🔢 Pure Mathematics & Algebra

1. Solve the equation:
x4−4x3+6x2−4x+1=0x^4 - 4x^3 + 6x^2 - 4x + 1 = 0
(Hint: Recognize patterns in binomial expansions.)

2. Prove that for all real xx,
x2+1x2≥2x^2 + \frac{1}{x^2} \geq 2

3. Find all real solutions of
1x+1x+1=1\frac{1}{x} + \frac{1}{x+1} = 1

4. How many integers between 1 and 1000 are divisible by neither 2 nor 5?

5. Let f(x)=x3−3x+1f(x) = x^3 - 3x + 1.
Determine the number of real roots and sketch the graph.

📐 Geometry & Coordinate Geometry

1. Given a triangle with vertices A(0,0),B(4,0),C(0,3)A(0,0), B(4,0), C(0,3),
find the equation of the median from point CC.

2. A circle is tangent to the x-axis at (3,0)(3,0) and passes through (5,4)(5,4).
Find its center and radius.

3. Prove: The sum of the interior angles of a convex n-gon is (n−2)⋅180∘(n-2) \cdot 180^\circ

4. Find the area enclosed between
y=xy = \sqrt{x} and y=xy = x

📊 Functions & Graph Sketching

1. Sketch and describe the graph of
y=1x2−4y = \frac{1}{x^2 - 4}

2. Given f(x)=∣x−3∣+∣x+1∣f(x) = |x-3| + |x+1|,
find where f(x)f(x) is differentiable.

3. Let f(x)=sin⁡(x2)f(x) = \sin(x^2).
Is f(x)f(x) an even function, odd, or neither?

🧮 Sequences & Series

1. A sequence is defined by a1=2a_1 = 2,
an+1=3an+1a_{n+1} = 3a_n + 1.
Find a closed-form expression for ana_n.

2. Prove:
13+23+⋯+n3=(n(n+1)2)21^3 + 2^3 + \cdots + n^3 = \left( \frac{n(n+1)}{2} \right)^2

3. Determine the convergence or divergence of the series
∑n=1∞1n(n+1)\sum_{n=1}^{\infty} \frac{1}{n(n+1)}

🧠 Problem Solving / Logical Reasoning

1. A 100-square checkerboard has one square removed.
Can the board still be tiled with dominoes, each covering 2 adjacent squares?

2. Two numbers have a product of 2025 and sum of 90.
What are the numbers?

3. A positive integer has exactly 6 positive divisors.
How many such numbers less than 100 exist?

🔁 Calculus & Rate of Change (Introductory)

1. Find the tangent line to the curve y=x3−xy = x^3 - x at x=2x = 2

2. A particle moves such that
s(t)=t3−6t2+9ts(t) = t^3 - 6t^2 + 9t.
Find the time when the particle is at rest.

• algebraic fluency

• logical deduction

• problem-solving intuition

• mathematical maturity