UNIVERSITY OF KERALA First Degree Programme in Mathematics Model Question Paper Semester III MM 1341 Methods of Algebra and Calculus - 1 Time: Three hours Total Weight : 30 All the first 16 questions are compulsory. They carry 4 weightages in all. 1. Find the units in Z 2. Find an integer n> 6 such that Z/nZ has zero divisors. 3. Find the order of [2] in Z/5Z. 4. Define kernel of a ring homomorphism. 5. If g ◦ is the exponent of a finite abelian group G then for any a in G , a g◦ = ...... 6. If H is a subgroup of a finite group G, then (order of H )×(index of H in G)=...... State whether True or False(Questions 7 and 8) 7. The set N of natural numbers is a subgroup of the group Z of integers under addition. 8. If R and S are integral domains then R × S is also an integral domain. 9. Find the unit vector oppositely directed to 6ˆ ı - 4ˆ +2 ˆ k. 10. If u.( v × w) = 3 find ( w × v).u 11. Determine whether the lines L 1 and L 2 are parallel. L 1 : x =5+3t , y =4 - 2t , z = -2+3t L 2 : x = -1+9t , y =4 - 6t , z =3+8t 12. Identify the quadric surface 4z = x 2 +4y 2 . 13. Find the equation for the surface that results when the circular paraboloid z = x 2 + y 2 is reflected about the plane y = z . 14. Convert the rectangular coordinates (4, -4 √ 3, 6) in to cylindrical coordinates. 15. Determine whether r(t)= te -t ˆ ı +(t 2 - 2t)ˆ + cos πt ˆ k is a smooth function of t. 16. If r(t)= t ˆ ı + 1 2 t 2 ˆ + 1 3 t 3 ˆ k is the position vector of a particle, find the velocity of the particle at time t = 1.
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
UNIVERSITY OF KERALA
First Degree Programme in Mathematics
Model Question Paper
Semester III MM 1341 Methods of Algebra and Calculus - 1
Time: Three hours Total Weight : 30
All the first 16 questions are compulsory. They carry 4 weightages in all.
1. Find the units in Z
2. Find an integer n > 6 such that Z/nZ has zero divisors.
3. Find the order of [2] in Z/5Z.
4. Define kernel of a ring homomorphism.
5. If g◦ is the exponent of a finite abelian group G then for any a in G , ag◦ = ......
6. If H is a subgroup of a finite group G, then (order of H)×(index of H in G)=......
State whether True or False(Questions 7 and 8)
7. The set N of natural numbers is a subgroup of the group Z of integers under addition.
8. If R and S are integral domains then R× S is also an integral domain.
9. Find the unit vector oppositely directed to 6ı− 4 + 2k.
10. If ~u.(~v × ~w) = 3 find (~w × ~v).~u
11. Determine whether the lines L1 and L2 are parallel.
L1 : x = 5 + 3t , y = 4− 2t , z = −2 + 3t
L2 : x = −1 + 9t , y = 4− 6t , z = 3 + 8t
12. Identify the quadric surface 4z = x2 + 4y2.
13. Find the equation for the surface that results when the circular paraboloid z = x2 + y2 is
reflected about the plane y = z.
14. Convert the rectangular coordinates (4,−4√
3, 6) in to cylindrical coordinates.
15. Determine whether ~r(t) = te−tı + (t2 − 2t) + cos πt k is a smooth function of t.
16. If ~r(t) = tı+ 12t2+ 1
3t3k is the position vector of a particle, find the velocity of the particle
at time t = 1.
Answer any 8 questions from among the questions 17 to 28.
These questions carry 1 weightage each.
17. Let M be the ring of 2×2 matrices with real entries under the addition and multiplication
of matrices. Show by examples that M is not commutative and that M has zero divisors.
18. Prove that a ring homomorphism f is one to one if and only if Ker(f) = {0}.
19. For any integer n, prove that n111 ≡ n(mod 11).
20. Let G = U13, the group of units of Z/13Z. Find the subgroup of G generated by [5].
21. Find all solutions of x2 ≡ 1(mod 35).
22. If a bug walks on the sphere x2 + y2 + z2 + 2x− 2y − 4z − 3 = 0 how close and how far
can it get from the origin?
