JEECUP / UPJEE (Polytechnic) Group A: Engineering & Technology Diploma

Real Numbers and Polynomials

• HCF and LCMHCF(a, b) × LCM(a, b) = a × b (for two numbers)
• Quadratic zerosFor ax² + bx + c: α + β = −b/a, αβ = c/a
• Cubic zerosFor ax³ + bx² + cx + d: α+β+γ = −b/a, αβ+βγ+γα = c/a, αβγ = −d/a
• Usefulα² + β² = (α + β)² − 2αβ

💡 The HCF × LCM product rule works for two numbers only, not three.

Linear and Quadratic Equations

• a₁x+b₁y+c₁=0, a₂x+b₂y+c₂=0Unique: a₁/a₂ ≠ b₁/b₂; No solution: a₁/a₂ = b₁/b₂ ≠ c₁/c₂; Infinite: all three ratios equal
• Quadratic formulax = (−b ± √(b² − 4ac)) / 2a
• DiscriminantD = b² − 4ac: D > 0 two real distinct, D = 0 equal, D < 0 no real roots

💡 "Equal roots" means D = 0; solve for k directly.

Arithmetic Progressions

• nth termaₙ = a + (n − 1)d
• SumSₙ = n/2 [2a + (n − 1)d] = n/2 (a + l)
• Term from sumaₙ = Sₙ − Sₙ₋₁

💡 Counting numbers divisible by k between two values is just an AP: n = (last − first)/k + 1.

Coordinate Geometry

• Distanced = √((x₂ − x₁)² + (y₂ − y₁)²)
• Section (m:n)((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n))
• Midpoint((x₁ + x₂)/2, (y₁ + y₂)/2)

💡 In the section formula, m multiplies the SECOND point.

Trigonometry

• Identitiessin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ
• Complementssin(90° − θ) = cos θ, tan(90° − θ) = cot θ
• sin values0°, 30°, 45°, 60°, 90° → 0, ½, 1/√2, √3/2, 1
• tan values0°, 30°, 45°, 60° → 0, 1/√3, 1, √3 (90° not defined)

💡 Write sin values as √0/2, √1/2, √2/2, √3/2, √4/2 to remember the whole table.

Heights and Distances

• Basictan θ = height / horizontal distance
• Two points (30°, 60°), gap dh = d·√3/2 when both points are on the same side

💡 Always draw the figure; angle of depression from the top equals angle of elevation from the bottom.

Circles and Areas

• CircleArea = πr², Circumference = 2πr
• Sector (angle θ)Area = (θ/360°) × πr², Arc length = (θ/360°) × 2πr
• TangentTangent ⟂ radius at point of contact; tangents from an external point are equal

💡 Use π = 22/7 when the radius is a multiple of 7; the options usually are built that way.

Surface Areas and Volumes

• CylinderV = πr²h, CSA = 2πrh, TSA = 2πr(r + h)
• ConeV = ⅓πr²h, CSA = πrl, l = √(r² + h²)
• Sphere / hemisphereV = (4/3)πr³ / (2/3)πr³, SA = 4πr² / CSA 2πr², TSA 3πr²
• Melting-recastingVolume before = volume after

💡 In recasting questions, cancel π on both sides before multiplying anything.

Statistics and Probability

• Mean (grouped)x̄ = Σfᵢxᵢ / Σfᵢ
• Medianl + ((n/2 − cf)/f) × h
• Empirical3 Median = Mode + 2 Mean
• ProbabilityP(E) = favourable / total, P(not E) = 1 − P(E)

💡 Two dice give 36 outcomes, one deck gives 52 cards: write the total first.