) For a given data, the mean and S.D. of 100 observations were obtained are 40 and 5.1 respectively. Later it was found that an observation was wrongly written as 50 instead of 40. Find the true mean and S.D.
b) If the p.d.f. of x is and zero elsewhere. Find k and S.D. of x.
Differentiate w.r.t.
.
) Find .
Which of the following statements are true or false? Give a short proof or a counter-example in support of your answer.
i) The mean of a binomial distribution, when and
is
.
ii) Mean deviation is minimum about median.
iii) The domain of 6x3 - 7y3 + 4xy, where it is continuous is .
iv) .
v) The CDF of any distribution satisfies and non-decreasing.
Give a real life situation problem, which is mathematically translated into
.
Also, explain how this linear system models your problem.
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Using the method of substitution, obtain the solution set in , of the following:
i)
ii)
iii)
) Find the values of for which ai is a solution of
. Also find all the roots of this equation.
b) Find all the roots of 3i - 3. Also show any one of them in an Argand diagram.
) Consider the linear system
Give the two reasons for Cramer's Rule being applicable for solving this system. Also use the rule to solve the linear system.
Give the following:
i) a matrix;
ii) the transpose of the matrix in (i) above;
iii) a system of linear equations represented by , where A is the matrix in (ii) above.
In the context of your IGNOU studies, give the following:
i) an example of an implication;
ii) the converse of your statement in (i) above;
iii) the contrapositive of your statement in (i) above;
iv) a statement using ;
v) a statement using .
Express the following situation in a Venn diagram:
In a survey of 60 women, it is found that 25 have studied upto Class 12 only, 10 have studied till Class 10 only, 26 got scholarships, 9 of those studying till Class 12 got scholarships, 8 of those studying till Class 10 got scholarships, and 11 had completed their BA degree.
See Answer →Given any two sets C and D, under what conditions on them will and
have the same number of elements? Give reasons for your answer.
If A and B are the set of even integers and set of odd integers, respectively, find and
.
Using the discriminant, give the nature of the roots of . Also solve the equation.
b) Find the cubic equation whose roots are the cubes of the roots of .
c) Obtain the resolvent cubics, by Descartes’ method and by Ferrari’s method, of the equation . Are the cubics the same? Further, use either method to obtain the roots of this equation.
Show that .
b) Let . Show that
Which of the following statements are true? Justify your answers. (This means that if you think a statement is false, give a short proof or an example that shows it is false. If it is true, give a short proof for saying so. For instance, to show that ‘{1, padma, blue} is a set’ is true, you need to say that this is true because it is a well-defined collection of 3 objects.)
i) Eliminating z from ,
and
gives
.
ii) The roots of are given by
.
iii) .
iv) Given any n positive numbers in , the product of their harmonic mean and their arithmetic mean is 1.
v) If A and B are two sets such that is empty, then either
or
.
vi) For any .
vii) The geometrical representation of the set is a point.
viii) Any finite set is a subset of .
ix) Every biquadratic equation has at least one real root.
x) The converse of the statement, ‘Every student of MTE-04 has completed FST-01’, is ‘Every student of FST-01 has completed MTE-04’.
Which of the following statements are true? Justify your answers. (This means that if you think a statement is false, give a short proof or an example that shows it is false. If it is true, give a short proof for saying so. For instance, to show that ‘{1, padma, blue} is a set’ is true, you need to say that this is true because it is a well-defined collection of 3 objects.)
i) Eliminating z from ,
and
gives
.
ii) The roots of are given by
.
iii) .
iv) Given any n positive numbers in , the product of their harmonic mean and their arithmetic mean is 1.
v) If A and B are two sets such that is empty, then either
or
.
vi) For any .
vii) The geometrical representation of the set is a point.
viii) Any finite set is a subset of .
ix) Every biquadratic equation has at least one real root.
x) The converse of the statement, ‘Every student of MTE-04 has completed FST-01’, is ‘Every student of FST-01 has completed MTE-04’.
) Does the function
satisfy the requirement of Schwarz's theorem at (1, 1)? Justify your answer. (b) Locate and classify the stationary points of the following:
(i)
(ii)
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