## Loyola College B.Sc. Mathematics Nov 2003 Algebra, Anal. Geometry, Calculus & Trigonometry Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – MATHEMATICS

# MT – 1500/MAT 500 – ALGEBRA, ANAL. GEOMETRY, CALCULUS & TRIGONOMETRY

01.11.2004                                                                                                           Max:100 marks

1.00 – 4.00 p.m.

SECTION – A

Answer ALL Questions.                                                                                (10 x 2 = 20 marks)

1. If y = sin (ax + b), find yn.
2. Show that in the parabola y2 = 4ax, the subnormal is constant.
3. Prove that cos h2x = cos h2x + sin h2
4. Write the formula for the radius of curvature in polar co-ordinates.
5. Find the centre of the curvature xy = c2 at (c, c).
6. Prove that .
7. Form a rational cubic equation which shall have for roots 1, 3 – .
8. Solve the equation 2x3 – 7x2 + 4x + 3 = 0 given 1+is a root.
9. What is the equation of the chord of the parabola y2 = 4ax having (x, y) as mid – point?
10. Define conjugate diameters.

SECTION – B

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

1. Find the nth derivative of cosx cos2x cos3x.
2. In the curve xm yn = am+n , show that the subtangent at any point varies as the abscissa of the point.
3. Prove that the radius of curvature at any point of the cycloid

x = a (q + sin q) and  y = a  (1 – cos q) is 4 a cos .

1. Find the p-r equation of the curve rm = am sin m q.
2. Find the value of a,b,c such that .
3. Solve the equation

6x6 – 35x5 + 56x4 – 56x2 + 35x – 6 = 0.

1. If the sum of two roots of the equation x4 + px3 + qx2 + rx + s = 0 equals the sum of the other two, prove that p3 + 8r = 4pq.
2. Show that in a conic, the semi latus rectum is the harmonic mean between the segments of a focal chord.

SECTION -C

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

1. a) If y = , prove that

(1 – x2) y2 – xy1 – a2y = 0.

Hence show that (1 – x2) yn+2 – (2n +1) xyn+1 – (m2 + a2) yn = 0.                     (10)

1. Find the angle of intersection of the cardioid r = a (1 + cos q) and r = b (1 – cos q).

(10)

1. a) Prove that  = 64 cos6 q – 112 cos4q + 56 cos2q –                                       (12)

1. b) Show that (8)
2. a) If  a + b + c + d = 0, show that

.                               (12)

1. b) Show that the roots of the equation x3 + px2 + qx + r = 0 are in Arithmetical

progression if 2 p3 – 9pq + 27r = 0.                                                                             (8)

1. a) 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.                     (8)

1. b) P and Q are extremities of two conjugate diameters of the ellipse and S is

a focus.  Prove that PQ2 – (SP – SQ)2 = 2b2.                                                              (12)

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## Loyola College B.Sc. Mathematics April 2012 Algebra, Analy. Geo., Calculus & Trigonometry Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – APRIL 2012

# MT 1500 – ALGEBRA, ANALY. GEO., CALCULUS & TRIGONOMETRY

Date : 28-04-2012              Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

PART – A

Answer ALL the questions:                                                                      (10 X 2 = 20 Marks)

1. Find the nth derivative of .
2. Find the slope of the straight line .
3. Write the formula for the radius of curvature in Cartesian form.
4. Define Cartesian equation of the circle of the curvature.
5. If ,are the roots of the equation x3+px2+qx+r=0. Find the value of .
6. Diminish the roots x4+x3-3x2+2x-4 =0 by 2.
7. Evaluate
8. Prove that
9. Define Pole and Polar of a ellipse.
10. In the hyperbola 16x2-9y2 = 144, find the equation of the diameter conjugate to the diameter x =2y.

PART – B

Answer any FIVE questions:                                                                                  (5 X 8 = 40 Marks)

1. Find the nth derivative of .
2. Find the angle between the radius vector and tangent for the curve at

.

1. Solve the equation x3-4x2-3x+18=0 given that two of its roots are equal.
2. Solve the equation x4-5x3+4x2+8x-8=0 given that 1-is a root.
3. Expand in terms .
4. Separate real and imaginary parts .
5. P and Q are extremities of two conjugate diameters of the ellipse and S is a focus. Prove that
6. The asymptotes of a hyperbola are parallel to 2x+3y=0 and 3x-2y =0 . Its centre is at (1,2) and it passes through the point (5,3). Find its equation and its conjugate.

PART – C

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

1. (a) If , show that

(b) Prove that the sub-tangent at any point on                is constant ant the subnormal is

(10 +10)

1. (a) Find the radius of curvature at any point on the curve

(b) Show that the evolute  of the cycloid   is another

cycloid .                                                                                                                     (10+10)

1. (a) Solve 6x5+11x4-33x3-33x2+11x+6=0.

(b)  Find by Horner’s method, the roots of the equation  which lies between 1 and 2

correct to two decimal places.                                                                                                                             (10+10)

1. (a) Prove that

(b) Prove that the Product of the perpendicular drawn from any point on a hyperbola to its

asymptotes is constant.                                                                                                   (10+10)

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## Loyola College B.Sc. Mathematics Nov 2012 Algebra, Analy. Geo., Calculus & Trigonometry Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc., DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – NOVEMBER 2012

# MT 3501/3500 – ALGEBRA, CALCULUS AND VECTOR ANALYSIS

Date : 02-11-2012              Dept. No.                                        Max. : 100 Marks

Time : 9.00 – 12.00

PART – A

ANSWER ALL THE QUESTIONS:                                                                                                (10 x 2 =  20)

1. Evaluate .
2. Evaluate .
3. Eliminate the arbitrary constants from .
4. Find the complete solution for
5. Find , if .
6. Prove that div , where is the position vector.
7. Find L(Sin2t).
8. Find .
9. Find the number and sum of all the divisors of 360.
10. State Fermat’s theorem.

PART – B

ANSWER ANY FIVE QUESTIONS:                                                                                  (5 x 8 = 40)

1. Change the order of integration and evaluate the integral  .
2. Express  in terms of Gamma functions and evaluate the integral .
3. Solve
4. Solve
5. Find .
6. Find .
7. Show that
8. Show that is divisible by 22.

PART – C

ANSWER ANY TWO QUESTIONS                                                                                               (2x 20 = 40)

1. (a) Evaluate taken over the positive quadrant of the circle .

(b)  Prove that

1. (a) Solve

(b) Solve (y+z)p + (z+x)q = x+y.

21.(a)  Verify Stoke’s theorem for  taken over the upper half surface of

the  sphere  x2+y2 +z2 = 1, z 0 and the boundary curve C, the x2+y2  = 1, z=0.

(b)  State and prove Wilson’s Theorem.

1. Using Laplace transform solve the equation given y(0) = 0 , y1(0)= -1.

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