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GSEB HSC JULY 2008 : Mathematics

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This Question Paper contains 8 Printed Pages. 0 5 0 (E) (JULY, 2008) [Maximum Marks : 100 Time : 3.00 Hours] Instructions : 1. Answer all the questions. 2. Write your answers according to the instructions given. 3. Begin each question from a new page. SECTION - A Given below are 15 multiple choice questions, each carrying ONE mark. Write the serial number [ (A) or (B) or X) or (D) ] which you feel is the correct 15 answer of the questions. 1. In A ABC, if A(1, -6), B(-5, 2) and the centroid is G(-2, 1), then Co-ordinates of vertex C are ? (-2,6) (B) (D) (-2, 7) (A) a (B) b (C) l a - b (D) (A) (- 2, 1) (C) (3, 2) 2. d{(a,O),(0,b)}=? a2 + b2 3. . The t point of Parabola y2'= 20 x is ? (t E R) (5t2, 4t) (B) (A) (5t, 4 t2 ) (C) (D) (5t2, lot ) (t, 2t) If y = 2x + c touches a parabola y2 = 16x, then value of c is ... (A) (C) 050(E ) 2 8 (B) -2 (D) 12- [ 1] P.T.O. 5. The equation of director circle of ellipse 9 2 + 16 2 =1 is ... (A) x2+ y2 = 9 (B) x2+ y2 = 16 (C) x2+ y2 = 25 (D) x2+ y2 7 6. The eccentricity of hyperbola x2 - y2 = 144 is ... (A) 21 (B) (C) , (D) 7. For non-null vectors U, E R3 are distinct vectors, then (axb) (cxd)is... a-ca-dl (A) Ib c (B) b dl 1 6-j 6 -j I a c a d a d a c' (C) 6-i b c (D) 64 b c a d a c 8. The projection of a = (1,1,1) on b = (2, 2,1) is ... (A) 5 (2, 2,1) (B) (1,3,2) 9 (C) (0, 0, 1) (D) 9 1(1,3,2) 9. The direction of a line passing through points (3, 2) 1) and (5, 6, 7) is ... (A) (8, 8, 8) (B) (2, 4, 3) (C) (4, 3, 2) (D) (2, 4, 6) 10. The perpendicular distance between 6x - 3y + 2z = 1 and 12x-6y+4z=21 is... (A) 63 17 (B) (C) 12 7 (D) 11. The centre of sphere (A) 6 31 19 14 IFI` - F -(2,4,6)+5 = 0 is... (2, 4, 6) (B) (1, 2, 3) (C) (2, 1, 3) (D) (2, 3, 5) 050(E) [2) 12. N (a , 6) form of the set jx l 1: (A) N (1, 3) (C) N(3, 1) 13. For <3, xe R}is... (B) N(2, 3) (D) N (-1, 3) a> 0, dy = dx (A) (B) y (C) (D) x VIy y 14. f 2 1 dx? x +4x+5 (A) tan1(x+5)+c (B) tan1(x+4)+c (C) tan1(x+2)+c (D) tan1(5x+4) c 4( __ 2 -1 +1 1l x (A) log dx =? 17 2 (B) (C) 2log 117 1 log 17 2 (D) None of these SECTION - B Instruction : In the following 16 to 30 questions each carries 1-1 mark. Answer your.questions as requirement. 15 16. If a line (a + 3) x + (a2 - 9) y + (a - 3) = 0 passes through origin, then find the value of a. OR Find K ; if the following lines 2x-5y + 3 = 0 5x-9y +K=0 and x - 2y + 1 = 0 are concurrent. 17. Find the equation of parabola whose focus is S(4, 0) and equation of its directrix is x+4=0. 050(E ) [3] P.T.O. 18. Find the tangents to the parabola y2 = 8x that is perpendicular to the line x+2y+5=0. 19. Prove that (x-y)x(x+y)=2 (xxy). 20. Obtain the cosine formula for a triangle by using vectors. 21. If the equation r 2 - r (2 , 1,1)+ 3 = 0 represents a sphere, then find its radius. 22. Obtain equation of a sphere having extremities of its diameter are (1, 1, 1) and (2, 2, 1). kx -1, x < 2 23. Find K if f (x) = x x >_ 2 is continuous at x = OR Obtain li m (2006)x + (2005)x - 2 x->0 x 1 24. Prove f (x) = e x is decreasing function for x # 0. 25. Find the approximate value of 28 . 26. Verify Rolle's theorem for f(x) = x2, 27. Evaluate f log x dx x . OR Evaluate : f [sin2 x + sin 2x]ex dx. x E [.-2, 2]. TC sin x ) dx Show that f x f (sin x)dx = 2 f f ( ) 0 0 29. Solve the differential equation x dx = y + 2 . 