Maths

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.


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.


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.


First Degree Programme in Chemistry


Semester III Complementary Course for ChemistryMM 1331.2 Mathematics - III

Theory of Equations and Vector Analysis



A 1. If α, β, γ... are the roots of f(x) = 0 then the equation whose roots are kα, kβ, kγ... is

(a) f(−x) = 0 (b) f(xk) = 0 (c) f(kx) = 0 (d) f( 1

x) = 0

2. The second term of f(x) = a◦xn +a1x

n−1 + · · ·+an = 0 can be removed by diminishing

the roots by

(a)−an

na◦(b)

−a◦na1

(c)−a1

na◦(d)

a1

na◦3. Define a complete equation.

4. State Descarte’s rule of signs.

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)



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)



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)



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)


First Degree Programme in Economics


Semester III Complementary Course for EconomicsMM 1331.5 Mathematics for Economics - III



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.



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+ · · ·



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.



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+ · · ·


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.

Classes f12-15 516-19 1020-23 2224-27 2528-31 2032-35 1236-39 5

38. Given the following data

X1 20 25 15 20 26 24X2 3.2 6.5 2.0 0.5 4.5 1.5X3 4.0 5.2 7.5 2.5 3.5 1.5

Obtain the least squares equation to predict X1 values from those of X2 andX3.

39. Evaluate the integral∫∞−∞

sin axx2+1

dx.


First Degree Programme in Geology


Semester III MM 1331.3 Complementary Course for Geology

Mathematics III (Analytic Geometry, Complex Numbers and Abstract Algebra)

Time: Three hours

All the first 16 questions are compulsory. Four consecutive questions beginning with the first form a

bunch. Each bunch carries 1 weight.

1. Write down the condition that the line y = mx+ c be a tangent to the parabola y2 = 8x.

2. State the rotation equations in Cartesian coordinates.

3. State Kepler’s third law.

4. Find the slopes of the asymptotes of the rectangular hyperbola y2 − x2 = 1.

5. Write down the polar form of the complex number 1 + i√3.

6. What is the area of the triangle whose vertices are represented by the complex numbers z, iz and

z + iz in the Argand plane?

7. If ω is a complex nth root of unity, what is the value of ω2010 + (ω2)2010

?

8. State de Moivre’s theorem.

9. If z is a complex number, what is the value of cosh2z − sinh2z?

10. What are the elements of the Klein group? Is the group abelian?

11. Show by an example that if H and K are subgroups of a group G, H ∪K need not be a subgroup

of G.

12. What are the generators of the cyclic group of Z under addition?

13. What do you mean by an integral domain?

14. Is Zn under addition and multiplication modulo n a field for all n?

If not, state the condition on n.

15. Is R a vector space over C?

16. What is the dimension of the vector space Fn[x] of all polynomials of degreee less than or equal

to n over the field of reals.

Answer any 8 questions from among the questions 17 to 28. These questions carry 1 weight each.

17. Let an x′y′ coordinate system be obtained by rotating an xy coordinate system through an angle

of θ = 300. Find the x′y′ coordinates of the point whose xy coordinates are (1,−√3).

18. Describe the graph of the equation y2 − 8x− 6y − 23 = 0.

19. As illustrated in the accompanying figure, a parabolic arch spans a road 40 feet wide. How high

is the arch if a centre section of the road 20 feet wide has a minimum clearance of 12 feet?

12m

12m

20m

40m

20. Find the equation of the hyperbola with vertices (±2, 0) and foci (±3, 0).

21. Prove that

(

1 + cos θ + i sin θ

1 + sin θ + i cos θ

)

n

= cosnθ + i sin nθ, where n is a positive integer.

22. Find all the values of (1 + i)1

4 .

23. Express cos 6θ in terms of powers of cos θ.

24. If sin(A+ iB) = x+ iy, prove thatx2

cosh2B+

y2

sinh2B= 1.

25. Show that a nonempty subset H of a group G is a subgroup of G iff ab−1 also belongs to H

whenever a, b ∈ H.

