## 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 2008 Algebra, Calculus And Vector Analysis Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

 XZ 8

THIRD SEMESTER – APRIL 2008

# MT 3501 – ALGEBRA, CALCULUS AND VECTOR ANALYSIS

Date : 26-04-08                  Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

SECTION – A

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

1. Evaluate .
2. If find the Jacobian of x and y with respect to r and .
3. Solve
4. Find the complete solution of

.

1. Find  at (2,0,1) for .
2. State Stoke’s theorem.
3. Evaluate ë (Sinh at).
4. Evaluate ë-1.
5. Find the sum of all divisors of 360.
6. Compute (720).

SECTION – B

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

1. By the changing the order of integration evaluate

1. Express  interms of Gamma function and evaluate .
2. Obtain the complete and singular solutions of .
3. Solve.
4. Find  if
5. Evaluate (i) ë   (ii) ë
6. Find  ë-1
7. Show that if x and y are both prime to the prime n, then xn-1-yn-1 is divisible by n. Deduce that x12-y12 is divisible by 1365.

SECTION – C

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

1. a) Evaluate over the tetrahedron bounded by the planes  and the coordinate planes.
1. b) Show that .
2. c) Using gamma function evaluate.
1. a) Solve
1. b) Solve the following by Charpit’s method

1. c) Solve
1. a) Verify Green’s theorem for  where C is the region bounded by y=x and y=x2.
1. b) Show that 18!+1 is divisible by 437.
1. a) State and prove Wilson’s theorem.
1. b) Solve given using Laplace

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## Loyola College B.Sc. Mathematics April 2008 Algebra, Calculus & Vector Analysis Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

 XZ 7

THIRD SEMESTER – APRIL 2008

# MT 3500 – ALGEBRA, CALCULUS & VECTOR ANALYSIS

Date : 26-04-08                  Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

PART – A

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

1. Show that G (n+1) = n G(n).
2. Show that
3. Form the partial differential equation by eliminating the arbitrary function from .
4. Solve:
5. Show that is solenoidal.
6. Show that curl
7. Find  ë .
8. Find ë .
9. Define Euler’s function.
10. Find the number of integer, less than 600 and prime to it.

PART – B

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

1. Show that.
2. Show that é
3. Solve:
4. Find the general integral of
5. Find the directional derivative of xyz-xy2z3 at(1,2,-1) in the direction

of

1. If find where C is the curve y=2x2 from (0,0) to (1,2).
2. Find ë  if

for

1. With how many zeros does         end.

PART – C

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

1. a) Evaluate
1. b) Evaluate over the region in the positive octant for which .
1. a) Find the complete integral of using charpits method.
1. b) If where is a constant vector and is the position vector of a point show that curl .
1. a) Verify Stoke’s theorem for

where S is the upper half of the sphere  and C its boundary.

1. b) Find (i) ëùand (ii) ëù
1. a) Solve using Laplace transforms

given that

and at t = 0.

1. b) Find the highest power of 11 in     .

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## Loyola College B.Sc. Mathematics April 2008 Algebra, Anal.Geo & Calculus – II Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

# XZ 54

SECOND SEMESTER – APRIL 2008

# MT 2500 – ALGEBRA, ANAL.GEO & CALCULUS – II

Date : 23/04/2008                Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

PART – A

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

1. Evaluate
2. Write the value of
3. Is exact?
4. Solve
5. State Raabe’s test.
6. Define uniform convergence of a sequence.
7. Find the Coefficient of in the expansion of
8. Write down the last term in the expansion of
9. Write the intercept and normal forms of the equation of a plane.
10. Find the Centre and radius of the sphere

PART – B

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

1. Evaluate
2. Solve
3. Test the Convergence of
4. Find the sum to infinity of the series
5. Sum the series
6. If a, b, c denote three Consecutive integers, show that
1. The foot of the perpendicular drawn form the origin to the plane is (12,-4,-3); find the equation of the plane.
2. Find the equation to the sphere through the four points (0,0,0), (a,0,0), (0,b,0), (0,0,c) and determine its radius.

PART – C

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

1. a) Evaluate
1. b) Find the area of the cardioid (12+8)
1. a) Prove that the series

is convergent if and

1. b) Sum the series
1. Find the image of the point (1,3,4) in the plane . Hence prove that the image of the line is .
2. Through the circle of intersection of the sphere and the plane two spheres and  are drawn to touch the place . Find the equations of the spheres.

