Algebra Class Quiz

Number Systems

• HCF(a, b) × LCM(a, b) = a × b

Algebra

Polynomials

• For zeroes of quadratic polynomial p(x) = ax² + bx + c, a ≠ 0:
• Sum of zeroes = -b/a
• Product of zeroes = c/a
• For zeroes of quadratic polynomial p(x) = ax³ + bx² + cx + d, a ≠ 0:
• Sum of zeroes = -b/a
• Product of zeroes = -d/a
• Sum of product of zeroes taken two at a time = c/a
• Identities:
• (a + b)² = a² + b² + 2ab
• (a - b)² = a² + b² - 2ab
• a² - b² = (a + b)(a - b)
• (a + b)³ = a³ + b³ + 3a²b + 3ab²
• (a - b)³ = a³ - b³ - 3a²b + 3ab²
• (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
• a³ - b³ = (a - b)(a² + b² + ab)
• a³ + b³ = (a + b)(a² + b² - ab)
• a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca)
• If a + b + c = 0, then a³ + b³ + c³ = 3abc

Linear Equations in Two Variables

• a1/a2 ≠ b1/b2: intersecting, consistent, 1 solution
• a1/a2 = b1/b2 ≠ c1/c2: parallel, inconsistent, 0 solution
• a1/a2 = b1/b2 = c1/c2: coincide, infinite solutions, consistent
• Methods for solving linear equations:
• Substitution Method
• Elimination Method
• Reduction Method

Quadratic Equations

• ax² + bx + c = 0, where 'a' ≠ 0
• Quadratic Formula: x = (-b ± √b² - 4ac) / 2a
• Discriminant (D) = b² - 4ac
• If D > 0, then two distinct real roots
• If D = 0, then two equal roots
• If D < 0, then no real roots

Arithmetic Progression

• General form of AP: a, a + d, a + 2d, a + 3d,...
• an = a + (n-1)d
• Sn = [2a + (n-1)d]/2
• Sn = n/2[a + l]
• Sn = Sn-1 + an
• If a, b, c are in AP then 2b = a + c

Coordinate Geometry

• Distance Formula: √(x2 - x1)² + (y2 - y1)²
• Distance of point (p, x) from origin: √p² + x²
• Section Formula: {(m1x2 + m2x1)/(m1 + m2), (m1y2 + m2y1)/(m1 + m2)}
• Midpoint Formula: {(x1 + x2)/2, (y1 + y2)/2}

Trigonometry

Trigonometric Identities

• Complementary angles:
• sin θ = cos (90° - θ)
• cos θ = sin (90° - θ)
• tan θ = cot (90° - θ)
• Other identities:
• tan θ = sin θ / cos θ
• cot θ = cos θ / sin θ
• sec θ = 1/cos θ
• cosec θ = 1/sin θ
• cot θ = 1/tan θ
• sin²θ + cos²θ = 1
• tan²θ + 1 = sec²θ
• 1 + cot²θ = cosec²θ
• sin²θ / cos²θ = 1/cos²θ
• sin²θ / sin²θ = 1/sin²θ

Surface Area and Volume

Cube

• Perimeter: 12l
• Base Area: l²
• LSA: 4l²
• TSA: 6l²
• Volume: l³
• d: √3 l

Cuboid

• Perimeter: 4(l + b + h)
• Volume: l × b × h
• Base Area: lb
• LSA: 2h(l + b)
• TSA: 2(lb + bh + lh)
• d: √(l² + b² + h²)

Cylinder

• Base Area: πr²
• CSA: 2πrh
• TSA: 2πr(h + r)
• Volume: πr²h

Cone

• Base Area: πr²
• CSA: πrl
• TSA: πr(l + r)
• Volume: 1/3πr²h

Sphere

• TSA: 4πr²
• Volume: 4/3πr³

Hemisphere

• Base Area: πr²
• CSA: 2πr²
• TSA: 3πr²
• Volume: 2/3πr³

Statistics

Mean

• Direct Method: mean = Σfmi / Σfi
• Assumed Mean Method: mean = a + (Σfidi / Σfi)