Let z = a + ib be a complex number. The conjugate of z, denoted as �, is the complex number a – ib, i.e., � = a – ib.
Also Z* � = | Z |2
Or Z–1 = � / | Z |2 ( Useful to find inverse of a complex number)
Numerical: Find the conjugate of (3 + 4i )
Solution: Conjugate � = 3-4i
Numerical: Find inverse of (3 + 4i )
Z–1 = � / | Z |2 = (3 – 4i)/5 = 3/5 – 4/5i
Argand Plane & Polar representation
Complex numbers can represented in 2 forms
Argand Plane
Polar Representation
Argand Plane
The complex number x + iy can be represented geometrically as the unique point P(x, y) in the XY-plane and vice-versa. Plane with complex number assigned to each of its point is called complex or Argand plane.
Let’s plot some points on the graph.
Note: Modulus of the complex number is distance between point P(x, y) to the origin O (0, 0)
Polar representation
Let point P represent z = x + iy.
Let x = r cos θ , y = r sin θ and therefore, z = r (cos θ + i sin θ).
Here – π < θ ≤ π
Point P is uniquely determined by the ordered pair of real numbers (r, θ), called the polar coordinates of the point P.
Numerical: Represent the complex number z =1+ i √3 in the polar form.
Solution: let z =1+ i √3 = r(cos θ + i sin θ)
r=| z | = (a2 + b2 )1/2 = ((1)2 + (√3)2)1/2 = 2
Comparing real parts of z =1+ i √3 = r(cos θ + i sin θ) = 2(cos θ + i sin θ)
1 = 2 cos θ
or cos θ = ½
or cos θ = π/3
Therefore, polar representation will be z = r(cos θ + i sin θ) = 2(cos π/3 + i sin π/3)
Quadratic Equation
We have seen of real numbers in the cases where discriminant is non-negative, i.e., ≥ 0,
Let us consider the following quadratic equation: ax2 + bx + c = 0 with real coefficients a, b, c and a ≠ 0.
One key basis for mathematical thinking is deductive reasoning. In contrast to deduction, inductive reasoning depends on working with different cases and developing a conjecture by observing incidences till we have observed each and every case. Thus, in simple language we can say the word ‘induction’ means the generalisation from particular cases or facts.
Statement: A sentence is called a statement, if it is either true ot false.
Motivation: Motivation is tending to initiate an action. Here Basis step motivate us for mathematical induciton.
Principle of Mathematical Induction: The principle of mathematical induction is one such tool which can be used to prove a wide variety of mathematical statements. Each such statement is assumed as P(n) associated with positive integer n, for which the correctness for the case n = 1 is examined. Then assuming the truth of P(k) for some positive integer k, the truth of P (k+1) is established.
Working Rule: Step 1: Show that the given statement is true for n = 1. Step 2: Assume that the statement is true for n = k. Step 3: Using the assumption made in step 2, show that the statement is true for n = k + 1. We have proved the statement is true for n = k. According to step 3, it is also true for k + 1 (i.e., 1 + 1 = 2). By repeating the above logic, it is true for every natural number
Principle of Mathematical Induction Mathematical induction is one of the techniques, which can be used to prove a variety of mathematical statements which are formulated in terms of n, where n is a positive integer.
Let P(n) be given statement involving the natural number n such that (i) The statement is true for n = 1, i.e. P(1) is true. (ii) If the statement is true for n = k (where k is a particular but arbitrary natural number), then the statement is also true for n = k + 1 i.e. truth of P(k) implies that the truth of P(k + 1). Then, P(n) is true for all natural numbers n.
Angle is a measure of rotation of a given ray about its initial point. The original ray is called the initial side and the final position of ray after rotation is called terminal side of the angle. The point of rotation is called vertex. If the direction of rotation is anti-clockwise, the angle is said to be positive and if the direction of rotation is clockwise, then the angle is negative.
When a ray OA starting from its initial position OA rotates about its end point 0 and takes the final position OB, we say that angle
AOB (written as ∠ AOB) has been formed. The amount of rotation from the initial side to the terminal side is Called the measure of the angle.
Positive and Negative Angles
An angle formed by a rotating ray is said to be positive or negative depending on whether it moves in an anti-clockwise or a clockwise direction, respectively.
In this system, a right angle is divided into 90 equal parts, called degrees. The symbol 1° is used to denote one degree. Each degree is divided into 60 equal parts, called minutes and one minute is divided into 60 equal parts, called seconds. Symbols 1′ and 1″ are used to denote one minute and one second, respectively.
i.e., 1 right angle = 90° 1° = 60′ 1′ = 60″
2. Centesimal System (French System)
In this system, a right angle is divided into 100 equal parts, called ‘grades’. Each grade is subdivided into 100 min and each minute is divided into 100 s.
