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


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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