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## Loyola College B.Sc. Mathematics Nov 2008 Algebra, Calculus And Vector Analysis Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034        LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034B.Sc. DEGREE EXAMINATION – MATHEMATICSTHIRD SEMESTER – November 2008MT 3501 – ALGEBRA, CALCULUS AND VECTOR ANALYSIS
Date : 06-11-08 Dept. No.         Max. : 100 Marks                 Time : 9:00 – 12:00                                               PART – A        (10 × 2 = 20 marks)
1. Evaluate   .2. What is  ?3. Find the complete integral of  q = 2yp2
4. Write down the complete integral of  z = px + qy + pq.
5. Find the constant k, so that the divergence of the vector   is zero.
6. State Gauss Divergence theorem.
7. Find L(cos23t).8. Find  .9. Find Φ(360).
10. Find the highest power of 5 in 79!

PART – B         (5 × 8 = 40 marks)
11. Change the order of integration and evaluate  .12. Prove that β(m,n+1 )+ β(m+1,n) = β(m,n).
13. Solve p tanx + q tany = tanz.
14. If   are irrotational, prove that
(a)   is  solenoidal.
(b)Find the unit vector normal to the surface   z = x2 + y2 – 3 at (2,-1,2).                              (4+4)15. Evaluate   by Stokes Theorem where   & C is the boundary of the        triangle with vertices (0,0,0), (1,0,0) and (1,1,0).
16. Find  (a) L(te-t sint).
(b)L(sin3t cosh2t).
17. Find  .18. (a) If N is an integer, prove that N5-N is divisible by 30.                                                            (6+2)
(b)State Fermat’s Theorem.
PART – C        (2 × 10 = 20 marks)
19. (a) Evaluate   over the positive octant of the sphere x2+y2+z2 = a2
(b)Establish β(m,n) =  .                                                                                 (10+10)
20. (a) Solve  .
(b) Solve by Charpit’s Method, pxy + pq + qy = yz.                                                           (10+10)
21. (a) Verify Green’s theorem for   where C is the boundary of the              region  x=0, y=0, x+y=1.
(b) Evaluate   .                                                                                         (10+10)22. (a) Using Laplace Transform, solve   given that y(0)=1, y`(0)=0..
(b) Using Wilson’s Theorem, prove that 10!+111   0 mod 143.                                               (12+8)

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## Loyola College B.Sc. Mathematics Nov 2008 Algebra, Calculus & Vector Analysis Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

# AB 07

THIRD SEMESTER – November 2008

# MT 3500 – ALGEBRA, CALCULUS & VECTOR ANALYSIS

Date : 06-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

SECTION-A

Answer all questions:                                                                        (10 x 2=20)

1. Evaluate .
2. If .
3. Obtain the partial differential equation by eliminating
4. Solve
5. If j =j  at  (1, -2, -1).
6. State Stoke’s theorem.
7. Find .
8. Find .
9. Find the number and sum of all the divisors of 360.
10. Find the number of integers less that 720 and prime to it.

SECTION-B

Answer any five questions:                                                              (5 x 8=40)

1. Evaluate  where R is the region bounded by the curves and .
2. Express interms of Gamma function and evaluate .
3. Solve .
4. Solve .
5. Show that .
6. (a) Find .

(b) Find .

1. Find .
2. Show that (18) is divisible by 437.

SECTION-C

Answer any two questions:                                                                          (2 x 20=40)

1. (a) Change the order of integration  and evaluate the integral.

(b) Evaluate  taken over the volume bounded by the plane .

(c) Evaluate .

1. (a) Solve

(b) Find a complete integral of .

1. Verify Gauss divergence theorem for  for the cylinderical region S given by .
2. Solve .

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## Loyola College B.Sc. Mathematics April 2009 Algebra, Calculus & Vector Analysis Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE Examination – Mathematics

Third Semester – OCT/NOV 2010

MT 3501/MT 3500 – Algebra, Calculus and Vector Analysis

Date & Time:                                Dept. No.                                                    Max. : 100 Marks

PART – A

Answer ALL questions.                                                                                                  (10 ´ 2 = 20)

1. Evaluate
2. Find the Jacobian of the transformation x = u (1 + v) ; y = v (1 + u).
3. Find the complete solution of z = xp + yq + p2 – q2.
4. Solve
5. For , find div at (1, -1, 1)
6. State Green’s theorem.
7. What is L(f¢¢ (t))?
8. Compute
9. Find the sum and number of all the divisors of 360.
10. Define Euler’s function f(n) for a positive integer n.

PART – B

Answer any FIVE questions                                                                                          (5 ´ 8 = 40)

1. Evaluate by changing the order of the integration.
2. Express in terms Gamma functions.
3. Solve z2( p2+q2 + 1 ) = b2
4. Solve p2 + q2 = z2(x + y).
5. Find
6. Find
7. Prove that
8. Show that 18! + 1 is divisible by 437

PART – C

Answer any THREE questions.                                                                               (2 ´ 20 =40)

1. (a) Evaluate  taken through the positive octant of the sphere x2 + y2 + z2 = a2.

