Quantitative Aptitude: Advance Math¶

Advance Math carries significant weightage in both SSC Tier-I and Tier-II (Geometry, Mensuration, Trigonometry, and Algebra).

1. Algebra¶

• \((a + b)^2 = a^2 + 2ab + b^2\)
• \((a - b)^2 = a^2 - 2ab + b^2\)
• \(a^2 - b^2 = (a - b)(a + b)\)
• \((a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)\)
• \(a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)\)
• If \(a + b + c = 0\), then \(a^3 + b^3 + c^3 = 3abc\)
• If \(x + \frac{1}{x} = k\):
• \(x^2 + \frac{1}{x^2} = k^2 - 2\)
• \(x^3 + \frac{1}{x^3} = k^3 - 3k\)

2. Geometry¶

Triangles¶

• Incenter (\(I\)): Intersection of angle bisectors (\(\angle BIC = 90^\circ + \frac{\angle A}{2}\)).
• Circumcenter (\(O\)): Intersection of perpendicular bisectors (\(\angle BOC = 2\angle A\)).
• Orthocenter (\(H\)): Intersection of altitudes (\(\angle BHC = 180^\circ - \angle A\)).
• Centroid (\(G\)): Intersection of medians (divides median in \(2:1\)).

Circles¶

• Alternate segment theorem.
• Tangent Secant Theorem: If \(PT\) is tangent and \(PAB\) is secant, then \(PT^2 = PA \times PB\).

3. Trigonometry¶

• \(\sin^2 \theta + \cos^2 \theta = 1\)
• \(1 + \tan^2 \theta = \sec^2 \theta \implies \sec^2 \theta - \tan^2 \theta = 1\)
• \(1 + \cot^2 \theta = \csc^2 \theta \implies \csc^2 \theta - \cot^2 \theta = 1\)
• Complementary Angles: If \(A + B = 90^\circ\):
• \(\sin A = \cos B\)
• \(\tan A \cdot \tan B = 1\)
• \(\sin^2 A + \sin^2 B = 1\)

4. Mensuration 2D & 3D¶