Model Question Paper Maths Karnataka : www.pue.kar.nic.in Pre University / PU

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Document Described : Maths Question Paper, Karnataka Question Paper

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

http://pue.kar.nic.in/PUE/support_ht...htm#youranchor
http://pue.kar.nic.in/PUE/PDF_files/...r/2010M/35.pdf

Answer all the ten questions : 10 × 1 = 10
1. If a | b and b | c then prove that a | c.
4. If the vectors ?a = 2 ^i + ^j + ^k and ?b = 3 ^i + 4 ^j – ^k , find ?a . ?b
5. Find the values of h and k for the equation
kx 2 + 2hxy + 4y 2 – 2x + 3y – 7 = 0 to represent a circle.
6. Write the eccentricity of the conic section represented by the parametric
equations x = at 2 and y = 2at.
8. Find the amplitude of [ 1 + cos ? + i sin ? ] .

PART – B :
Answer any ten questions : 10 × 2 = 20
11. Find the sum of all positive divisors of 72.
13. Write the composition table for G = { 2, 4, 6, 8 } under multplication modulo 10 and find the identity element.
14. Find the area of the triangle whose two adjacent sides are determined by the vectors 2 ^i + 3 ^j + 2 ^k and 4 ^i + 5 ^j + 3 ^k .
15. Show that the two circles x 2 + y 2 – 6x – 2y + 1 = 0 and x 2 + y 2 + 2x – 8y + 13 = 0 touch each other externally.
16. Find the centre of the hyperbola 9x 2 – 4y 2 + 18x – 8y – 31 = 0 .
20. A particle is travelling in a straight line whose distance is given by
s = 4t 3 – 6t 2 + t – 7 units. Find the velocity of the particle after
t = 2 seconds.

PART – C :
I. Answer any three questions : 3 × 5 = 15
23. Find the G.C.D. of 506 and 1155 and express it in the form of
506 (a) + 1155 (b) ( where a and b are integers ). Also show that the
expression is not unique. 5
b) Solve by Cramer’s Rule : 2x – 3y = 5, 7x – y = 8. 2
25. Prove that the set M of all 2 × 2 matrices with elements of real numbers form an Abelian group with respect to addition of matrices. 5
26. a) Find the vector of magnitude 12 units which is perpendicular to
both the vectors ?a = 4 ^i – ^j + 3 ^k and ?b = – 2 ^i + ^j – 2 ^k . 3
b) If ?a = 5 ^i + 2 ^j – 3 ^k and ?b = 4 ^i + 2 ^j + 5 ^k are two vectors, find the projection of ?a on ?b . 2

II. Answer any two questions : 2 × 5 = 10
27. a) Derive the condition for the line y = mx + c to be a tangent to
the circle x 2 + y 2 = a 2 in the form c 2 = a 2 ( ) 1 + m 2 . 3
b) Find the radical axis of the circles 3x 2 + 3y 2 – 9x + 6y – 1 = 0
and 2x 2 + 2y 2 – 8x + 16y – 3 = 0. 2
28. a) Find the equations of the parabolas whose directrix is x + 2 = 0,
axis is y = 3 and the length of the latus rectum is 8 units. 3
b) Find the eccentricity of the hyperbola
x 2 – 3y 2 – 4x – 6y – 11 = 0. 2
29. a) If cos – 1 x + cos – 1 y + cos – 1 z = p, then prove that x 2 + y 2 + z 2 + 2xyz = 1. 3
b) Find the general solution of the equation cos 2 ? + cos ( 2 ? ) = 2. 2

III. Answer any three of the following questions : 3 × 5 = 15
31. a) If y = e m sin – 1 x , prove that ( ) 1 – x 2 y 2 – xy 1 – m 2 y = 0. 3
b) Find the angle between the curves y 2 = 4x and x 2 = 2y – 3
at the point ( 1, 2 ) . 2
32. a) Find the length of sub-normal to the curve x 3 y 2 = a 5 at any
point on it. Also show that length of sub-tangent varies directly as abscissa at that point. 3
b) Evaluate )
? ( sec x ( 1 + tan x )
e – x dx. 2
34. Find the area of the circle x 2 + y 2 = 6 by integration method. 5

PART – D :
Answer any two of the following questions : 2 × 10 = 20
36. a) State and prove De Moivre’s theorem for rational index. 6
b) Prove by vector method
sin ( A – B ) = sin A cos B – cos A sin B. 4
37. a) Show that rectangle of maximum perimeter which can be inscribed in
a circle of radius a is a square of side a 2 . 6
b) Find the general solution of
( ) 3 – 1 cos x + ( ) 3 + 1 sin x = 2 . 4

PART – E :
Answer any one of the following questions : 1 × 10 = 10
39. a) Find the cube roots of the complex number 3 – i and express them in an Argand diagram. 4
b) Find the equation of the chord of the circle x 2 + y 2 + 4x – 6y – 9 = 0 bisected at ( 0, 1 ). 4
c) Find the number of incongruence solutions and the incongruence solution of 2x = 2 ( mod 6 ). 2

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