# LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034.

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FourTH SEMESTER – APRIL 2003

## ST 4201 / sTA 201  –  MATHEMATICAL STATISTICS

28.04.2003

9.00 – 12.00                                                                                                     Max : 100 Marks

### PART – A                                       (10´ 2=20 marks)

1. Two dice are thrown. What is the probability that the sum of the numbers on the two dice is eight?
2. The probability that a customer will get a plumbing contract is and the probability that he will get an electric contract is 4/9. If the probability of getting at least one is 4/5,determine the probability that he will get both.
3. Consider 2 events A and B such that and . Verify whether the given statement is true (or) false. .
4. Define i)  independent events and ii)  mutually exclusive events.
5. State any four properties of a distribution function.
6. The random variable X has the following probability function
 X = x 0 1 2 3 4 5 6 7 P (X=x) 0 k 2k 2k 3k k2 2k2 7k2+k

Find k.

1. Let f (x) =

0    ;   else where

Find E(X).

1. Let X ~ B (2, p) and Y~B (4, p). If P , find P.
2. Define consistent estimator.
3. State Neyman – Pearson lemma.

### PART – B                                         (5´ 8=40 marks)

1. A candidate is selected for three posts. For the first post three are three candidates, for the

second there are 4 and for the third there are 2. What are the chances of his getting

1. i) at least one post and  ii)  exactly one post?
2. Three boxes contain 1 white, 2 red, 3 green ; 2 white, 3 red, 1 green and 3 white, 1 red, 2 green balls. A box is chosen at random and from it 2 balls are drawn at random. The balls so drawn happen to be white and red. What is the probability that they have come from the second box?
3. Find the conditional probability of getting five heads given that there are at least four heads, if a fair coin is tossed at random five independent times.
4. Derive the mean and variance of hyper-geometric distribution.
5. Let X be a random variable having the p.d.f

f(x) =

Find the m.g.f. of X and hence obtain the mean and variance of X.

1. If X is B(n,p), show that E= p and E.
2. Let X be  N(m,s2).  i)  Find b so that
3. ii) If P (X < 89) =0.90 and P(X < 94) =0.95, find m and s2.
4. If X and Y are independent gamma variates with parameters m and n respectively,

Show that  ~ .

### PART – C                                         (2´20=40 marks)

1. If the random variables x1 and x2 have the joint  p.d.f

f  (x1 ,x2) =

i ) find the conditional mean of X1 given  X2 and  ii)  the  correlation coefficient

between  X1 and X2.

1. a)  Find all the odd and even order  moments of Normal distribution.
2. Let (X,Y) have a bivariate normal distribution. Show that each marginal distribution

in normal.

1. a) Derive the p.d.f of F- variate with (n1,n2) d.f.
2. Find the g.f of exponential distribution.
• a) Let X1, X2, …. Xn  be a  random sample of size n from N (q,1) . Show that the sample

mean is an unbiased estimator of the parameter q.

1. Write a short note on:
2. i) null hypothesis ii) type I and type II errors iii)    standard error
3. iv) one -sided and two -sided tests.

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# LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034.

B.Sc. DEGREE EXAMINATION – MATHEMATICS

SECOND SEMESTER – APRIL 2003

## MT 2500 / MAT 501  –  ALGEBRA ANALYTICAL GEOMETRY, cALCULUS – II

23.04.2003

9.00 – 12.00                                                                                                    Max : 100 Marks

### PART – A                                       (10´ 2=20 marks)

Answer ALL questions. Each question carries TWO marks.

1. Prove that
2. Evaluate
3. State cauchy’s root test for convergence of a given series.
4. Show that
5. If Y= show that
6. Solve
7. Solve where
8. Evaluate
9. Find the equation to the plane through the point (3,4,5) and parallel to the plane
10. Find the equation of the sphere with centre (-1, 2, -3) and radius 3 units.

### Answer any FIVE questions. Each question carries EIGHT marks.

1. If n =   prove that   .
2. Evaluate .
3. Sum the series .
4. Sum the series .
5. Solve .
6. Solve .
7. Test for convergence of the series .
8. Find the perpendicular distance from P (3, 9, -1) to the line .

### PART – C                                         (2´20=40 marks)

Answer any TWO questions. Each question carries TWENTY marks.