3. Circular System (Radian System) In this system, angle is measured in radian.
A radian is the angle subtended at the centre of a circle by an arc, whose length is equal to the radius of the circle.
The number of radians in an angle subtended by an arc of circle at the centre is equal to arc/radius.
Relationships
(i) π radian = 180° or 1 radian (180°/π)= 57°16’22” where, π = 22/7 = 3.14159 (ii) 1° = (π/180) rad = 0.01746 rad (iii) If D is the number of degrees, R is the number of radians and G is the number of grades in an angle θ, then
(iv) θ = l/r where θ = angle subtended by arc of length / at the centre of the circle, r = radius of the circle.
Trigonometric Ratios
Relation between different sides and angles of a right angled triangle are called trigonometric ratios or T-ratios
Trigonometric (or Circular) Functions
Let X’OX and YOY’ be the coordinate axes. Taking 0 as the centre and a unit radius, draw a circle, cutting the coordinate axes at A,B, A’ and B’, as shown in the figure.
Now, the six circular functions may be defined as under (i) cos θ = x (ii) sin θ = y (iii) sec θ = 1/x, x ≠ 0 (iv) cosec θ = 1/y, y ≠ 0 (v) tan θ = y/x, x ≠ 0 (vi) cot θ = x/y, y ≠ 0
Domain and Range
Range of Modulus Functions
sin θ|≤ 1, |cos θ| ≤ 1, |sec θ| ≥ 1, |Cosec θ| ≥ 1 for all values of 0, for which the functions are defined.
Trigonometric Identities
An equation involving trigonometric functions which is true for all those angles for which the functions are defined is called trigonometrical identity. Some identities are
Sign of Trigonometric Ratios
Trigonometric Ratios of Some Standard Angles
Trigonometric Ratios of Some Special Angles
Trigonometric Ratios of Allied Angles
Two angles are said to be allied when their sum or difference is either zero or a multiple of 90°.
The angles — θ, 90° ± θ, 180° ± θ, 270° + θ, 360° —θ etc., are angles allied to the angle θ, if θ is measured in degrees.
Trigonometric Periodic Functions
A function f(x) is said to be periodic, if there exists a real number T> 0 such that f(x + T)= f(x) for all x. T is called the period of the function, all trigonometric functions are periodic.
Maximum and Minimum Values of Trigonometric Expressions
Trigonometric Ratios of Compound Angles
The algebraic sum of two or more angles are generally called compound angles and the angles are known as the constituent angle. Some standard formulas of compound angles have been given below.
CARTESIAN PRODUCT OF SETS: Given two non-empty sets A and B, the set of all ordered pairs (x, y), where x ∈ A and y ∈ B is called Cartesian product of A and B; symbolically, we write A × B = {(x, y) | x ∈ A and y ∈ B}. Example- A = {1, 2, 3} and B = {4, 5}, then A × B = {(1, 4), (2, 4), (3, 4), (1, 5), (2, 5), (3, 5)} and B × A = {(4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3)}. Two ordered pairs are equal, if and only if the corresponding first elements are equal and the second elements are also equal. If there are p elements in A and q elements in B, then there will be pq elements in A × B, i.e., if n(A) = p and n(B) = q, then n(A × B) = pq. If either P or Q is the null set, then P × Q will also be empty set, i.e., P × Q = φ If A and B are non-empty sets and either A or B is an infinite set, then so is A × B. A × A × A = {(a, b, c): a, b, c ∈ A}. Here (a, b, c) is called an ordered triplet.
Ordered Pair An ordered pair consists of two objects or elements in a given fixed order.
Equality of Two Ordered Pairs Two ordered pairs (a, b) and (c, d) are equal if a = c and b = d.
Cartesian Product of Two Sets For any two non-empty sets A and B, the set of all ordered pairs (a, b) where a ∈ A and b ∈ B is called the cartesian product of sets A and B and is denoted by A × B. Thus, A × B = {(a, b) : a ∈ A and b ∈ B} If A = Φ or B = Φ, then we define A × B = Φ
Note:
A × B ≠ B × A
If n(A) = m and n(B) = n, then n(A × B) = mn and n(B × A) = mn
If atieast one of A and B is infinite, then (A × B) is infinite and (B × A) is infinite.
Relations A relation R from a non-empty set A to a non-empty set B is a subset of the cartesian product set A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B. The set of all first elements in a relation R is called the domain of the relation B, and the set of all second elements called images is called the range of R.
