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GSEB HSC MARCH 2008 : Mathematics

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This Question Paper contains 8 Printed Pages. 0 5 0 (E) (MARCH, 2008) [Maximum Marks : 100 Time : 3.00 Hours] 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 Given below are 1 to 15 multiple choice questions, each carrying ONE mark. 15 Write the letter of the correct option (A) or (B) or (C) or (D). 1. Find the value of a, if P(2, 3) is circumcentre of the triangle with vertices A(a, 6), B(5, 1) and C(4, 6). (B) 1 (A) - 4 (D) 0 (C) 4 2. Find a if a line x+ y+1 = 0 is converted in the form of a line xcos a+ysin cc =p. (A) 4 (B) 37t 4 (D) 7it 4 5 7r (C) 4 3. If the circle x2+ y2+ 4x+ Ky - 4 = 0 touches both the axes , then find out K. (A) (C) 8 2 (B) (D) 4 1 4. Obtain the equation of a Parabola having focus (0, -2) and the equation of directrix is y = 2 and (0, 0) is the vertex of Parabola. (A) x2=-8y (B) y2= 8x (C) x2 = 8y (D) y2 = - 8x 050(E) [1] P.T.O. 5. Find the radius of a director - circle of an ellipse 4x2 + 9y2 = 36. (A) (C) 6. If a (B) 10 13 (D) 5 10, b= 2 and a .6 =12, then find l ax b - (A) (C) 12 16 (B) (D) 14 18 7. Find magnitude of projection of vector i + + k on j . (A) -1 (B) 0 (C) 1 (D) 2 8. Find the measure of the angle between plane r .(1, 2, 1) = 1 and 2 1 1 (A) 6 (B) It (C) 4 9. Find lim (D) None of them (1 +x)) -1 x---)o (A) x 0 (B) 1 (C) Y (D) None of them 1 10. Find ^1- cosxl2 d tan-1 dx 1+cosx) (A) 0 ;It<x<27t. (B) Y (C) - Y2 (D) .1 11. Find c applying Rolle's theorem to f (x) = 1+ sin x , x E [0, 7t] (B) / 4 (D) / 2 [2] 0 12. 1 2 dx Evaluate : 1+x (B) (A) (C) 7t 6 (D) 2Tt 3 12 3 13. Find the area of the region bounded by the curve y =tan x, X-axis and the lines x =0 andx= 4. (A) log 2 (B) 2 log 2 (C) 2 log 2 (D) 2 log 2 d2y 14. Determine the degree of the differential equation d2 2 + 3 dy 2 = x21og dx2 dx (dx) (A) 1 (B) 2 (C) 0 (D) not defined 15. A stone falls from a tower of height 40 m. What will be its velocity, when it reaches on the land ? 28 m/s (B) (A) 14 m/ s 7 m/s (D) (C) 21 m/s SECTION - B Answer the following 15 questions. (No. 16 to 30) Each question carries ONE mark. 15 16. Find the point A on the X-axis which is at the distance of 5 units from point B(2, -3). 17. Obtain the equation of a circle which touches the X-axis, given that the equations of lines containing two of the diameters of the circle are 3x-2y-5=0and x+y-5=0. 18. Find the focus of a Parabola y2 + 6y - 2x + 5 = 0. 050(E ) [31 P.T.O. 19. The equations of the asymptotes of Hyperbola are 3x + 4y = 2 and 4x - 3y = 2. Find the eccentricity. 20. Find the unit vector in the direction of vector (1, 2, 3). 21. Find the area of a Parallelogram, if its diagonals are 2i + k and i + j + k - 22. Represent the equation of line 3 - x = 2 - y = 1- Z in the vector-form. 134 23. Find the length of a chord, cut by sphere x2 + y2 + z2 - x - y - z = 0 on any axis. f (x)-1 24. If f '(x) = f (x) and f ( O) = 1, then find out the value of lim x-^0 x 25. Evaluate : f x4x(1+logx)dx, x>0. 26. Evaluate : ex A. k 12x+8, if 1:5 x<-2 27. If f f(x)dx=47 ; f(x)= 1 6x , if 2Sx<k then find k . 28. Find the length of sub tangent of y = e" . 29. If a distance of 150 cm . is travelled in 30 seconds with an initial velocity of 10 cm/s , find the constant accleration (retardation). 