⭐ Sabse Important Points · एक नज़र में
- Quadratic Formula: For ax^2 + bx + c = 0, roots are x = [-b ± √(b^2 - 4ac)] / (2a).
- Determinant of 2x2 Matrix: For A = [[a, b], [c, d]], det(A) = ad - bc.
- Discriminant (D = b^2 - 4ac): Determines nature of roots: D > 0 (real and distinct), D = 0 (real and equal), D < 0 (complex).
- Inverse of 2x2 Matrix: A^{-1} = (1/det(A)) * [[d, -b], [-c, a]], provided det(A) ≠ 0.
- Sum and Product of Roots: If α and β are roots, then α + β = -b/a and αβ = c/a.
- Cramer's Rule: For system ax+by=e, cx+dy=f, x = (ed - bf)/(ad - bc), y = (af - ec)/(ad - bc).
Introduction · परिचय
Algebra and Matrices form a core part of the NDA Mathematics syllabus. This topic covers fundamental algebraic identities, quadratic equations, sequences and series, and the basics of matrices and determinants. A strong grasp of these concepts is essential for solving problems quickly and accurately in the exam.
Quadratic Equations and Algebraic Identities
A quadratic equation is of the form ax^2 + bx + c = 0, where a ≠ 0. The roots can be found using the quadratic formula. Algebraic identities are standard formulas that help simplify expressions and solve equations rapidly.
📐 Formula / Rule Box · सूत्र याद रखो
Quadratic Formula
x = [-b ± √(b^2 - 4ac)] / (2a)
Sum of Roots
α + β = -b/a
Product of Roots
αβ = c/a
Quadratic Formula
For ax^2 + bx + c = 0, roots are x = [-b ± √(b^2 - 4ac)] / (2a).
⭐ Key Points · याद रखने वाली बातें
- Discriminant (D = b^2 - 4ac): Determines nature of roots: D > 0 (real and distinct), D = 0 (real and equal), D < 0 (complex).
- Sum and Product of Roots: If α and β are roots, then α + β = -b/a and αβ = c/a.
- Important Identities: (a ± b)^2 = a^2 ± 2ab + b^2; a^2 - b^2 = (a - b)(a + b); (a ± b)^3 = a^3 ± 3a^2b + 3ab^2 ± b^3.
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: Find the roots of 2x^2 - 7x + 3 = 0.
✅ Solution: x = 3 and x = 1/2
💡 Explanation: Using quadratic formula: a=2, b=-7, c=3. D = (-7)^2 - 4*2*3 = 49 - 24 = 25. Roots = [7 ± √25]/(4) = [7 ± 5]/4. So x = (7+5)/4 = 3 and x = (7-5)/4 = 1/2.
Matrices and Determinants
A matrix is a rectangular array of numbers. Determinants are scalar values that can be computed from square matrices and are used to solve systems of linear equations, find inverses, and determine properties like singularity.
📐 Formula / Rule Box · सूत्र याद रखो
Determinant 2x2
det([[a, b], [c, d]]) = ad - bc
Inverse 2x2
A^{-1} = (1/det(A)) [[d, -b], [-c, a]]
Cramer's Rule
For system ax+by=e, cx+dy=f, x = (ed - bf)/(ad - bc), y = (af - ec)/(ad - bc).
| Matrix | Determinant |
|---|---|
| Rectangular array of numbers | Scalar value for square matrices |
| Addition/subtraction possible for same order | Only defined for square matrices |
| Multiplication requires compatible orders | Multiplicative property: det(AB)=det(A)det(B) |
⭐ Key Points · याद रखने वाली बातें
- Determinant of 2x2 Matrix: For A = [[a, b], [c, d]], det(A) = ad - bc.
- Inverse of 2x2 Matrix: A^{-1} = (1/det(A)) * [[d, -b], [-c, a]], provided det(A) ≠ 0.
- Properties: det(AB) = det(A)det(B); det(A^T) = det(A); If any row/column is zero, det = 0.
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: Find the inverse of A = [[2, 3], [1, 4]].
✅ Solution: A^{-1} = (1/5) * [[4, -3], [-1, 2]]
💡 Explanation: det(A) = 2*4 - 3*1 = 8 - 3 = 5. Using formula: A^{-1} = (1/5)[[4, -3], [-1, 2]].
Practice Questions · खुद हल करके देखो
पहले खुद try karo, phir “Answer dekho” dabakar check karo. 👇
Q1If α and β are roots of x^2 - 5x + 6 = 0, find α^2 + β^2.
👀 Answer dekho
✅ Answer: 13
💡 Explanation: α+β=5, αβ=6. α^2+β^2 = (α+β)^2 - 2αβ = 25 - 12 = 13.
Q2Find the determinant of matrix A = [[1, 2], [3, 4]].
👀 Answer dekho
✅ Answer: -2
💡 Explanation: det(A) = 1*4 - 2*3 = 4 - 6 = -2.
Q3Solve for x: 3x - 2 = 7.
👀 Answer dekho
✅ Answer: x = 3
💡 Explanation: 3x = 9 ⇒ x = 3.
⚠️ Common Mistakes · ये गलतियाँ बिल्कुल मत करना
- Forgetting to check the discriminant before assuming real roots: Always compute D = b^2 - 4ac to determine nature of roots.
- In matrix inverse, forgetting to divide by determinant: The formula requires multiplying by 1/det(A); missing this leads to wrong answer.
- Misapplying sum/product of roots: Ensure the quadratic is in standard form ax^2+bx+c=0 before using α+β = -b/a and αβ = c/a.
⚡ Quick Revision · Exam से पहले 2 मिनट में दोहराओ
- Quadratic Formula: x = [-b ± √(b^2 - 4ac)]/(2a) – memorize and use for all quadratic equations.
- Determinant 2x2: det([[a,b],[c,d]]) = ad - bc – quick computation for inverse and Cramer's rule.
- Sum and Product of Roots: α+β = -b/a, αβ = c/a – useful for symmetric expressions without solving.