Model Question Paper Mathematics II Karnataka : www.pue.kar.nic.in Pre University/ PUC
Organization : Karnataka Department of Pre-University Education
Exam : II PU Mathematics
Document type : Question Paper
Website : pue.kar.nic.in
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Karnataka PUE Mathematics Question Paper
New Scheme ( For Students studied during the Year 2005-2006 )
Time : 3 Hours ] [ Max. Marks : 90
Related : PUC/ Pre University Biology II Model Question Paper : www.pdfquestion.in/5966.html
Instructions : i) The question paper has four Parts – A, B, C and D. Answer all the parts.
ii) Part – A carries 10 marks, Part – B carries 20 marks,
Part – C carries 40 marks and Part – D carries 20 marks.
PART – A
Answer all the ten questions : 10 × 1 = 10
1. Find the number of incongruent solutions for 6x = 3 ( mod 15 ).
2. If the matrix has no inverse, find x.
3. In a group G = { 1, 2, 3, 4 } under multiplication modulo 5 find ( 3 × 4 – 1 ) – 1 .
4. If the vectors – a = 3^i + ^j – 2^k and – b = ^i + – ^j –
5. Find the centre of the circle 4x 2 + 4y 2 + 4x + 2y + 1 = 0.
6. If the line x + y + 2 = 0 touches the parabola y 2 = 8x, find the point of contact.
7. Find the value of sec – 1 ( – 2 ) .
8. Express ( 1 + i ) 2 3 – i in x + iy form.
9. Differentiate y = a 4 log a x w.r.t. x.
10. Evaluate : ) – ( x 2 – 1x 2 + 1 dx.
PART – B
Answer any ten questions : 10 × 2 = 20
11. If a/b and a/c then prove that a/b + c.
12. Solve by Cramer’s Rule : 3x + 2y = 84x – 3y = 5.
13. Prove that H = { 0, 3 } is a sub-group of the group G = { 0, 1, 2, 3, 4, 5 }
14. Find the volume of the parallelopiped whose coterminous edges are
15. Find k for which the circles 2x 2 + 2y 2 – 18x + 6y – 7 = 0 and
3x 2 + 3y 2 + 4x + ky + 3 = 0 intersect orthogonally.
16. Find
the eccentricity of the ellipse if its minor axis is equal to distance
PART – C
I. Answer any three questions : 3 × 5 = 15
23. a) Find the number of all positive divisors and the sum of all such
b) Find the remainder when 71 × 73 × 75 is divided by 23. 2
24. Prove that a 2 + 1 ab ac ab b 2 + 1 bc ac bc c 2 + 1 = 1 + a 2 + b 2
25. If Q + is the set of all positive rationals, prove that ( Q + ‚ – ) is
26. a) If – a = ^i + ^j + ^k , – b = ^i + 2^j + 3^k and
b) If cos a, cos ß and cos – are direction cosines of the vector
27. a) Find the equation of tangent to the circle
x 2 + y 2 + 2gx + 2fy + c = 0 at the point ( x ) 1 ‚ y 1 on it. 3
b) Find the equation of the circle two of whose diameters are
28. a) Find the eccentricity and equations to directrices of the ellipse
4x 2 + 9y 2 – 8x + 36y + 4 = 0. 3
b) Find the equations of the asymptotes of the hyperbola.
PART – D
Answer any two of the following questions : 2 × 10 = 20
35. a) Define ellipse and derive standard equation to the ellipse
b) State Caley-Hamilton theorem. Verify the Caley-Hamilton theorem for
36. a) Find the fourth roots of the complex number – 1 + i 3 and
37. a) An inverted circular cone has depth 12 cms and base radius 9 cms.
38. a) Prove that )
b) Solve the differential equation ( y 2 + y ) dx + ( x 2 + x ) dy = 0.
March / April, 2007
Part – A :
1. Find an integer x, satisfying 5x ≡ 4 ( mod 13 ).
2. If →a = 2 ^i + 3 ^j and →b = 3 ^i + 4 ^j , find the magnitude of →a + →b .
3. Find the equation of the directrix of the parabola y 2 = – 8x.
4. If a ≡ b ( mod m ) and n > 1 is a positive divisor of m, prove that a ≡ b ( mod n ).
5. Define the binary operation, on a non-empty set S. Give an example to show that, on Z, the operation ✳, defined by a ✳ b = a b , is not binary.
6. Find the angle between the vectors 2 ^i – 2 ^j + ^k and 2 ^i – ^j – 2 ^k .
7. Integrate sin 3x cos x with respect to x.
8. Form the differential equation of the family of straight lines passing through the origin of Cartesian plane.
9. Find the G.C.D. of 408 and 1032 using Euclidean algorithm. Express it in two ways in the form 408m + 1032n where m, n are integers
10. A man 6 feet tall moves away from a source of light 20 feet above the ground level and his rate of walking being 4 miles/hour. At what rate, is the length of the shadow changing ? At what rate is the tip of the shadow moving ?
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