23. Use vectors to prove that the diagonals of a parallelogram are perpendicular if the sides
are equal in length.
24. A force ~F = 4ı− 6 + k Newtons is applied to a point that moves a distance of 15 metres
in the direction of the vector ı + + k. How much work is done?
25. Show that the lines L1 and L2 intersect and find their point of intersection.
L1 : x = 2 + t, y = 2 + 3t, z = 3 + t
L2 : x = 2 + t, y = 3 + 4t, z = 4 + 2t
26. Find the arc length of the graph of ~r(t) = 3 cos tı + 3 sin t + tk. (0 ≤ t ≤ 2π)
27. Find the displacement and distance travelled if ~r(t) = t2ı + 13t3k over the time interval
1 ≤ t ≤ 3.
28. Find two elevation angles that will enable a shell, fired from ground level with a muzzle
speed of 800 ft./sec. to hit a ground target 10,000 ft. away.
Answer any 5 questions from among the questions 29 to 36.
These questions carry 2 weightages each.
29. Prove that Z/mZ is a field if and only if m is a prime.
30. Show thatn5
5+
n3
3+
7n
15is an integer for any n.
31. Find the least non negative a congruent to 569(mod 71). Verify that 5a ≡ 1(mod 71).
32. Using Cayley’s theorem describe a group homomorphism from U8 to S4.
33. (a) Find the points where the curve ~r(t) = t ı + t2 − 3t k intersect the plane
2x− y + z = −2
(b) Find the acute angle that the tangent line to the curve makes with the normal to
the plane at each point of intersection.
34. (a) Find the arc length parametrization of the line x = 1 + t , y = 3 − 2t , z = 4 + 2t
that has the same direction as the given line and has reference point (1,3,4).
(b) Find the point on the line in part (a), that is 25 units from the reference point in
the direction of increasing parameter.
35. Find the point(s) on the curve 4x2 + 9y2 = 36 where the radius of curvature is minimum.
36. (a) Prove that each plannet moves in a plane through the centre of the sun.
(b) Using Newton’s law of universal gravitation and second law of motion, show that
the acceleration vector ~a =−GM
r3~r where ~r is the position vector of the particle
moving in a central force field.
Answer any two questions from among the questions 37 to 39.
These questions carry 4 weightages each.
37. (a) State and prove the Chinese Reminder Theorem.
(b) Solve
x ≡ 7(mod 11)
x ≡ 6(mod 8)
x ≡ 10(mod 15)
Find the smallest nonnegative solution.
38. (a) State and prove the Lagrange’s Theorem on finite groups.
(b) If a is any element of a finite group G, prove that the order of a divides the number
of elements of G.
39. Let L1 and L2 be the lines:
L1 : x = 1 + 4t , y = 5− 4t , z = −1 + 5t
L2 : x = 2 + 8t , y = 4− 3t , z = 5 + t
(a) Are the lines parallel?
(b) Do the lines intersect?
(c) Find the distance between the lines.
UNIVERSITY OF KERALA
First Degree Programme in Chemistry
Model Question Paper
Semester III Complementary Course for ChemistryMM 1331.2 Mathematics - III
Theory of Equations and Vector Analysis
Time: Three hours Total Weight : 30
All the first 16 questions are compulsory. They carry 4 weightages in all.
A 1. If α, β, γ... are the roots of f(x) = 0 then the equation whose roots are kα, kβ, kγ... is
B 5. Form a rational cubic equation whose roots are 1, 3−√
2 i.
6. Find the arc length parametrization of the curve x = t, y = t, z = t that has the same
direction as the given curve and has the reference point (0,0,0).
7. Find the unit tangent vector to the graph ~r(t) = ln t ı + t at t = e.
8. Find the directional derivative of f(x) =√
xy ey at P(1,1) in the direction of the
negative y-axis.
C 9. Suppose that during a certain time interval a proton has a displacement of ∆r =
0.7ı + 2.8 − 1.5k and its final positin vector is known to be ~r = 3.6k. What was the
initial position vector of the proton?
