MODEL QUESTION PAPER-4 II P U C MATHEMATICS
Instruction: 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, Part-E carries 10 marks. PART-A Answer all the ten questions. 10 1. The linear congruence 2. If * 3. In a group , find 2 has no solution.Why? is a scalar matrix, find x and y. . find n.
1=10
4. If the direction cosines of are
5. Find the length of the tangent from (1,2) to the circle6. What is the eccentricity of the conic whose parametric equations are 7. Find the value of 8. Find the amplitude of 9. Differentiate
12
.
with respect to x.
2
10. Evaluate
sec (3x) dx
0
PART- B Answer any ten questions. 11. Find the G.C.D. of 595 and 252. 12. Solve by Cramer¡¯s rule: 5x-2y=11 , 2x 3y=12. 13. On the set inverse of 8 in | | 14. Prove that | | | | | | 15. Find the center and radius of the circledescribed by x=3 5cos , y=-9 5s . 16. Prove that difference of the focal distance of any point on the hyperbola is equal to 17. Solve for , 18. Express 19. in form of x iy. ( ) . . of positive rationales, is defined by .Find the identity and 10 2=20
.
20. The sum of two numbers is12. Find the numbers when their Product is maximum. 21. Evaluate 22. Form the differential equation representing family of straightlines passing through the origin. PART - C . Answer any three questions: 23. Find the number of all positive divisorsand the sum of all positive divisors of 29645. 24. State the Cayley-Hamiton theorem. Find the Eigen values for the matrix * .Hence find . * . are coplanar. 2 3 2 25. a) Prove that the cube roots of unity form an Abelian group under multiplication. b) Let M be the set of (2 2) non-singular matrices over Z and Does exist? If so, find 26.a) Show that the points . 3 5=15 5 5 3
b) Find the sine of the angle between the vectors 2i j k and j k 3i.
1 Contributed by RP Bhadrashetty, Lecturer in Mathematics, SUJM PU college, Harapanahalli
. Answer any two questions. 27. a) Derive the length of thetangent from the external point
2 5=10 to the
circle . b) Find k, for which thetwo circles 2 x2 2y2 kx-y 4=0 and 3x2 3y2 2x ky 2=0 cuts orthogonal. 28. a) Find the vertex, focus and directrix of the conic . b) Find the equation of the tangent at 29. a) If sin-1x sin-1y sin-1z = , show that b) Solve the equation 2sin = 1, where .. Answer any three of the following questions. 30. a) Differentiate w.r.t. x from the first principle. b) ( ) . 31. a) . b)Show that for the curve y=ax the sub tangent is a constant. 32. a) b) 33. a) b) . . . . on the hyperbola 3x2-4y2=12. ¨C
3
2 3 2 3 2 3 5=15 3 2 3 2 3 2 32 5 2 10=20 (a >b). 6 4
34. Find the area enclosed between the parabolas PART-D Answer any two of thefollowing questions: 35. a) Define an ellipse as a locus and derive its equation in standard form b) Show that | 36. a) | .
6 b) Prove by vector method 37. a) A man 6ft tall moves away from a source of light 20ft above the ground level and his rate of walking being 4miles/hr. At what rate, is the length of the shadow changingAt what rate is the tip of shadow moving b) Find general solution of 38. a) b) Solve: . PART-E Answer any one of the following questions:39. a) Find the cube roots of and represent them in an Argand diagram. b) Find the length of the common chord of the intersecting circles x2 y2 x2 y2 . c) Find the remainder when is divided by 19. 40. a) Let be three vectors of magnitude 3, 4 and 5 respectively. If each one is perpendicular to the sum of the other two vectors, prove that| | ¡Ì . b) Evaluate c) Differentiate w.r.t.x. @@@@@@@@@@@@@@@@ 1 10=10 4 4 2
¡Ì ¡Ì ¡Ì
4
6 4 6 4
.
4 4 2
2
