Mathematics & AlgorithmsTutorials

Dr. S.G. Tewari

Probability

• From five statisticians and six economists a committee consisting of three statisticians and two economists is to be formed. How many different committees can be formed if– No restrictions are imposed?– Two particular statisticians must be on the

committee?– One particular economist can not be on the

committee?

Cont…

• A die is tossed. If the number is odd, what is the probability that it is prime?

• Police plan to enforce speed limits by using radar traps at 4 different locations within the city limits. The radar traps at each of these locations L1, L2, L3, L4 are operated for 40%, 30%, 20% and 30% of the time. If a person who is speeding on his way to work has probabilities of 0.2, 0.1, 0.5 and 0.2 respectively of passing through these locations, what is the probability that he will be fined (for over speed)?

Contd…

• In a certain college 25% of boys and 10% of girls are studying mathematics. The girls constitute 60% of the student body. – What is the probability that mathematics is being

studied?– If a student is selected at random and is found to be

studying mathematics, find the probability that the student is • a girl?• a boy?

• A businessman goes to hotels X, Y, Z 20%, 50%, 30% of the time, respectively. It is known that 5%, 4%, 8% of the rooms in X, Y, Z hotels have faulty plumbing. – Determine the probability that the businessman

goes to hotel with faulty plumbing – What is the probability that businessman’s room

having faulty plumbing is assigned to hotel Z?

• Assume that 50% of all APIIT students are good in mathematics. Determine the probabilities that among 18 APIIT students– Exactly 10– At least 10– At most 8– At least 2 and at most 9

• Are good in mathematics.

• Out of 800 families with 5 children each, how many would you expect to have– 3 boys– 5 girls– Either 2 or 3 boys.

• Assume equal probabilities for boys and girls.

• Determine the probability of getting 9 exactly twice in 3 throws with a pair of fair dice.

• If X be a binomially distributed random variable with E(X)=2 and Var(X)=4/3, find the distribution of X.

• If z is normally distributed with mean 0 & variance 1. find– P(z ≥ -1.64)– P(z ≥ 1)– P(-1.96 ≤ z ≤ 1.96)– P(z ≤ 1)

• Determine the minimum mark a student must get in order to receive an A grade if the top 10% of the students are awarded A grades in an examination where the mean mark is 72 and standard deviation is 9.

• When the mean of marks was 50% and S.D. 5% then 60% of the students failed in an examination. Determine the ‘grace’ marks to be awarded in order to show that 70% of the students passed. Assume that the marks are normally distributed.

• A university awards distinction, first class, second class, third class or pass class according as the student gets 80% or more; 60% or more; between 45% and 60%; between 30% and 45%; or 30% or more marks respectively. If 5% obtained distinction and 10% failed, determine the percentage of students getting second class. Assume that marks X are normally distributed.

• Tea bags labeled as containing 2g of tea leaves. In actual fact, the mass of tea leaves per bag is normally distributed with a mean of 2.05g and standard deviation 0.05g– If one tea bag is selected randomly, find the probability that the

tea bag is• More than 1.9g• Less that 1.98g• More than 1.92g, given that the tea bag is less than 2.08g

• Calculate the expected number of tea bags which contain 1.95g to 2.10g of tea leaves in a box of 100 tea bags, round you answer to the nearest integers.

• If 10% of the truck drivers on road are drunk determine the probability that out of 400 drivers randomly checked– At most 32– More than 49– At least 35 but less than 47 drivers are drunk on

the road.

• A pair of dice is rolled 180 times. Determine the probability that a total of 7 occurs– At least 25 times– Between 33 and 41 times inclusive– Exactly 30 times.

• Determine the probability that by guess-work a student can correctly answer 25 to 30 questions in a MCQ consisting of 80 questions. Assume that in each question with four choices, only one choice is correct and student has on knowledge.

Contd…

• A distributor of bean seeds determines from extensive tests that 5% of large batch of seeds will not germinate. He sells the seeds in packets of 200 and guarantees 90% germination. Determine the probability that a particular packet will violate the guarantee.

