MODEL PAPER
II PUC MATHEMATICS(35)
Time : 3 hours PART - A Max. Marks : 100
Answer ALL the questions.
1. Find the number of solutions of 5x ¡Ô3(mod 15) 2.If a =2i+3j+6k and =i-2j+2k, find the projection of a on b b 0 2x6 3. If is a skew symmetric matrix, find x. 3x4 0 4. If ab =a+b-5 a,b I, find the identity. 5. Find the position of the point (2, -3) with respect to the circle x 2+y 2-2x-6y-11=0 6. Find theaxis of the parabola y2+x-2y+4=0 7. Evaluate cos[2sin-1(1/3)]
10X1=10
8. Prove that 9. If f(x)=tan 10. Evaluate
1 1
3 3 cos i sin 7 7
2
7 2
=i
x 1
then find f '(1)
¡Ò x 2 sin x3 dx
PART -B
Answer any TEN questions 10X2=20 11. If p is a prime number and p/ab then prove that p/a or p/b 12. Find the value of ¦Ë so that the vectors 3i-j- ¦Ëk, i-j+k and 2i-4j+3k are coplanar A 13. If A(adjA)=|A|I =(adjA)A , Prove that (adjA)-1 = ¨OA¨O 14. In a group G if (ab)-1 = a -1 b -1 Show that G is an abelian. 15. Show that radical axis of two circles is perpendicular to line joining the centers of circles x2 y2 16. Any tangent to the ellipse =1 makes intercepts h and k on the co-ordinate axes. Show a2 b2 a 2 b2 2 =1 that 2 h k 12 1 5 sin1 = 17. If sin find x. x x 2 18. Find the modulus and amplitude of complex number 2-2i 3 dy 19. If ex +ey =ex+y , Prove that= -ey-x dx 20. If y2 =8kx andxy=2p intersect orthogonally, Prove that p2 =128k4 1 21. Integrate w.r.t x 6x x 2 22. Form the differential equation of the circle passing through the origin and having their centre on the line y=2x PART-C Answer any THREE questions 3X5=15 23. Find theGCD of 495 and 675 and find kand l such that 495k+675l andshow that k and l are not unique. 5 24. a)Prove that sin(A+B) =sinAcosB +cosA.sinBby vector method 3 b)Show that (4 +3 ) x (2 +3 )_=6( x ) a a a 2 b b b 25. Prove that the set of all +ve rational numbers forms an abelian group w.r.t multiplication * defined by
a*b=ab/6 and hence solve x*3-1 =2. 26. Solve by matrixmethod: 3x+2y-z=4 , x-y+4z=11, 2x+y-z=1. 5 5
Answer any TWO questions
2X5=10
27. Find the equation of the circle with center on 2x+3y=7 and cutting orthogonally circles x2 +y2 -10x-4y+21=0and x2 +y2 -4x-6y+11=0 5 28. Find the the centre, eccentricity and ends of the latus rectum of the hyperbola 9x 2 -16y 2-18x-64y+89=0 b)Find k if y=3x+k touches the ellipse 4x2 +9y2 =36 29. a)Find sin(cos-1 1/3 - sin-1 2/3) b)Find the general solution of cos 2x +cos3x=0 32 3 2 3X5=15 3 3 2 3
Answer any THREE questions
30. a)Differentiate sinax by first principles b)Differentiate log a x with reference to log xa 31. a)If y=cos m ¦È, x=cos¦È, Prove that (1-x2)y2-xy1+m2y=0
x2 b)Integrate w.r.t x 6 4x 1 3cosxsinx 32. a)Integrate w.r.tx 4cosx3sinx
b)Find the angle between the curves x2 +y2 +3x-8=0 and x2 +y2 =5 at (1,2) 33. a)Differentiate b)Evaluate
2
tan1
13x 6x w.r.t sin1 13x 19x 2
2 3 2 5
sin 3 x ¡Ò cos 3 x sin3 x 0 PART-D
34. Find the area enclosed between the parabola x2 =4y and the line x=4y-2
Answer any TWO questions
35. a)Prove that [ ¡Á , c ¡Á]=[ ]2 a b b¡Á c a abc b)If =i+j-k, =i-3j+k, =3i-4j+2k find ¡Á¡Á a c a b c b
b)Prove that 3 3 4
¨O
x y z
x 2 y z y 2 z x z 2 x y
¨O
=(x-y)(y-z)(z-x)(x+y+z)
36. a)Derive the equation of the hyperbola in the standard form
x2 y2 =1 , Write equation to the locus a2 b2 x2 y2 =1 and write the of the point of intersection of perpendicular tangents to the hyperbola a2 b2
name of the equation.. 6 b)Findthe general solution of trigonometric equation 3 tanx= 2 secx-1 4 37. a)A man 6 feet in height moves away at a uniform rate of 4m.p.h. From a source of light which is 20 feet above the ground. Find the rate at which the shadow lengthens and the rate at which the tip of his shadow is moving. 6 b)Solve the equation (y2 +y)dx +(x2+x)dy=0 given that y=2 and x=1 4 38. a)If cis =cis =cis =0, Prove that i) cos 3 +cos 3 + cos 3 =3 cos( + + ) sin 3 + sin3 + sin 3 =3 sin( + + ) ii) cos2+cos2 +cos2 =sin2 +sin2+sin2 =
2
3 2
6
b)Show that
¡Ò log sinx dx
0
=
log2 2 PART -E
4
Answer any ONE question
39. a)Find all the fourth roots of 1 i 3 b)Find the length of the common chord of intersecting circles x2 +y2 -4x-5=0 and x2 +y2 -2x+8y+9=0 c)Find the remainder when 520 is dividedby 7 40. a)Show that maximum rectangle that can be inscribed in a circle is a square b)Evaluate ¡Ò cosec 3 x d x c)Find the order and degree of Differential equation4 4 2 4 4
1 2
d2 y dy = 1 2 dx d x
[ ]
2
