Top 15 Class 10 CBSE Math Competency Questions & Solutions (2024)

Understanding CBSE Competency-Based Questions

Competency-based questions in CBSE Class 10 Math assess conceptual understanding and problem-solving skills beyond rote learning. These questions require applying mathematical principles to real-world scenarios, analyzing relationships, and justifying solutions logically. Based on CBSE’s official guidelines, these 15 questions represent high-probability exam patterns for 2024.

Quadratic Equations: Prime Factorization Approach

Problem: Express the quadratic polynomial (n^2 + 3n - 4) as a product of two prime factors where (n) is a natural number. Verify that (n) is an even composite number.

Solution:

1. Factorize the polynomial:
   (n^2 + 3n - 4 = (n + 4)(n - 1))
2. For prime factors:
   • When (n = 3):
     Factors are (3 + 4 = 7) (prime) and (3 - 1 = 2) (prime)
   • (n = 3) is natural and even composite (since 4 is the smallest even composite)
3. Validation:
   (n = 1) gives factors 5 and 0 (invalid).
   (n = 2) gives 6 and 1 (6 not prime).

Key Insight: Test natural numbers systematically to satisfy dual conditions.

Zeroes and Coefficients Relationship

Problem: If (p) and (q) are zeroes of (2x^2 + 5x - 4), evaluate ((1 - p)(1 - q)) without finding (p) and (q).

Solution:

1. Use polynomial properties:
   (p + q = -\frac{5}{2}), (pq = -2)
2. Expand the expression:
   ((1 - p)(1 - q) = 1 - (p + q) + pq = 1 - (-\frac{5}{2}) + (-2) = 1 + 2.5 - 2 = 1.5)
3. Expert Tip: Leverage (\alpha + \beta) and (\alpha\beta) relationships to bypass direct root calculation.

Linear Equations: Dependency Check

Problem: Which pair of linear equations has a unique solution?
a) (x - y = 5) and (3x - 3y = 15)
b) (2x - 3y = 12) and (4x - 6y = 24)

Analysis:

• Option (a) represents dependent equations (second is (3 \times) first), yielding infinite solutions.
• Option (b) has parallel lines ((\frac{a_1}{a_2} = \frac{b_1}{b_2}
  eq \frac{c_1}{c_2})), no solution.
• Correct answer: Neither (trick question). Unique solutions require consistent, independent equations.

Arithmetic Progression: Sum Validation

Problem: Sum of first (p) terms of an AP is (6p^2 - 2p). Find the 5th term.

Solution:

1. (S_p = 6p^2 - 2p)
   (S_1 = a_1 = 6(1)^2 - 2(1) = 4)
2. Common difference (d):
   Coefficient of (p^2) in quadratic (S_p) is (\frac{d}{2} \implies 6 = \frac{d}{2} \implies d = 12)
3. 5th term: (a_5 = a_1 + 4d = 4 + 4(12) = 52)

Pro Tip: For (S_p = Ap^2 + Bp), (A = \frac{d}{2}) and (B = a - \frac{d}{2}).

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Advanced Problem-Solving Techniques

Coordinate Geometry: Rhombus Properties

Problem: Vertices of rhombus PQRS: (P(2, -3)), (Q(6, 5)), (R(-2, 1)). Find intersection point of diagonals and vertex (S).

Solution:

1. Diagonals bisect at midpoint of PR:
   Midpoint (= \left(\frac{2 + (-2)}{2}, \frac{-3 + 1}{2}\right) = (0, -1))
2. For vertex (S(x, y)):
   Midpoint of QS is ((0, -1)):
   (\frac{6 + x}{2} = 0 \implies x = -6)
   (\frac{5 + y}{2} = -1 \implies y = -7)
   (\therefore S(-6, -7))

Rule: Diagonals of rhombus bisect at 90°; use midpoint formulas.

Polynomial Applications

Problem: Polygon exterior angles form an AP with smallest angle (8^\circ) and common difference (4^\circ). Find number of sides.

Solution:

1. Sum of exterior angles (= 360^\circ)
2. (S_n = \frac{n}{2} [2 \times 8 + (n - 1)4] = 360)
   (\implies \frac{n}{2} [16 + 4n - 4] = 360)
   (\implies n(2n + 6) = 360)
   (\implies 2n^2 + 6n - 360 = 0)
   (\implies n^2 + 3n - 180 = 0)
   (\implies (n + 15)(n - 12) = 0)
   (\therefore n = 12) (sides)

Concept: Sum of exterior angles is always (360^\circ), regardless of polygon type.

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Actionable Resources and Checklist

Practice Checklist:

1. Solve quadratic equations using factorization and discriminant analysis
2. Verify polynomial properties using (\alpha + \beta) and (\alpha\beta)
3. Test linear equation pairs for consistency/dependency
4. Apply AP sum formulas to real-world sequences
5. Use coordinate geometry formulas midpoints and diagonals

Recommended Tools:

• NCERT Exemplar Problems: Builds foundational problem-solving skills
• RS Aggarwal Solutions: Ideal for varied difficulty levels
• CBSE Competency Focus: Practice official competency-based questions

"When solving quadratic equations, which step do you find most challenging? Share your experience below!"

Final Insight: Competency-based questions test logical reasoning, not just memorization. Regular practice of diverse problem types ensures mastery.

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Solutions derived from CBSE’s competency-focused materials and expert pedagogical analysis. Always validate answers with algebraic principles.