1. Algebra Fundamentals โ
USTET Math emphasizes solid algebraic foundations. Make sure you can handle these core concepts confidently!
Laws of Exponents
| Rule | Formula | Example |
|---|---|---|
| Product Rule | a^m * a^n = a^(m+n) | 2^3 * 2^4 = 2^7 = 128 |
| Quotient Rule | a^m / a^n = a^(m-n) | 5^6 / 5^2 = 5^4 = 625 |
| Power Rule | (a^m)^n = a^(mn) | (3^2)^3 = 3^6 = 729 |
| Zero Exponent | a^0 = 1 (a โ 0) | 100^0 = 1 |
| Negative Exponent | a^(-n) = 1/a^n | 2^(-3) = 1/8 |
| Fractional Exponent | a^(m/n) = n-th root of a^m | 8^(2/3) = (cube root of 8)^2 = 4 |
Solving Equations
Linear Equations
- Isolate variable on one side
- ax + b = c โ x = (c-b)/a
- Check by substituting back
Quadratic Equations
- Factor if possible
- Quadratic formula: x = (-b ยฑ sqrt(b^2-4ac)) / 2a
- Discriminant: b^2-4ac (determines # of solutions)
Discriminant Guide
- b^2 - 4ac > 0: Two real solutions
- b^2 - 4ac = 0: One real solution (repeated)
- b^2 - 4ac < 0: No real solutions (complex)
2. Polynomials & Factoring ๐ข
Master these factoring techniques - they appear in nearly every USTET math section!
Factoring Methods
| Method | Pattern | Example |
|---|---|---|
| GCF | Factor out common terms | 6x^2 + 9x = 3x(2x + 3) |
| Difference of Squares | a^2 - b^2 = (a+b)(a-b) | x^2 - 16 = (x+4)(x-4) |
| Perfect Square Trinomial | a^2 ยฑ 2ab + b^2 | x^2 + 6x + 9 = (x+3)^2 |
| Sum of Cubes | a^3 + b^3 = (a+b)(a^2-ab+b^2) | x^3 + 8 = (x+2)(x^2-2x+4) |
| Difference of Cubes | a^3 - b^3 = (a-b)(a^2+ab+b^2) | x^3 - 27 = (x-3)(x^2+3x+9) |
Polynomial Operations
- Addition/Subtraction: Combine like terms (same variable and exponent)
- Multiplication: FOIL for binomials, distribute for larger polynomials
- Division: Long division or synthetic division
- Remainder Theorem: f(a) = remainder when f(x) is divided by (x-a)
3. Geometry Essentials ๐
USTET includes fundamental geometry concepts. Memorize these formulas!
2D Shapes - Area & Perimeter
| Shape | Area | Perimeter |
|---|---|---|
| Rectangle | A = length x width | P = 2(l + w) |
| Square | A = side^2 | P = 4s |
| Triangle | A = (1/2) x base x height | P = a + b + c |
| Circle | A = pi x r^2 | C = 2 x pi x r |
| Trapezoid | A = (1/2)(b1 + b2) x h | P = sum of all sides |
| Parallelogram | A = base x height | P = 2(a + b) |
3D Shapes - Volume & Surface Area
| Shape | Volume | Surface Area |
|---|---|---|
| Cube | V = s^3 | SA = 6s^2 |
| Rectangular Prism | V = lwh | SA = 2(lw + lh + wh) |
| Cylinder | V = pi x r^2 x h | SA = 2(pi)r^2 + 2(pi)rh |
| Sphere | V = (4/3)(pi)r^3 | SA = 4(pi)r^2 |
| Cone | V = (1/3)(pi)r^2 h | SA = (pi)r^2 + (pi)rs |
4. Trigonometry ๐
Trig is essential for USTET! Master SOH-CAH-TOA and the unit circle.
SOH-CAH-TOA
SOH
sin(ฮธ) = Opposite / Hypotenuse
CAH
cos(ฮธ) = Adjacent / Hypotenuse
TOA
tan(ฮธ) = Opposite / Adjacent
Special Angles to Memorize
| Angle | sin | cos | tan |
|---|---|---|---|
| 0ยฐ | 0 | 1 | 0 |
| 30ยฐ | 1/2 | sqrt(3)/2 | sqrt(3)/3 |
| 45ยฐ | sqrt(2)/2 | sqrt(2)/2 | 1 |
| 60ยฐ | sqrt(3)/2 | 1/2 | sqrt(3) |
| 90ยฐ | 1 | 0 | undefined |
Key Trigonometric Identities
- Pythagorean: sin^2(ฮธ) + cos^2(ฮธ) = 1
- Reciprocal: csc = 1/sin, sec = 1/cos, cot = 1/tan
- Quotient: tan = sin/cos, cot = cos/sin
- Co-function: sin(90-ฮธ) = cos(ฮธ), cos(90-ฮธ) = sin(ฮธ)
5. Word Problems & Applications ๐
USTET includes practical math problems. Here's how to approach them!
Distance-Rate-Time
D = R x T
Distance = Rate x Time
Rearrange as needed: R = D/T, T = D/R
Work Problems
1/T = 1/A + 1/B
T = combined time, A & B = individual times
Common Problem Types
- Age Problems: Set up equations relating current and past/future ages
- Mixture Problems: Concentration x Volume = Amount of pure substance
- Percentage: Part = Percent x Whole (remember to convert %)
- Interest: Simple: I = Prt, Compound: A = P(1 + r/n)^(nt)
Example: Work Problem
Problem: Pipe A fills a tank in 6 hours. Pipe B fills it in 4 hours. How long to fill together?
Solution:
Rate A = 1/6 tank/hour, Rate B = 1/4 tank/hour
Combined rate = 1/6 + 1/4 = 2/12 + 3/12 = 5/12 tank/hour
Time = 1 / (5/12) = 12/5 = 2.4 hours or 2 hours 24 minutes
6. USTET Math Tips ๐ฏ
Exam Day Strategies
- Read Carefully: USTET problems can be tricky in wording
- Show Work: Even mentally, track your steps to avoid careless errors
- Check Units: Make sure your answer has the right units
- Estimate First: Eliminate obviously wrong choices
- Time Management: Don't spend too long on one problem
Common Mistakes to Avoid
- Forgetting to distribute negative signs
- Mixing up formulas (area vs perimeter)
- Calculator errors (even on simple operations)
- Not simplifying final answers
- Misreading the question (what's being asked?)
7. Practice Questions ๐
Q1: Simplify (x^3 * x^4) / x^2
= x^(3+4-2) = x^5. Answer: x^5
Q2: Factor x^2 - 5x - 14
Find factors of -14 that add to -5: -7 and 2. Answer: (x - 7)(x + 2)
Q3: If sin(ฮธ) = 3/5 in Q1, find cos(ฮธ)
sin^2 + cos^2 = 1 โ cos^2 = 1 - 9/25 = 16/25 โ cos = 4/5 (positive in Q1). Answer: 4/5