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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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034B.Sc. DEGREE EXAMINATION – MATHEMATICSFIRST SEMESTER – APRIL 2007MT 1500 – ALG.,ANAL.GEOMET. CAL. & TRIGN. – I
Date & Time: 24/04/2007 / 1:00 – 4:00 Dept. No. Max. : 100 Marks
SECTION –AAnswer all:                                                                              2 x 10 = 20
1. If y = a cos5x + b sin5x, show that  . 2. Write down the nth derivative of eax. 3. What is the formula for radius of curvature in parametric form.                  4. Find the sub tangent and the sub normal for y = 3×3. 5. If x = sin 2ө and y = cos 2ө, find  . 6. Determine the quadratic equation having 3-2i  as one of its roots. 7. Derive the relation sin ix = i sinh x. 8. Separate into real and imaginary parts for cos (x+iy). 9. Define conjugate diameters.10. Write the polar form of the conic.
SECTION –BAnswer any five:                                                                              5x 8 = 40
11. Find the nth derivative of sin2x sin4x sin6x.12. Find the angle of intersection of the cardioids r = a(1+cosө) and r = b(1-cosө).13. Find the lengths of the sub tangent and the sub normal at the point (a, a)      for the curve y = x3+ 3x+4.  14. Show that the roots of the equation x3+px2+qx+r =0 are in A.P if    2p3-9pq+27r =0.15. Solve the equation 6×5+11×4-33×3-33×2+11x+6= 0.
16. Prove that  = 7 – 56 sin2ө + 11 2 sin4ө – 64 sin6ө.                        17. Prove that 32 cos6ө = cos6ө + 6 cos4ө +15 cos2ө + 10.                    18. Prove that the product of the focal distances of a point on an ellipse is equal to the           square of the semi-diameter which is conjugate to the diameter through the point.

 

 

 

SECTION –CAnswer any two:                                                                              2x 20 = 40
19. State and prove Leibnitz theorem and prove that (1-x2)y2 –xy1+m2y = 0 and       (1-x2) yn+2 –(2n+1)xyn+1+(m2-n2)yn = 0  for  y = sin( msin-1x).                                                                                                                                               (P.T.O)20 a) Find the evolute of the ellipse  .     b) Find the p-r equation of rm = am sinm ө.                                                (10+10)
21 a) Find the equation whose roots are the roots of the equation          x4-x3-10×2+4x+24 = 0 increased by 2 and hence solve the equation.      b) Find the sum of the fourth power of the roots of the equation             x3-2×2+x+1 = 0.                                                                                   (10+10)                                                                                          22 a) Prove that  .
b) Prove that the tangent to a rectangular hyperbola terminated by its asymptotes          is bisected at the point of contact and encloses a triangle of constant area.                                                                                                                                                                                                                                                                                       (10+10)

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

XZ 1

FIRST SEMESTER – APRIL 2008

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

 

 

 

Date : 07/05/2008                Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

PART – A

Answer ALL the questions.                                                             (10 x 2 = 20 marks)

1. Find the n^th derivative of e^ax.
2. Prove that the subtangent to the curve is of constant length.
3. Find the coordinates of the center of curvature of the curve at .
4. What is the curvature of a (i) circle (ii) straight line.
5. Determine the quadratic equation having (3-2i) as a root.
6. If are the roots of the equatim . Show that .
7. Prove that .
8. Prove that .
9. Find the pole of the line  with respect to the parabola y²=4ax.
10. If are the eccentricities of a hyperbola and its conjugate,

prove that .

 

PART – B

Answer any FIVE questions.                                                           (5 x 8 = 40 marks)

 

1. At which point is the tangent to the curve  parallel to the line
2. Final the angle at which the radius vector cuts the curve .
3. Prove that the radius of curvature at any point of the cycloid

and is .

1. Show that if the roots of the equation are in arithmetic progression then .
2. If  show that .

 

1. If  prove that

1. i)
2. ii)

1. Find the locus of poles of chords of the parabola which subtend a right angle at the focus.
2. Find the equation of a rectangular hyperbola referred to its asymptotes as axes.

 

PART – C

Answer any TWO questions.                                                           (2 x 20 = 40 marks)

1. a) If prove that and

.

1. b) Find the (p,r) –equation of the curve and hence show that the radius of curvature at any point varies as the cube of the focal distance.