• The average number of phone calls/minute coming into switch board between 2 and 4 PM is 2.5. Determine the probability that during one particular minute there will be – 0– 1– 2– 3– 4 or fewer– More than 6– At most 5– At least 20 calls

• Fit a Poisson distribution to the following data:

Xi 0 1 2 3 4

Observed frequencies fi

30 62 46 10 2

• Determine the probability that 2 of 100 books bound will be defective if it is known that 5% of books bound at this bindery are defective. – Use B.D.– Use Poisson approximation to B.D.

• The probability of a person having an accident in a certain period of time is 0.0003. For a population of 7500 people, draw a histogram showing the probabilities of 0, 1, 2, 3, 4, 5 and 6 people having an accident in this period.

• Two shipments of computers are received. The first shipment contains 1000 computers with 10% defectives and the second shipment contains 2000 computers with 5% defectives. One shipment is selected at random. Two computers are found good. Find the probability that the two computers are drawn from the first shipment.

Complex

• If z1 = 1 – i and z2 = 7 + i, find the modulus and principal arguments for z1 – z2

z1z2*

1 2

1 2

z zz z

• Locate the points

• In the Argand diagram and show that these four points form a square.

1 2 3 49 , 4 13 , 8 8 , 3 4z i z i z i z i

• If Z1 =1− 3i, Z2=−2+ 5i and Z3=−3− 4i, determine in a+ ib form:

1 2

1

3

1 2

1 2

1 2 3

( )

( )

( )

( )

a Z ZZbZZ ZcZ Z

d Z Z Z

• Solve the equations:

( )2 3( )( 2 ) ( 3 ) 2 3

3 4( ) 0 ,1 3

a i a ibb x i y y i x i

iy y ic x y Rix x y

• Express the following complex numbers in polar form:

( )3 4 ( ) 3 4( ) 3 4 ( )3 4a i b ic i d i

• Convert (a) 4 30∠ ◦ (b) 7 −145∠ ◦ into a+ib form, correct to 4 significant figures.

• Simplify

• Evaluate, in polar form: 2 30∠ ◦ +5 −45∠ ◦ −4 120∠ ◦.

2

4

2 1 3( ) ( )1 21

ia b iii

• Determine the moduli and arguments of the complex roots

(3+4i)1/3

(−2+i)1/4

• Determine the two square roots of the complex number (5 + 12i) in polar and cartesian forms and show the roots on an Argand diagram.

• Express the roots of (−14+ 3i)−2/5 in polar form.• Find all solutions of the equation,

4 2(1 4 ) 4 0z i z i

• Determine in polar and cartesian forms– (a) [3 41∠ ◦]4 – (b) (−2−i)5.

• Change (3−4i) into – (a) polar form, – (b) exponential form.

• Convert 7.2ei1.5 into rectangular form

• Express z=2e1+iπ/3 in Cartesian form and polar form.• When displaced electrons oscillate about an

equilibrium position the displacement x is given by the equation:

– Determine the real part of x in terms of t, assuming (4mf − h2) is positive.

24

2 2

mf hht j tm m a

x Ae

• Determine the domain in the z-plane represented by

( )3 4 5

( ) Im( ) 6

( ) ( )4 2

a z

b z

c amp z

• Find the locus of z when

• Is purely imaginary. 2

z iz

• Simplify

2 3

9 5

cos5 sin 5 cos7 sin 7

cos 4 sin 4 cos sin

i i

i i

• If then and

express cos3 in the terms of cos3 and cos express sin4 in terms of cos4 and cos2

cos sinz i 1 2cosnnz nz

1 2 sinnnz i nz

• Solve z4 + 1 = 0 and locate the roots in the Argand diagram.

