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GSEB HSC JULY 2009 : Mathematics

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0 5 0 (E) (JULY, 2009) [Maximum Marks : 100 Time : 3.00 Hours] Instructions : 1. Answer all the questions. 2. Write your answers according to the instructions given below with the questions. 3. Begin each section on a new page. SECTION - A Given below are 1 to 15 multiple choice questions . Each carries one mark. Write the letter (A), (B), (C) or (D) in your answer book of the alternative which you feel is the correct answer of the questions. 1. The origin be shifted to (- 2, 3) so that the new co-ordinates of ......... would be (3,-2). (A) (-1, 1) (B) (1, 1) (C) (1, -1) (D) (-1, -1) 2. For all a, b, c E R, 2a + 3b + 5c = 0, the line ax + by + c = 0 passes through fixed point .......... (a2 + b2 # 0) (A) (2, 3) (B) (-2, -3) (C) C 2 5 3) (D) (25 , 5) 5 3. Circle x2 + y2 - tax - lay +a 2 = 0, a # 0 ............. (A) passes through origin. (B) touches only X-axis. (C) touches only Y- axis . (D) touches both the axes. 050(E ) [17] P.T.O. 4. One of the end point of the focal chord of Parabola y2 16x is 4 , 2 then the other end point is ........... . 1) 4 1 (B) ( -4 , - 2 (C) (64, - 32) (D) (- 64, 32) 2 2 5. Equation of a tangent to 3 - 2 =1 and parallel to y = x is .......... . (A) x-y+1 =0 (B) x + y - 1 = 0 (C) 6. If (D) x-y+2= 0 IxI =IyI=Ix-yI (A) then I z x+y+2=0 x +yl= ............ (B) IxI ( C) 3 x (D) 3 Ixl 7. For a parallelogram ABCD, AB = a and BC = b , then its area = ...... (A) (C) 1 2 a xb I (d +b)x(a -b)I (D) None of these 8. A plane cuts axes at A, B, C such that the centroid of LABC is (1, 3, 1), the equation of this plane is ............. z (A) x+y+z=3 (B) x +y+- =3 3 1 3 1 3 1 (D) None of these (C) 3x + 3y + z = 3 9. x E N* (-2, S) = f (x) e N(9, 0.01), then the maximum value of 8 is ...... , where f (x) = 5 - 2x. (A) 0.001 (B) 0.005 (C) 0.009 (D) None of these 050(E) [18] 10. d If d f ( x) = g(x), then d -1 (B) (D) 11. f {sin (log x) + cos (log x)} dx = ........ + c. (A) x sin (log x) (B) x cos (log x) cos (log x) (C) sin (log x) (D) 1 C dx=.......... +c. log 2) - x2 i IN x (A) sin-1 1 log 2 (B) (C) sin-1 (D) None of these 2a 13. - sin- 1 x log 2 f(x) l dx = ........... o f (x) + f (2a - x) (A) a (B) a (C) 2 -a -a (D) 14. Degree of a differential equation d2y dx2 2 2 3 1 =(y+dx)2 (A) 1 (B) 2 (C) 3 (D) is .............. . 4 050(E) [19] P.T.O. 15. A particle is projected vertically upward with a velocity of 24.5 m/sec., then velocity of that particle after 2 sec. is ........... m/sec. (A) 4.9 (B) -4.9 (C) -14.7 (D) 14.7 SECTION - B Answer the following 16 to 30 questions. Each carries one mark. 16. Find the incentre of the triangle whose vertices are (--,f3-,1), ( 0, 0), (0,2). 17. Obtain the location of point (a cos a, a sin a) in the plane relative to a circle x2+y2=r2, where aE(-mt) 7t], lal <r, a#0. 18. There is a point on the Parabola y2 = 8x whose Y- coordinate is two times the X- coordinate. If this point is not the vertex of the Parabola, find that point. 19. Let L and L' be the feet of perpendicular drawn from the foci S and S' 22 respectively to the tangent at any point P(x, y) of the ellipse + 6 =1, then find SL S'L' . OR Find the measure of eccentric angle of point (-2, - 2'J) on the ellipse 2x2 + y2 = 16. 20. If a, 0,Y are the direction angles of the vector i , then find the value of cos 2a + cos 2 0 + cos 2Y 21. Force 2i + 2j + 2k is applied at B(1, 2, 3); find the torque around A(-1) 2, 0). 22. Find the equation of the line through (4, 3, 2) and parallel to the line x-10 y-2 _ z-1 15 5 3 23. If the position vectors of the end points of a diameter of a sphere are 4i and 2. j , find the Cartesian equation of the Sphere. 050(E) [20) 24. The formula connecting the periodic time T and length 1 of a pendulum is T = 2 7 . If there is an error of 2% in measuring the length 1, what will be the percentage error in T ? 25. Discuss the validity of Rolle's Theorem for f (X) = 1 X4, x E [-1, 1]. OR The radius of a right circular cone is constant. If there is an error Sh in measuring its height, what will be the error in measurement of its volume ? e x -1 26. Evaluate : J e + f0 27. Obtain the value of 7r sin3 x cos3x dx. OR n+1 3 If f f (x) dx = n3 then find the value of f f(x) dx. n -3 28. Obtain the differential equation representing the family of curves y = a cos-1 x + b, where a and b are arbitrary constants. 