1. a) Evaluate I =                                                                                           (10)
2. b) Find the reduction formulae for In =                                                  (10)
3. a) Solve by variation of parameter method.                                         (10)
4. b) Solve (10)
5. a) Sum the series (10)

1. b) Sum the series (10)
2. a) Show that the series  is convergent when k is greater than unity and

divergent when k is equal to or less than unity.                                                           (10)

1. Find the equation of the sphere which passes through the circle

and touch the plane            (10)

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# B.Sc. DEGREE EXAMINATION – MATHS / CHEMISTRY

II SEMESTER – NOVEMBER 2003

### PH 203 / 205 / 403 – GENERAL PHYSICS  II

08.11.2003                                                                                                                              100 Marks

## 1.00 – 4.00

PART – A

Answer All questions                                                                         (10 x 2 = 20 marks)

1. What is a Zone plate?
2. Give the geometry of a Nicol prism
3. Define specific rotatory power of an optically active substance
4. State Gauss’s law in differential form
5. Three capacitors of capacitance values 1 mF, 2 mF and 3 mF are arranged in series. What is the effective capacitance?
6. Define the ampere, the unit of current.
7. Distinguish between amplitude and frequency modulations.
8. What are the charge carriers in semiconductor devices?
9. Give the truth table of the NAND gate
10. List any four properties of X-rays.

#### PART – B

Answer any FOUR questions                                                        (4 x 7 ½  = 30 marks)

1. Prove the rectilinear propagation of light by Fresnel’s theory of half-period zones
2. Derive an expression for the loss of energy on sharing of charges between two capacitors.
3. Find the magnetic field at any point due to an infinitely long wire carrying current.
4. State and prove De Morgan’s theorems.
5. Discuss the theory and production of X-rays with a neat diagram.

#### PART – C

Answer any FOUR questions                                                      (4 x 12 ½ = 50 marks)

1. Explain the theory of production and of analysis of different types of polarized beams.
2. Using Gauss’s law, determine the intensity of electric field due to (i) a charged sphere and (ii) a line charge.
3. Derive an expression for the intensity of magnetic field along the axis of a current carrying circular coil.
4. Explain the working of a two-stage RC coupled amplifier with a circuit diagram. Also explain the frequency response of the amplifiers.
5. Discuss with necessary theory the working of Bragg spectrometer.

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# B.Sc. DEGREE EXAMINATION  –  MATHEMATICS

Fourth  SEMESTER  – NOVEMBER 2003

# ST 4201/STA 201 MATHEMATICAL  STATISTICS

14.11.2003                                                                                        Max: 100 Marks

9.00 – 12.00

SECTION A

Answer ALL the questions.                                                      (10 ´ 2 = 20 Marks)

1. Define an event and probability of an event.
2. If A and B any two events, show that P (AÇBC) = P(A) – P(AÇB).
3. State Baye’s theorem.
4. Define Random variable and p.d.f of a random variable.
5. State the properties of distribution function.
6. Define marginal and conditional p.d.fs.
7. Examine the validity of the given Statement “X is a Binomial variate with

mean 10 and S.D  4”.

1. Find the d.f of exponential distribution.
2. Define consistent estimator.

SECTION B                          (5 ´ 8 = 40 Marks)

1. An urn contains 6 red, 4 white and 5 black balls.  4 balls are drawn at random.

Find the probability that the sample contains at least one ball of each colour.

1. Three persons A,B and C are simultaneously shooting. Probability of A hit the

target is  ;  that for B is    and for C is  Find   i)  the probability that

exactly one of them will hit the target ii) the probability that at least one of them

will hit the target.

1. Let the random variable X have the p.d.f

Find P( ½ < X <  ¾) and    ii) P ( – ½ < X< ½).

1. Find the median and mode of the distribution

.

1. Find the m.g.f of Poisson distribution and hence obtain its mean and variance.

1. If X and Y are two independent Gamma variates with parameters m and g

respectively,  then show that    Z =  ~ b (m,g).

1. Find the m.g.f of Normal distribution.
2. Show that the conditional mean of Y given X is linear in X in the case of bivariate normal distribution.

## SECTION – C

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

1. Let X1and X2 be random variables having the joint p.d.f

Show that the conditional means are

(10+10)

1. If f (X,Y) has a trinomial distribution, show that the correlations between

X and Y is   .

1. i)    Derive  the p.d.f of ‘t’ distribution with ‘n’ d.f
2. ii) Find all odd order moments of Normal distribution.                       (15+5)
3. i) Derive the p.d.f of ‘F’ variate with (n1,n2) d.f                                     (14)

1. ii) Define   i)   Null and alternative Hypotheses                                      (2)
2. ii) Type I and Type II errors.                                                (2)

and         iii)   critical region                                                                   (2)

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## 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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