CBSE Class 10 · Boards 2027 · NCERT 2026-27 syllabus
NCERT Exemplar Solutions Class 10 Maths Chapter 4 Quadratic Equations
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Boards begin February 2027
Pre-boards are in December
Quadratic Equations is in Algebra, which carries 20 of the 80 marks in the 2026-27 CBSE Maths paper
If Zeba were younger by 5 years than what she really is, then the square of her age (in years) would have been 11 more than five times her actual age. What is her age now?
Step 1Let Zeba's actual age be x years. Her 'younger by 5 years' age = (x − 5) years.
Step-by-step solution
Let Zeba's actual age be x years. Her 'younger by 5 years' age = (x − 5) years.
At present Asha's age (in years) is 2 more than the square of her daughter Nisha's age. When Nisha grows to her mother's present age, Asha's age would be one year less than 10 times the present age of Nisha. Find the present ages of both Asha and Nisha.
Step 1Let Nisha's present age = x years. Then Asha's present age = x² + 2 years.
Step-by-step solution
Let Nisha's present age = x years. Then Asha's present age = x² + 2 years.
After (Asha's_present − x) = (x² + 2 − x) years, Nisha will be x² + 2 (Asha's present age). At that time, Asha's age = (x² + 2) + (x² + 2 − x) = 2x² − x + 4.
Had Ajita scored 10 more marks in her Mathematics test out of 30 marks, 9 times these new marks would have been the square of her actual marks. How many marks did she get in the test?
Step 1Let Ajita's actual marks be x. 'If she had scored 10 more' = x + 10. The condition gives 9(x + 10) = x².
Step-by-step solution
Let Ajita's actual marks be x. 'If she had scored 10 more' = x + 10. The condition gives 9(x + 10) = x².
CBSE board pattern: Assertion-Reason and case-based questions
Written in the CBSE board format. These are not numbered NCERT/Exemplar questions; some case studies are built on textbook problems. Neither NCERT book has Assertion-Reason questions: they, and case-based questions, are a CBSE board-paper format, here on this chapter's 2026-27 syllabus, with solutions by LazyTopper.
CBSE board pattern · Assertion-Reason1 mark
Assertion (A): A quadratic equation with all-rational coefficients can have both roots irrational.
Reason (R): If b² − 4ac is a non-square positive rational, then by the quadratic formula the roots involve √D and are irrational (in conjugate pairs).
Assertion (A): If the coefficient of x² and the constant term of a quadratic equation have opposite signs, then the quadratic equation has real roots.
Reason (R): For ax² + bx + c = 0, if a and c have opposite signs then ac < 0 ⇒ −4ac > 0 ⇒ D = b² − 4ac > 0.
Step 1If a and c have opposite signs, then ac < 0 so −4ac > 0.
Step-by-step solution
If a and c have opposite signs, then ac < 0 so −4ac > 0.
Therefore D = b² − 4ac = b² + (positive quantity) ≥ positive ⇒ D > 0.
D > 0 ⇒ two distinct real roots. Assertion TRUE; Reason TRUE.
Reason directly proves Assertion → (A).
Answer:Option (A) — both true and Reason explains Assertion.
At t minutes past 2 pm, the time needed by the minute hand of a clock to show 3 pm was found to be 3 minutes less than t²/4 minutes.
(i) Write a quadratic equation in t.
(ii) Solve to find the value of t.
Step 1At t minutes past 2 pm, time remaining until 3 pm = (60 − t) minutes.
Step-by-step solution
At t minutes past 2 pm, time remaining until 3 pm = (60 − t) minutes.
A train travels at a certain average speed for a distance of 63 km, and then a distance of 72 km at an average speed of 6 km/h more than its original speed. The total journey takes 3 hours.
(i) Form a quadratic equation in the original speed x km/h.
(ii) Find the original average speed.
Step 1Let original average speed = x km/h. Time for first stretch = 63/x; time for second stretch = 72/(x + 6).
Step-by-step solution
Let original average speed = x km/h. Time for first stretch = 63/x; time for second stretch = 72/(x + 6).
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Questions from the NCERT Class 10 Mathematics Exemplar Problems (the CBSE board-pattern section is in the board format, not numbered book questions); the solutions are written by LazyTopper. LazyTopper is an independent platform and is not affiliated with CBSE or NCERT.