Quadratic Equations — Previous Year Questions (Class 10 Mathematics)
Quadratic equations are a cornerstone of CBSE Class 10 Maths, testing factorization, the quadratic formula, and discriminant analysis. Expect 2–5 mark ques
TL;DR: Quadratic equations are a cornerstone of CBSE Class 10 Maths, testing factorization, the quadratic formula, and discriminant analysis. Expect 2–5 mark…
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Quadratic equations are a cornerstone of CBSE Class 10 Maths, testing factorization, the quadratic formula, and discriminant analysis. Expect 2–5 mark ques
Quadratic Equations — Previous Year Questions with Solutions
Q (2022, 2 marks): Solve the quadratic equation 2x^2 - 7x + 3 = 0 by factorization method.
Answer: 2x^2 - 7x + 3 = 0
We need two numbers that multiply to (2)(3) = 6 and add to -7.
Those numbers are -6 and -1.
2x^2 - 6x - x + 3 = 0
2x(x - 3) - 1(x - 3) = 0
(2x - 1)(x - 3) = 0
So x = 1/2 or x = 3
Final Answer: x = 1/2, 3
Q (2023, 2 marks): Find the value of k for which the quadratic equation x^2 + kx + 9 = 0 has equal roots.
Answer: For equal roots, discriminant = 0
For equation x^2 + kx + 9 = 0: a = 1, b = k, c = 9
Discriminant = b^2 - 4ac = 0
k^2 - 4(1)(9) = 0
k^2 - 36 = 0
k^2 = 36
k = ±6
Final Answer: k = 6 or k = -6
Q (2021, 3 marks): A two-digit number is such that the product of its digits is 18. If 63 is subtracted from the number, the digits interchange their places. Find the number.
Answer: Let the two-digit number be 10a + b where a and b are digits.
Given: ab = 18 ... (1)
Also: (10a + b) - 63 = 10b + a
10a + b - 63 = 10b + a
9a - 9b = 63
a - b = 7 ... (2)
From (2): a = b + 7
Substitute in (1): (b + 7)b = 18
b^2 + 7b - 18 = 0
(b + 9)(b - 2) = 0
Since b is a digit, b = 2
So a = 9
The number is 92.
Verification: 9 × 2 = 18 ✓, 92 - 63 = 29 ✓
Final Answer: 92
Q (2022, 3 marks): Solve 1/(x-1) + 1/(x-2) = 1/6 for x ≠ 1, 2.
Answer: 1/(x-1) + 1/(x-2) = 1/6
LCM of (x-1) and (x-2) is (x-1)(x-2)
[(x-2) + (x-1)] / [(x-1)(x-2)] = 1/6
(2x - 3) / (x^2 - 3x + 2) = 1/6
6(2x - 3) = x^2 - 3x + 2
12x - 18 = x^2 - 3x + 2
x^2 - 15x + 20 = 0
Using quadratic formula: x = [15 ± sqrt(225 - 80)] / 2 = [15 ± sqrt(145)] / 2
Final Answer: x = (15 + sqrt(145))/2 or x = (15 - sqrt(145))/2
Q (2023, 3 marks): If one root of the equation 3x^2 - kx + 2 = 0 is 1/3, find the other root and the value of k.
Answer: Since 1/3 is a root, it satisfies the equation:
3(1/3)^2 - k(1/3) + 2 = 0
3(1/9) - k/3 + 2 = 0
1/3 - k/3 + 2 = 0
1 - k + 6 = 0 (multiply by 3)
k = 7
Now for equation 3x^2 - 7x + 2 = 0, using product of roots:
α × β = c/a = 2/3
(1/3) × β = 2/3
β = 2
Final Answer: Other root = 2, k = 7
Q (2021, 2 marks): If the discriminant of 5x^2 + 4x + p = 0 is -4, find the value of p.
Answer: For equation 5x^2 + 4x + p = 0:
a = 5, b = 4, c = p
Discriminant = b^2 - 4ac = -4
16 - 4(5)(p) = -4
16 - 20p = -4
-20p = -20
p = 1
Final Answer: p = 1
Frequently Asked Questions
What is the difference between factorization and quadratic formula?
Factorization works when roots are rational and easily factorable. The quadratic formula x = [-b ± sqrt(b^2 - 4ac)] / 2a works for all quadratic equations. Use the formula when factorization is tedious.
When do quadratic equations have no real roots?
When the discriminant (b^2 - 4ac) is negative. In that case, roots are complex/imaginary.
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