LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034  B.Sc. DEGREE EXAMINATION – MATHEMATICS

AA 01

FIRST SEMESTER – NOV 2006

                        MT 1500 – ALG.,ANAL.GEOMET. CAL. & TRIGN. – I

(Also equivalent to MAT 500)

 

 

Date & Time : 01-11-2006/1.00-4.00           Dept. No.                                                       Max. : 100 Marks

 

 

SECTION –A

Answer all:                                                                                    2 x 10 = 20

  

1. If y = A e^mx+B e^-mx , show that y_{2 =} m²y.
2. Write down the n^th derivative of .
3. If x = at² and y = 2at , find .
4. Prove that the sub tangent to the curve y = a^x is of constant length.
5. Determine the quadratic equation having 1+ as one of its roots.
6. Calculate the sum of the cubes of the roots of equation x⁴+2x+3 = 0.
7. Prove that cos ix = cosh x.
8. Separate sin (x+iy) into real and imaginary parts.
9. Define conjugate diameters.
10. Write the angle between the asymptotes of the hyperbola

 

SECTION –B

Answer any five:                                                                              5x 8 = 40

 

11. Find the n^th derivative of .
12. Find the angle at which the radius vector cuts the curve .
13. Show that the parabolas and intersect at right

angles.

14 Solve the equation 6x⁴-13x³-35x²-x+3= 0 given that is a root of it.

 

15. Solve the equation x⁴-2x³-21x²+22x+40= 0 given the roots are in A.P.

 

16. Prove that cos 6ө in terms of sin ө.
17. Prove that 32 sin⁶ө = 10 -15 cos2ө + 6 cos4ө -cos6ө.
18. If P and D are extremities of conjugate diameters of an ellipse, prove that the locus of middle point of PD is .

 

 

SECTION –C

Answer any two:                                                                                  2x 20 = 40

 

19. State and prove Leibnitz theorem and , prove that

.

 

20 a) Find the evolute of the parabola y²= 4ax.

1. b) Prove that p-r equation of r= a(1+ cos ө) is p² =  .

(10+10)

 

21 a) Find the real root of the equation  x³+6x-2 = 0.

1. b) If a+b+c+d = 0 , prove that

.

(10+10)

 

22 a) Separate tan^-1(x + iy) into real and imaginary parts.

 

1. b) Derive the polar equation of a conic.

(10+10)

 

 

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