Which one among the following is indicated by the following, figure ?

1. Basics of Algebraic Expressions (basic)

At its heart, Algebra is the language of logic used to describe relationships between quantities. Unlike arithmetic, which deals with fixed numbers, algebra uses variables (usually letters like x, y, or a) to represent values that can change or are unknown. When we combine these variables with constants (fixed numbers) using mathematical operations like addition or multiplication, we create an algebraic expression. For instance, in economic modeling, we often use the intercept form of a linear equation, Y = a + bX, where 'a' and 'b' are constants that define the starting point and the rate of change respectively Macroeconomics (NCERT class XII 2025 ed.), Determination of Income and Employment, p.58. This allows us to solve for unknowns by substituting one value into another, a method fundamental to quantitative problem-solving Macroeconomics (NCERT class XII 2025 ed.), Determination of Income and Employment, p.53. One of the most powerful tools in algebra is the Identity. An identity is an equality that holds true for any value assigned to its variables. A classic example is the binomial expansion: (a + b)² = a² + 2ab + b². Geometrically, this isn't just a formula; it represents the area of a large square with side length (a + b). If you partition this square, you find it consists of one smaller square (a²), another smaller square (b²), and two identical rectangles (ab). This principle of representing the same total value through different expressions is a core concept used even in complex fields like national income accounting to ensure different methods of calculation yield the same result Macroeconomics (NCERT class XII 2025 ed.), National Income Accounting, p.23. Understanding the building blocks of these expressions helps us translate word problems into solvable math. We categorize these expressions based on their structure:

Component	Definition	Example (in 3x + 5)
Variable	A symbol for a number we don't know yet.	x
Coefficient	The number multiplying the variable.	3
Constant	A fixed value that does not change.	5

Sources: Macroeconomics (NCERT class XII 2025 ed.), Determination of Income and Employment, p.58; Macroeconomics (NCERT class XII 2025 ed.), Determination of Income and Employment, p.53; Macroeconomics (NCERT class XII 2025 ed.), National Income Accounting, p.23

2. Standard Algebraic Identities (basic)

In quantitative aptitude, an algebraic identity is an equality that holds true regardless of the values assigned to its variables. Unlike a standard equation which might only be true for specific values (like x + 2 = 5), an identity represents a fundamental equivalence between different ways of expressing the same mathematical relationship. In your broader studies, such as in Macroeconomics, you will see identities used to show that different methods of calculating GDP (like the product or income method) are essentially different expressions of the same variable Macroeconomics (NCERT class XII 2025 ed.), National Income Accounting, p.23. One of the most vital identities to master is the Square of a Sum: (a + b)² = a² + 2ab + b². To understand this from first principles, imagine a large square with a side length of (a + b). The total area of this square is naturally (a + b)². If you divide this large square into sections based on the lengths 'a' and 'b', you find it is composed of:

• One square with area a²
• One square with area b²
• Two identical rectangles, each with an area of ab

When you sum these four parts together, you get a² + 2ab + b², providing a perfect geometric proof of the algebraic formula. Mastering these identities is not just about memorization; it's about computational efficiency. For instance, if you need to find the square of 103, instead of long multiplication, you can view it as (100 + 3)². Applying the identity gives you 100² + 2(100)(3) + 3², which is 10,000 + 600 + 9 = 10,609. This logic of using formulas to simplify complex calculations is a recurring theme in advanced studies, including spatial analysis in geography where specific formulas are used to determine relative values Geography of India, Majid Husain, Spatial Organisation of Agriculture, p.17.

Sources: Macroeconomics (NCERT class XII 2025 ed.), National Income Accounting, p.23; Geography of India, Majid Husain, Spatial Organisation of Agriculture, p.17

3. Properties of Inequalities (intermediate)

In quantitative aptitude, Inequalities are expressions that compare two values or algebraic terms using symbols like >, <, ≥, or ≤. Unlike equations where we find a single point of balance, inequalities describe a range of possibilities. The most fundamental property to master is the Addition and Subtraction Rule: you can add or subtract the same number from both sides of an inequality without changing its direction. For instance, if a > b, then a + c > b + c. This logic underpins how we analyze gaps in various fields, from basic algebra to the measurement of income distribution in an economy through tools like the Lorenz Curve, which geometrically represents the distance between actual income distribution and perfect equality Indian Economy, Nitin Singhania (ed 2nd 2021-22), Poverty, Inequality and Unemployment, p.45.

The "Golden Rule" of inequalities—and where most students make mistakes—concerns Multiplication and Division. If you multiply or divide both sides by a positive number, the inequality sign stays the same. However, if you multiply or divide by a negative number, the inequality sign must flip. For example, if 10 > 5, multiplying by -1 gives us -10 and -5; since -10 is less than -5, the sign must reverse (-10 < -5). This behavior is different from linear equations and is vital when solving variables in competitive exams.

Geometrically, we can visualize the relationship between magnitudes through areas. Just as the binomial expansion (a + b)² = a² + 2ab + b² can be proven by partitioning a large square into smaller rectangles and squares, inequalities often deal with how these areas compare. In economic terms, we distinguish between natural inequalities (based on inherent traits) and socially-produced inequalities (created by systems and opportunities) Political Theory, Class XI (NCERT 2025 ed.), Equality, p.36-37. Understanding the mathematical properties of inequalities allows us to quantify these social gaps and understand concepts like the Gini Coefficient, which uses the area between the line of perfect equality and the Lorenz curve to measure economic disparity Indian Economy, Nitin Singhania (ed 2nd 2021-22), Poverty, Inequality and Unemployment, p.45.

