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GSEB HSC MARCH 2009 : Mathematics

8 pages, 65 questions, 2 questions with responses, 3 total responses,

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This Question Paper contains 8 Printed Pages. 050 ( E ) (MARCH, 2009) Time : 3.00 Hours] [Maximum Marks : 100 Instructions : 1. All the questions are compulsory. 2. Write your answers according to the instructions given below with the questions. 3. Begin each section from a new page. SECTION - A Questions from 1 to 15 are multiple choice questions, each carrying ONE mark. Write the letter of the correct option (A), (B), (C) or (D) in your answer book from 1 15 the alternatives. 1. Measure of the angle between lines x = 1 and Jx + y - 4 = 0 is ..... 2% (B) 6 (C) 7n/ (D) 2 (A) 2. If the lines 5x - ky 7 = 0 and 2x + 3y + 5 = 0 are mutually perpendicular to each other, then k = ....... 10 (A) - 15 (C) - 2 3 (B) 3 15 (D) 2 3. The equation of the line passing through the diametric points of the circle (x - 2)2 + (y-3)2 = 25 is .... (A) 2x+y=5 (B) (x-2) + (y-3) = 25 (D) x+2y=10 (C) x+y=5 4. Parametric equation of y2 = 12x are ............... ( t E R). (A) (6t, 3t2) ( B) (4t2, 3t) (C) (4t2, 8t) ( D) (3t2, 6t) 050(E ) [17] P.T.O. 5. Measure of the angle between asymptotes of x2 - y2 = 1 is ....... 7C (A) (C) 71 4 (B) 3 (D) 2 If Y=(2,0), y =(3,O ), (A) = then I x+y ........ (B) < (C) > (D) None of them. 7. Projection of i in the direction of j is ........ (A) o (C) (B) (D) k x-5 y-3 z+1 _ 8. If the directions of x -1 _ y+2 _ z-3 and 11c c -2 are same , then c = .......... (A) - 2 (B) (C) 4 (D) 9. The radius of the Sphere x2 + (A) (C) 10. lim x (^-2 x-oo 4 2 +z2-2x-2y -2z-1=0 is ..... (B) (D) J 13 1) - ........... (A) log 2 x (B) loge e (C) loge 2 (D) loge x 11. If x = 3t2 - 6t + 5 and v 0, then t= ........ (A) 0 (B) 1 (C) 5 (D) -1 12. The rate of increasing volume of the Sphere w.r.t., its surface area is ....... (A) 2r (B) (C) (D) 050(E) [18] 2 / 2 13. f sinx +cosx 22 dx=.......... +c. (A) (x-cosx) (B) (x + sin x) (C) (x + Cos X) (D) (x -sinx) 14. The area of the region bounded by y = tan x, X axis, x = 0 and x = is ........... units. (A) 2 log 2 (B) 2 log 2 (C) log 2 (D) 2 15. The order and degree of the differential equations d = 1 + dy dx2 dx are ....... (A) 4,4 (B) 2,4 (C) 4, 2 (D) 2,2 SECTION - B Answer the following questions from S. Nos. 16 to 30. Each question carries ONE mark. 15 16. Find the point P(a, b) on the line, whose parametric equations are x = 2t + 1 and y = 1 - t , t c= R, so that a + b = 1. 17. Find the Cartesian equation of the Circle whose parametric equations are x=-1+2sinO andy= 1+2cos0, 0e(-71,'K)18. For the Parabola y2 - 8x, obtain length of the Latus rectum and the end points of latus rectum. 19. Find the measure of eccentric angle for Ellipse x2 2 + =1 at (- 6, 4) which belongs to it. 100 25 OR Find the equation of the Ellipse, whose vertex is ( 5, 0) and foci ( 4, 0 050(E ) [19] P.T.O. 20. Find direction angles of Y=(3,0,-4). 21. Decide the direction for vectors opposite or different. x = (1, 3, 1), y = (2) 6, - 2) whether same, 22. Find the displacement of a particle 3i + 21 5k due to the force 2i - j - k and find the work done. 23. Find the perpendicular distance from P ( 1, 2, 3) to line x-6 y-7 z-7 -2 24. Obtain the equation of the Plane passing through (3, 4, 2), (2, 2, -1) and (7, 0, 6). 25. Obtain 26. Find : d (xx ) d I sin-1 a I where a # 0. 1 27. Find : f sin3 x cos4 x dx . 1 OR Find the value of f 2 - x2 dx 0 28. Obtain f [ea log x +e x log a 1 29. Find the area of the region bounded by curve x = y, X- axis and the lines x=2 andx=3. 30. Find the differential equation for the family of the lines y = mx + c, where m, c are arbitrary constant. 050(E) [20] SECTION - C Answer the following questions from Nos. 31 to 40. Each question carries TWO marks. Answer as directed in the questions. 