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MathsMediumClass 11

Range of Quadratic Expression

Quadratic Equations

7

JEE Qs

8%

Hard

60

min

Always visualize the graph of the parabola and its vertex to correctly determine the range, especially for restricted domains.

šŸ§® Key Formulas

f(x) = ax^2 + bx + c
Discriminant D = b^2 - 4ac
Vertex coordinates = (-b/(2a), -D/(4a))
If a > 0, Range of f(x) for x āˆˆ R is [-D/(4a), infinity)
If a < 0, Range of f(x) for x āˆˆ R is (-infinity, -D/(4a)]

āœ… Key Points for JEE

  • 1The range of a quadratic expression f(x) = ax^2 + bx + c (a ā‰  0) over real numbers R is determined by the y-coordinate of its vertex, -D/(4a), and the sign of the leading coefficient 'a'.
  • 2If 'a' > 0, the parabola opens upwards, indicating a minimum value at the vertex (-b/(2a), -D/(4a)), and no maximum value. The range is [-D/(4a), āˆž).
  • 3If 'a' < 0, the parabola opens downwards, indicating a maximum value at the vertex (-b/(2a), -D/(4a)), and no minimum value. The range is (-āˆž, -D/(4a)].
  • 4For a restricted domain [x1, x2], the range is found by evaluating f(x1), f(x2), and f(-b/(2a)) (only if -b/(2a) lies within [x1, x2]), and then identifying the minimum and maximum among these relevant values.

āš ļø Common Mistakes

  • āœ•Incorrectly identifying whether the parabola opens upwards or downwards, leading to errors in determining minimum vs. maximum values.
  • āœ•Making calculation errors while finding the vertex coordinates (-b/(2a), -D/(4a)).
  • āœ•Failing to consider the vertex's y-coordinate when the domain is restricted, or not checking if the vertex lies within the restricted domain.

šŸ“ Practice Questions

See all

Q22.The roots of the quadratic equation 3x2 āˆ’px + q = 0 are 10th and 11th terms of an arithmetic progression with common difference 32 . If the sum of the first 11 terms of this arithmetic progression is 88 , then q āˆ’2p is equal to -.

2025Ā·NumericalMedium

Q13. The number of real solution(s) of the equation x2 + 3x + 2 = min{|x āˆ’3|, |x + 2|} is : (1) 1 (2) 0 (3) 2 (4) 3

2025Ā·MCQMedium

Q17.Let Ī±Īø and Ī²Īø be the distinct roots of 2x2 + (cos Īø)x āˆ’1 = 0, Īø āˆˆ(0, 2Ļ€). If m and M are the minimum and the maximum values of Ī±4Īø + Ī²4Īø , then 16(M + m) equals : (1) 24 (2) 25 (3) 17 (4) 27

2025Ā·MCQHard

Q22.If the equation a(b āˆ’c)x2 + b(c āˆ’a)x + c(a āˆ’b) = 0 has equal roots, where a + c = 15 and b = 365 , then a2 + c2 is equal to

2025Ā·NumericalMedium

Q7. Let the line passing through the points (āˆ’1, 2, 1) and parallel to the line xāˆ’12 = y+13 = 4z intersect the line yāˆ’3 x+2 3 = 2 = zāˆ’41 at the point P . Then the distance of P from the point Q(4, āˆ’5, 1) is (1) 5 (2) 5āˆš5 (3) 5āˆš6 (4) 10

2025Ā·MCQMedium

Q13.The sum, of the squares of all the roots of the equation x2 + |2x āˆ’3| āˆ’4 = 0, is (1) 3(3 āˆ’āˆš2) (2) 6(3 āˆ’āˆš2) (3) 6(2 āˆ’āˆš2) (4) 3(2 āˆ’āˆš2)

2025Ā·MCQMedium

NCERT Chapters

  • Class 11 Maths Ch 5: Quadratic Equations
  • Class 11 Maths Ch 2: Relations and Functions