Sources: Indian Economy, Nitin Singhania (ed 2nd 2021-22), Poverty, Inequality and Unemployment, p.45; Political Theory, Class XI (NCERT 2025 ed.), Equality, p.36-37

4. Geometric Series and Infinite Sums (intermediate)

In the world of quantitative aptitude, a Geometric Series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio (r). For example, in the sequence 1, 1/2, 1/4, 1/8..., the first term (a) is 1 and the common ratio (r) is 0.5. While an arithmetic progression grows by adding a constant, a geometric progression grows (or shrinks) exponentially. This concept is vital in economics, particularly when understanding how population grows or how money circulates through a banking system Geography of India, Contemporary Issues, p.49.

The most fascinating aspect of these series is the Infinite Sum. You might wonder: how can adding an infinite number of terms result in a finite number? This happens only when the common ratio is a fraction between 0 and 1 (0 < r < 1). As we progress, the terms become so infinitesimally small that they effectively vanish. The formula to find this sum (S) is remarkably elegant: S = a / (1 - r). This formula is the mathematical backbone of the Money Multiplier effect, where an initial injection of money into the economy creates successive rounds of spending and income, each a fraction of the previous one Macroeconomics, Money and Banking, p.51.

To visualize this geometrically, imagine a large square. If you divide it into smaller parts corresponding to the terms of a series, the sum of those parts will never exceed the area of the original square if the ratio is managed correctly. In financial terms, this explains why an initial deposit of ₹100, with a reserve ratio of 20% (leaving 80% or 0.8 to be lent out), doesn't create infinite money, but rather a specific, predictable total increase in the money supply Indian Economy, Money and Banking, p.159.

Concept	Arithmetic Progression (AP)	Geometric Progression (GP)
Growth Pattern	Additive (e.g., 2, 4, 6, 8)	Multiplicative (e.g., 2, 4, 8, 16)
Infinite Sum	Always tends to Infinity	Finite if the ratio is less than 1

Sources: Geography of India, Contemporary Issues, p.49; Macroeconomics, Money and Banking, p.51; Indian Economy, Money and Banking, p.159

5. Geometric Representation of Algebra (exam-level)

In quantitative aptitude, algebra is often viewed as a collection of abstract rules, but it has a powerful geometric foundation. By representing algebraic identities as physical areas, we can transform rote memorization into intuitive understanding. The most fundamental example is the expansion of the binomial (a + b)² = a² + 2ab + b². Geometrically, this identity represents the total area of a large square with a side length of (a + b). While algebra helps us find equilibrium in markets or calculate GDP aggregates Microeconomics (NCERT class XII 2025 ed.), Market Equilibrium, p.74, geometry provides the visual proof of why these identities hold true. To visualize (a + b)², imagine a large square where each side is divided into two segments: one of length a and one of length b. By drawing internal lines from these division points, the large square is partitioned into four distinct sub-regions. This decomposition allows us to see the components of the algebraic expression as tangible spaces:

Algebraic Component	Geometric Representation	Area
a²	A square with side length a	a × a
b²	A square with side length b	b × b
2ab	Two identical rectangles with sides a and b	2 × (a × b)

When you sum these areas together, you perfectly fill the large square of side (a + b). This principle of geometric decomposition is a recurring tool in mathematics, used even to explain complex concepts like the sum of infinite geometric series in economic models Macroeconomics (NCERT class XII 2025 ed.), Chapter 3: Money and Banking, p. 51. Understanding that the '2ab' term represents two specific rectangular regions prevents the common mistake of thinking (a + b)² is simply a² + b².

Sources: Microeconomics (NCERT class XII 2025 ed.), Market Equilibrium, p.74; Macroeconomics (NCERT class XII 2025 ed.), Chapter 3: Money and Banking, p.51; Macroeconomics (NCERT class XII 2025 ed.), National Income Accounting, p.23

6. Solving the Original PYQ (exam-level)

Now that you have mastered the building blocks of area-based proofs and algebraic identities, this question brings those concepts together in a visual format. In our previous modules, we discussed how an identity is not just a string of symbols but a representation of spatial logic. The figure associated with this problem represents a large square with a side length of (a + b). By breaking down this total area into its constituent parts, we move from a abstract visualization to a concrete mathematical expression.

To arrive at the correct answer, follow the geometric logic: the area of a square is calculated as side squared, so a square with side (a + b) has a total area of (a + b)². If you look at the internal partitioning of such a figure, you will find it consists of one square with area a², another smaller square with area b², and two identical rectangles with area ab. Summing these areas together—a² + ab + ab + b²—perfectly matches the expansion in (C) (a + b)² = a² + 2ab + b². This geometric proof is a fundamental way to visualize the binomial expansion.

It is important to recognize why the other options are classic UPSC distractors. Option (A) describes an infinite geometric series, which, as noted in Macroeconomics (NCERT class XII 2025 ed.), usually requires a figure showing a sequence of diminishing shapes rather than a single partitioned square. Options (B) and (D) involve inequalities; while they are mathematically valid properties, they describe relative values rather than a decomposition of area. The trap here is choosing an option that is mathematically true but does not actually match the spatial structure of the figure provided.