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Practice Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables HOTS MCQs with detailed solutions, NCERT-based concepts, board exam insights, shortcut tricks, and competency-based questions for CBSE Board Exams.
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Chapter Information
Subject: Mathematics
Class: 10
Chapter: Pair of Linear Equations in Two Variables
Question Type: HOTS MCQs
Difficulty Level: Board Exam (Higher Order Thinking)
Based On: NCERT Latest Syllabus
Introduction:
Higher Order Thinking Skills (HOTS) questions require students to analyze mathematical situations, compare different solving methods, and apply concepts rather than simply perform calculations. In the latest CBSE pattern, students are expected to identify the most efficient method, understand the nature of solutions, and interpret mathematical statements correctly.
These questions improve logical reasoning, analytical ability, and confidence in solving unfamiliar problems. Regular practice of HOTS questions helps students perform well not only in board examinations but also in scholarship and competitive examinations.
What You Will Learn?
✔ Analytical Thinking
✔ Selecting the Best Method
✔ Nature of Solutions
✔ Mathematical Reasoning
✔ Competency-Based Concepts
✔ Board-Level Problem Solving
Why This Topic Is Important?
HOTS questions develop conceptual understanding and prepare students for the latest competency-based examination pattern.
Exam Relevance
✔ CBSE Board Exams
✔ State Board Exams
✔ School Assessments
✔ Scholarship Exams
✔ Olympiad Foundation Practice
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Q1.
Without solving the equations, determine the nature of the solution:
$$4x+5y=9$$
$$8x+10y=18$$
Question Difficulty:
🟡 Medium
A) One Unique Solution
B) No Solution
C) Infinitely Many Solutions
D) Cannot be Determined
Answer:
$$\text{Infinitely Many Solutions}$$
Useful Formula for this Question:
$$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$$
Concept Behind This Question:
Compare the coefficients instead of solving the equations.
Solution:
$$\frac{4}{8}=\frac{5}{10}=\frac{9}{18}=\frac12$$
All three ratios are equal.
Hence, the equations represent the same straight line.
Therefore, the correct answer is:
$$\text{Infinitely Many Solutions}$$
Shortcut / Smart Trick
Always simplify the ratios before deciding.
Board Exam Insight
This type of question frequently appears as a direct MCQ.
Exam Tip:
Do not waste time solving the equations if ratio comparison is sufficient.
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Q2.
Without solving, identify the nature of the solution:
$$5x-2y=7$$
$$10x-4y=15$$
Question Difficulty:
🟡 Medium
A) One Unique Solution
B) No Solution
C) Infinitely Many Solutions
D) Exactly Two Solutions
Answer:
$$\text{No Solution}$$
Useful Formula for this Question:
$$\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}$$
Concept Behind This Question:
Equal slopes but different intercepts indicate parallel lines.
Solution:
$$\frac{5}{10}=\frac{-2}{-4}=\frac12$$
but
$$\frac{7}{15}\ne\frac12$$
Therefore, the lines are parallel.
Hence, there is:
$$\text{No Solution}$$
Shortcut / Smart Trick
Compare only the ratios first.
If the first two are equal, immediately check the third.
Board Exam Insight
Students often miss the sign while comparing ratios.
Exam Tip:
Always simplify negative ratios carefully.
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Q3.
Which method is generally the fastest for solving
$$x+y=14$$
$$x-y=6$$
Question Difficulty:
🟢 Easy
A) Graphical Method
B) Elimination Method
C) Cross Multiplication Method
D) Graph Plotting
Answer:
$$\text{Elimination Method}$$
Useful Formula for this Question:
Opposite coefficients → Addition.
Concept Behind This Question:
Choose the method requiring the least calculation.
Solution:
The coefficients of
$$y$$
are
$$+1$$
and
$$-1$$
Therefore,
adding the equations eliminates
$$y$$
immediately.
Hence, the elimination method is the fastest.
Shortcut / Smart Trick
Look for coefficients
$$+1$$
and
$$-1$$
before starting.
Board Exam Insight
Method-selection questions are becoming common.
Exam Tip:
The fastest correct method usually saves valuable examination time.
————————————————–
Q4.
A student solved a pair of equations using substitution even though one variable already had equal and opposite coefficients.
Which method would have been more efficient?
Question Difficulty:
🟡 Medium
A) Graphical Method
B) Elimination Method
C) Cross Multiplication Method
D) No Method
Answer:
$$\text{Elimination Method}$$
Useful Formula for this Question:
Opposite coefficients → Add directly.
Concept Behind This Question:
Choose the most efficient algebraic method.
Solution:
When opposite coefficients already exist,
the elimination method requires fewer steps.
Therefore,
the correct answer is:
$$\text{Elimination Method}$$
Shortcut / Smart Trick
Always inspect the equations before deciding the method.
Board Exam Insight
Efficiency-based questions test mathematical reasoning.
Exam Tip:
The shortest correct method is generally preferred.
————————————————–
Q5.
Which statement is always true?
Question Difficulty:
🟡 Medium
A) Every pair of linear equations has one solution.
B) Every pair of linear equations has infinitely many solutions.
C) Every pair of linear equations has either one, none, or infinitely many solutions.
D) Every pair of linear equations has exactly two solutions.
Answer:
$$\text{Every pair of linear equations has either one, none, or infinitely many solutions.}$$
Useful Formula for this Question:
Possible outcomes:
One
None
Infinite
Concept Behind This Question:
Understand all possible cases.
Solution:
Depending on the coefficient ratios,
a pair of linear equations may have
one solution,
no solution,
or infinitely many solutions.
Therefore,
option
$$\text{C}$$
is correct.
Shortcut / Smart Trick
Remember the three possible cases only.
Board Exam Insight
Concept-based questions like this are common in competency papers.
Exam Tip:
Never assume that every system has a unique solution.
————————————————–
Important Formulas and Concepts
Unique Solution
$$\frac{a_1}{a_2}\ne\frac{b_1}{b_2}$$
No Solution
$$\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}$$
Infinitely Many Solutions
$$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$$
————————————————–
FAQs
Q. Why should we compare ratios before solving?
Answer:
It quickly tells the nature of the solution without lengthy calculations.
Q. Which algebraic method is usually fastest?
Answer:
It depends on the coefficients of the given equations.
Q. Are HOTS questions calculation-based?
Answer:
Not always. Many HOTS questions test reasoning and conceptual understanding.
————————————————–
Common Mistakes Students Make
❌ Ignoring ratio comparison.
❌ Choosing a lengthy method unnecessarily.
❌ Forgetting negative signs while comparing ratios.
❌ Assuming every pair has one solution.
————————————————–
Quick Revision Notes
✔ Compare coefficient ratios first.
✔ Choose the easiest solving method.
✔ Revise all three solution conditions.
✔ Think logically before calculating.
✔ Save time by avoiding unnecessary steps.
————————————————–
Conclusion:
This HOTS Practice Set strengthens conceptual understanding and analytical thinking related to Pair of Linear Equations in Two Variables. These questions prepare students for the latest CBSE competency-based examinations while improving speed and accuracy.
Related links
- Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables part-9
- Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables part-8
- Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables part-7
- Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables part-6
- Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables part-5
- Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables part-4
- Class 10 Maths Chapter 3 Pair of Linear Equations in Two variables part-3
- Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables part-2
- Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables – Part 1
- Class 10 Maths Chapter 2 Polynomials (Graphical Zeroes) – Part 13
- Class 10 Maths Chapter 2 Polynomials MCQ – Part 5
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