Mathematics Content (VI-VIII)

UTET Paper II — Mathematics and Science

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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. Expect 15-20 questions from this content area. 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.

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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.

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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

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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**

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**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**

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**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**

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**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%**

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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.

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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.**