Welcome to Part 1 of our new 8-part series on Chapter 4, Quadratic Equations. This post contains the top 25 Multiple Choice Questions (MCQs) to help you master concepts like roots, discriminant, and nature of roots.
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Top 25 MCQs - Quadratic Equations
Question 1: Standard Form
Which of the following is a quadratic equation?
LHS = x³ - x²
RHS = (x - 1)³ = x³ - 3x² + 3x - 1
x³ - x² = x³ - 3x² + 3x - 1
Subtracting x³ from both sides: -x² = -3x² + 3x - 1
2x² - 3x + 1 = 0
This is of the form ax² + bx + c = 0, so it is a quadratic equation.
Question 2: Roots of Equation
The roots of the quadratic equation x² - 3x - 10 = 0 are:
x² - 5x + 2x - 10 = 0
x(x - 5) + 2(x - 5) = 0
(x - 5)(x + 2) = 0
x = 5 or x = -2.
Question 3: Discriminant
The discriminant of the quadratic equation 2x² - 4x + 3 = 0 is:
Here a=2, b=-4, c=3.
D = (-4)² - 4(2)(3)
D = 16 - 24 = -8.
Question 4: Nature of Roots
If the discriminant of a quadratic equation is less than zero (D < 0), then the roots are:
Question 5: Value of k (Equal Roots)
Find the value of k for which the quadratic equation 2x² + kx + 3 = 0 has two real equal roots.
b² - 4ac = 0
k² - 4(2)(3) = 0
k² - 24 = 0
k² = 24 => k = ±√24 = ±2√6.
Question 6: Maximum Value
The maximum number of roots for a quadratic equation is:
Question 7: Equation Formation
Which of the following equations has 2 as a root?
For (c): 2(2)² - 7(2) + 6 = 8 - 14 + 6 = 0.
Since LHS = RHS, 2 is a root.
Question 8: Word Problem (Numbers)
The sum of a number and its reciprocal is 10/3. The number is:
x + 1/x = 10/3
(x² + 1)/x = 10/3 => 3x² + 3 = 10x
3x² - 10x + 3 = 0
3x² - 9x - x + 3 = 0
3x(x-3) -1(x-3) = 0
(3x-1)(x-3) = 0
x = 3 or x = 1/3. Both satisfy the condition.
Question 9: Reciprocal Roots
If one root of the equation 4x² - 2x + (k-4) = 0 is the reciprocal of the other, then k is:
Product of roots (c/a) = (k-4)/4 = 1.
k - 4 = 4
k = 8.
Question 10: Roots Nature
The roots of the equation x² + x + 1 = 0 are:
D = (1)² - 4(1)(1) = 1 - 4 = -3.
Since D < 0, the equation has no real roots.
Question 11: Value of p
If -5 is a root of the quadratic equation 2x² + px - 15 = 0, then p is:
2(-5)² + p(-5) - 15 = 0
2(25) - 5p - 15 = 0
50 - 15 - 5p = 0
35 = 5p => p = 7.
Question 12: Word Problem (Rect)
The perimeter of a rectangle is 82 m and its area is 400 m². The breadth of the rectangle is:
Area = l*b = 400.
(41-b)b = 400 => b² - 41b + 400 = 0.
(b-25)(b-16) = 0. So b=16 or 25. Since length > breadth, b = 16 m.
Question 13: Perfect Square
For what value of k is the polynomial 9x² + 30x + k a perfect square?
b² - 4ac = 0
(30)² - 4(9)(k) = 0
900 - 36k = 0
36k = 900 => k = 25.
(Check: 9x² + 30x + 25 = (3x + 5)²).
Question 14: Real Roots Condition
The equation (x+1)² - x² = 0 has number of real roots equal to:
x² + 2x + 1 - x² = 0
2x + 1 = 0
x = -1/2.
This is a linear equation, so it has only 1 real root.
Question 15: Value of √6...
The value of √6 + √6 + √6 + ... is:
Squaring both sides: x² = 6 + x
x² - x - 6 = 0
(x - 3)(x + 2) = 0
x = 3 or x = -2. Since value must be positive, x = 3.
Question 16: Positive Root
The positive root of √(3x² + 6) = 9 is:
3x² + 6 = 81
3x² = 75
x² = 25
x = ±5. The positive root is 5.
Question 17: Altitude of Triangle
The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, the other two sides are:
x² + (x-7)² = 13² (Pythagoras theorem)
x² + x² - 14x + 49 = 169
2x² - 14x - 120 = 0 => x² - 7x - 60 = 0
(x-12)(x+5) = 0. So x = 12.
Base = 12, Altitude = 12-7 = 5.
Question 18: Roots (Irrational)
The roots of the equation 4x² + 4√3x + 3 = 0 are:
D = b² - 4ac = (4√3)² - 4(4)(3)
D = (16 × 3) - 48
D = 48 - 48 = 0.
Since D = 0, roots are real and equal.
Question 19: Opposite Signs
If the roots of ax² + bx + c = 0 are of opposite signs, then:
Question 20: Sum of Squares
The sum of squares of two consecutive natural numbers is 313. The numbers are:
(a) 12² + 13² = 144 + 169 = 313. Correct.
Question 21: Quadratic Formula
Which of the following is the correct quadratic formula for ax² + bx + c = 0?
Question 22: Value of c
If x = 1 is a common root of ax² + ax + 3 = 0 and x² + x + b = 0, then the value of ab is:
Put x=1 in first eq: a(1) + a(1) + 3 = 0 => 2a = -3 => a = -3/2.
Put x=1 in second eq: 1 + 1 + b = 0 => b = -2.
Value of ab = (-3/2) × (-2) = 3.
Question 23: Non-Real Roots
The equation 2x² + 5x + 4 = 0 has:
D = 5² - 4(2)(4) = 25 - 32 = -7.
Since D < 0, there are no real roots.
Question 24: Difference of Roots
If the difference of the roots of the equation x² - 5x + c = 0 is 1, then c is equal to:
α + β = 5, αβ = c.
Given |α - β| = 1.
(α - β)² = (α + β)² - 4αβ
1² = 5² - 4c
1 = 25 - 4c => 4c = 24 => c = 6.
Question 25: Reciprocal Roots Sum
If the roots of x² + px + 12 = 0 are in the ratio 1:3, then p is:
Product: α(3α) = 12 => 3α² = 12 => α² = 4 => α = ±2.
Sum: α + 3α = -p => 4α = -p => p = -4α.
If α = 2, p = -8. If α = -2, p = 8.