26. If G is a group, show that G is abelian iff (ab)2 = a2b2, for all a, b ∈ G.

27. Give an example of a commutative ring that is not an integral domain.

28. Show that V = {(x1, x2, 0) : x1, x2 ∈ R} is a subspace of the vector space R over R.

Answer any 5 questions from among the questions 29 to 36. These questions carry 2 weights each.

29. Sum to infinity the series: 1 +cos θ

1!+

cos 2 θ

2!+

cos 3 θ

3!+ . . .

30. Sum to infinity the series: sin θ − 1

2sin 2 θ + 1

3sin 3 θ − . . .

31. Prove that the line tangent to the ellipsex2

a2+

y2

b2= 1 at the point (x0, y0) has the equation

xx0

a2+

yy0

b2= 1.

32. Identify and sketch the curve 153x2 − 192xy + 97y2 − 30x− 40y − 200 = 0

33. Find the eccentricity, foci, centre and the equations of the directrices of the ellipse

2x2 + 3y2 = 1.

34. Prove that a subgroup of a cyclic group is cyclic.

35. Define an operation ∗ on the set of real numbers R by a ∗ b = a+ b+ ab. Is R a group under this

operation? If so, what is the identity element?

36. Define a vector space. If V1 and V2 are subspaces of a vector space V , prove that V1 ∩ V2 is a

vector space. Also, show by an example that V1 ∪ V2 need not be a vector space.

Answer any 2 questions from among the questions 37 to 39. These questions carry 4 weights each.

37. (a) Express sin4θ cos2θ in terms of cosines of multiples of θ.

(b) Separate tan−1(α+ iβ) into real and imaginary parts.

38. (a) Find the equation of the asymptotes of the curve x2 − y2 − 4x+ 8y − 21 = 0

(b) Sketch the graph of r =2

1 + 2 sin θin polar coordinates.

39. (a) Define a field with an example.

(b) Prove that every field is an integral domain.

(c) Is every integral domain a field?

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First Degree Programme in Physics and Computer Applications


Semester III MM 1331.6 Complementary Course for Physics andComputer Applications

Mathematics III (Vector Differentiation, Coordinate Systems,Abstract Algebra and Fourier Transforms)

Time: Three hours

All the first 16 questions are compulsory. Four consecutive questions beginning with the first

form a bunch. Each bunch carries 1 weight.

1. If ~r(t) = tan−1t ı + t cos t −√

t k, calculate ~r ′(t)

2. Find the unit tangent vector to the graph of ~r(t) = 5 cos t ı + 5 sin t at t = π/3

3. Find the directional derivative of f(x, y) =√

xy at P (1, 4) in the direction of a vector

making the counterclockwise angle θ = π/3 with the X-axis.

4. If the electric potential at a point (x, y) in the xy - plane is given by V (x, y) = e−2x cos 2y,

calculate the electric intensity vector at (π/4, 0).

5. The value of (∇× ~r,∇ · ~r) is

(a) (0, 3) (b) (0,-3) (c) (3,0) (d) (-3,0)

6. Evaluate

∫ 1

−1

∫ 2

0

∫ 1

0

(x2 + y2 + z2) dx dy dz

7. State Pappu’s theorem.

8. Evaluate

∫ π/2

0

∫ π/2

0

∫ 1

0

ρ3 sin φ cos φ dρ dφ dθ

9. Show by an example that if H and K are subgroups of a group G, H ∪ K need not be a

subgroup of G.

10. What are the generators of the cyclic group of Z under addition?

11. What do you mean by an integral domain?

12. Is Zn under addition and multiplication modulo n a field for all n?

If not, state the condition on n.

13. Is R a vector space over C?

14. Write down the Euler’s formulas for Fourier coefficients of a periodic function with period

2π.

15. State whether True or False: The Fourier series of an even periodic function is a sine series.

16. State whether f(x) = ex is a periodic function. (4× 1 = 4 weights)



17. Evaluate ∇× (~a × ~r) if ~a is a constant vector and ~r is the position vector of an arbitrary

point in 3-space.