Note:
A relation may be represented either by the Roster form or by the set of builder form, or by an arrow diagram which is a visual representation of relation.
If n(A) = m, n(B) = n, then n(A × B) = mn and the total number of possible relations from set A to set B = 2mn
Inverse of Relation For any two non-empty sets A and B. Let R be a relation from a set A to a set B. Then, the inverse of relation R, denoted by R-1 is a relation from B to A and it is defined by R-1 ={(b, a) : (a, b) ∈ R} Domain of R = Range of R-1 and Range of R = Domain of R-1.
Functions A relation f from a set A to set B is said to be function, if every element of set A has one and only image in set B. In other words, a function f is a relation such that no two pairs in the relation have the first element.
Real-Valued Function A function f : A → B is called a real-valued function if B is a subset of R (set of all real numbers). If A and B both are subsets of R, then f is called a real function.
Some Specific Types of Functions Identity function: The function f : R → R defined by f(x) = x for each x ∈ R is called identity function. Domain of f = R; Range of f = R
Constant function: The function f : R → R defined by f(x) = C, x ∈ R, where C is a constant ∈ R, is called a constant function. Domain of f = R; Range of f = C
Polynomial function: A real valued function f : R → R defined by f(x) = a0 + a1x + a2x2+…+ anxn, where n ∈ N and a0, a1, a2,…….. an ∈ R for each x ∈ R, is called polynomial function.
Rational function: These are the real function of type f(x)g(x), where f(x)and g(x)are polynomial functions of x defined in a domain, where g(x) ≠ 0.
The modulus function: The real function f : R → R defined by f(x) = |x| or
for all values of x ∈ R is called the modulus function. Domaim of f = R Range of f = R+ U {0} i.e. [0, ∞)
Signum function: The real function f : R → R defined by f(x) = |x|x, x ≠ 0 and 0, if x = 0 or
is called the signum function. Domain of f = R; Range of f = {-1, 0, 1}
Greatest integer function: The real function f : R → R defined by f (x) = {x}, x ∈ R assumes that the values of the greatest integer less than or equal to x, is called the greatest integer function. Domain of f = R; Range of f = Integer
Fractional part function: The real function f : R → R defined by f(x) = {x}, x ∈ R is called the fractional part function. f(x) = {x} = x – [x] for all x ∈R Domain of f = R; Range of f = [0, 1)
Algebra of Real Functions Addition of two real functions: Let f : X → R and g : X → R be any two real functions, where X ∈ R. Then, we define (f + g) : X → R by {f + g) (x) = f(x) + g(x), for all x ∈ X.
Subtraction of a real function from another: Let f : X → R and g : X → R be any two real functions, where X ⊆ R. Then, we define (f – g) : X → R by (f – g) (x) = f (x) – g(x), for all x ∈ X.
Multiplication by a scalar: Let f : X → R be a real function and K be any scalar belonging to R. Then, the product of Kf is function from X to R defined by (Kf)(x) = Kf(x) for all x ∈ X.
Multiplication of two real functions: Let f : X → R and g : X → R be any two real functions, where X ⊆ R. Then, product of these two functions i.e. f.g : X → R is defined by (fg) x = f(x) . g(x) ∀ x ∈ X.
Quotient of two real functions: Let f and g be two real functions defined from X → R. The quotient of f by g denoted by fg is a function defined from X → R as
Set A set is a well-defined collection of objects.
Representation of Sets There are two methods of representing a set
Roster or Tabular form In the roster form, we list all the members of the set within braces { } and separate by commas.
Set-builder form In the set-builder form, we list the property or properties satisfied by all the elements of the sets
Set is a collection of well defined objects which are distinct from each other. Sets are usually denoted by capital letters A, B,C,… and elements are usually denoted by small letters a, b,c,…
If ‘a’ is an element of a set A, then we write a ∈ A and say ‘a’ belongs to A or ‘a’ is in A or ‘a’ is a member of A. If ‘a’ does not belongs to A, we write a ∉ A.
Standard Notations
• N : A set of natural numbers. • W : A set of whole numbers. • Z : A set of integers. • Z+/Z– : A set of all positive/negative integers. • Q : A set of all rational numbers. • Q+/Q– : A set of all positive/ negative rational numbers. • R : A set of real numbers. • R+/R–: A set of all positive/negative real numbers. • C : A set of all complex numbers.
Methods for Describing a Set
(i) Roster/Listing Method/Tabular Form In this method, a set is described by listing element, separated by commas, within braces.
e.g., A = {a, e, i, o, u}
(ii) Set Builder/Rule Method In this method, we write down a property or rule which gives us all the elements of the set by that rule.
e.g.,A = {x : x is a vowel of English alphabets}
Types of Sets
1. Finite Set A set containing finite number of elements or no element.
2. Cardinal Number of a Finite Set The number of elements in a given finite set is called cardinal number of finite set, denoted by n (A).
