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      Karnataka 2nd PUC MATHEMATICS (March, 2009) Question Paper


      Posted Date: 16-Aug-2010  Last Updated:   Category: Question Papers    
      Author: Member Level: Diamond    Points: 10 (Rs 1)


      Karnataka Pre University Board 2nd year MATHEMATICS March, 2009 Question Paper.



      Code No. 35

      March, 2009
      MATHEMATICS

      Time : 3 Hours 15 Minutes ] [ Max. Marks : 100

      Instructions : i) The question paper has five Parts – A, B, C, D and E.Answer all the parts.
      ii) Part – A carries 10 marks, Part – B carries 20 marks,
      Part – C carries 40 marks, Part – D carries 20 marks and
      Part – E carries 10 marks.

      PART – A
      Answer all the ten questions : 10 × 1 = 10
      1. Find the least positive remainder when 7 30 is divided by 5.
      2. If 4 x + 2 2x – 3 x + 1is a symmetric matrix, find x.
      3. Define a subgroup.
      4. Find the direction cosine of the vector 2 ^ i – 3 ^ j + 2 ^ k .
      5. If the radius of the circle x 2 + y 2 + 4x – 2y – k = 0 is 4 units, then
      find k.
      6. Find the equation of the parabola if its focus is ( 2, 3 ) and vertex is
      ( 4, 3 ).
      7. Find the value of sin 1 2 cos – 1 ( – 1 ) 
      8. If 1, ω, ω 2 are the cube roots of unity, find the value of ( ) 1  ω + ω 2 
      9. Differentiate 3 x sinh x w.r.t. x.
      10. Integrate
      1 – cos 2x
      1 + cos 2x
      w.r.t. x.

      PART – B

      Answer any ten questions : 10 × 2 = 20

      11. If a ≡ b ( mod m ) and n is a positive divisor of m, prove that
      a ≡ b ( mod n ).
      12. Without actual expansion show that
      43    1    6
       
      35    7    4
       
      17    3    2
        = 0.
      13. Is G = { 0, 1, 2, 3 } , under ⊗ modulo 4 a group ? Give reason.
      14. Find the equation of two circles whose diameters are x + y = 6 and
      x + 2y = 4 and whose radius is 10 units.11 Code No. 35
      15. Find the area of the parallelogram whose diagonals are given by the
      vectors 2 ^ i –
       ^ j +
       ^ k and 3 ^ i + 4 ^ j –
       ^ k .
      16. Find the eccentricity of the ellipse ( a > b ), if the distance between the
      directrices is 5 and distance between the foci is 4.
      17. Solve cot – 1 x + 2 tan – 1 x = 5π
      6 .
      18. Find the least positive integer n for which
      1 + i
      1 – i
      n = 1.
      19. If y = ( )  x +  1 + x 2 m, prove that ( )   1 + x 2  dy/dx – my = 0.
      20. Show that for the curve y = be x athe subnormal varies as the square of
      the ordinate y.
      21. Evaluate ⌡ ⌠1elog e x dx .
      22. Find the order and degree of the differential equation
      1 + dy/dx22= d 2 y dx 2 .