1. a) Find the equation of the evolute of the parabola .

1. b) Solve .

1. a) Find the real root of to two places of decimals using Horner’s method.

1. b) Evaluate .

1. a) Prove that .

1. b) Derive the equation of the tangent at the point whose rectorial angle is on the conic .

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AB 01

   B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – November 2008

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

 

 

 

Date : 10-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

PART – A

 

Answer all questions:                                                                        (10 x 2 = 20 marks)

 

1. If find .
2. In the curve , prove that the subtangent is of constant length.
3. Write the formula for radius of curvature in parametric form.
4. Define evolute.
5. If is a root of find the other root.
6. Define a reciprocal equation.
7. Prove that .
8. Find .
9. Find the equation of the chord of the parabola having the mid point at .
10. Define a rectangular hyperbola.

 

PART – B

 

Answer any FIVE questions:                                                           (5 x 8 = 40 marks)

 

11. Show that in the curve the subnormal varies as the cube of the ordinate.
12. Show that the radius of curvature at any point of the catenary .
13. If where  find the minimum value of u.
14. Find the radius of curvature of the cardivid
15. Solve the equation given that  is a root of it.
16. Solve the reciprocal equation
17. Express interms of .
18. Derive the polar equation of a conic.

 

PART – C

 

Answer any TWO questions:                                                           (2 x 20 = 40 marks)

 

19. a) If show that (1-
20. b) Find the evolute of the parabola
21. a) Solve given that it has two pairs of equal roots.
22. b) Find the positive root of the equation correct to two places of decimals, using Horner’s method.
23. a) Prove that 64
24. b) Prove that
25. a) Sum the series .
26. b) Show that in a conic the semilatus rectum is the harmonic mean between the segments of a focal chord.

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

   B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – NOVEMBER 2010

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

 

 

 

Date : 10-11-10                     Dept. No.                                                 Max. : 100 Marks

Time : 1:00 – 4:00

 

PART – A

Answer ALL the questions.                                                                                                (10 x 2 = 20)

 

1. Find y_n when
2. Show that, in the parabola y²=4ax,  the subtangent at any point is double the abscissa.
3. Find the radius of curvature of xy=30 at the point (3,10).
4. Define evolutes.
5. Form equation given that 3+2c is a root.
6. If α,β,γ, are the roots of the equation x³+px²+qx+r=0 find the value of ∑α².
7. Evaluate
8. Prove that cosh =
9. Find the polar of the point (3,4) with respect the parabola y²=4ax.
10. Define conormal and concyclic points.

 

PART  –  B

 

Answer any FIVE questions.                                                                                               (5 x 8 = 40)

 

11. Show that in the curve hy²=(x+a)³ the square of the subtangent varies as the subnormal.
12. Find the radius of curvature at the point ‘t’ of the curve

x=a(cost+tsint); y=a(sint-tcost).

13. Find the coordinates of the centre of curvature at given point on the curve y=x²;
14. Solve the equation x⁴+2x³-5x²+6x+2=0 given that 1+is a root of it.
15. Find the real root of the equation x³+6x-2=0 using Horner’s method.
16. Expand sin³θ cos⁴θ in terms of sines of multiples of θ.
17. If sin(θ+iφ) =tanα + isecα , prove that cos2 θ cosh2φ =3.
18. Show that the area of the triangle formed by the two asymptotes of the rectangular hyperbola xy=c² and the normal at (x₁,y₁) on the hyperbola is .

PART – C

Answer any TWO questions.                                                                                            (2 x 20 = 40)

 

19. a) Prove that if y=sin(msin^-1x), then (1-x²)y₂-xy₁₊m²y=0.
20. b) Show that the evolute of the cycloid x=a(θ – sinθ);y=a(1-cosθ) is another cycloid.

 

20. a) solve 2x⁶-9x⁵+10x⁴-3x³+10x²-9x+2=0.
21. b) If α is a root of the equation x³+x²-2x-1=0 show that α² -2 is also a root.

 

21. a) if u=log tan show that tanh   = tan  and θ = -i log tan
22. b) sum to infinity the series
23. a) Find the locus of mid points of normal chords to the ellipse
24. b) Find the polar of the point (x₁, y₁) with respect to the parabola y²=4ax.

 

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