18. A bug, starting at the reference point (1,0,0) of the curve ~r(t) = 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?

19. If ~F (x, y) = g(y)y ı− 2xg(y) is conservative, find the nonzero function g(y).

20. Determine the constant c so that the vector ~F = (2x + 3y)~i + (2z − 6y)~j + (az + 5x)~k is

solenoidal

21. Sketch the graph of r =5

3 + 3 sin θin polar co-ordinates.

22. Let G be a wedge in the first octant cut from the cylindrical solid y2 + z2 = 1 by the planes

y = x and x = 0. Evaluate

∫ ∫ ∫G

zdV .

23. Evaluate the integral

∫ a

0

∫ √a2−x2

0

∫ a2−x2−y2

0

x2dz dy dx (a > 0).

24. Show that a nonempty subset H of a group G is a subgroup of G iff ab−1 also belongs to

H whenever a, b ∈ H.

25. If G is a group, show that G is abelian iff (ab)2 = a2b2, for all a, b ∈ G.

26. Give an example of a commutative ring that is not an integral domain.

27. Show that V = {(x1, x2, 0) : x1, x2 ∈ R} is a subspace of the vector space R over R.

28. Find the Fourier transform of f(x) =

k if 0 < x < a

0 otherwise

(8× 1 = 8 weights)



29. Write down 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 from the

above result.

30. A shell has a muzzle speed of 800 ft/s. The barrel makes an angle of 45◦ with the horizontal

and the barrel opening is assumed to be at ground level. Find the parametric equations

for the shell’s trajectory, the maximum height reached by the shell, the horizontal distance

travelled by the shell and the speed of the shell at impact.

31. The planet Pluto has eccentricity e = 0.249 and semimajor axis a = 39.5AU . Choose a

polar co-ordinate system with the center of the Sun at the pole and find a polar equation of

Pluto’s orbit in that co-ordinate system. Also find the period T in years and the perihelion

and aphelion distances.

32. Find the volume and the position of the centre of gravity of the tetrahedron bounded by

the planex

a+

y

b+

z

c= 1 and the coordinate planes.

33. Prove that a subgroup of a cyclic group is cyclic.

34. Define an operation ∗ on the set of real numbers R by a ∗ b = a + b + ab. Is R a group

under this operation? If so, what is the identity element?

35. Define a vector space. If V1 and V2 are subspaces of a vector space V , prove that V1 ∩ V2 is

a vector space. Also, show by an example that V1 ∪ V2 need not be a vector space.

36. Find the Fourier series expansion of f(x) = x2 with period 2l on (−l, l).

(5× 2 = 10 weights)



37. a) Show that ∇2f(r) =2f ′(r)

r+ f ′′(r).

b) A heat-seeking particle is located at the point (2,3) on a flat metal plate, whose temper-

ature at a point (x, y) is T (x, y) = 10−8x2−2y2. Find an equation for the trajectory of

the particle if it moves continuously in the direction of maximum temperture increase.

38. (a) Define a field with an example.

(b) Prove that every field is an integral domain.

(c) Is every integral domain a field?

39. Find the Fourier series expansion of f(x) if f(x) =

−π if − π < x < 0

x if 0 < x < π

Deduce that1

12+

1

22+

1

32+ · · · = π2

8

(2× 4 = 8 weights)


First Degree Programme in Physics


Semester III Complementary Course for PhysicsMM 1331.1 Mathematics - III

Theory of Equations, Differential Equations and Theory of Matrices



1. Find the real root of the equation x3 + 6x + 20 = 0; one root being 1 + 3i.

2. If α, β, γ are the roots of the equation x3 + 3x + 5 = 0 then find Σαβ.

3. What is standard reciprocal equation.

4. Find the maximum possible number of real roots of the equation x7 +x4 +10x3− 28 = 0.

5. Find the value of k such that the equation of the roots of x3− 6x2 + kx + 64 = 0 may be

in GP.

6. Form the equation whose roots are negative of the roots of x4 − 4x3 + 6x2 − x + 2 = 0.

7. Solve√

1− y2dx +√

1− x2dy = 0.

