1. Algebra Fundamentals ➕

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 ≠ 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 → x = (c-b)/a
Check by substituting back

Quadratic Equations

Factor if possible
Quadratic formula: x = (-b ± 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 🔢

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 ± 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 📐

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 📊

Trig is essential for USTET! Master SOH-CAH-TOA and the unit circle.

SOH-CAH-TOA

SOH

sin(θ) = Opposite / Hypotenuse

CAH

cos(θ) = Adjacent / Hypotenuse

TOA

tan(θ) = Opposite / Adjacent

Special Angles to Memorize

Angle	sin	cos	tan
0°	0	1	0
30°	1/2	sqrt(3)/2	sqrt(3)/3
45°	sqrt(2)/2	sqrt(2)/2	1
60°	sqrt(3)/2	1/2	sqrt(3)
90°	1	0	undefined

Key Trigonometric Identities

Pythagorean: sin^2(θ) + cos^2(θ) = 1
Reciprocal: csc = 1/sin, sec = 1/cos, cot = 1/tan
Quotient: tan = sin/cos, cot = cos/sin
Co-function: sin(90-θ) = cos(θ), cos(90-θ) = sin(θ)

5. Word Problems & Applications 📝

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 🎯

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 📋

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(θ) = 3/5 in Q1, find cos(θ)

sin^2 + cos^2 = 1 → cos^2 = 1 - 9/25 = 16/25 → cos = 4/5 (positive in Q1). Answer: 4/5

Mathematics