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


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

Code No. 35

June/July, 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. 3x ≡ 2 ( mod 6 ) has no solution. Why ?
2. If direction cosines of → a are 1 3 2 3 and n, find n.
3. On I ( the set of all integers ), and operation ✳ is defined by a ✳ b = a b ,
∀ a, b ∈ I. Examine whether ✳ is binary or not on I.
4. A and B are square matrices of the same order and | A | = 4, | B | = 5.
Find | AB |.
5. Given two circles with radii r 1 , r 2 and having d as the distance
between their centres, write the condition for them to touch each other
externally.
6. Find the sum of the focal distances of any point on 4x 2 + 9y 2 = 36.
7. Evaluate sin – 1 ( sin 130° ) .Code No. 35 10
8. Find the least positive integer n for which 1 – i 1 + i n= 1.
9. Given the function f ( x ) = | x |, find L f
  l
( 0 ) .
10. Evaluate ⌡ ⌠– π/4 π/4( )  sin 3 x + cos x  dx .

PART – B

Answer any ten questions : 10 × 2 = 20

11. If ca ≡ cb ( mod m ) and c, m are relatively prime then prove that
a ≡ b ( mod m )
12. For the matrix A = cos θ sin θ– sin θ   cos θ verify that AA lis symmetric.
13. Define a semigroup. Examine whether { 1, 2, 3, 4 } is a semigroup under
'addition modulo 5' ( )+ 5  .
14. On Q + ( set of all +ve rationals ) , an operation ✳ is defined by
a ✳ b = ab 3 , ∀ a, b ∈ Q + . Find the identity element and a – 1 in Q + .
15. If λ ^ i +
 ^ j + 2 ^ k , 2 ^ i – 3 ^ j + 4 ^ k and ^ i + 2 ^ j – ^ k are coplanar, find λ.
16. Find the equation of the circumcircle of the triangle formed by
( 0, 0 ), ( 3, 0 ) & ( 0, 4 ).11 Code No. 35
17. Solve tan – 1 x = sin – 1
1 2 – cot – 1 13
18. Show that the real and imaginary parts of 5e i tan – 1 4 are 3, 4
respectively.
19. If y = sin – 1 x – 1
x + 1 + sec – 1 x + 1x – 1
prove that dy dx = 0.
20. At any point on the curve x m y n = a m + n , show that the subtangent
varies as the abscissa of the point.
21. Evaluate ⌡ ⌠ 
 
[ sin ( log x ) + cos ( log x ) ] dx.
22. Form the differential equation of the family of circles touching y-axis at
origin.

PART – C

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

23. a) Define GCD of two integers a and b. Find the GCD of 275 and
726. 3
b) Find the number of positive divisors of 252 by writing it as the
product of primes ( prime power factorisation ). 2Code No. 35 12
24. Solve by matrix method : 2x – y = 10
x – 2y = 2
Also, verify that the coefficient matrix of this system satisfies Cayley-
Hamilton theorem. 5
25. Prove that a non-empty subset H of a group G, is a subgroup of G, if
∀ a, b ∈ H, ab – 1 ∈ H. Hence prove that, if H and K are subgroups
of a group G then H

I K also, is a subgroup of G. 5
26. a) Given
→ a = 2^ i + ^ j + ^ k ,
→ b = ^ i + 2^ j – ^ k , find a unit
vector perpendicular to
→ a and coplanar with
→ a and
→ b . 3
b) If
→ a +
→ b +
→ c =
→ 0 , prove that
→ a ×
→ b =
→ b ×
→ c =
→ c ×
→ a .
2

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

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 each other
orthogonally. 3
b) ( 1, 2 ) is the radical centre of three circles. One of the circles is
x 2 + y 2 – 2x + 3y = 0. Examine whether the radical centre
lies inside or outside all the circles. 213 Code No. 35
[ Turn over
28. a) Given the equation of the conic
9x 2 + 4y 2 – 18x + 16y – 11 = 0, find its centre and the area
of its auxiliary circle. 3
b) Obtain the equation of the directrix of the parabola x = 2t
 2 ,
y = 4t. 2
29. a) If sin – 1 x + sin – 1 y + sin – 1 z =
π
2 , prove that
x 2 + y 2 + z 2 + 2xyz = 1. 3
b) Find the general solution of tan 2θ tan θ = 1. 2

III. Answer any three of the following questions : 3 × 5 = 15

30. a) Differentiate sin 2x w.r.t. x from first principle. 3
b) Differentiate ( sin x )
 log x w.r.t x. 2
31. a) Differentiate cos – 1 ( )  4x 3 – 3x  w.r.t. cos – 1 ( )  1 – 2x 2  . 3
b) Show that the curves y = 6 + x – x 2 and y ( x – 1 ) = x + 2
touch each other at ( 2, 4 ). 2
32. a) If y = sin ( )  m cos – 1 x  , prove that
( )  1 – x 2  y 2 – xy 1 + m 2 y = 0. 3
b) Evaluate ⌡⌠ 1 x ( )  x 5 + 1 dx.
33. a) Integrate
sin x + 18 cos x
3 sin x + 4 cos x
w.r.t. x. 3
b) Evaluate ⌡  ⌠1 – x 1 + x dx. 2
34. Find the area of x 2 + y 2 = 6 by integration. 5

PART – D

Answer any two of the following questions : 2 × 10 = 20

35. a) Derive a condition for y = mx + c to be a tangent to the hyperbola
x 2 a 2 – y 2 b 2 = 1. Also, find the point of contact. Using the condition
derived, find the equations of tangents to
x 216 – y 212 = 1, which are
parallel to x – y + 5 = 0. 6
b) Prove that
1     a     a 2 + bc
 
1     b     b 2 + ca
 
1     c     c 2 + ab
  = 2 ( a – b ) ( b – c ) ( c – a ). 4
36. a) State De Moivre's theorem. Prove it for positive and negative integral
indices. Using it prove that
Z 10 – 1
Z 10 + 1
= i tan 5θ if
Z = cos θ + i sin θ. 6
b) Find the general solution of cos 2θ = 2 ( cos θ – sin θ ) . 415 Code No. 35
37. a) The volume of a sphere increases at the rate of 4π c.c./sec. Find the
rates of increase of its radius and surface area when its volume is
288 π c.c. Also find (i) the change in volume in 5 secs, (ii) rate of
increase of volume w.r.t. radius when the volume is 288 π c.c. 6
b) Obtain the equations of parabolas having ( 1, 5 ) and ( 1, 1 ) as ends
of the latus rectum. 4
38. a) Prove that ⌡  ⌠0π
x dx a 2 cos 2 x + b 2 sin 2 x= π 2 2ab . 6
b) Find the particular solution of xy ( )  1 + x 2 
dy dx– y 2 = 1, given
that, when x = 1, y = 0. 4

PART – E

Answer any one of the following questions : 1 × 10 = 10

39. a) If | |  
→ a + 
→ b + 
→ c   = | |  
→ a + 
→ b – 
→ c   , find the angle between
→ a +
→ b and
→ c . 4
b) Among all right-angled triangles of a given hypotenuse, show that the
triangle which is isosceles has maximum area. 4
c) Find the fourth roots of 16 cis
π
2 . 2Code No. 35 16
40. a) If 2 150 × 3 12 × 135 ≡ a ( mod 7 ), find the least positive remainder
when a is divided by 7. 4
b) Given the circles 2 ( )  x 2 + y 2  – 12x – 4y + 10 = 0 and
x 2 + y 2 + 5x – 13y + 16 = 0, find the length of their common
chord.  4
c) Evaluate ⌡⌠02x2 – x + x dx. 2


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