3. Infinite Set A set containing infinite number of elements.
4. Empty/Null/Void Set A set containing no element, it is denoted by (φ) or { }.
5. Singleton Set A set containing a single element.
6. Equal Sets Two sets A and B are said to be equal, if every element of A is a member of B and every element of B is a member of A and we write A = B.
7. Equivalent Sets Two sets are said to be equivalent, if they have same number of elements.
If n(A) = n (B), then A and B are equivalent sets. But converse is not true.
8. Subset and Superset Let A and B be two sets. If every element of A is an element of B, then A is called subset of B and B is called superset of A. Written as A ⊆ B or B ⊇ A
9. Proper Subset If A is a subset of B and A ≠ B, then A is called proper subset of B and we write A ⊂ B.
10. Universal Set (U) A set consisting of all possible elements which occurs under consideration is called a universal set.
11. Comparable Sets Two sets A and Bare comparable, if A ⊆ B or B ⊆ A.
12. Non-Comparable Sets For two sets A and B, if neither A ⊆ B nor B ⊆ A, then A and Bare called non-comparable sets.
13. Power Set (P) The set formed by all the subsets of a given set A, is called power set of A, denoted by P(A).
14. Disjoint Sets Two sets A and B are called disjoint, if, A ∩ B = (φ).
Venn Diagram
In a Venn diagram, the universal set is represented by a rectangular region and a set is represented by circle or a closed geometrical figure inside the universal set.
Operations on Sets
Union of Sets
The union of two sets A and B, denoted by A ∪ B is the set of all those elements, each one of which is either in A or in B or both in A and B.
Intersection of Sets
The intersection of two sets A and B, denoted by A ∩ B, is the set of all those elements which are common to both A and B.
If A1, A2,… , An is a finite family of sets, then their intersection is denoted by
Complement of a Set
If A is a set with U as universal set, then complement of a set, denoted by A’ or Ac is the set U – A .
Difference of Sets
For two sets A and B, the difference A – B is the set of all those elements of A which do not belong to B.
Symmetric Difference
For two sets A and B, symmetric difference is the set (A – B) ∪ (B – A) denoted by A Δ B.
Laws of Algebra of Sets
For three sets A, B and C
(i) Commutative Laws A ∩ B = B ∩ A A ∪ B = B ∪ A
(ii) Associative Laws (A ∩ B) ∩ C = A ∩ (B ∩ C) (A ∪ B) ∪ C = A ∪ (B ∪ C)
(iii) Distributive Laws A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
(iv) Idempotent Laws A ∩ A = A A ∪ A = A
(v) Identity Laws A ∪ Φ = A A ∩ U = A
(vi) De Morgan’s Laws
(a) (A ∩ B) ′ = A ′ ∪ B ′ (b) (A ∪ B) ′ = A ′ ∩ B ′ (c) A – (B ∩ C) = (A – B) ∩ (A- C) (d) A – (B ∪ C) = (A – B) ∪ ( A – C)
(vii) (a) A – B = A ∩ B’ (b) B – A = B ∩ A’ (c) A – B = A ⇔A ∩ B= (Φ) (d) (A – B) ∪ B= A ∪ B (e) (A – B) ∩ B = (Φ) (f) A ∩ B ⊆ A and A ∩ B ⊆ B (g) A ∪ (A ∩ B)= A (h) A ∩ (A ∪ B)= A
(viii) (a) (A – B) ∪ (B – A) = (A ∪ B) – (A ∩ B) (b) A ∩ (B – C) = (A ∩ B) – (A ∩ C) (c) A ∩ (B Δ C) = (A ∩ B) A (A ∩ C) (d) (A ∩ B) ∪ (A – B) = A (e) A ∪ (B – A) = (A ∪ B)
(ix) (a) U’ = (Φ) (b) Φ’ = U (c) (A’ )’ = A (d) A ∩ A’ = (Φ) (e) A ∪ A’ = U (f) A ⊆ B ⇔ B’ ⊆ A’
Important Points to be Remembered
• Every set is a subset of itself i.e., A ⊆ A, for any set A. • Empty set Φ is a subset of every set i.e., Φ ⊂ A, for any set A. • For any set A and its universal set U, A ⊆ U • If A = Φ, then power set has only one element i.e., n (P(A)) = 1 • Power set of any set is always a non-empty set. Suppose A = {1, 2}, thenP(A) = {{1}, {2},{1,2}, Φ}.(a) A ∉ P(A) (b) {A} ∈ P(A) • (vii) If a set A has n elements, then P(A) or subset of A has 2n elements. • (viii) Equal sets are always equivalent but equivalent sets may not be equal. The set {Φ} is not a null set. It is a set containing one element Φ.