10. The value of (∇× ~r,∇ · ~r).(a) (0, 3) (b) (0,-3) (c) (3,0) (d) (-3,0)
11. The value of
∫C
f(x, y)dx along any line segment C parallel to the y-axis is
(a) 1 (b) -1 (c) 0 (d) No conclusion can be drawn.
12. Using Stoke’s theorem find the value of
∫C
~r · d~r where C is a simple closed curve in
2-space.
D 13. If V is the volume enclosed by a surface S, then find the value of
∫∫C
~r · ~ndS.
14. Determine the constant a so that the vector ~F = (x + 2y)ı + (3y + 2z) + (2y − az)k is
solenoidal.
15. Give an example, from Physics, of an inverse-square field ~F (~r) in 3-space.
16. Evaluate ∇(~a · ~r) if ~a is a constant vector and ~r is the position vector of an arbitrary
point in 3-space.
(4× 1 = 4 weights)
Answer any 8 questions from among the questions 17 to 28.
These questions carry 1 weightage each.
17. Transform x3 − 9x2 + 5x + 12 = 0 in to an equation lacking the second term.
18. Transform x3 − 7
3x2 +
11
36x− 25
72= 0 in to an equation with intgral coefficients.
19. Show that x5 − 2x2 + 7 = 0 has at least two imaginary roots.
20. Solve x4 − 8x3 + 17x2 − 8x + 1 = 0.
21. A bug, starting at the reference point ( 1, 0, 0) of the curve ~r = cos t ı + sin t + t k
walks up the curve for a distance of 10 units. What are the bug’s final co-ordinates?
22. Define the inverse square law and show that such a field is solenoidal.
23. Find the value of n for which rn~r is irrotational.
24. Evaluate ∇× (~a×~r) if ~a is a constant vector and ~r is the position vector of an arbitrary
point in 3-space.
25. Using the Green’s theorem evaluate
∫C
f(x)dx + g(y)dy where C is an arbitrary simple
closed curve in an open connected set D. What do you infer about the vector field~F (x, y) = f(x)ı + g(y).
26. Evaluate the flux of the vector field ~F (x, y, z) = zk across the outward oriented sphere
x2 + y2 + z2 = a2.
27. Find the nonzero function f(x) such that ~F (x, y) = f(x)y ı− 2xf(x) is conservative.
28. Find the work done by the force field ~F (x, y) = xy ı + x2 on a particle that moves
along the parabola x = y2 from (0,0) to (1,1).
(8× 1 = 8 weights)
Answer any 5 questions from among the questions 29 to 36.
These questions carry 2 weightages each.
29. Solve 2x3 + x2 − 7x− 6 = 0, given that difference between the roots is 3.
30. Solve 4x3 − 24x2 + 23x + 18 = 0, given that the roots are in Arithmetic Progression.
31. If α, β and γ are the roots of x3 + px2 + qx + r = 0, form the equation whose roots are
β + γ − 2α, γ + α− 2β and α + β − 2γ.
32. Using the vector equation of the ellipsex2
a2+
y2
b2= 1 find the curvature of the ellipse at
the end points of the major and minor axes. Deduce the curvature of the circle.
33. Suppose that a particle moves through 3-space so that its position vector at time t is
~r = t2ı + t− t3k. Find the scalar tangential and normal components of acceleration at
time t. Also find the vector tangential and normal components of acceleration at time
t = 1.
34. Prove that curl(grad φ)=0 where φ is a scalar point function.
35. Evaluate the surface integral
∫∫σ
xzdS where σ is the part of the plane x + y + z = 1
that lies in the first octant. What happens if the integrand is xy ?
36. Check whether ~F (x, y) = yexy ı + xexy is conservative or not. If it is so find the
corresponding scalar potental.
(5× 2 = 10 weights)
Answer any two questions from among the questions 37 to 39.
These questions carry 4 weightages each.
37. a) Solve x4 + 3x3 + x2 − 2 = 0
b) Find the condition that the roots of the equation ax3 + 3bx2 + 3cx + d = 0 may be
in G.P.
