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


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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
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