Results on Number of Elements in Sets
• n (A ∪ B) = n(A) + (B)- n(A ∩ B) • n(A ∪ B) = n(A)+ n(B), if A and B are disjoint. • n(A – B) = n(A) – n(A ∩ B) • n(A Δ B) = n(A) + n(B)- 2n(A ∩ B) • n(A ∪ B ∪ C)= n(A)+ n(B)+ n(C)- n(A ∩ B) – n(B ∩ C)- n(A ∩ C)+ n(A ∩ B ∩ C) • n (number of elements in exactly two of the sets A, B, C) = n(A ∩ B) + n(B ∩ C) + n (C ∩ A)- 3n(A ∩ B ∩ C) • n (number of elements in exactly one of the sets A, B, C) = n(A) + n(B) + n(C) – 2n(A ∩ B) – 2n(B ∩ C) – 2n(A ∩ C) + 3n(A ∩ B ∩ C) • n(A’ ∪ B’)= n(A ∩ B)’ = n(U) – n(A ∩ B) • n(A’ ∩ B’ ) = n(A ∪ B)’ = n(U) – n(A ∪ B) • n(B – A) = n(B)- n(A ∩ B)
Ordered Pair
An ordered pair consists of two objects or elements in a given fixed order.
Equality of Ordered Pairs Two ordered pairs (a1, b1) and (a2, b2) are equal iff a1 = a2 and b1 = b2.
Cartesian Product of Sets
For two sets A and B (non-empty sets), the set of all ordered pairs (a, b) such that a ∈ A and b ∈ B is called Cartesian product of the sets A and’ B, denoted by A x B. A x B={(a,b):a ∈ A and b ∈ B}
If there are three sets A, B, C and a ∈ A, be B and c ∈ C, then we form, an ordered triplet (a, b, c). The set of all ordered triplets (a, b, c) is called the cartesian product of these sets A, B and C.
i.e., A x B x C = {(a,b,c):a ∈ A,b ∈ B,c ∈ C}
Properties of Cartesian Product
For three sets A, B and C
• n (A x B)= n(A) n(B) • A x B = Φ, if either A or B is an empty set. • A x (B ∪ C)= (A x B) ∪ (A x C) • A x (B ∩ C) = (A x B) ∩ (A x C) • A x (B — C)= (A x B) — (A x C) • (A x B) ∩ (C x D)= (A ∩ C) x (B ∩ D) • If A ⊆ B and C ⊆ D, then (A x C) ⊂ (B x D) • If A ⊆ B, then A x A ⊆ (A x B) ∩ (B x A) • A x B = B x A ⇔ A = B • If either A or B is an infinite set, then A x B is an infinite set. • A x (B’ ∪ C’ )’ = (A x B) ∩ (A x C) • A x (B’ ∩ C’ )’ = (A x B) ∪ (A x C) 2elements in common. • If ≠ B, then A x B ≠ B x A • If A = B, then A x B= B x A • If A ⊆ B, then A x C = B x C for any set C.
Question 1. Describe in brief the three challenges faced by democracy. (2014 D) Answer:
Foundational challenge. It relates to making the transition to democracy and then instituting democratic government. It involves bringing down the existing non-democratic regime, keeping military away from controlling government and establishing a sovereign and functional State.
Challenge of expansion. It involves applying the basic principle of democratic government across all the regions, different social groups and various institutions. It pertains to ensuring greater power to local governments, extension of federal principle to all the units of the federation, inclusion of women and minority groups, etc. Most established democracies, e.g., India and US, face the challenge of expansion.
Challenge of deepening of democracy. This challenge involves strengthening of the institutions and practices of democracy. It means strengthening those institutions that help people’s participation and control in the government. It aims at bringing down the control and influence of rich and powerful people in making governmental decisions.
Question 2. What do you mean by foundational challenge in democracy? What values can help to overcome this challenge? (2014 OD) Answer: Transition to democratic institutions from non-democratic regimes, separation of military from governing authority, establishing a sovereign and a functional state can be some of the foundational challenges in democracies. The values that may help overcome them are:
honesty
equality
freedom
Question 3. Describe in brief the three challenges faced by democracy. Answer:
Foundational challenge. It relates to making the transition to democracy and then instituting democratic government. It involves bringing down the existing non-democratic regime, keeping military away from controlling government and establishing a sovereign and functional State.
