Hardest Math Questions for Class 9, 10, 11, and 12 📚

Looking for the toughest math problems to challenge your skills? Here's a collection of the hardest questions for students of classes 9, 10, 11, and 12. These questions are designed to test your understanding of core concepts and help you master advanced topics. Whether you're preparing for exams or just love solving math, these problems are for you!

Class 9: Hardest Math Questions 🔢

• Algebraic Identities: Simplify and prove:
  (a³ + b³ + c³ - 3abc) = (a + b + c)(a² + b² + c² - ab - bc - ac), where \(a + b + c = 0\).
• Geometry: Prove that in a cyclic quadrilateral, the sum of opposite angles is 180°.
• Trigonometry: If \(\sin A + \cos A = \sqrt{2} \cos A\), find the value of \(\tan A\).
• Quadratic Equations: The roots of \(x² - px + q = 0\) are in the ratio 1:2. Find \(p\) and \(q\).
• Surface Area & Volume: A hemispherical bowl with a radius of 7 cm is polished at ₹5 per cm². Find the total cost.

Class 10: Hardest Math Questions 🔢

• Polynomials: If \(p(x) = 2x³ + 3x² - 5x + 6\) is divided by \(x - 2\), find the remainder.
• Pair of Linear Equations: Solve for \(x\) and \(y\):
  \(\frac{1}{x} + \frac{1}{y} = 5\), and \(\frac{1}{x} - \frac{1}{y} = 3\).
• Probability: Two balls are drawn from a box with 5 red, 4 blue, and 3 green balls. What is the probability both balls are the same color?
• Arithmetic Progression: Find the sum of all three-digit numbers divisible by 7.
• Circles: Prove the tangent drawn from an external point to a circle is equal to \(\sqrt{(distance² - radius²)}\).

Class 11: Hardest Math Questions 📐

• Limits: Evaluate: \(\lim_{x \to 0} \frac{\sin³ x}{x³}\).
• Trigonometry: Prove: \(\sin⁶ A + \cos⁶ A = 1 - 3\sin² A\cos² A\).
• Permutation & Combination: In how many ways can 5 boys and 5 girls sit in a row if no two girls sit together?
• Probability: Three balls are drawn from a bag containing 4 white, 5 black, and 6 red balls. Find the probability that at least one is black.
• Conic Sections: Find the equation of a parabola with focus \((2, 3)\) and directrix \(x - 4 = 0\).

Class 12: Hardest Math Questions 📏

• Integration: Evaluate: \(\int \frac{x³}{\sqrt{1-x²}} dx\).
• Matrices: Prove: \(A² - 6A + 7I = 0\), where \(A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}\).
• Probability: Two dice are thrown. Find the probability the product of the numbers is a perfect square.
• Linear Programming: A manufacturer produces products \(A\) and \(B\), with profits of ₹20 and ₹30 respectively. Maximize profit with a labor constraint of 120 hours.
• Differential Equations: Solve: \(\frac{dy}{dx} + y \tan x = \sin x\).

Why These Questions Matter? 🤔

These challenging math problems test advanced concepts and help build analytical skills necessary for competitive exams and higher studies. By solving them, students can improve their confidence and problem-solving abilities.

FAQs About Hard Math Problems ❓

Q: Are these questions based on the CBSE syllabus? 📘
A: Yes, these questions are aligned with the CBSE and NCERT syllabus for Classes 9 to 12.

Q: How can I solve these difficult problems? 🤔
A: Start with understanding the concepts thoroughly and practice similar problems to build your skills.

For more such problems, visit NCSOLVE!