(b)  Show that

1. (a) Solve (p2 + q2) y = qz.

(b)  Solve (x2 – y2)p + (y2 – zx)q = z2 – xy

1. (a) Verify Gauss divergence theorem for taken over the region bounded by the planes x = 0, x = a, y = 0 y = a, z = 0 and z = a.

(b)  State and prove Fermat’s theorem.

1. (a) Using Laplace transform solve  given that .

(b)  Show that if n is a prime and r < n, then (n – r)!  (r – 1)! + (-1)r – 1 º 0 mod n.

## Loyola College B.Sc. Mathematics April 2011 Algebra, Calculus And Vector Analysis Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – APRIL 2011

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

Date : 12-04-2011              Dept. No.                                                    Max. : 100 Marks

Time : 1:00 – 4:00

PART – A

Answer ALL questions.                                                                                                 (10 ´ 2 = 20)

1. Evaluate
2. Find when u = x2 – y2; v = x2 + y2
3. Solve
4. Find the complete integral of z = px + qy +p2q2
5. Find grad f if f = xyz at (1, 1, 1)
6. Evaluate divergence of the vector point function
7. Find L[sin2 2t]
8. Find
9. Find the sum of all divisors of 360.
10. Find the remainder when 21000 divisible by 17.

PART – B

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

1. Change the order of integration and evaluate
2. Express in terms of Gamma functions and evaluate
3. Solve p2 + pq = z2
4. Solve xp + yq = x
5. Show that the vectoris irrotational.
6. Evaluate: (a) L[cos 4t sin 2t]                   (b) L[e-3t sin2t]
7. Find
8. Show that 18! + 1 is divisible by 437.

PART – C

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

1. (a) Evaluate where the region V is bounded by x + y+ z = a (a > 0),
x = 0; y = 0; z = 0
.

(b)  Evaluate  where R is the region in the positive quadrant for which
x + y £ 1.

(c)  Show that

1. (a) Solve (x2 + y2 + yz)p + (x2 + y2 – xz)q = z(x+y)

(b)  Find the complete integral and singular integral of p3 + q3 = 8z

1. (a) Solve y¢¢ + 2y¢ – 3y = sin t given that y(0) = y¢(0) = 0

(b)  State and prove the Weirstrass inequality.

1. (a) State and prove Wilson’s theorem.

(b)  Verify Green’s theorem in the XY plane for where C is the closed curve in the region bounded by y = x; y = x2.

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## Loyola College B.Sc. Mathematics April 2011 Algebra, Anal.Geo & Calculus – II Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

SECOND SEMESTER – APRIL 2011

# MT 2501/MT 2500 – ALGEBRA, ANAL.GEO & CALCULUS – II

Date : 08-04-2011              Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

PART – A

Answer ALL questions:                                                                                                     (10 x 2 = 20)

1. Evaluate .
2. Evaluate .
3. Solve: .
4. Solve: .
5. Prove that the series is convergent.
6. Test for convergency the series .
7. Find the general term in the expansion of .
8. Prove that the coefficient of in the expansion of is .
9. Find the equation of the sphere which has its centre at the point and touches the

plane  .

1. Find the distance between the parallel planes and

PART – B

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

1. Prove that .
2. If ( n being a positive integer), prove that .

Also evaluate  and .

1. Solve: .

1. Solve
2. Test for convergency and divergency the series
3. Show that the sum of the series .
4. Show that if

1. Find the equation of the plane passing through the points

.

PART – C

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

1. a) Evaluate (10 marks)
2. b) Find the area and the perimeter of the cardiod .                    (10 marks)
3. a) Solve: . (10 marks)
4. b) Discuss the convergence of the series for

positive values of .                                                                                       (10 marks)

21.a) Show that the error in taking  as  an approximation to  is

approximately equal to  when  is small.                                                   (10 marks)

1. b) show that (10 marks)

1. a) A sphere of constant radius passes through the origin and meets the axes in A, B, C.

Prove that the centroid of the triangle ABC lies on the sphere

(10 marks)

1. b) Find the shortest distance between the lines

.

Also find the equation of the line of shortest distance.                              (10 marks)

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## Loyola College B.Sc. Mathematics April 2012 Algebra,Analytical Geometry And Calculus Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc., DEGREE EXAMINATION – MATHEMATICS

SECOND SEMESTER – APRIL 2012

MT 2501/MT2500- ALGEBRA,ANALYTICAL GEOMETRY AND CALCULUS

Date: 16-04-2012           Dept. No.                                                         Max. : 100 marks

Time: 9.00 – 12.00

PART-A

1. Evaluate .
2. Find x dx.
3. Define exact differential equations.
4. Solve (D22D + 1) y = 0.
5. Show that the series is convergent.
6. State Cauchy root test for convergence of a series.
7. Find the coefficient of in the expansion of 1 +  +  +  + …
8. Prove that
9. Find the direction cosines of the line joining the points (3,-5,4) and (1,-8,-2).
10. Find the angle between the planes and .