Challenge of expansion. It involves applying the basic principle of democratic government across all the regions, different social groups and various institutions. It pertains to ensuring greater power to local governments, extension of federal principle to all the units of the federation, inclusion of women and minority groups, etc. Most established democracies, e.g., India and US, face the challenge of expansion.
Challenge of deepening of democracy. This challenge involves strengthening of the institutions and practices of democracy. It means strengthening those institutions that help people’s participation and control in the government. It aims at bringing down the control and influence of rich and powerful people in making governmental decisions.
Question 4. Explain with examples why some laws that seek to ban something are not very successful in politics. (2011 D) Answer: Law has an important role to play in political reform. Carefully devised changes in law can help to discourage wrong political practices and encourage good ones. But legal constitutional changes by themselves are not effective, until carried out by political activists, parties, movements and politically conscious citizens. Any legal change must carefully look at what results it will have on politics. Sometimes it can be counter-productive.
For example, many states have banned people who have more than two children from contesting panchayat elections. This has resulted in denial of democratic opportunity to many poor men and women.
The best laws are those which empower the people to carry out democratic reforms. The Right to Information Act is a good example that supplements the existing laws. “Any law for political reforms is a good solution but who will implement it and how”—is the question. It is not necessary that the legislators will pass legislations that go against the interests of the political parties and MPs.
Question 5. “Legal constitutional changes by themselves cannot overcome challenges to democracy.” Explain with example. (2015 D, 2013 D, 2011 D) Or How are the challenges to democracy linked to the possibility of political reforms? Explain. Or Suggest any five political reforms to strengthen democracy. (2014 D) Answer: As legal constitutional changes by themselves cannot overcome challenges to democracy, democratic reforms need to be carried out mainly by political activists, parties, movements and politically conscious citizens. (i) Any legal change must carefully look at what results it will have on politics. Generally, laws that seek a ban on something are rather counter-productive; For example, many states have debarred people who have more than two children from contesting Panchayat elections. This has resulted in denial of democratic opportunity to many poor women, which was not intended. The best laws are those which empower people to carry out democratic reforms; for example, the Right to Information Act which acts as a watchdog of democracy by controlling corruption.
(ii) Democratic reforms are to be brought about principally through political parties. The most important concern should be to increase and improve the quality of political participation by ordinary citizens.
(iii) Any proposal for political reforms should think not only about what is a good solution, but also about who will implement it and how. Measures that rely on democratic movements, citizens’ organizations and media are likely to succeed.
Question 6. Explain the ‘foundational challenge’ of democracy by stating three points. (2011 D) Answer:
Foundational challenge relates to making the transition to democracy and then instituting democratic government. It involves establishing a sovereign and functional state.
It involves bringing down the existing non-democratic regime, keeping military away from controlling government and establishing a civilian control over all governmental institutions by holding elections.
It involves the recognition of people’s choice and opportunity to change rulers, recognise people’s will. In countries like Myanmar political leader Suu Kyi has been kept under house arrest for more than 20 years. Thus, in this case, foundational challenge recognizes the need to release political leaders and recall them from exile and holding of multiparty elections.
Question 7. Explain ‘the challenge of deepening of democracy’ by stating three points. (2012 D, 2014 OD) Answer: The challenge of deepening of democracy:
This challenge involves strengthening of the institutions and practices of democracy. It means strengthening those institutions that help people’s participation and control in the government. The challenge lies in realising the expectations of the people in a democracy. It is possible that some significant decisions may take place through consensus but challenging moments in democracy usually involve conflict between those groups who have power and those who aspire for a share in power. In Bolivia, the water struggle was a challenge of deepening of democracy.
The challenge of deepening of democracy is faced by every nation in one form or another. It aims at bringing down the control and influence of the rich and powerful people in making governmental decisions. The need is for individual freedom and dignity to have legal and moral force.
Question 8. How are some countries of the world facing the ‘challenge of expansion of democracy’? Explain with examples. (2012 D, 2012 OD) “Most of the established democracies are facing the challenge of expansion.” Support the statement with examples. (2016 D) Answer: Most of the established democracies face the challenge of expansion. This involves applying the basic principle of democratic government across all the regions, different social groups and various institutions. Ensuring greater power to local government, extension of federal principle to all the units of federation, inclusion of women and minority groups, etc. falls under this challenge. This means less and less decisions should remain outside the arena of democratic control. Most of the countries including India and the US face this challenge.
Question 9. Explain with examples how do some countries face foundational challenge of democracy. (2013 OD) Answer:
Foundational challenge relates to making the transition to democracy and then instituting democratic government. It involves establishing a sovereign and functional state.
