Karnataka 2nd PUC MATHEMATICS (June 2008) Question Paper


Karnataka Pre university 2 year Mathematics Question Paper.

Code No. 35

June, 2008
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 number of incongruent solutions of 9x ≡ 21 ( mod 30 ).
2. Evaluate
4321     4322
4323     4324.
3. In a group
( )
 Z 6 ‚ + mod 6  , find 2 +
6 4 – 1 + 6 3 – 1.
4. Find the position vector of the point P which is the mid-point AB where
the position vectors of A and B are
 ^ i +
 ^ j + 2 ^ k and 3 ^ i – 3 ^ j + 2 ^ k .
5. Find the equation to a circle whose centre is ( a, 0 ) and touching the y-
axis.
6. Find the equation to directrix of ( x + 1 )
 2 = – 4 ( y – 3 ).
7. Find the value of cos – 1 ( sin 330° ) .Code No. 35 10
8. If 1, ω, ω 2 are the cube roots of unity, find the value of ( )  1 + ω – ω 2 2.
9. If y = e  x + x  e , find
dy dx.
10. Evaluate ⌡  ⌠
 e x 1 + tan x
cos x dx.

PART – B

Answer any ten questions. 10 × 2 = 20

11. Find the G.C.D. of 352 and 891.
12. Find the characteristic roots of the matrix
1     4
 
3     2
  .
13. Prove that a group of order three is Abelian.
14. Find the volume of the parallelopiped whose co-terminus edges are the
vectors
 ^ i + 3 ^ j + 2 ^ k , 2 ^ i –
 ^ j + 3 ^ k and ^ i + ^ j + ^ k .
15. Find the equation to the parabola whose focus is ( 3, 2 ) and its directrix
is x = 1.
16. Prove that
sin  2 tan – 1 1 – x 1 + x = 1 – x 2
17. Find the equation of a circle passing through the origin, having its centre
on the line y = x and cutting orthogonally the circle
x 2 + y 2 – 4x – 6y + 10 = 0.
18. Prove that ( 1 – i )
 9 = 16 – 16i.
19. If y = log e
1 – cos x 1 + cos x, then prove that dy/dx = 2 cosec x.
20. Find the point on the curve y 2 = x the tangent at which makes an angle
of 45° with the x-axis.
21. Evaluate ⌡ ⌠01 x ( 1 – x )7 dx.
22. Form the differential equation by eliminating the arbitrary constant
( y – 2 )2 = 4a ( x + 1 ).

PART – C

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

23. a) Find the number of positive divisors and sum of all such
positive divisors of 756. 3
b) If a/bc and ( a, b ) = 1, then prove that a/c. 2Code No. 35 12
24. Solve by matrix method :
3x + y + 2z = 3
2x – 3y – z = – 3
x + 2y + z = 4. 5
25. Prove that the set z of integers is an Abelian group under binary
operation ✳ defined by a ✳ b = a + b + 3, ∀ a, b ∈ z. 5
26. a) If
→ a = ^ i – 2 ^ j – 3 ^ k ,
→ b = 2 ^ i + ^ j – ^ k and
→ c = ^ i +3 ^ j – 2 ^ k , find a unit vector perpendicular to
→ a and in thesame plane on → b and → c . 3
b) Find the area of a parallelogram whose diagonals are the vectors
2 ^ i + ^ j + ^ k and ^ i – 2 ^ j + 3 ^ k . 2

II. Answer any two questions : 2 × 5 = 10

27. a) Find the length of the tangent from the point ( ) x 1 ‚  y 1  to the
circle x 2 + y 2 + 2gx + 2fy + c = 0. 3
b) Find the equations of tangent to the circle
x 2 + y 2 – 2x – 4y – 4 = 0, which are perpendicular to
3x – 4y + 6 = 0. 213 Code No. 35
[ Turn over
28. a) Find the focus and equation to the directrix of the ellipse
9x 2 + 5y 2 – 36x + 10y – 4 = 0. 3
b) Find the equation to the hyperbola in the standard form
x 2 a 2 – y 2b 2 = 1, given that length of latus rectum = 14 3 ande = 4 3. 2
29. a) If tan – 1 x + tan – 1 y + tan – 1 z = π2 , prove that xy + yz + zx = 1. 3
b) Find the general solution of sin 2 θ – cos 2θ = 5 4 . 2

III. Answer any three of the following questions : 3 × 5 = 15
30. a) Differentiate a x w.r.t. x by first principles. 3
b) If y = tan – 1 4x 4 – x 2  , prove that dy dx=4 4 + x 2 . 2
31. a) If y = ( )  sin – 1 x 2+ ( )  cos – 1 x 2 prove that 1 – x 2  y 2 – xy 1 – 4 = 0. 3 b) If x = 3 sin 2θ + 2 sin 3θ, and y = 2 cos 3θ – 3 cos2θ ,prove that dy/dx= – tan θ2
32. a) Prove that in the curve y = e x a the subnormal varies as the
square of the ordinate and subtangent is constant. 3
b) Evaluate ⌡  ⌠0π/2 sin x . cos x 1 + sin 4 x dx. 2
33. a) Evaluate ⌡  ⌠2 – 3 tan x 1 + 2 tan x dx. 3
b) Evaluate ⌡⌠1  1 + e x   ( )  1 – e – x dx. 2
34. Find the area of the ellipse 9x 2 + 16y 2 = 144 by integration. 5

PART – D

Answer any two of the following questions : 2 × 10 = 20
35. a) Define hyperbola as a locus and derive the standard equation of the
hyperbola in the form x 2a 2 – y 2b 2 = 1. 6
b) Prove that 1     a      a 2a 2    1      aa     a 2    1= ( )  a 3 – 1  2
36. a) If cos α + cos β + cos γ = 0 = sin α + sin β + sin γ , prove that
i) cos 2α + cos 2β + cos 2γ = 0
sin 2α + sin 2β + sin 2γ = 0
ii) cos 2 α + cos 2 β + cos 2 γ = 3 2 sin 2 α + sin 2 β + sin 2 γ 2 . 6
b) Prove that [ ]  
→ a ×  → b  
→ b ×  → c   
→ c  ×  → a  = [ ]  → a  → b  → c   2 . 4
37. a) The surface area of a sphere is increasing at the rate of 8 sq.cm/sec.
Find the rate at which the radius and the volume of the sphere are
increasing when the volume of the sphere is
500 π
3 c.c. 6
b) Find the general solution of sin θ + sin 2θ + sin 3θ = 0. 4
38. a) Prove that ⌡  ⌠0
π/2 cos 2 x1 + sin x cos x dx = π 3 3. 6
b) Find the general solution of the differential equation
xy dy/dx = 1 + y 21 + x 2 ( )  1 + x + x 2  . 4


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