38. A shell is fired from ground level with a muzzle speed of 320 ft/s and elevation angle 60◦.
Find the parametric equations for the shell’s trjectory, the maximum height reached,
the horizontal distance travelled and the speed of the shell at impact.
39. Consider the function ~F (x, y, z) = (x2− yz)ı + (y2− zx) + (z2− xy)k over the volume
enclosed by the rectangular parallelopiped 0 ≤ x ≤ a, 0 ≤ y ≤ b and 0 ≤ z ≤ c. Verify
Gauss’s divergence theorem for ~F .
(2× 4 = 8 weights)
UNIVERSITY OF KERALA
First Degree Programme in Economics
Model Question Paper
Semester III Complementary Course for EconomicsMM 1331.5 Mathematics for Economics - III
Time: Three hours Total Weight : 30
All the first 16 questions are compulsory. They carry 4 weightages in all.
1. If f ′(x) =1
x2, find f(x).
2. Find the antiderivative of1√x
.
3. Evaluate
∫ 1
0
dx
1 + x.
4. If
∫ 2
−1
f(x)dx = 3 and
∫ 5
2
f(x)dx = −1 find
∫ 5
−1
f(x)dx.
5. If
∫ 4
1
f(x)dx = 2 and
∫ 4
1
g(x)dx = 10 find
∫ 4
1
[3f(x)− g(x)]dx.
6. Find the total cost function if it is known that the cost of zero output is c and that the
marginal cost of an output x is ax + b.
7. Evaluate
∫log x dx
8. Find
∫sin x
cos2 xdx
9. Find the area bounded by the parabola y = x2 + 1 and the x-axis between x = 0 and
x = 2.
10. Evaluate
∫exdx
1 + ex
11. Find the sum of the series 1 +1
2+
1
4+
1
8+ · · ·
12. Determine whether the series 1 + 22 + 23 + · · · is convergent or divergent.
13. The sum of n terms of a series isn
2n + 1. Find the sum to infinity of the series.
14. Write down an infinite series for1
e.
15. Find the domain of the function f(x) = 1 + x + x2 + x3 + · · ·
16. State whether the following is true or false.
The function f(x) = x13 has a Maclaurin series.
Answer any 8 questions from among the questions 17 to 28.
These questions carry 1 weightage each.
17. Sketch the region whose area is represented by the definite integral
∫ 4
1
2dx and hence
evaluate it.
18. Evaluate
∫ 9
4
2x√
xdx.
19. Integrate with respect to x : (a)ax + b√
x(b)
1
1− sin x
20. Evaluate tha following by substituition method.
(a)
∫x2dx
(x3 + 5)2(b)
∫(log x)2
x
21. If the marginal cost function is f ′(q) = 2 + 3√
q +5√
q, find the total cost function when
f(1) = 21.
22. Prove that, if Y is the constant stream of yield and r is the rate of interest, then the
Capitalisation is given byY
r.
23. Find the sum of the following infinite series.
(a) 1− 1
2+
1
4− 1
8+ · · ·
(b) 2 +√
2 + 1 +1√2
+1
2+ · · ·
24. Expand f(x) = sin x by Taylor’s formula about x = 0.
25. Sum to infinity the series∞Σ
n=1
1
(n + 1)!
26. Show that log(n + 1) = log n +1
n− 1
2.1
n2+
1
3.1
n3− 1
4.1
n4+ · · ·
27. Write the series with the sum of n termsn
n + 1. Find the sum to infinity of the series.
28. Find the sum of the series
1 +3
4+
3.5
4.8+
3.5.7
4.8.12+ · · ·
Answer any 5 questions from among the questions 29 to 36.
These questions carry 2 weightages each.
29. Evaluate
∫x2exdx.
30. Evaluate (a)
∫2x4dx
1 + x10(b)
∫e√
x
√x
dx
31. The price elasticity of a demand curve x = f(p) is of the form a− bp (a,b are constants).
Find the demand curve.
32. If the marginal revenue function is pm =ab
(x + b)2− c, show that p =
a
x + b− c is the
demand law.
33. Show that the sum to infinity of the series1.2
1!+
2.3
2!+
3.4
3!+ · · · is 3e.