20 31. Two of the vertices of a triangle are (1, -6) and (-5, 2). The centroid of the triangle is (-2, 1). Find the third vertex of the triangle. Also find the area of this triangle. OR Prove that not both co-ordinates of all the vertices of an equilateral triangle can be rational numbers. 32. Prove that y = x + 3 is a tangent to the Parabola y2 = 12x. Also find co-ordinates of its point of contact. 33. If the difference between measures of the eccentric angles of P and Q is and H if PQ cuts intercepts c and d on the axes, then prove that a2 b2 c2 + d2 2, where P and Q are on the Ellipse. 34. For the rectangular Hyperbola x2 - y2 = 9, consider the tangent at (5, 4). Find the area of the triangle, which this tangent makes with two asymptotes. 35. Find the point of intersection of the lines z+3 L: x-3 y+2 = z+1 and M: x x_ ; y=-1. 23 1 -1 =1 36. Get the radius and the centre of the Circle that is formed by the intersection of Sphere x2 + y2 + z2 = 25 and the Plane 2x + 2y + z = 12. 37. For y =(tan x + xtanx , find dx 38. Curves x2 y = 1 and a5 y = x3 are intersecting perpendicularly. Prove that a6 = 6. 050(E ) [21) P.T.O. 39. If initial velocity of Projectile is 28 m/s. and horizontal range is 40 m., then find measure of angle of Projection. OR When acceleration is constant and instantaneous speed is 22 m/s, the particle covers a distance of 10320 m. in 60 seconds, then find the acceleration. 40. Obtain f tan3x dx. OR x+sinx Obtain f dx. 1+cosx SECTION - D Answer the following questions from S. Nos. 41 to 50. Each question carries THREE marks. Answer as directed in the questions. 30 41. For A(1, 2) and B(2, 1), find the point dividing AB from A's side in n equal parts. From this, find the coordinates of the point of trisection. 42. Find the equation of the Circle which passes through (1, -2) and (4, -3) and having a centre lies on the line 3x + 4y = 7. OR Find the equation of the Circle, if common chord of the circles x2 + y2 - 4x = 0 and x2 + y2 - 6y = 0 as the diametre. 43. Find a unit vector in R3 making an angle of measure Y with each of the vector (1, -1, 0) and (0, 1, 1). 44. Prove that centroid and incentre for the equilateral triangle are same. With this, if A(6, 4, 6), B(12, 4, 0), and C(4, 2, -2), then find the incentre for triangle ABC by using vectors. 050(E) [22] 45. Find the foot of the perpendicular and equation of perpendicular line passing through (2, -1, 2) to plane 2x - 3y + 4z = 44. OR Find the equation of the Plane passing through the line of intersection of the planes 3x - 4y+ 5z = 10 and 2x + 2y - 3z = 4 and parallel to the line x=2y=3z. 46. A water tank is in the form of an inverted cone. The radius and the height of the cone are 10 cm. and 20 cm. respectively. If water is poured in the tank at the rate of 5 cm3 / second, then how fast the height of the water will be increasing when its depth is 15 cm. ? OR Prove that the length of the part of the tangent intercepted between the axes coordinate of the Curve x% + yY = aY is constant. (a > 0) 2 47. Evaluate :. f 2x -1 1 dx . 48. Find the area of the region bounded by curves y = x2 and y = x + 2. OR Find the area of the region bounded by curves y2 = 4x and x2 = 4y. 49. Evaluate 50. Find 050(E ) dy + 2Y dx x = ex lim ax (a+ly +1 X-41 x-1 [23] P.T.O. SECTION - E Answer the following questions from S.Nos. 51 to 54. Each question carries FIVE marks. 20 51. Find the equation of a circum-circle of the triangle formed by the lines x+y=6, 2x+ y=4 and x+2y=5. OR Find the co - ordinates C and D for the square ABCD, if A(-1, 3) and B(2, -2). 52. Find : 7, - cos-1 x lim x->-1+ x +1 m 53. If 2x = yY + y -1 fl ; x > then prove that y2 + xyl = m2y 54. Evaluate f cos 2x cos 4x cos 6x dx. OR Evaluate f (4x+3)1x2 -4x+13 dx. 050(E) [24] 2009

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