29. A body projected in vertical direction attains maximum height 16 m. Find its initial velocity. 4 u2 sin2 a 30. Range of a projectile is times its maximum height 2 . Find measure g of angle of projection. SECTION - C Answer the following questions from-31 to 40. Each carries TWO marks, as directed in the question. 31. The equation of a perpendicular bisector of AB is 5x + 2y - 18 = 0, if A is (- 3, 2); then find the co-ordinates of the midpoint of AB. OR Find the co-ordinates of the foot of the perpendicular from A(a, 0) to the line a y=mx+ -; m#0. m 050(E) [21] P.T.O. 32. Find the locus of point P such that the slopes of the tangents drawn from P to a Parabola have (i) constant sum (ii) constant non zero product. OR Find the co-ordinates of the points of contact of the tangents drawn from (1, 5) to the Parabola y2 = 24x. 33. If the difference between measures of the eccentric angles of P and Q is 2 and a2 b2 H if PQ cuts intercepts c and d on the axes, prove that 2 + 2 c2 d 34. Find the equation of a curve from every point of which the tangents to the 22 x - y =1 intersect at right angles. 144 36 OR If the chord of the Hyperbola joining P(a) and Q(13) on the hyperbola subtends Hyperbola a right angle at the centre C(0, 0); prove that a2 + b2 sina sino = 0. 35. If d#0, 6+c # 0 and d + b + c # 0 ; show that coplanar. a, b + c, a + b + c are 36. The dot product with i + j + k of the unit vector having the same direction as the vector sum of 2i + 4j 5k and X i + 2j + 3k is 1, find X . 37. Find the equation of the sphere passing through the point 0(0, 0, 0), A(- a, b, c), B(a, - b, c), C(a, b, - c). 38. If y = tan -,(2 + 3x ), then find dx . OR -x)2 then prove that (1- x2)y2 - xyl = 2. If y = (cos' 39. Obtain the intervals in which function f (x) = x3 - 6x2 - 36x + 2 in increasing and decreasing. 050(E) [22] J 1 1 40. Evaluate : J x2 1-x)2 dx. 0 SECTION - D Answer the following questions from 41 to 50, each carrying THREE marks as directed in the question. H 41. If A is (- 2, 1) and B is (1, - 7); find a point on AB such that 5AP = 3AB. OR 1 If P(at2, 2at), Q(t2 , ta) and S(a, 0) are three points, show that SP + Q is independent of t. 42. Find the co-ordinates of points which are at minimum and maximum distance from the point (- 7, 2) on the circle x2 +Y2_ 10x - 14y - 151 = 0. OR Find the equation of the circle that touches the Y-axis and passes through (- 2) 1) and (- 4, 3). 43. Prove by using vectors that the perpendicular bisectors of the sides of a triangle are concurrent. 44. Find the measure of the angle between two lines if their direction cosines 1,m,n satisfy 1+m+n=0, 12-m2 +n2 = 0. 45. Obtain the foot of perpendicular, perpendicular distance and equation of perpendicular line from A(2, 3, 2) on r (1, - 2, 1) = - 5 . n 1 4 . In ii" V V. .3 n--+6 r = 1 16x2 +8r-3 47. By using mean value theorem for log(1 + x) in [0, x], prove that 11 0 < log(1 + x) x<1 where x > 0. OR The slope of the tangent at the point (1, 1) on the curve xy + ax + by = 2 is 2, find a and b. 050(E ) [23] P.T.O. 48. Evaluate : J sin_ --1 - Cos_--' dx, sin ^x + cos r 6x+7 49. Evaluate : J ^(4 x) (5-x) dx (x < 4) 50. Solve : dy + 4xy _ 1 dx x2 + 1 (x2 +1)2 SECTION - E Answer the following questions from 51 to 54, each carrying FIVE marks. 51. The lines x - 2y + 2 = 0, 3x - y + 6 = 0 and x - y = 0 contain the three sides of a triangle. Determine the co-ordinates of the orthocentre without finding the co-ordinates of the vertices of the triangle. OR Find the equation of the line passing through (NF3, -1) if its perpendicular distance from the origin is . 52. Find lim 25 15 : x-^1 x25-1 - x 15 _ 1 53. If f (x + y) = f (x ). f (y), then find f'(3); where f (x) = log (e + x ), x > 0. r 1 -1 54. Evaluate : J sin 0 2x 1 + x2 OR Prove that the area of the region bounded by the circle x2 +Y 2= 16 and the Parabola y2 = 6x is 3 (4n + ^) . 050(E ) [241 JULY, 2009

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Additional Info : Gujarat Secondary Education Board (GSEB) Higher Secondary Certificate (HSC) - Science Stream - English Medium
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