Linear and Quadratic Equations (Class 10)

What this worksheet covers

This is an eighteen-question practice worksheet on linear and quadratic equations for Class 10, written to the pattern of the SOF International Mathematics Olympiad (IMO). It carries 24 marks and is designed to be attempted in 45 minutes, split across three sections — Mathematical Reasoning, Everyday Mathematics and Reasoning and Achievers Section. Every question is multiple choice with exactly one correct option, and every one comes with a fully worked solution rather than just an answer key — so a child who gets one wrong can see where the reasoning went astray instead of only learning that it did.

Skills your child practises

• Applied problem-solving on linear and quadratic equations at Class 10 level
• Working accurately against a time limit, as the real paper demands
• Mathematical Reasoning questions in the olympiad format
• Everyday Mathematics and Reasoning questions in the olympiad format
• Achievers Section questions in the olympiad format
• Reading worked solutions to find the step that went wrong

How to use it

1. 1There are 18 questions in this worksheet. All questions are compulsory.
2. 2Each question has four options (A), (B), (C) and (D). Exactly one option is correct.
3. 3Questions 1–15 carry 1 mark each. Questions 16–18 (Achievers Section) carry 3 marks each.
4. 4There is no negative marking. Use of a calculator is not allowed.
5. 5Rough work may be done on the blank space at the side of each page.

Sample questions from this worksheet

A preview of what's inside Linear and Quadratic Equations (Class 10) — the PDF has the full set, and every question comes with a worked solution like these.

1. 1. A pair of linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 has a unique solution when _______.

   • Aa₁ = a₂
   • B{a₁/a₂} = {b₁/b₂} ≠ {c₁/c₂}
   • C{a₁/a₂} = {b₁/b₂} = {c₁/c₂}
   • D{a₁/a₂} ≠ {b₁/b₂}correct
   Solution: When the coefficient ratios of x and y differ, the two lines have different slopes and must cross at exactly one point, giving a unique solution. Equal ratios throughout give coincident lines, and equal coefficient ratios with a different constant ratio give parallel lines.

2. 2. Solving 2x + y = 8 and x − y = 1 gives _______.

   • Ax = 4, y = 0
   • Bx = 2, y = 4
   • Cx = 3, y = 2correct
   • Dx = 1, y = 6
   Solution: Adding the two equations eliminates y: 3x = 9, so x = 3. Substituting into x − y = 1 gives y = 2. Hence x = 3 and y = 2, which satisfies 2(3) + 2 = 8.

3. 3. If one root of x² + kx + 12 = 0 is 3, then the value of k and the other root are _______.

   • Ak = −7, other root 4correct
   • Bk = 7, other root 4
   • Ck = −7, other root −4
   • Dk = −4, other root 7
   Solution: Substituting x = 3: 9 + 3k + 12 = 0, so 3k = −21 and k = −7. The product of the roots is {12/1} = 12, so the other root is 12 ÷ 3 = 4. Hence k = −7 and the other root is 4, and their sum 7 matches −k.