re is 9 nd the equtions of lines contining two of the dimeters of the circle re - y + 1 = 0 nd + y - 5 = Find the eqution of prbol which psses through (, ) nd is symmetric bout X - is. The verte of the prbol is t the origin. 19. Find the eccentricity of the ellipse, the length of whose ltus - rectum is hlf the length of mjor is. Obtin the eqution of the uiliry circle nd director circle of the ellipse 16 + y 9 = In R, find the unit vector orthogonl to (, ). 1. Find the volume of the tetrhedron V - ABC if V is (,, - ), A (,, ) B (,, 1 ) nd C ( 1,, - 1 ).. Find the eqution of the line through (,, ) nd prllel to the line y z - 10 = = Find the centre nd rdius of the sphere + y + z - - y + 8z - 1 = 0.. If f ( ) = log 7, then find f ( 7 ). 5. Rdius of circulr metl plte when heted increses by %. If its rdius is 10 cm, find the increse in its re Apply Rolle s theorem to f ( ) = sin + cos + 1,, nd find c. 6. Using the formul [ f ( ) + f ' ( ) ] e d = e f ( ) + c, 1 find log e e d, > 0 Find d. e If k d =, then find k Solve sec. tn y d + sec y. tn dy = 0.

4 Time : hours 0 - Mthemtics - Mrch 007 Mrks : 100 Pg - 9. If initil velocity of projectile is 8 m/s nd horizontl rnge is 0 m, find the mesure of ngle of projection. 0. A prticle moves on line nd its distnce from fied point t time t is, where = t + t. Find velocity nd ccelertion t t = 1. S EC T I O N C Answer the following 1 to 0 questions s directed. Ech question crries two mrks Find prmetric equtions of the lines pssing through A (, - 1 ), B ( 0, ). Also write BA - AB s sets. If the distnces from the origin to the lines sec θ + y cosec θ = nd cos θ - y sin θ = cos θ re p nd p respectively, prove tht p + p =.. If the line + y + 16 = 0 is tngent to the prbol y = K, find K nd the point of contct. One end - point of focl chord of the prbol y = 16 is (, 8 ). Find the other end - point nd the length of the focl chord. y. If tngent to + = 1 intersects the mjor is t T nd minor is t N, nd if b b C is the centre, then prove tht + = 1. CT CN. If the chord of the hyperbol joining P ( θ ) nd Q ( φ ) on the hyperbol subtends right ngle t the centre C ( 0, 0 ), then prove tht + b sin θ. sin φ = 0. Obtin the equtions of the tngents to the hyperbol 5 - y = 5 from the point ( 0, ). 5. Find unit vector orthogonl to (, 1, 1 ) nd ( 1,, ). 6. If y nd nd y re unit vectors, show tht y is lso unit vector. 7. Get the rdius of the circle tht is formed by the intersection of the sphere + y + z = 5 nd the plne + y + z = 1.

5 Time : hours 0 - Mthemtics - Mrch 007 Mrks : 100 Pg If f ( ) =, f ( ) = 1, g ( ) = - 1 nd g ( ) =, then find f ( ). g ( ) - g ( ). f ( ) lim. - If - y = 1, then prove tht d y y + 1 = 0. d 9. (, ) lies on y = + b. Slope of tngent t (, ) is. Find nd b. e ( 1 + ) 0. Evlute : d. sin ( e ) S E C T I O N D Answer the following 1 to 50 questions s directed. Ech question crries mrks If P ( t, t ), Q t independent of t -, t nd S (, 0 ) re the points, show tht Find the incentre of the tringle, whose vertices re (, 1 ), ( 1, 5 ) nd ( -, 1 ) is SP SQ. Get the eqution of the circle touching both the es nd lso touching the line + y - 6 = 0 in the first qudrnt. Show tht the circles + y y - 90 = 0 nd + y - - 8y + 60 = 0 touch ech other eternlly.. Forces (, 5, 6 ) nd ( - 1,, 1 ) ct on prticle s result of which the prticle moves from A (, -, - ) to B ( 6, 1, - ). If the unit of force is newton nd distnce is mesured in meters, find the work done.. Find the perpendiculr distnce from A ( 1, 0, ) to the line r = (, 7, 1 ) + k (1,, - ), k R. Also find the foot of the perpendiculr. 5. Get the eqution of the plne pssing through ( 1,, ) nd (, - 1, ) nd perpendiculr to the plne + y + z = 7. + b, > 1 6. f ( ) = 11, = 1. f is continuous t = 1. Find nd b. 5 - b, < 1

6 Time : hours 0 - Mthemtics - Mrch 007 Mrks : 100 Pg Two trins strt from the sme plce. One trvels towrds south t speed of 50 km/h nd nother trvels towrds west t speed of 60 km/h. Find the distnce between them fter two hours. Divide 6 into two prts such tht the sum of their cubes is minimum. 8. Obtin d s the limit of sum Obtin sin θ sin θ + cos θ 0 dθ. dy 50. Solve differentil eqution = + y. d S E C T I O N E Answer the following 51 to 5 questions. Ech question crries 5 mrks A line psses through (, - ) nd the length of the perpendiculr segment from the origin to this line is. Find the eqution of the line. Find the eqution of the line pssing through (, ) nd contining line - segment of length between the lines + y = nd + y = Find lim cos. cos. 5. If = ( cos θ + θ sin θ ), y = ( sin θ - θ cos θ ), then prove tht sec θ y =, θ 0,, 0. θ d 5. Evlute Evlute d, where >. -

/ 3, then (A) 3(a 2 m 2 + b 2 ) = 4c 2 (B) 3(a 2 + b 2 m 2 ) = 4c 2 (C) a 2 m 2 + b 2 = 4c 2 (D) a 2 + b 2 m 2 = 4c 2

/ 3, then (A) 3(a 2 m 2 + b 2 ) = 4c 2 (B) 3(a 2 + b 2 m 2 ) = 4c 2 (C) a 2 m 2 + b 2 = 4c 2 (D) a 2 + b 2 m 2 = 4c 2 SET I. If the locus of the point of intersection of perpendiculr tngents to the ellipse x circle with centre t (0, 0), then the rdius of the circle would e + / ( ) is. There re exctl two points on the