8. Solve y′′ − 5y′ + 4y = 0.

9. Solvedy

dx=

y + 2

x− 1

10. Find the particular integral of y′′ − y = ex.

11. Find the particular integral of y′′ + 4y = cos x.

12. Are the following matrices row equivalent?1 0 0 0

0 5 3 7

0 4 9 10

0 0 0 0

and

1 0 0 0

0 13

5

7

50 4 9 10

0 0 0 0

13. Check whether the matrix

1 1 −3

2 16 1

0 0 4

is singular.

14. Find the inverse of the matrix

3 0 0

0 1 0

0 0 2

.

15. Find the rank of the matrix

(1 3 0

0 0 1

).

16. Define diagonalizable matrix.



17. If α, β, γ are the roots of the equation x3 +qx+r = 0 then find (i) Σ1

β + γ, (ii) Σ(β−γ)2.

18. Solve the equation x3 − 9x2 + 14x + 24 = 0, two of whose roots are on the ratio 3 :2.

19. Solve the equation x3 − 12x− 65 = 0 by Cardan’s method.

20. Form a rational cubic equation whose roots are 2 and 3 + i.

21. The stress in thick cylinders is given by rdp

dr+ 2p = 2C where C is a constant. Find p in

terms of r.

22. Solvedy

dx+ 2y tan x = sin x.

23. Solve xdy

dx+ y = x3y4.

24. Solved2y

dx2− 2

dy

dx+ y = x2.

25. If A =

[cos α sin α

− sin α cos α

], find An.

26. If A =

[1 −1

2 3

]show that A2 − 4A + 5I = 0.

27. Find the eigen values of the matrix

(1 3

2 1

).

28. Find the rank of the matrix A =

1 1 −1

1 2 1

−1 1 3

by reducing it to the echelon form.



29. Solve the equation x4 + 8x3 + 6x2 − 8x− 7 = 0 by Ferraris method.

30. Solve the equation 6x6 − 35x5 + 56x4 − 56x2 + 35x− 6 = 0.

31. Find the orthogonal trajactories of the family of co-axial circles x2 + y2 + 2λx + C = 0

where λ is the parameter.

32. Solve y2 + x2 dy

dx= xy

dy

dx

33. Solve (D2 − 3D + 2)y = e3x cos x.

34. Test the consistency and solve:

5x + 3y + 7z = 4

3x + 26y + 2z = 9

7x + 2y + 10z = 5

35. Let A be an n × n matrix with n distinct eigen values. Prove that the corresponding

eigen vectors are linearly independent.

36. Find the eigen vectors of the matrix A =

3 −5 −4

−5 −6 −5

−4 −5 3



37. (a) If θ is a root of the equation x3 + x2 − 2x− 1 = 0 show that θ2 − 2 is another root.

(b) If α, β, γ are the roots of the equation x3−6x2 +11x+k = 0 find the equation whose

roots are α− 2β + γ, β− 2γ + α, γ− 2α + β. Also find k so that the roots are in AP.

38. (a) A rocket with weight 10 lbs is fixed vertically upwards starting from rest. The effect

of the gradual burning of the charge is to produce an acceleration15 g

10− tft / sec2

at time t. The charge is consumed in 5 seconds. What is the velocity of the rocket

at time t.

(b) Solve x2 d2y

dx2− x

dy

dx+ y = cos log x.

39. Prove that

−9 4 4

−8 3 4

−16 8 7

is diagonalizable and find the diagonal form.


First Degree Programme in Statistics


Semester III MM 1331.4 Complementary Course for Statistics

Mathematics III: Integration and Complex Numbers

Time: Three hours

All the first 16 questions are compulsory. Four consecutive questions beginning with the first

form a bunch. Each bunch carries 1 weight.

1. Define the average value of a continuous function f(x) defined on [a, b].

2. Suppose that a particle moves so that its velocity at time t is V (t) = t2− 2t m/s. Find the

displacement of the particle during the time interval 0 ≤ t ≤ 3.

