Mathematics Content (VI-VIII)
UTET Paper II — Mathematics and Science
Overview
Mathematics content for Classes VI-VIII forms a substantial portion of UTET Paper II. This section tests your command over upper-primary mathematical concepts as prescribed in NCERT textbooks. Questions are direct applications of formulas, theorems and problem-solving techniques rather than rote memorisation.
The scope covers seven major domains: Number System, Algebra, Geometry, Mensuration, Commercial Mathematics, Data Handling, and Symmetry with Practical Geometry. Mastery requires both conceptual clarity and computational accuracy — most questions involve multi-step calculations where one error cascades through the entire solution.
For Uttarakhand TET specifically, questions align closely with NCERT Class VI-VIII textbooks. Focus on standard problem types and ensure you can solve them within 60-90 seconds each.
Key Concepts
- Integers extend the number line in both directions — operations with negative numbers follow sign rules: same signs give positive product, different signs give negative product.
- Rational numbers are quotients p/q where q ≠ 0 — every integer is rational (denominator = 1), and rational numbers are dense (infinite rationals exist between any two rationals).
- Exponents represent repeated multiplication — laws of exponents (aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, a⁰ = 1) apply only when bases are equal.
- Linear equations have degree one — solving involves isolating the variable using inverse operations while maintaining equality on both sides.
- Angle sum property of triangles is 180° — for quadrilaterals it is 360°; exterior angle equals sum of non-adjacent interior angles.
- Mensuration connects 2D and 3D measurement — area is square units, volume is cubic units; surface area of 3D shapes requires summing component face areas.
- Ratio compares like quantities, proportion equates two ratios — unitary method converts to per-unit value first.
- Mean = Sum ÷ Count; Median = Middle value (ordered); Mode = Most frequent value — these three measures describe data centrally but behave differently with outliers.
Formulas / Key Facts
Number System
- Product of integers: (+)(+) = +, (−)(−) = +, (+)(−) = −
- aᵐ × aⁿ = aᵐ⁺ⁿ
- aᵐ ÷ aⁿ = aᵐ⁻ⁿ
- (aᵐ)ⁿ = aᵐⁿ
- a⁻ⁿ = 1/aⁿ
- a⁰ = 1 (a ≠ 0)
Algebra
- (a + b)² = a² + 2ab + b²
- (a − b)² = a² − 2ab + b²
- a² − b² = (a + b)(a − b)
- (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
Geometry
- Sum of angles in triangle = 180°
- Sum of angles in quadrilateral = 360°
- Exterior angle = Sum of two non-adjacent interior angles
- Pythagoras theorem: Hypotenuse² = Base² + Perpendicular²
Mensuration
- Area of triangle = ½ × base × height
- Area of circle = πr²; Circumference = 2πr
- Surface area of cuboid = 2(lb + bh + hl)
- Volume of cuboid = l × b × h
- Surface area of cylinder = 2πr(r + h)
- Volume of cylinder = πr²h
Commercial Mathematics
- Profit% = (Profit/CP) × 100
- SI = (P × R × T)/100
- CI = P(1 + R/100)ⁿ − P
Data Handling
- Mean = Σxᵢ/n
- Probability = Favourable outcomes / Total outcomes
Worked Examples
Example 1: Exponents Simplify: (2³ × 2⁵) ÷ 2⁴
Step 1: Apply multiplication rule — 2³ × 2⁵ = 2³⁺⁵ = 2⁸ Step 2: Apply division rule — 2⁸ ÷ 2⁴ = 2⁸⁻⁴ = 2⁴ Step 3: Calculate — 2⁴ = 16
Answer: 16
Example 2: Linear Equation Solve: 3x − 7 = 2x + 5
Step 1: Bring variable terms to one side — 3x − 2x = 5 + 7 Step 2: Simplify — x = 12
Answer: x = 12
Example 3: Mensuration Find the volume of a cylinder with radius 7 cm and height 10 cm. (Use π = 22/7)
Step 1: Apply formula — V = πr²h Step 2: Substitute — V = (22/7) × 7² × 10 Step 3: Calculate — V = (22/7) × 49 × 10 = 22 × 7 × 10 = 1540 cm³
Answer: 1540 cm³
Example 4: Commercial Mathematics A shopkeeper buys an article for ₹400 and sells it for ₹460. Find profit percentage.
Step 1: Calculate profit — Profit = SP − CP = 460 − 400 = ₹60 Step 2: Apply formula — Profit% = (60/400) × 100 = 15%
Answer: 15%
Common Mistakes
Wrong: Adding exponents when bases are different (2³ × 3² = 6⁵). Fix: Exponent laws apply only when bases are identical. Calculate each power separately, then multiply: 8 × 9 = 72.
Wrong: Forgetting to transpose the sign when moving terms across the equals sign. Fix: When a term crosses the equals sign, its operation reverses: +5 becomes −5, ×3 becomes ÷3.
Wrong: Using diameter instead of radius in circle formulas. Fix: Always check whether the given measurement is radius or diameter. If diameter d is given, use r = d/2.
Wrong: Confusing surface area with volume — using square units for volume or cubic units for area. Fix: Area is always in square units (cm²), volume in cubic units (cm³). Surface area of 3D objects is still in square units.
Wrong: Calculating mean when asked for median, or vice versa. Fix: Mean requires summing all values; median requires arranging in order first. Read the question carefully.
Wrong: In probability, counting outcomes incorrectly — especially in dice and card problems. Fix: A standard die has 6 faces (1-6), a deck has 52 cards (13 each of 4 suits). List favourable outcomes explicitly before dividing.
Quick Reference
- Sign rule for multiplication: Like signs → positive, unlike signs → negative.
- Exponent zero rule: Any non-zero number raised to power zero equals 1.
- Triangle angle sum = 180°; Quadrilateral = 360°.
- Volume of cuboid = l × b × h; Cylinder = πr²h.
- SI formula: (Principal × Rate × Time) ÷ 100.
- Mean = Total sum ÷ Number of observations.