2 30. Write down the order of the differential equation i2 +3y=O. SECTION - C Instruction : In the following questions 31 to 40, 20 each question carries 2 marks. 31. Let Abe (3, -1) and B (0, 4). If P (x, y)E AB , obtain the maximum and minimum values of 3y - x. OR Find the equations of lines containing the diagonals of the rectangle formed by the lines x = 2, x = -1, y = 6 and y =- 2. 32. Find the maximum and minimum distances of points on the circle x2+ y2- 4x - 2y - 20 = 0 from the point (10, 7). OR Prove that for every value of K, the circle 2x2+ 2y2-12x + Ky + 18 = 0 touches the X axis. 33. Find the equation of Ellipse passing through the points (1, 4) and (- 6, 1). 34. Find the measure of angle between the asymptotes of hyperbola 3x2 -2 Y2 = 35. Find a unit vector orthogonal to (2, 1, 1) and (1, 2, 3) 36. Find the area of a parallelogram if its diagonals are 2i + k and i + j + 050(E ) [51 P.T.O. 37. Obtain : lim 10 + cos x - 3 x->n (n -x)2 OR Obtain : lim (1- x) tan (itx) x->1 2 n / I1 38. Find : lim I r =1 ^4r2 -1) sin 2x dx 39. Find : f 2 2 2 2 m sin x - n cos x x 1 1_x2 40. Evaluate : f 1 + x2 0 OR Show that : f dx it _ O 2+cosx 3,f3SECTION - D Instructions : Given below are 41 to 50 questions. Each question carries 3 marks. Write your answer carefully. 30 41. If G and I are respectively the centroid and incentre of the triangle whose vertices are A(-2, -1), B(1, -1) and C(1, 3), find IG. 42. If circle x2+ y2+ 2x + f y + K = 0 touches both the axes, then find f and K. 43. If x+y+z' = 0, then provethat xxy= yxz=zxx. OR If the vectors ( a, 1, 1), (1 , b, 1) and ( 1, 1, c) are coplaner vectors, then show that 1-a + 1-bb 1 +1c 050(E) 1. [6] 44. Find the shortest distance between the lines x+1yz x' _ = z and 123 45. Find the vector and cartesian equation of plane and distance from origin to the plane which passes through points A(1, 1, 0), B(0, 1, 1) and C(1, 0, 1). (1 + mxr - (1 + nxr 46. Obtain : lim x-40 x2 ; m,nE N. 47. If y = a cos ( log x) + b sin (log x ), then prove that x2y2+ xyl + y = 0. 48. Using the mean value theorem, prove that 1 tan-1 x - tan-1 y 1 2 +y2 x-y 1 1 +x (x y 0). OR Show that curves y = ax3 and x2 +3 Y2 = b2 are orthogonal curves. (a:0, b# 0). 49. Solve the differential equation : x x - y+x sin =0. (Y)' 50. If the time is taken for horizontal range R is T, prove that angle of projection 2 has measure tan-1 gT 2R OR Velocity of a projectile at the maximum height is 5 times its velocity at half the maximum height. Prove that angle of projection has measure 3 050(E ) [7] P.T.O. SECTION E Instructions : Each question carries 5 marks of the following 51 to 54 questions. 20 Answer the following questions. 51. In AABC, C is (4, -1). The line containing the altitude from A is 3x+ y+ 11= 0 and the line containing the median AD through A is x+ 2y + 7 = 0. Find the equations of lines containing the three sides of the triangle. OR Find the equation of the line that passes through the point of intersection of 3x - 4y + 1= 0 . and 5x + y -1= 0 and that cuts off intercepts of equal magnitude on the two axes. 52. f (x) = ex , x > 0 log(x+e) ;x<0 If f continuous at x = 0 ? It is differentiable at x = 0 ? Why ? 53. Obtain f dx sinx+secx 4 54. Obtain : f x3 dx as the limit of a sum. OR V Prove that f x. sec x dx = 7C log (.12- + 1). 0 1+tanx 2- 050(E ) [ 8] JULY, 2008

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