Calicut University 2007 B.Sc Computer Science SECOND YEAR -ANALYTIC GEOMETRY AND CLCULUS - Question Paper

C31392

C 31392    (Pages 2)    Name.Jr]ASJMfiL..'.V.'i^/i..............

Reg. No..................................

SECOND YEAR B.Sc. DEGREE EXAMINATION, MAY 2007

Part IIIMathematics (Subsidiary)

Paper IANALYTIC GEOMETRY AND CALCULUS (2001 admissions)

Time: Three Hours    Maximum : 65 Marks

Unit I (Analytic Geometry)

* i?" .    (Maximum Marks 20)

\ ' 1. Find the Cartesian and cylindrical co-ordinates of the point whose spherical co-ordinate is

S'fcf-f)- 0'

n    -    '    (6 marks)

y\

,,. |*⁰ 2. What does the equation become when it is transferred to parallel axes through the point (a, b-c). The equation is 0c a)² + (y fe)² = c². _t>c³'^JSrj

6/.-g ;a_r-!j_iqfn^as^k-^s\-

, 3. Fm3 the vertex, focus, axis of symmetry and directrix of the parabola y² + 2y + 4x - if = 0. ^J

.    *    (6 marks)

4. Find the Co-ordinates of the centre, foci and equation to the directions of the ellipse :

^f-    a** + 4y-12.-*₊4 = o.,-4}, _<+ -4)

   (6 marks)

/s*j

₇. po ⁵'    ^{Find the} equation of the sphere described on the line joining (3, 4, 5) and (1, 2, 3) as diameter.

3 '^b    - 4'*-    = O    (3 marks)

6.    Discuss and sketch the surface x² +y² = 4z. Lfopkc.

.    ¹    (3 marks)

Unit II (Differential Calculus)

(Maximum Marks 30)

A*¹" .    j

⁰' ⁷- Given the value of the hyperbolic function sinhx = - -, find the value of other five hyperbolic

3

irfc

(5 marks)

'' 8. Find where y = cos h (4 - 3x). 3 ^A    '    _{(3 marks)}

Turn over

2    C 31392

- ' dy

9- Find where y = secA"¹ (sin 2*). *~2- (Urt**- S-^-*- ccr5P,v. 7o    (3 marks)

A otrS **- Y    '

'    i- .    r -+_i_

_fy iO. Find the tt¹¹* derivative of _fa _ ₂, ₊ CjJ. [jpp** +

A*'    .    b         _N/v - 3<'0^p'^_

"11. Using Leibnitzs theorem find the n^th derivative ofx³ log (1 + x). -0 (r~M⁽ j " C* (6 xjarks) r \^r    -3 ^,1'" -f- 3 * C*7~r

_v<\12. Verify Rolls theorem for /U) = * (x - 1) on the interval (0, 1).    (e. marks)

13. Find the points of inflextion of the curve y = x³ - 9x² + 7x - 6.    (6 marks)

<9²U (9²U    jy

14.    Verify that    for = sin -¹ --    (6 marks)

15.    Verify Eulers theorem when u - x³ + y³ + 3x²z.    (5 marks)

Unit DI (Integral Calculus)

(Maximum Marks 15)

¹

,\ 16. Use the trapezoidal rule to evaluate the integral ofyOc) from 0 to from the table below :

2t

x 2it 3it 4 it 5tt 6it    , , _/L q 'li

XI    0        <2 I b *r I

12 12 12 12 12 12

y(x) : -00000 -25882 -50000 -70711 -86603 -96593 1.0000

(5 marks)

2 2 x y

17. Find the area of an elliptic quadrant of + s; 1-    (5 marks)

a b*

qH8. Find the volume of the solid formed by the revolution about the major axis of nn ellipse with axes 2a and 36.

(5 marks)

19. Find the surface area of a sphere of radius a.    (5 marks)

2 3

.20. Evaluate j / (y² - 3a*) <y dx.    (5 marks)

   ** 1 -3

Attachment: calicut-university-2007-b-sc-computer-science-31738-1.jpg calicut-university-2007-b-sc-computer-science-31738-2.jpg

Earning:   Approval pending.