• Standard form:
  ax + by + c = 0
• Solution is the point where two lines intersect.

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2. Quadratic Equations

Standard Form:

ax² + bx + c = 0

Roots Formula (Quadratic Formula):

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}x=2a−b±b2−4ac​​

Nature of Roots:

• If D = b² – 4ac > 0 → Real & distinct
• If D = 0 → Real & equal
• If D < 0 → Imaginary

Factorization Method:

Find two numbers whose sum = b and product = ac

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3. Arithmetic Progression (AP)

• nth Term:

an=a+(n−1)da_n = a + (n – 1)dan​=a+(n−1)d

• Sum of n Terms:

Sn=n2[2a+(n−1)d]S_n = \frac{n}{2}[2a + (n – 1)d]Sn​=2n​[2a+(n−1)d]

• Sum when 1st and last term known:

Sn=n2(a+l)S_n = \frac{n}{2}(a + l)Sn​=2n​(a+l)

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4. Coordinate Geometry

Distance Formula:

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}d=(x2​−x1​)2+(y2​−y1​)2​

Section Formula (Internal Division):

Point dividing (x₁, y₁) & (x₂, y₂) in ratio m:n (mx2+nx1m+n,my2+ny1m+n)\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)(m+nmx2​+nx1​​,m+nmy2​+ny1​​)

Midpoint Formula:

(x1+x22,y1+y22)\left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right)(2×1​+x2​​,2y1​+y2​​)

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5. Linear Inequations

• Solution represented on number line.
• For inequality reversal: multiply/divide by negative → sign flips.

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6. Probability

P(E)=Number of favourable outcomesTotal outcomesP(E) = \frac{\text{Number of favourable outcomes}}{\text{Total outcomes}}P(E)=Total outcomesNumber of favourable outcomes​

• 0 ≤ P(E) ≤ 1
• P(E) + P(Ē) = 1

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7. Polynomials

• For quadratic polynomial ax² + bx + c:
  • Sum of roots (α + β):
  −ba-\frac{b}{a}−ab​
  • Product of roots (αβ):
  ca\frac{c}{a}ac​
• Factor Theorem:
  If P(a) = 0 → (x – a) is a factor.
• Remainder Theorem:
  P(x) ÷ (x – a) → remainder = P(a)

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8. Linear Programming (Basics)

• Objective function: Z = ax + by
• Constraints are inequations.
• Feasible region determined by graph.

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9. Simultaneous Linear Equations

Substitution / Elimination

• Solve for one variable and substitute in the other equation.

Cross-Multiplication Method:

For
a₁x + b₁y + c₁ = 0
a₂x + b₂y + c₂ = 0 x(b1c2−b2c1)=y(c1a2−c2a1)=1(a1b2−a2b1)\frac{x}{(b_1c_2 – b_2c_1)} = \frac{y}{(c_1a_2 – c_2a_1)} = \frac{1}{(a_1b_2 – a_2b_1)}(b1​c2​−b2​c1​)x​=(c1​a2​−c2​a1​)y​=(a1​b2​−a2​b1​)1​

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10. Real Numbers

Laws:

• Euclid Division Algorithm:

a=bq+r,  0≤r<ba = bq + r,\ \ 0 \le r < ba=bq+r,  0≤r<b

• HCF × LCM = Product of numbers

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11. Matrices (if included in your board)

• Addition/Subtraction: element-wise
• Scalar multiplication: multiply each element
• Multiplication:

(A⋅B)ij=∑aikbkj(A \cdot B)_{ij} = \sum a_{ik}b_{kj}(A⋅B)ij​=∑aik​bkj​

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