3. Evaluate

∫sin x

cos2 xdx

4. Evaluate

∫sin

√x√

xdx

5. Evaluate

∫ π4

0

tan2x sec2x dx

6. Solve the initial value problemdy

dx=

x + 1√x

, y(1) = 0.

7. Find the area bounded by the parabola y = x2 +1 and the x-axis between x = 0 and x = 2.

8. Evaluate

∫ 4

2

∫ 1

0

x2yz dx dy

9. State an integral formula for finding the arc length of a smooth curve y = f(x) over the

interval [a, b].

10. Express area of a region R in the xy-plane as a double integral.

11. Express

∫ 2

0

∫ √x

0

f(x, y) dx dy as an equivalent integral with the order of integration re-

versed.

12. Write down the polar form of the complex number 1 + i√

3.

13. What is the area of the triangle whose vertices are represented by the complex numbers z,

iz and z + iz in the Argand plane?

14. If ω is a complex nth root of unity, what is the value of ω2010 + (ω2)2010

?

15. State de Moivre’s theorem.

16. If z is a complex number, what is the value of cosh2z − sinh2z?


These questions carry 1 weight each.

17. A projectile is fired vertically upward from the ground level with an initial velocity of 112

m/s.

(a) Find the velocity at t = 3 s and at t = 5 s.

(b) How high will the projectile rise?

(c) Find the speed of the projectile when it hits the ground.

18. Find the equation of the curve that satisfies the following conditions. At each point (x, y)

on the curve the slope equals the square of the distance between the point and the y-axis.

The point (-1,2) is on the curve.

19. Suppose tha the position function of a particle moving along a coorddinate line is s(t) =

6t2 + 1. Find the average velocity of the particleover the time interval 1 ≤ t ≤ 4 by

integration. Verify the result by computing it algebraically.

20. Find the displacement and distance travelled during the time interval 0 ≤ t ≤ 3 if the

velocity function is v(t) = t3 − 3t2 + 2t.

21. Find the area of the region enclosed by x = y2 and y = x + 2.

22. A ball is hit directly upward with an initial velocity of 49 m/s and is struck at a point that

is 1 m. above the ground. Assuming that the free fall model applies, find how high the ball

travels.

23. Evaluate

∫xe−2x dx

24. Evaluate

∫ √tan x sec4x dx

25. Prove that

(1 + cos θ + i sin θ

1 + sin θ + i cos θ

)n

= cos nθ + i sin nθ, where n is a positive integer.

26. Find all the values of (1 + i)14 .

27. Express cos 6θ in terms of powers of cos θ.

28. If sin(A + iB) = x + iy, prove thatx2

cosh2B+

y2

sinh2B= 1.


These questions carry 2 weights each.

29. Express sin4θ cos2θ in terms of cosines of multiples of θ.

30. Separate tan−1(α + iβ) into real and imaginary parts.

31. Find the sum of the series : 1 +1

2cos x +

1

4cos 2x +

1

8cos 3x + · · ·

32. Evaluate

∫ 2

0

∫ 1

y/2

ex2

dx dy by changing the order of integration.

33. Find the perimeter of the astroid x23 + y

23 = a

23 .

34. Find the area of the region inside the circle r = 5 sin θ and outside the limacon r = 2+sin θ.

35. Find the volume of the solid that results when the region enclosed by the curve

y = sec x, x =π

4, x =

π

3, y = 0

is revolved about the x-axis.

36. Use cylindrical coordinates to evaluate

∫ 3

−3

∫ √a−x2

−√

a−x2

∫ a−x2−y2

0

x2 dz dy dx.


These questions carry 4 weights each.

37. Find the area of the surface that is generated by revolving the portion of the curve y = x2

between x = 1 and x = 2 about the x-axis.

38. (a) Express sin4θ cos2θ in terms of cosines of multiples of θ.

(b) Separate tan−1(α + iβ) into real and imaginary parts.

39. Use a double integral to find the volume of the tetrahedron bounded by the coordinate

planes and the plane z = 4− 4x− 2y.

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