Caselets

Caselets – Advanced Quantitative Aptitude for the Uttarakhand Patwari / Lekhpal Programme

Target audience: Students preparing for the Uttarakhand Patwari / Lekhpal exam who want a deep, rigorous treatment of the Caselets section of the Quantitative Aptitude paper.
Approach: First‑principle derivation, systematic methodology, edge‑case analysis, and a challenging practice problem with full solution.
Prerequisites: Algebra (linear equations & inequalities), basic number theory, combinatorics, and elementary optimisation (linear programming).

1. What is a Caselet?

2. The Mathematical Core of a Caselet

2.1 Modelling as a System of Equations

1. Identify Variables

   • Assign a symbol to every unknown quantity that the problem could ask about.
   • Keep track of units (hectares, rupees, days).

2. Translate Text into Relations

   • Equality → linear or polynomial equation.
   • Ratio → proportion equation ( \frac{a}{b} = \frac{c}{d} ).
   • Difference → subtraction or absolute value.
   • Inequality → constraint ( \leq, \geq ).

3. Write a Complete System

   • Include all constraints.
   • Check for redundancy: sometimes two equations are linear combinations of each other.

4. Solve

   • If square system (same number of independent equations as variables), use substitution or elimination.
   • If over‑determined, check consistency (sum of left sides equals sum of right sides).
   • If underdetermined, look for extremal solutions (min/max) or additional context from the narrative.

2.2 When Linear Algebra Suffices

• Linear case: All relations are linear.
• Matrix formulation: ( A\mathbf{x} = \mathbf{b} ).
• Rank test: If (\text{rank}(A) = \text{rank}([A|\mathbf{b}]) = n) → unique solution.
• Determinant: If ( \det(A) \neq 0 ) → unique solution.

2.3 Non‑Linear or Integer Constraints

• Quadratic terms: e.g., area (= \text{length} \times \text{width}).
• Product constraints: e.g., two workers together cover 120 acres in 8 days → ( \frac{120}{8} = 15) acres/day → (r_1 + r_2 = 15).
• Integer restrictions: When variables represent counts (people, days, plots).
• Modular arithmetic: e.g., “each plot must contain an odd number of trees”.

2.4 Optimization in Caselets

• Min/max problems: “Find the minimum number of days required”.
• Linear programming:
  [ \begin{aligned} \text{Maximize } & c^T \mathbf{x} \ \text{s.t. } & A\mathbf{x} \leq \mathbf{b}, \ & \mathbf{x} \geq 0 \end{aligned} ]
• Corner‑point rule: Since the feasible region is a polyhedron, optimum lies at a vertex; test integer vertices if variables must be integer.

3. Systematic Solution Strategy (Step‑by‑Step)

Tip: For *“minimum /