Time : 3 hours 02 - Mathematics - July 2006 Marks : 100 Pg - 1 Instructions : S E CT I O N - A
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1 Time : 3 hours 0 Mathematics July 006 Marks : 00 Pg Instructions :. Answer all questions.. Write your answers according to the instructions given below with the questions. 3. Begin each section on a new page. S E CT I O N A Given below are to 5 multiple choice questions. Each carries one mark. Write the serial number ( a or b or c or d ) in your answer book of the alternative which you feel is the correct answer of the question.. d [ ( l 7 l, 8 ), ( l 7 l, 3 ) ] =? ( a ) 5 ( b ) ( c ) 5 ( d ). The Cartesian equation of the line passing through ( 5, 6 ) and ( 3, 6 ) is ( a ) y 6 = 0 ( b ) y + 6 = 0 ( c ) 5 = 0 ( d ) + 3 = 0 3. The equation of the circle touching the y ais and having its centre at ( 3, 4 ) is ( a ) + y y + 6 = 0 ( b ) + y 6 + 8y + 9 = 0 ( c ) + y 6 8y + 9 = 0 ( d ) + y 6 + 8y + 6 = 0 4. The end points of the latus rectum for parabola = 6y are 3 3 ( a ) ± 3, ( b ), 3 ( c ) 3, 3 ( d ) 3 ± 3, 5. Measure of the angle between asymptotes of 4 y = 9 is 4 ( a ) tan 4 ( b ) tan 4 ( c ) ( d ) tan Which is a unit vector? ( a ) ( cosα, sinα ) ( b ) (sinα, cosα ) ( c ) (, ) ( d ) (cosα, sinα ) 7. = (, ) and y = (, 0 ), then cos ^ y = ( a ) ( b ) 0 ( c ) ( d ) y 8. Measure of the angle between + y + z = and r = ( 0, 0, 0 ) + k (,, ), k R is ( a ) ( b ) ( c ) ( d ) 6 3 4
2 Time : 3 hours 0 Mathematics July 006 Marks : 00 Pg 9. The plane r. (,, ) = touches the sphere + y + z 4y + z 3 = 0, then the point of contact is ( a ) (, 4, ) ( b ) (, 4, ) ( c ) (, 4, ) ( d ) none of these 0. 4 e e lim 4 4 ( a ) 4e ( b ) =? e 4 ( c ) 4e ( d ) log e 4. The derivative of sin with respect to cos is ( a ) ( b ) ( c ) 0 ( d ) none of these. Radius of a circular metal plate, when heated, increases by %. If its radius is 0 cm., then the increase in its area is ( a ) 4 cm ( b ) 4 cm ( c ) 0 cm ( d ) cm 3. 0 l l = ( a ) ( b ) ( c ) ( d ) none of these 4. The degree and order of the differential equation d y dy = + are ( a ) 6 and ( b ) 3 and ( c ) and ( d ) and 5. A body projected in vertical direction attains maimum height 50 m. Its velocity at 5 m height is ( a ) 7 0 m/s ( b ) 7 0 m/s ( c ) 7 0 m/s ( d ) 490 m S E C T I O N B Answer the following 6 to 30 questions. Each question carries one mark In which ratio does the ais divide the line segment joining A ( 3, 5 ) and B (, 6 )? 7. Obtain the equation of the circle which has a diagonal of rectangle formed by =, =, y = 3 and y =. Obtain the equation of a circle with radius 5/, if it passes through (, ) and (, 4).
3 Time : 3 hours 0 Mathematics July 006 Marks : 00 Pg 3 8. There is a point on the parabola y = whose coordinate is two times the y coordinate. If this point is not the verte of the parabola, find the point. y 9. Find the parametric equation of the director circle of the ellipse + = Find a unit vector orthogonal to both (,, ) and ( 3,, ).. Find the projection of (,, ) on (,, ).. Find the perpendicular distance of the point P ( 4, 5, 3 ) from the line 5 y + z 6 = = d 3 ( ) Find d 006 loge e 3. Find ( sin ). e 4. Evaluate Find the area of the region bounded by the curve y = cos, ais and the lines = 0 and =. 6. Evaluate tan sec Evaluate Evaluate ( cosec sec + ), l l. 8. Obtain the differential equation representing all the lines of family y = m + c ( where m and c are arbitrary constants ). 9. If the distance of a particle eecuting rectilinear motion is from a fied point at time t, where = t 3 9t + t + 8, then when will the velocity become Two balls are thrown vertically upwards with velocities 9.6 m/s and 9.8 m/s. Find the height of the second ball, when the first ball attains maimum height.