PART –B

1. Evaluate
2. Find the surface area of the solid formed by revolving the cardiodabout the initial line.
3. Solve
4. Solve .
5. Examine the convergence of
6. Assuming that the square and the higher powers of x may be neglected

show that

1. Sum to infinity the series
2. Find the shortest distance between the linesand and the equation of the line.

PART – C

ANSWER ANY TWO QUESTIONS:                                                                         (2×20= 40)

1. (a) Evaluate.

(b)  Find the length of the curve  between the points given by and.

1. (a) Solve

(b) Solve the following equation by the method of variation of parameter:

.

1. (a) Test the convergence of the series

(b)  Find the equation of the sphere passing through the points (2,3,1),(5,-1, 2),
(4,3,-1) and (2,5,3).

1. (a) Show that

(b)  Sum the series

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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, Calculus And Vector Analysis Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – NOVEMBER 2012

# MT 1503 – ANALYTICAL GEOMETRY OF 2D,TRIG. & MATRICES

Date : 10/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

PART – A

Answer all questions:                                                                                           (10 x 2 = 20)

1. Write down the expression of cos in terms of cosθ and sinθ.
2. Give the expansion of sinθin ascending powers of θ.
3. Express sin ix and cosix in terms of sin hx and coshx.
4. Find the value of log(1 + i).
5. Find the characteristic equation of A = .
6. If the characteristic equation of a matrix is , what are its eigen values?
7. Find pole of lx + my + n = 0 with respect to the ellipse
8. Give the focus, vertex and axis of the parabola
9. Find the equation of the hyperbola with centre (6, 2), focus (4, 2) and e = 2.
10. What is the polar equation of a straight line?

PART – B

Answer any five questions.                                                                                 (5 X 8 = 40)

1. Expandcos in terms of sinθ .
2. If sinθ = 0.5033 show thatθ is approximately .
3. Prove that .
4. If tany = tanα tanhβ ,tanz = cotα tanhβ, prove that tan (y+z) = sinh2βcosec2α.
5. Verify Cayley Hamilton theorem for A =
6. Prove that the eccentric angles of the extremities of a pair of semi-conjugate diameters of an ellipse differ by a right angle.
7. Find the locus of poles of all tangents to the parabola with respect to

1. Prove that any two conjugate diameters of a rectangular hyperbola are equally inclined to the asymptotes.

PART – C

Answer any two questions:                                                                                (2 X 20 = 40)

1. (i) Prove that .

(ii) Prove that .

1. (i) Prove that if

(ii) Separate into real and imaginary parts tanh(x + iy).

1. Diagonalise A =
2. (i) Show that the locus of the point of intersection of the tangent at the extremities of a pair of

conjugate diameters of the ellipse is the ellipse

(ii) Show that the locus of the perpendicular drawn from the pole to the tangent to the circle r = 2a

cosθ  isr = a(1+cosθ).

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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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## Loyola College B.Sc. Mathematics Nov 2012 Algebra, Anal.Geo & Calculus – II Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – NOVEMBER 2012

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

Date : 08/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

PART – A

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

1. Write the nth derivative of
2. If y = a show that
3. Define the evolute of a curve.
4. Find the p-r equation of the curve r = a sin q.
5. Determine the quadratic equation having 3 – 2 i as a root.
6. Diminish the roots of by 2.
7. Show that
8. Express in locus of logarithmic function.
9. Define a rectangular hyporbola.
10. Write down the angle between the asymptotes of the hyperbola

PART – B

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

1. Show that in the parabola the subtangent at any point is double the abscissa and the subnormal is a constant.
2. Find the radius of curvature at the point ‘O’ on
3. Show that if the roots of
4. Find the p-r equation of the curve with respect to the focus as the pole.
5. Separate into real and uniaguinary parts.
6. Find the sum of the series
7. Find the locus of poles of ale Laugets to with respect to
8. Derive the polar equation of a comic.

PART – C

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

1. a) If prove that
2. b) Show that r = a sec2 and r = b cosec2 intersect at right angles.
3. a) Find the minimum value of
4.        b) Find the radius of  curvature of .
5. a) Solve: given that the roots are in geometric progression.

1. b) Solve: .

1. a) Express cos8q in locus of power of sinq.

1. b) If e1 and e2 are the eccentricities of a hyperbola and its conjugate show that .

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