100 Must-Solve Gaokao Math Questions
With Step-by-Step Explanations and Model Answers
Section 1: Algebra & Functions
Question 1
Solve for x in the equation:
x² - 5x + 6 = 0
Solution:
This is a standard quadratic equation. We factor it:
x² - 5x + 6 = (x - 2)(x - 3) = 0
Set each factor to 0:
x-2=0→x=2
x-3=0→x=3
Answer: x = 2 or x = 3
Question 2
Solve:
|2x - 4| = 6
Solution:
Split into two cases:
Case 1: 2x - 4 = 6 → 2x = 10 → x = 5
Case 2: 2x - 4 = -6 → 2x = -2 → x = -1
Answer: x = 5 or x = -1
Question 3
Given f(x) = (2x + 3)/5, find the inverse function f⁻¹(x).
Solution:
Let y = f(x) = (2x + 3)/5
Solve for x:
5y = 2x + 3 → 2x = 5y - 3 → x = (5y - 3)/2
,Swap x and y:
f⁻¹(x) = (5x - 3)/2
Answer: f⁻¹(x) = (5x - 3)/2
Question 4
Simplify:
((9x³)/(27x⁷))²
Solution:
Simplify inside:
9/27 = 1/3, x³/x⁷ = x⁻⁴
So: (1/3 * x⁻⁴)² = 1/9 * x⁻⁸
Answer: 1/(9x⁸)
Question 5
Solve the inequality:
x² - 4x + 3 < 0
Solution:
First solve x² - 4x + 3 = 0 → (x - 1)(x - 3) = 0 → x = 1, 3
Test intervals:
Between x = 1 and x = 3, expression is negative.
Answer: 1 < x < 3
, Section 2: Geometry & Trigonometry
Question 6
Find the area of a triangle with base 10 cm and height 6 cm.
Solution:
Area = 1/2 * base * height = 1/2 * 10 * 6 = 30 cm²
Answer: 30 cm²
Question 7
In a right triangle, if sin(θ) = 3/5, find cos(θ).
Solution:
Using Pythagorean identity: cos²θ = 1 - sin²θ = 1 - (9/25) = 16/25 → cos( θ) = 4/5
Answer: 4/5
With Step-by-Step Explanations and Model Answers
Section 1: Algebra & Functions
Question 1
Solve for x in the equation:
x² - 5x + 6 = 0
Solution:
This is a standard quadratic equation. We factor it:
x² - 5x + 6 = (x - 2)(x - 3) = 0
Set each factor to 0:
x-2=0→x=2
x-3=0→x=3
Answer: x = 2 or x = 3
Question 2
Solve:
|2x - 4| = 6
Solution:
Split into two cases:
Case 1: 2x - 4 = 6 → 2x = 10 → x = 5
Case 2: 2x - 4 = -6 → 2x = -2 → x = -1
Answer: x = 5 or x = -1
Question 3
Given f(x) = (2x + 3)/5, find the inverse function f⁻¹(x).
Solution:
Let y = f(x) = (2x + 3)/5
Solve for x:
5y = 2x + 3 → 2x = 5y - 3 → x = (5y - 3)/2
,Swap x and y:
f⁻¹(x) = (5x - 3)/2
Answer: f⁻¹(x) = (5x - 3)/2
Question 4
Simplify:
((9x³)/(27x⁷))²
Solution:
Simplify inside:
9/27 = 1/3, x³/x⁷ = x⁻⁴
So: (1/3 * x⁻⁴)² = 1/9 * x⁻⁸
Answer: 1/(9x⁸)
Question 5
Solve the inequality:
x² - 4x + 3 < 0
Solution:
First solve x² - 4x + 3 = 0 → (x - 1)(x - 3) = 0 → x = 1, 3
Test intervals:
Between x = 1 and x = 3, expression is negative.
Answer: 1 < x < 3
, Section 2: Geometry & Trigonometry
Question 6
Find the area of a triangle with base 10 cm and height 6 cm.
Solution:
Area = 1/2 * base * height = 1/2 * 10 * 6 = 30 cm²
Answer: 30 cm²
Question 7
In a right triangle, if sin(θ) = 3/5, find cos(θ).
Solution:
Using Pythagorean identity: cos²θ = 1 - sin²θ = 1 - (9/25) = 16/25 → cos( θ) = 4/5
Answer: 4/5
