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MathsMediumClass 11

Graph of Quadratic ā€” Sign analysis, range

Quadratic Equations

7

JEE Qs

8%

Hard

60

min

Master the interplay between 'a', 'D', and the vertex to quickly determine the shape, x-intercepts, and range of a quadratic expression for sign analysis and solving inequalities.

šŸ§® Key Formulas

f(x) = ax^2 + bx + c
Discriminant D = b^2 - 4ac
Roots x = (-b Ā± sqrt(D)) / 2a
Vertex coordinates (-b/2a, -D/4a)
Axis of symmetry x = -b/2a

āœ… Key Points for JEE

  • 1The sign of 'a' determines the opening of the parabola: 'a > 0' means upward (minimum value), 'a < 0' means downward (maximum value).
  • 2The discriminant D determines the number of real roots (x-intercepts): D > 0 (two distinct real roots), D = 0 (two equal real roots/parabola touches x-axis), D < 0 (no real roots/parabola never crosses x-axis).
  • 3For sign analysis, the quadratic expression's sign changes at its real roots. If D > 0, between the roots, f(x) has the opposite sign to 'a'. Outside the roots, f(x) has the same sign as 'a'.
  • 4If D <= 0, the quadratic expression never changes sign and always has the same sign as 'a' for all real x.
  • 5The range of a quadratic function is determined by its vertex and opening direction: if a > 0, Range = [-D/4a, āˆž); if a < 0, Range = (-āˆž, -D/4a].

āš ļø Common Mistakes

  • āœ•Forgetting to consider the sign of 'a' when determining the opening direction of the parabola or the sign of the quadratic expression.
  • āœ•Incorrectly applying sign analysis for cases where D <= 0 (no real roots or coincident roots), leading to incorrect sign changes.
  • āœ•Confusing the role of D (nature of roots) with -D/4a (y-coordinate of the vertex, which determines the extremum of the range).

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 10 Mathematics Chapter 4: Quadratic Equations
  • Class 11 Mathematics Chapter 2: Relations and Functions