      PART – C

      I. Answer any three questions : 3 × 5 = 15

      23. Find the G.C.D. of a = 495 and b = 675 using Euclid Algorithm.
      Express it in the form 495 ( x ) + 675 ( y ). Also show that x and y
      are not unique where x, y ∈ z. 5
      24. Solve the linear equations by matrix method : 5
      3x + y + 2z = 3
      2x – 3y – z = – 3
      x + 2y + z = 4
      25. a) On the set of rational numbers, binary operation ✳ is defined by
      a ✳ b = a 2 + b 2 , a, b ∈ R, show that ✳ is commutative
      and associative. Also find the identity element. 3
      b) If a is an element of the group ( G, ✳ ), then prove that
      ( )  a – 1 
       – 1
      = a. 2
      26. a) Find the sine of the angle between the vectors
       ^ i – 2 ^ j + 3 ^ k
      and 2 ^ i +
       ^ j +
       ^ k . 3
      b) Show that the vectors
       ^ j + 2 ^ k ,
       ^ i – 3 ^ j – 2 ^ k and –
       ^ i + 2 ^ j
      form the vertices of the vectors of an isosceles triangle. 2
      II. Answer any two questions : 2 × 5 = 1013 Code No. 35
      [ Turn over
      27. a) Derive the condition for the two circles
      x 2 + y 2 + 2 g 1 x + 2 f
       1 y + c 1 = 0 and
      x 2 + y 2 + 2 g 2 x + 2 f
       2 y + c 2 = 0 to cut orthogonally. 3
      b) Show that the radical axis of the two circles
      2x 2 + 2y 2 + 2x – 3y + 1 = 0 and
      x 2 + y 2 – 3x + y + 2 = 0 is perpendicular to the line joining
      the centres of the circles. 2
      28. a) Find the ends of latus rectum and directrix of the parabola
      y 2 – 4y – 10x + 14 = 0. 3
      b) Find the value of k such that the line x – 2y + k = 0 be a
      tangent to the ellipse x 2 + 2y 2 = 12. 2
      29. a) If tan – 1 x + tan – 1 y + tan – 1 z = π, show that
      x + y + z – xyz = 0. 3
      b) Find the general solution of tan 4θ = cot 2θ. 2
      III. Answer any three of the following questions : 3 × 5 = 15
      30. a) Differentiate tan x w.r.t. x from the first principle. 3
      b) If y = tan – 1 2 + 3x 2 3 – 2x 2  , prove that dy dx= 2x 1 + x 4 . 2
      31. a) If y = cos ( )  p sin – 1 x  , prove that Code No. 35 14
      ( )  1 – x 2  y 2 – xy 1 + p 2 y = 0. 3
      b) Find the equation of the normal to the curve y = x 2 + 7x – 2 at
      the point where it crosses y-axis. 2
      32. a) Integrate e 3x 3 + tan x cos x w.r.t. x. 3
      b) Find the angle between the curves 4y = x 3 and y = 6 – x 2
      at ( 2, 2 ). 2
      33. a) If x m y n = ( x + y ) m + n , prove that x y dx = y. 3
      b) Integrate 17 – 6x – x 2 w.r.t. x. 2
      34. Find the area between the curves y 2 = 6x and x 2 = 6y. 5

      PART – D

      Answer any two of the following questions : 2 × 10 = 20
      35. a) Define hyperbola as a locus and hence derive the equation of the
      hyperbola in the form
      x 2
      a 2 –
      y 2
      b 2 = 1. 6
      b) Show that
      b 2 + c 2 ab ac
      ba c 2 + a 2 bc
      ca cb a 2 + b 2
      = 4a 2 b 2 c 2 . 4
      36. a) If cos α + 2 cos β + 3 cos γ = 0, sin α + 2 sin β + 3 sin γ = 0,
      show that i) cos 3α + 8 cos 3β + 27 cos 3γ = 18 cos ( α + β + γ )15 Code No. 35
      ii) sin 3α + 8 sin 3β + 27 sin 3γ = 18 sin ( α + β + γ ).
      6
      b) Prove that [ ]  
      → a + 
      → b     
      → b + 
      → c      
      → c  + 
      → a  = 2 [ ]  
      → a 
      → b 
      → c   . 4
      37. a) The volume of a sphere is increasing at the rate of 4π c.c./sec. Find
      the rate of increase of the radius and its surface area when the
      volume of the sphere is 288π c.c. 6
      b) Find the general solution of 3 tan x = 2 sec x – 1. 4
      38. a) Show that ⌡ ⌠0π/4log ( 1 + tan x ) dx = π8 log 2. 6
      b) Solve the differential equation tan y dydx = sin ( x + y ) + sin ( x – y ). 4

      PART – E

      Answer any one of the following questions : 1 × 10 = 10
      39. a) Find the cube roots of 3 – i 3 and find their continued product. 4
      b) Show that ( )  
      → a ×  
      → b 
       2
      =
      → a
       2

      → b
       2
      – ( )  
      → a . 
      → b 
       2
      c) Find the length of the chord of the circle
      x 2 + y 2 – 6x – 2y + 5 = 0 intercepted by the line x – y + 1 = 0. 2
      40. a) Evaluate ⌡⌠03x + 2x + 2 +  5 – x dx. 4
      b) Show that among all the rectangles of a given perimeter, the square
      has maximum area. 4
      c) Differentiate sec ( 5x )
       0 w.r.t. x. 2


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