• Show that

• Find the nth root of unity.• Solve the following equations,

4 232sin cos cos6 2cos 4 cos 2 2

4

6

1 0

0

z

z i

• Determine the region in complex plain represented by

( )1 2 3

( ) Re( ) 3

( ) ( )6 3

a z i

b z

c amp z

• Verify triangle inequality for z1=2+3i, z2=4-i

• Find the locus of

1 1 3z z

Vectors

• For the vector compute,

• Determine if the sets of vectors are parallel or not

2, 4,1 , 6,12, 3

4,10 , 2, 9

a b

a b

2,4

13 , , 22

a

a a a

• Find a unit vector that points in the same direction as

• Compute the dot product for each of the following

5,2,1w

5 8 , 2

0,3, 7 , 2,3,1

v i j w i j

a b

• Determine the angle between

• Determine if the following vectors are parallel, orthogonal, or neither.

3, 4, 1 , 0,5,2a b

6, 2, 1 , 2,5,2

1 12 ,2 4

a b

u i j v i j

• A constant force of F=10i+2j −k newtons displaces an object from A=i+j +k to B=2i−j +3k (in metres). Find the work done in newton metres.

• Calculate the work done by a force F=(−5i+j +7k) N when its point of application moves from point (−2i−6j +k) m to the point (i−j +10k) m.

• Determine the projection of

• Determine the projection of

2,1, 1 onto 1,0, 2b a

1,0, 2 onto 2,1, 1a b

• Determine the direction cosines and direction angles for

• A plane is defined by any three points that are in the plane. If a plane contains the points ,

Find a vector that is orthogonal to the plane.

2,1, 4a

(1,0,0), (1,1,1), (2, 1,3)P Q R

• Determine if the three vectors lie in the same plane or not.

• Find the volume of a parallelepiped whose edges are a=2i-3j+4k, b=i+2j-k, c=3i-j+2k

• Find the volume of the tetrahedron having the following vertices (2,1,8), (3,2,9), (2,1,4), (3,3,10).

1, 4, 7 , 2, 1,4 , 0, 9,18a b c

• Find the moment and the magnitude of the moment of a force of (i+2j −3k) newtons about point B having co-ordinates (0, 1, 1), when the force acts on a line through A whose co-ordinates are (1, 3, 4).

• The axis of a circular cylinder coincides with the z-axis and it rotates with an angular velocity of (2i − 5j + 7k) rad/s. Determine the tangential velocity at a point P on the cylinder, whose co-ordinates are ( j + 3k) metres, and also determine the magnitude of the tangential velocity.

• Determine the vector equation of the line through the point with position vector 2i+3j-k which is parallel to the vector i-2j+3k.

• Find the point on the line corresponding to λ=3 in the resulting equation of part (a).

• Express the vector equation of the line in standard Cartesian form.

• Determine the constant b such that the vectors 4i+2j-k, bi+j+k, 3i-j-5k are coplanar.

• Find the tangent line(s) to the parametric curve given by

• The parametric equations of a cycloid are . Find

5 3 24 , at (0,4)x t t y t

4 sin , 4 1 cosx y 2

2,dy d ydx dx

• Determine the x-y coordinates of the points where the following parametric equations will have horizontal or vertical tangents.

• Eliminate the parameter from the following set of parametric equations.

3 23 , 3 9x t t y t

2 , 2 1x t t y t

• Find the second derivative for the following set of parametric equations,

• Write down the equation of the line that passes through the points (2,-1,3) and (1,4,-3). Write down all three forms of the equation of the line.

5 3 24 ,x t t y t

• Determine if the line that passes through the point (0,-3,8) and is parallel to the line given by x=10+3t, y=12t and z=-3-t passes through the xz-plane. If it does give the coordinates of that point.

• Determine the equation of the plane that contains the points

• Determine if the plane given by –x+2z=10 and the line given by is parallel ?

(1, 2,0), (3,1,4), (0, 1,2)P Q R

5,2 ,10 4r t t

• Find the general formula for the tangent vector and unit tangent vector to the curve given by

• Find the vector equation of the tangent line to the curve given by

• Find the normal and binormal vectors for

2( ) 2sin 2cosr t t i tj tk

2( ) 2sin( ) 2cos( ) at 3

r t t i t j t k t

( ) ,3sin ,3cosr t t t t