It involves bringing down the existing non-democratic regime, keeping military away from controlling government and establishing a civilian control over all governmental institutions by holding elections. Eg: Nepal, Egypt, Pakistan.
In countries like Pakistan, democracy comes for and/ or remains for a short time and gets replaced by dictatorial rule.
It involves the recognition of people’s choice and opportunity to change rulers, recognize people’s will. In countries like Myanmar political leader Suu Kyi was kept under house arrest for more than 20 years. Thus, in this case, foundational challenge recognizes the need to release political leaders and recall them from exile and holding of multiparty elections.
Question 10. Analyse three major challenges before countries which do not have democratic form of governments. (2013 OD) Answer: Challenges faced by countries which do not have a democratic form of government:
These countries face the foundational challenge of making the transition to democracy and then instituting democratic government.
They also face the challenge of bringing down the existing non-democratic regime, and keeping the military away from controlling the government.
Such countries have to make great efforts to establish a sovereign and functional State
Question 1. Analyse any three values that make democracy better. (2017 D) Answer: We feel that democracy is a better form of government than any other form of government because:
Democracy promotes equality among citizens.
It enhances dignity of individual. It promotes dignity of women and strengthens the claims of the disadvantaged.
It improves the quality of decision making. There is transparency in a democracy.
It provides methods to resolve conflicts.
Democracy allows room to correct mistakes.
Question 2. On the basis of which values will it be a fair expectation that democracy should produce a harmonious social life? Explain. (2017 OD) Answer: No society can fully and permanently resolve conflicts among different groups. But we can certainly learn to respect these differences and evolve a mechanism to negotiate the differences. Belgium is an example of how successfully differences were negotiated among ethnic groups. Therefore, democracy is best suited to accommodate various social divisions as it usually develops a procedure to conduct their competition. But the example of Sri Lanka shows how distrust between two communities turned into widespread conflict. Thus, a democracy must fulfil the following conditions and be based on these values in order to achieve a harmonious social life—
Majority and minority opinions are not permanent. Democracy is not simply rule by majority opinion. The majority needs to work with minority so that government may function to represent the general view.
Rule by majority does not become rule by majority community in terms of religion or race or linguistic groups, etc.
Democracy remains democracy so long as every citizen has a chance of being in majority at some point of time. No individual should be debarred from participating in a democracy on the basis of religion, caste, community, creed and other such factors.
Question 3. Why do we feel that democracy is a better form of government than any other form? Explain. 2015OD Answer: Democracy is a better form of government than any other form because:
It is based on the idea of deliberation and negotiation. Thus the necessary delay in implementation.
Decisions are acceptable to people and are more effective.
A citizen has the right and the means to examine the process of decision-making. There is transparency in a democracy.
Democratic government is a legitimate government, people’s own government.
Ability to handle differences, decisions and conflicts is a positive point of democratic regimes.
Democracy has strengthened the claims of the disadvantaged and discriminated castes for equal status and equal opportunity.
Question 4. Why do we feel that democracy is a better form of government than any other form of government? Explain. (2012 OD) Or How do you feel that democracy is better than any other form of government? Explain. (2013 OD) Or “Democracy is more effective than its other alternatives.” Justify the statement. (2015 D) Answer: We feel that democracy is a better form of government than any other form of government because:
Democracy promotes equality among citizens.
It enhances dignity of individual. It promotes dignity of women and strengthens the claims of the disadvantaged.
It improves the quality of decision making. There is transparency in a democracy.
It provides methods to resolve conflicts.
Democracy allows room to correct mistakes.
Question 5. How do democracies accommodate social diversity? Explain with examples. (2011 OD, 2014 OD) Or Explain the conditions in which democracies are able to accommodate social diversities. (2012 D) Or “Democracies lead to peaceful and harmonious life among citizens”. Support the statement with suitable examples. (2013 OD) Answer: No society can fully and permanently resolve conflicts among different groups. But we can certainly learn to respect these differences and evolve a mechanism to negotiate the differences. Belgium is an example of how successfully differences were negotiated among ethnic groups. Therefore, democracy is best suited to accommodate various social divisions as it usually develops a procedure to conduct their competition. But the example of Sri Lanka shows how distrust between two communities turned into widespread conflict, and thus a democracy must fulfil the following two conditions in order to achieve a harmonious social life:
Majority and minority opinions are not permanent. Democracy is not simply rule by majority opinion. The majority needs to work with minority so that government may function to represent the general view.
Rule by majority does not become rule by majority community in terms of religion or race or linguistic groups, etc.
