Q1) If y = cot x, show that d2y/dx2 + 2y dy/dx = 0 (Marks 2)
Ans1) Here y = cot x
differentiating both sides w.r.t. x:-
dy/dx = -cosec2x
differentiating again w.r.t. x:-
d2y/dx2 = -d/dx (cosec2x)
d2y/dx2 = -2cosecx(-cosec x . cot x ) = +2cosec2x . cot x
or d2y/dx2 = -2 dy/dx . y
or d2y/dx2 + 2y dy/dx = 0.
A = 2 2
4 0 B = 6 2
find the matrix C such that A + B + C is a zero matrix. (Marks 2)
A = 2 2
4 0 B = 6 2
we want to find C such that A + B + C = 0
< => C = 0 – A – B = -A – B
= – 2 2
4 0 – 6 2
= -2 -2
-4 0 + -6 -2
= -8 -4
Q5) Construct a 3 x 2 matrix whose elements in the ith row and the jth coloumn are given by:-
aij = (3i + j)/2 (Marks 2)
Ans5) We want a 3 x 2 matrix
=> 1 i 3 and 1 j 2
where aij = (3i + j)/2
… a11 = (3 + 1)/2 = 2; a12 = (3 + 2)/2 = 5/2;
a21 = (6 + 1)/2 = 7/2; a22 = (6 + 2)/2 = 4;
a31 = (9 + 1)/2 = 5; a32 = (9 + 2)/2 = 11/2
… The reqd matrix is :-
Q5. Mention any two demerits of territorial representation. (1+1=2)
Ans. Territorial representation may be criticised on the following grounds:-
(a) It tends to promote localism and parochialism in its representatives, to the neglect of larger national interests.
(b) Due to restriction of choice of candidates from the same constituency, it has often thrown up men of inferior mind, with manipulative instincts for survival in power, rather than persons of experience, talent, vision and dedication.
Q6. What is secret ballot? Write its any one advantage. (1+1=2)
Ans. The secret ballot system is an ideal device to elect candidates in a democracy. It means that the voters can cast their votes for any candidates of their choice without disclosing their name. Thus the voter is free of fear and pressure of any kind. It makes the democratic process impartial and just.
Q7. Mention any two developmental functions of Zila Parishad. (2)
Ans. At the apex of the Panchayati Raj system is the Zila Parishad, located at the district level. Its developmental functions include:-
(1) to look after the development work of the district and secure the execution of developmental plans and projects, and other activities in the Blocks.
(2) to coordinate and consolidate the developmental plans prepared by the Block in the district and also to advise the state government on all matters relating to development activities in the district.
Q11. Define Liberty. Explain briefly its negative and positive aspects. (2+1+1=4)
Ans. The term liberty is derived from the Latin word ‘Liber’ which means free. It is a word of negative meaning denoting absence of restraint. It means freedom to do something or enjoy one’s powers, provided it does not injure the freedom of others. Its positive connotation is that it is a necessary condition of free and full development of our personality. It involves the opportunity
for many-sided growth. It includes the capacity to act, availability of an effective range of choices and spontaneity, that is, the ability to act in accordance with one’s own personality, without having to make a great effort at self-denial or self-control and without being subjected to external constraints.
Negative liberty means absense of restrains. An individual is considered as rational and only he knows what is his interest for the development of his personality. Therefore each individual should be given personal liberty with regards to his personal affairs and the society can not interfere in it. It also believes that there is no conflict between personal interest and social interest and by serving his own interest, an individual also serves the social interest. Personal liberty is thus regarded as a pre-condition of any social progress.
Positive liberty, on the other hand, is not the absense of restrains rather it is the presence of those socio-economic and political conditions without which liberty can not be realized. Here the object of liberty is the development of man as a social being. However, without proper opportunities and
social conditions, liberty can not be realized. Liberty of an individual must correspond with social welfare. The duty of the state, therefore, is to create positive conditions for the realization of liberty.