34. Prove that log x =x− 1
x + 1+
1
2
x2 − 1
(x + 1)2+
1
3
x3 − 1
(x + 1)3+ · · ·
35. Show that the Maclurin series expansion of log(1 + sin x) is
x− x2
2+
x3
6− x4
12+ · · ·
36. Find the fraction corresponding to the repeating decimal 0.151515....by expressing it as
an infinite geometric series.
Answer any two questions from among the questions 37 to 39.
These questions carry 4 weightages each.
37. Explain Domar’s model of public debt and national income and prove with usual notation
that the ratio of debt to income approachesα
r.
38. Evaluate
∫ 1
0
dx
1 + xby Simpson’s rule, dividing the interval (0,1) into four equal parts.
39. Find the sum of the series
1.3
4.8+
1.3.5
4.8.12+
1.3.5.7
4.8.12.16+ · · ·
UNIVERSITY OF KERALA
Third Semester B.Sc. Degree ExaminationFirst Degree Programme under CBCSSComplementary Course for Electronics
Model Question PaperMM1331.8 Mathematics III
Time: 3 Hours Weight: 30Section A
All the first 16 questions are compulsory. They carry 4 weights in all.
1. Distinguish between discrete and continuous random variables.
2. Distinguish between probability density function and distribution function ofa random variable.
3. Write down the probability density function of a Binomial Distribution.
4. Define expectation of a random variable.
5. Define moments.
6. Define moment generating function of a distribution.
7. Distinguish between correlation and regression.
8. Distinguish between population and sample.
9. What do you mean by a statistical hypothesis?
10. What is critical region?
11. Distinguish between large sample and small sample tests.
12. What is the confidence interval for the mean of a normal population N(µ, σ)when σ is known.
13. State Cauchy’s integral theorem.
14. Find the residue ofsinh z
z4at z = 0.
15. State residue theorem.
16. Find the value of the integral∫|z|=1
e−z
z3 dz.
Section B
Answer any 8 questions from among the questions 17 to 28. These questionscarry 1 weight each.
17. A random variable X has the density function
f(x) =
{k
1+x2 if −∞ < x < ∞0 elsewhere.
Determine k and the distribution function.
18. A coin is tossed until head appears. What is the expectation of the numberof tosses required?
19. Find the variance of the binomial distribution.
20. What are properties of Normal distribution?
21. What is the p.d.f of a uniform distribution. Find its distribution function.Draw the graphs of p.d.f and the corresponding distribution function.
22. What do you mean by principle of least squares?
23. What are regression lines? Why there are two regression lines?
24. A sample of 25 items were taken from a population with standard deviation10 and sample mean is found to be 65. Can it be regarded as sample from anormal population with µ = 60.
25. Discuss the uses of t distribution in the tests of significance.
26. State Taylor’s theorem. Expand 1/z2 as Taylor series about z = 2.
27. Obtain the Laurent’s series expansion of 1/(z−1)(z−2) valid in a neighborhoodof z = 1.
28. Evaluate∫
Cdz
(z2+4)2where C is |z − i| = 2.
Section C
Answer any 5 questions from among the questions 29 to 36. These questionscarry 2 weights each.
29. Define Beta distribution and find its mean.
30. Define normal distribution and find its mean.
31. If the mean and variance of a binomial distribution are 4 and 2 respectively.Find the probability of
(a) exactly two successes.
(b) at least two successes.
32. Derive poisson distribution as a limiting form of binomial distribution.
33. Prove that coefficient of correlation lies between −1 and 1.
34. In a sample 600 men from city A, 450 are found to be smokers. Out of 900 fromcity B, 450 are smokers. Do the data indicate that the cities are significantlydifferent with respect to prevalence of smoking.
35. Find the poles and residues ofz2 − 2z
(z + 1)2(z2 + 4).
36. If f(z) has a pole of order m at z = a, then prove that the residue of f(z) atz = a is
1
(m− 1)!limn→a
dm−1
dzm−1[(z − a)mf(z)]
Section D
Answer any 2 questions from among the questions 37 to 39. These questionscarry 4 weights each.
37. Fit an approximate normal curve to the following data and calculate the the-oretical frequencies.