Democracy remains democracy so long as every citizen has a chance of being in majority at some point of time. No individual should be debarred from participating in a democracy on the basis of religion, caste, community, creed and other such factors.
Question 6. Explain the ways in which democracy has succeeded in maintaining dignity and freedom of citizens. (2012 D) Or, “Democracy stands much superior to any other form of government in promoting dignity and freedom of the individual.” Support the statement with suitable examples. 20130D Answer: The passion of respect and freedom are the basis of democracy:
Economic disparity in society has been minimized to a great extent.
In many democracies women were deprived of their right to vote for a long period of time. After long struggle they achieved their right, respect and equal treatment.
Democracy in India has strengthened the claims of the disadvantaged and discriminated castes for equal states and opportunities, for example, SCs and STs.
In democracy all adult citizens have the right to vote.
Democracy evolves a mechanism that takes into account the differences and intrinsic attributes of various ethnic groups. In a democracy majority always needs to work taking into account the interest of the minority so that the minority do not feel alienated.
Question 7. How is democracy a better form of government in comparison with other forms of governments? Explain. (2016 D, 2014 D) Or, “There is an overwhelming support for the idea of democracy all over the world.” Support the statement. (2015 OD) Answer: Over a hundred countries of the world today claim and practice some kind of democratic politics.
They have formal constitutions, hold elections, have parties and they guarantee rights of citizens. Thus, in most countries, the democracy produces a government that is accountable to the citizens and responsive to the needs and expectations of the citizens.
No society can fully and permanently resolve conflict among different groups. But we can learn to respect these differences and evolve mechanisms to negotiate them. Democracy is best suited as it develops a procedure to conduct competitions. Belgium is a successful example of negotiating difference among ethnic population.
Passion for respect and freedom is the basis of democracy and has been achieved in various degrees in various democracies.
The support for democracy is overwhelming all over the world and is evident from South Asia, where the support exists in countries with democratic as well as undemocratic regimes.
People wish to be ruled by representatives elected by them as a democratic government is people’s own government and makes them believe that it is suitable for their country as it is a legitimate government.
Question 8. “Most destructive feature of democracy is that its examination never gets over.” Support the statement with appropriate arguments. (2011 D) Answer: Suitable arguments:
As people get some benefits of democracy, they ask for more.
People always come up with more expectations from the democratic set up.
They also have complaints against democracy.
More and more suggestions and complaints by the people is also a testimony to the success of democracy.
A public expression of dissatisfaction with democracy shows the success of the democratic project.
Question 9. “Democracy is seen to be good in principle but felt to be not so good in practice.” Justify the statement. (2013 D) Answer: If we look at some of the democratic policies being implemented in more than one hundred countries of the world, democracy seems to be good. For example, having a formal Constitution, holding regular elections, guaranteeing the citizens certain rights, working for the welfare of the people etc. make us advocate that democracy is good.
But if we look in terms of social situations, their economic achievements and varied cultures, we find a very big difference in most of the democracies. The vast economic disparities, social injustice based on discrimination, standard of life, sex discrimination, etc. create many doubts about the merits of democracy. Whenever some of our expectations are not met, we start blaming the idea of democracy. Since democracy is a form of government, it can only create conditions for achieving our goals if they are reasonable.
Question 10. “Democracy stands much superior in promoting dignity and freedom of the citizens”. Justify the statement. (2016 OD) Answer: Examples to illustrate that dignity and freedom of citizens are best guaranteed in a democracy:
(i) Dignity of women. Democracy recognizes dignity of women as a necessary ingredient of society. The one way to ensure that women related problems get adequate attention is to have more women as elected representatives. To achieve this, it is legally binding to have a fair proportion of women in the elected bodies. Panchayati Raj in India has reserved one-third seats in local government bodies for women. In March 2010, the Women’s Reservation Bill was passed in the Rajya Sabha ensuring 33% reservation for women in Parliament and State legislative bodies.
(ii) Democracy has strengthened the claims of disadvantaged and discriminated castes. When governments are formed, political parties usually take care that representatives of different castes and tribes find a place in it. Some political parties are known to favour some castes. Democracy provides for equal status and opportunities for all castes.
(iii) Democracy transforms people from the status of a subject into that of a citizen. A democracy is concerned with ensuring that people will have the right to choose their rulers and people will have control over the rulers. Whenever possible and necessary, citizens should be able to participate in decision-making that affects them all.
(iv) A citizen has the right and the means to examine the process of decision-making. There is transparency in a democracy like India. In October 2005, the Right to Information (RTI) law was passed which ensures all its citizens the right to get all the information about the functions of the government departments. In a democracy, people also have the right to complain about its functioning.
