Pre University Board 2009 P.U.C Physics, Chemistry, Maths & Biology " Maths " Code 35 - Question Paper
Code No. 35
[ Total No. of Printed Pages : 16
Total No. of Questions : 40 ]
March, 2009
nstmctioos : 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.
Answer all the ten questions : 10 x 1 = 10
1. Find the least positive remainder when 7 30 is divided by 5.
; 4 A' + 2 |
is a symmetric matrix, find x:
L 2x - 3 x + 1
3. Define a subgroup.
AAA
4. Kind the direction cosine of the vector 2 i -3j + 2k.
5. If the radius of the circle x2 + y2 + 4x-2y-#c = 0 is 4 units, then find k.
6. Find the equation of the parabola if its focus is ( 2. 3 ) and vertex is ( 4, 3 ).
7. Find the value of sin cos 1 ( - 1 )
Code No. 35 10
8. If 1. w, a> are the cubs roots of unity, find the value of (l-co + w2)6.
9. Differentiate 3 v sinh x w.r.t., x.
, . , , /1 - c os 2x
10. Integrate \ W'rt X' .......
FART - B
Answer any ten questions : 10 x 2 = 20
II. If a s b ! mod ? and i is a positive divisor of m, prove that
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as h { mod n ). | ||||||||||||
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11 Code No. 35
Find the eccentricity of the ellipse I a > b }, If the distance between the
directrices Is 5 and distance between the foci Is 4.
Solve cot" 1 x + 2 tan ~1 x = -g- .
Find the least positive integer n for which y~j If y = ( x + V 1 + x2) m , prove that ( V1" +x2 ) - my = 0.
Show that for the curve y = bea the subnormal varies as the square of the ordinate y.
e
Evaluate f log x dx.
i
J l&eX
Find the order and degree of the differential equation
. 3
1 +
dx J
Answer any three questions : 3x5=15
23. Find the G.C.D. of a = 495 and b = 675 using Euclid Algorithm. Express it in the form 495 { x } + 675 ( y ). Also show that x and y are not unique where x, y e z. 5
24. Solve the linear equations by matrix method :
3x + y + 2z = 3 2x - 3y - z = - 3 x + 2y + z = 4
25. a) On the set of rational numbers, binary operation * is defined t
a * b = Va2 + b2 . a, be R, show that * is commutative an associative. Also find the identity element.
b) If a is an element of the group ( G, * ). then prove that ( a ' 1 ) 1 = a.
26. a) Find the sine of the angle between the vectors t - 2 J + 3fc ant
2i+j+k. v
b) Show that the vectors j + 2/c, ( - 3j -2/c and - i + 2j fom the vertices of the vectors of an isosceles triangle. 2
II. Answer any two questions : 2 x 5 = 1C
27. a) Derive the condition for the two circles
x2 + y2 + 2g jX + 2 /, y + c i = 0 and
x2 + y2 + 2g2x+2f2y + c2 = 0 to cut orthogonally. 3 13 Code No. 35
b) Show that the radical axis of the two circles
2x2 + 2y 2 + 2x - 3y + 1=0 and
x2 + y2 ~ 3x + y + 2 = 0 is perpendicular to the line joining the
centres of the circles. 2
28. a) Find the ends of latus rectum and directrix of the parabola
y 2 - 4y - lOx + 14 = 0. 3
b) Find the value of k such that the line x-2t/ + k = 0bea
tangent to the ellipse x2 + 2y2 - 12. 2
29. a) If tan ' 1 x + tan ~ 1 y + tan 1 z = it, show that
x + y + z - xyz = 0. 3
b) Find the general solution of tan 40 = cot 20. 2
III. Answer any three of the following questions : 3x5=15
30. a) Differentiate tan x w.r.t. x from the first principle. 3
Tr - i I" 2 + 3*2 1 i.i- j. dy 0
b) If y = tan 1 2 Prove that dx = l~+x*
31. a) If y = cos ( p sin ~ 1 x ) , prove that
( 1 - x2 ) y 2 - xy j + p2 y = 0. 3
b) Find the equation of the normal to the curve y = x 2 + 7x - 2 at
the point where it crosses y-axis. 2
b) Find the angle between the curves 4y = x 3 and y = 6 - x 2
at ( 2, 2 ). 2
33. a) If xm yn ~ { x + y ) m + n , prove that x = y. 3
1
b) Integrate
w.r.t. x.
7 - 6x - x1
34. Find the area between the curves y 2 = 6x and x2 = 6y.
Answer any two of the following questions : 2 x 10 = 20
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35. a) Define hyperbola as a locus and hence derive the equation of the | ||||||||||||||||||||||||
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36. a) If cos a + 2 cos (j + 3 cos y = 0, sin a + 2 sin jl + 3 sin y = 0,
show that i) cos 3a + 8 cos 3p + 27 cos 3y - 18 cos ( a + p + y)
ii) sin 3a + 8 sin 3(5 + 27 sin 3y = 18 sin ( a + P + y )
6
4
+ a]= 2[ af2] .
c c
b) Prove that [ + 1) +
. a) The volume of a sphere is increasing at the rate of 4n c.c./sec. Find
the rate of increase of the radius and its surface area when the
volume of the sphere is 288k c.c. 6
b) Find the general solution of V3 tan x = V2 sec x - 1. 4
T
. a) Show that J log ( 1 + tan x) dx = g log 2. 6
o
b) Solve the differential equation
tan y = s*n ( x + y 1 + sin ( * - y ) 4
Answer any one of the following questions : 1x10=10
9. a) Find the cube roots of 3 - i V3 and find their continued product. 4
( * ,} 2 > f > > \ 2
b) Show that \ a x b ) = a b_ ~ \ a . b J . 4
c) Find the length of the chord of the circle
x2 + y2-6x-2y + 5 = 0 intercepted by the line x- y + I = 0. 2
| Turn over
Vx + 2
dx.
Vxr+ 2 + V5 - x
40. a) Evaluate
b] Show that among all the rectangles of a given perimeter, the square
has maximum area.
c) Differentiate sec ( 5x) 0 w.r.t. x.
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Attachment: |
| Earning: Approval pending. |
