Calicut University 2007 B.Sc Computer Science FIRST YEAR ,IL- - Question Paper

C 30817    (Pages 2)

Name.    . .;U.VJ.................

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

FIRST YEAR B.Sc. DEGREE EXAMINATION, MARCH/APRIL 2007

Part IIISubsidiary Mathematics Paper IANALYTIC GEOMETRY AND CALCULUS

(2001 Admissions)

Time : Three Hours    - Maximum : 65 Marks

Maximum marks from Unit I is 20,

Unit II is 30 and Unit III is 15.

Unit I (Analytic Geometry)

(Maximum Marks 20)

KFind the cylindrical and spherical co-ordinates of the point whose Cartesian co-ordinates are (0, 0, 1).

(6 marks)

S: Find the new equation of the line x - 3$,+ 6 = 0 when

' y 3- Find the equation of the parabola with focus (-% - 1 Hand vertex ( T, - X)    (6 marks)⁽

4>. Find the eccentricity, latus rectum, foci and directrices of the ellipse %² + 25y² ~ 225, (6 marks) 5. Find the equation of the sphere with centre at (1, -2, 3) and radius = 5 units.    (3 marks)

(X; Discuss the sketch the surface x^z + 9z² = 9.

Unit II (Differential Calculus)

(Maximum Marks 30)

29

Y._ Given the value cosh x = find the values of other five hyperbolic functions.    (5 marks)

dy    '^/V

8. .-Find where y = sech    (3 marks)

\

(6 marks) Tr "/

(3 marks)

&

/ . dy y9. Find ", wherey = sinh¹ (tan x).

ylO. Find the n.^th derivative of

(3 marks)

o

(5 marks)

W - Uh V "i~ ^lin| + 2 ^V*^Wj (i _j -    T ^,v*^{v ,/a}

Using Leibnitzs theorem; find the /I^th derivative of x² sin Zx. Verify Rolls theorem for fix) - (.t + 2)² {x - 3) in the interval {-2, 3J.

u i* ₄.

Find the points of inflexion of the curve y = x⁴ - 6x² + 8bc ~ 1.

C 30817

(6 marks) (6 marks) (6 marks)

(6 marks) (5 marks)

at. 2. ' 13.

2..

ifiu <?²u    ,____

Verify that    for u = sin (x + y) + log (x + y).

14.

15.

Verify Eulers theorem when u = x* ~ 4x³y + 3 xS - y*.

Unit III (Integral Ualculus) (Maximum 15)

<V V-

U

-\r>ply Simpsons rule to evaluate J    to two places of decimals by dividing tin;

four equal parts.

Find the area of the loop of the curve y² = x⁴ (x + 2).

Find the length of the curve 27 y² = 4s³ from x 0 to x 3.

Find the volume generated when the part of the curve y = <? sin x between x ~ 0 and .t about the .t-axis.

UV' rt

range into

(5 marks)

18.

(5 ixiarAii)

(5 marks) < jc revolves

(5 marks) (5 marks)

M²*²) r

0. Evaluate J J xy dx dy,

\A'

aV A _A(/i

Vi- "0 /

&

\

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