[७१.2

058/21

Total Number of Questions : 32

Time : 3.00 Hours Max. Marks : 200
‏قاع‎
‎1. Expand e 8] in powers of x—1. (2 Marks)
2
2. Differentiate Joos tdt with respect to ‘x’. (2 Marks)
1
3. If the normals at two points of the parabola | = 4x intersect on the curve, find the product of
the ordinates of the two points. (2 Marks)
4. A straight line and a conic are described in polar forms as ie 30056 + 5116 and =1+ecos0
respectively. If the line touches the conic at some point, find the eccentricity ‘e’ and identify
the conic. (2 Marks)
5. Find the values of log ( - 1) and log (- 1). (2 Marks)
6. A particle moving along the curve © has an instantaneous velocity 8 - cosec*t. Obtain the path ‏ع‎
‎described by the particle, given that it passes through the point (2, 3 (4 Marks)
7. Compute the area between the curve y = sin2x and the x-axis from x = 0 to x 27. (4 Marks)
7 siny
8. Evaluate | | രം. (4 Marks)
a vy
9. Find the eccentricity of the ellipse whose one pair of conjugate diameters are y = x + 3 and
3y+2x+5=0. (4 Marks)
10. Identify the points on the region 8 : 0 > × < 7, 055 1, where the complex function f(z) ಇ sinz
has a maximum value. (4 Marks)
11. Determine the range and kernel of the linear transformation T: R? ‏ج‎ R? defined by
T(x, ५, 2) =(x+z, x+y + 22, 2x+y + 32). (5 Marks)
12. Determine the volume of the cone cut from the unit solid sphere by the cone ‏ص‎ =F where (0,೪,0)
is any point in space in spherical coordinates. (5 Marks)
13. Find the partial differential equation satisfied by the set of all spheres of radius "ಗ with their centers on
the xy-plane. (5 Marks)
14. Construct all the distinct possible composition tables for the group (6, *), where G = {e, a, 0, ८), ‘e’
being the identity element for the binary composition '*'. (5 Marks)
15. Prove that all the values of i“ are real. (5 Marks)
16. Let 6 bea positively oriented simple closed contour in the complex plane and ൪ is a point inside 0.
3
Find the value of ಗ್ಗ 5 ds. (5 Marks)
ಜೀ
17. Prove that the function f(x) = sinx is uniformly continuous on [0, ௯). (5 Marks)
⋅⋅ ↥⊏ 1 13 1 x
⋅ ⋅⇟−↥⊟−−⇃∙≁− −⋯ −− =.
18. Sum the series: 1+ 2005 2405 8 २46 60560 —..., where 2 <0< 2 (7 Marks)

P.T.O.