4 Time : 3 hours 0 Mathematics July 006 Marks : 00 Pg 4 S EC T I O N C Answer the following 3 to 40 questions as directed. Each question carries two marks Prove by using slopes that A (, 8 ) B (, 6 ) and C ( 6, 0 ) are the vertices of a right triangle. Find the equation of the perpendicular bisector of AB where A is ( 3, ) and B is ( 7, 6 ). 3. For the parabola = y, find the area of the triangle whose vertices are the verte of the parabola and two end points of its latus rectum. 33. If the end points of a chord of the ellipse b + a y a b = 0 have eccentric angles with measure α and β, then prove that the equation of the line containing the chord is α + β y α + β α β cos + sin = cos. a b y 34. If the eccentricities of = ± are e and e respectively, then prove that a b e + e = e. e. If the chord of hyperbola joining P ( α ) and Q ( β ) on the hyperbola subtends a right angle at the centre C ( 0, 0 ), then prove that a + b sinα sinβ = Prove that : [ + y y + z z + ] = [ y z ]. 36. If, y, z are coplanar vectors, then prove that + y, y + z, and z + are coplanar. If ( + y ). ( y ) = 63 and l l = 8 l y l, then find l l. 37. Get the radius of the circle that is the intersection of the sphere + y + z = 49 and the plane + 3y z = If = a ( cosθ ), y = a ( θ sinθ ), θ [ 0, ], a 0, then find d y.
5 Time : 3 hours 0 Mathematics July 006 Marks : 00 Pg Verify Rolle s theorem for f ( ) = sin + cos, 0,. If it is applicable, find c. In which interval the function f ( ) = is increasing and in which interval is it decreasing? sin 40. Evaluate. + sin S E C T I O N D Answer the following 4 to 50 questions as directed. Each question carries 3 marks A is (, 0 ) and B is (, 0 ). If l AP PB l = 4, then find the equation of locus of P. Origin is circumcentre of triangle with vertices A (, tan ), B (, tan ) and C (, tan ) ( 0 < θ I < /, i > 0, i =,, 3 ). θ θ 3 3 θ3 y sin θ + sin θ + sin θ3 If the centroid of Δ ABC is (, y ), prove that =. cos θ + cosθ + cosθ3 4. If the equation 3 + ( 3 p ) y + qy p = 8pq represents a circle, find p and q. Also determine the centre and radius of the circle. 43. Forces measuring 5, 3 and unit act in the direction ( 6,, 3 ), ( 3,, 6 ), (, 3, 6 ) respectively. As a result, the particle moves from (,, 3 ) to ( 5,, ). Find the resultant force and work done. 44. Find the vector and Cartesian equations of the line passing through (,, 3 ) and perpendicular to the two lines y z r = ( 0, 0, 0 ) + K (,, ), K R and = =. 3 6 Find the measure of the angle between two lines, if their direction cosines l, m, n satisfy l + m + n = 0, l + m n = Find the vector and Cartesian equations of the plane containing the lines y z 5 r = (,, 3 ) + K (, 3, 4 ), K R and = = Find lim cos ( sin ). tan ( sin )
6 Time : 3 hours 0 Mathematics July 006 Marks : 00 Pg Prove that if > 0, then < tan < Obtain sin as the limit of a sum Prove that = log dy 50. Solve y = y +. If y ( ) = 0, then find the particular solution of the given differential equation. The population of a city increases at the rate of 3 % per year. How many years will it take for the population to double? S E C T I O N E Answer the following 5 to 54 questions. Each question carries 5 marks A is ( 4, 5 ) in Δ ABC and the lines 5 + 3y 4 = 0 and 3 + 8y + 3 = 0 contain two of the altitudes of the triangle. Find the coordinates of B and C. 5. If f ( ) = e e e + e, 0, f ( 0 ) =, then prove that f is not continuous at = 0. ( + m) n ( + n) m Find lim, 0 m, n N. 53. If = sin t and y = sin pt, then prove that ( ) d y dy + p y = Evaluate + 5e + 6e sec Evaluate. + cosec
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