30. If the maximum horizontal range is 200 m, find the minimum velocity for that. SECTION - C Answer the following 10 questions (31 to 40). Each question carries TWO marks. Do as directed : 20 31. A line passing through (2, 4) intersects the X-axis and Y-axis at A and B respectively. Find the equation of the locus of the mid-point of AB . 32. For the Parabola x2 = 12 y, find the area of the triangle, whose vertices are the vertex of the parabola and the two end-points of its latus rectum. 33. Find the equation of Ellipse, which is passing through the points (1, 4) and (- 6, 1). 34. Find the equation of Hyperbola for which the distance from one vertex to two foci are 9 and 1. OR Find the measure of angle between the asymptotes of hyperbola 3x2 -2 Y2 = 1. x y= x x z and x#0, then prove that 5 = z. 35. If x.y = 36. If d .b = a.c = 0, I a = b = Ic =1, then prove that a = 2 (bxc) where (b "c)=/. 6 37. Find the equation of a sphere given that its centre is (1, 1, 0) and that it touches the plane 2x + 2y + z + 5 = 0. 38. If .Y = tan - 1 5x 2 , then find 1-6x dx . OR f(x) = [x]. Is f continuous and differentiable at x = 1 ? 39. Find the measure of the angle between the curves y=sinxand y=cosx, 0< x<n. 050(E ) [5] P.T.O. 40. Obtain f tan x dx sin x cos x kit x# 2, tanx>O. OR Obtain f 4 1 4 dx. sin x + cos x SECTION - D Answer the following 10 questions (41 to 50). Each question carries 3 (THREE) marks. Do as directed. 30 41. A is (2,/2, 0) and B is ( 2,12-, 0). If AP-PB = 4, find the equation of locus of P 42. Find the equation of the incircle of the triangle formed by the following lines x=2, 4x +3y=5 and 4x-3y+13=0. OR Get the equation of the circle that passes through the origin and that cuts chords of length 5 on the lines y = x 43. Prove by vectors, that if the median on the base of a triangle is also altitude on the base, the triangle is isosceles. OR There are two forces (2, 5, 6) and (-1, 2, 1) that act on a particle and as a result of which the particle moves from A(4,-3,-2) to B(6, 1, -3). Find the work done. 44. Prove that the lines x - 1 = y-2 - z-3 and x-4 = y-1 = z intersect each 23452 other and also find the point of intersection. 45. Obtain the equation of a plane that passes through the points (2, 3, - 4) and (1, -1, 3), and that is parallel to X-axis. 050(E) , [6] 46. Find lim log x - 3 x->e3 x-e3 47. Prove that of all the rectangles having the same area, the square has minimum perimeter. OR y = ax3 + bx2 + cx + 5 touches X-axis at (-2, 0) and the slope of the tangent where it meets Y-axis is 3 , then find a, b, c. 48. Evaluate : . f log(1+x) dx. 0 (1+x)2 49. Find the area of the region bounded by the curve y = 2 1-x 2 and X-axis. OR Evaluate : 3 f e _x dx as a limit of the sum. 2 50. Solve the differential equation. x dy + y dx = xy dx, y(1)=1. SECTION - E Answer the following 4 questions (51 to 54 ). Each question carries FIVE marks . Do as directed 20 51. The equation of the line containing one of the sides of an equilateral triangle is x + y = 2 and one of the vertices of the triangle is (2, 3). Find the equations of lines containing the remaining sides of the triangle. OR A is (1, 3) in AABC and the lines x - 2y + 1 = 0 and y - 1 0 contain two medians of the triangle. Find the co-ordinates of B and C. 050(E ) [7] P.T.O. x'^ -1-n(x-1) 52. Find lim 2 ; x * 1. x-1 (x_1) 2 53. If y = log (1 + sin x), then prove that ey . _ + 1 = O . x2 54. Evaluate : f 2007x + 2008 12008x + 2007 dx. OR Evaluate : 1 dx. sinx+secx 050(E) [81 MARCH, 2008

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