These go beyond Exercise 4.2/4.3 into genuine HOTS (Higher Order Thinking Skills) territory — a general root-relationship proof, a reverse-engineering root problem, a fresh-numbers speed word problem, an assertion-reason question, and two further word problems with numbers not used in the textbook. See the standard-level Chapter 4 Solutions first if you haven’t already.
Extra Questions (HOTS Level): Quadratic Equations (Class 10 Maths Chapter 4)
Q1. (General proof) If α and β are the roots of the general quadratic equation ax²+bx+c = 0 (a≠0), prove that α+β = −b/a and αβ = c/a.
By the quadratic formula, the two roots are α = (−b+√D)/2a and β = (−b−√D)/2a, where D = b²−4ac.
Sum: α+β = [(−b+√D)+(−b−√D)]/2a = −2b/2a = −b/a.
Product: αβ = [(−b)²−(√D)²]/(2a)² = (b²−D)/4a² = (b²−(b²−4ac))/4a² = 4ac/4a² = c/a.
This general result (not tied to any specific numbers) is what lets you find the sum/product of roots without solving the equation at all.
Q2. (Reverse-engineering) One root of the equation 3x²−10x+k = 0 is the reciprocal of the other. Find k.
Let the roots be α and 1/α. Product of roots = α×(1/α) = 1. By Q1’s result, product of roots = c/a = k/3. So k/3 = 1 → k = 3.
Q3. (Fresh-numbers word problem) A car covers a distance of 300 km at a uniform speed. Had the speed been 5 km/h more, it would have taken 2 hours less for the same journey. Find the original speed of the car.
Let speed = x km/h. Time taken = 300/x hours. New time = 300/(x+5). Condition: 300/x − 300/(x+5) = 2.
300[(x+5)−x] = 2x(x+5) → 1500 = 2x²+10x → x²+5x−750 = 0.
D = 25+3000 = 3025 = 55². x = (−5+55)/2 = 25 (rejecting the negative root).
Original speed = 25 km/h.
Q4. (Assertion-Reason) Assertion (A): The equation x²+4x+4 = 0 has two equal real roots. Reason (R): A quadratic equation ax²+bx+c=0 has equal roots if and only if its discriminant b²−4ac = 0. Which is correct — are both A and R true, and if so, does R correctly explain A?
For x²+4x+4=0: D = 16−16 = 0, so the equation does have two equal real roots (x=−2, repeated). Assertion is true.
The Reason is the exact, general condition for equal roots and it directly explains why A holds. Reason is true and is the correct explanation of the Assertion.
Q5. (Multi-number word problem, fresh numbers) The sum of the squares of two consecutive even positive integers is 340. Find the integers.
Let the integers be x and x+2 (both even). x²+(x+2)² = 340 → 2x²+4x−336 = 0 → x²+2x−168 = 0.
D = 4+672 = 676 = 26². x = (−2+26)/2 = 12 (rejecting the negative root).
The integers are 12 and 14.
Q6. (Adjusted-quantity word problem) A shopkeeper buys a certain number of books for ₹80. If he had bought 4 more books for the same total amount, each book would have cost him ₹1 less. Find the number of books he bought.
Let the number of books = x, so cost per book = 80/x. With 4 more books: cost per book = 80/(x+4) = 80/x − 1.
80/x − 80/(x+4) = 1 → 80[(x+4)−x] = x(x+4) → 320 = x²+4x → x²+4x−320 = 0.
D = 16+1280 = 1296 = 36². x = (−4+36)/2 = 16 (rejecting the negative root).
He bought 16 books.
See also: Revision Notes | Formulas Handbook
