Sign in
12thPass logo
New chatPYQ LibraryDoubtsRank report
Sign in to see Recents

Your guest activity stays on this device

Sign in to save progress →
Sign in

Quadratic Equation and Inequalities question

2005 · Shift 0 · Q111
Guest · filters and generic practice availableBrowsing as a guest · PYQ filters and generic practice are available. Sign in only for personalised features and saved progress.
  1. PYQ Library
  2. /JEE Main
  3. /Mathematics
  4. /Quadratic Equation and Inequalities
  5. /2005 · Shift 0 · Q111

Quadratic Equation and Inequalities question

2005 · Shift 0 · Q111

JEE MainMathematicsQuadratic Equation and InequalitiesMCQ+4 / −1
If the roots of the equation x2−bx+c=0{x^2} - bx + c = 0x2−bx+c=0 be two consecutive integers, then b2−4c{b^2} - 4cb2−4c equals
  1. A
    −2-2−2
  2. B
    333
  3. C
    222
  4. D
    111
View written solutionFree

Correct answer: D

  1. Let the two consecutive integer roots be nnn and n+1n+1n+1.

  2. For the quadratic equation x2−bx+c=0,x^2 - bx + c = 0,x2−bx+c=0, by Vieta’s formulas:

    • Sum of roots =b= b=b
    • Product of roots =c= c=c

    Hence, b=n+(n+1)=2n+1b = n + (n+1) = 2n+1b=n+(n+1)=2n+1 and c=n(n+1).c = n(n+1).c=n(n+1).

  3. Now compute the discriminant expression: b2−4c=(2n+1)2−4n(n+1).b^2 - 4c = (2n+1)^2 - 4n(n+1).b2−4c=(2n+1)2−4n(n+1).

  4. Expand: (2n+1)2=4n2+4n+1 (2n+1)^2 = 4n^2 + 4n + 1(2n+1)2=4n2+4n+1 and 4n(n+1)=4n2+4n.4n(n+1) = 4n^2 + 4n.4n(n+1)=4n2+4n.

    Therefore, b2−4c=(4n2+4n+1)−(4n2+4n)=1.b^2 - 4c = (4n^2+4n+1) - (4n^2+4n) = 1.b2−4c=(4n2+4n+1)−(4n2+4n)=1.

  5. So the required value is 1.\boxed{1}.1​.

  6. Comparing with the options:

    • A: −2-2−2 ❌
    • B: 333 ❌
    • C: 222 ❌
    • D: 111 ✅

Therefore, the correct option is D.

PreviousNext

More from Quadratic Equation and Inequalities

  • If the roots of the equation x2−bx+c=0 be two consecutive integers, then b2−4c equals2005 · MCQ
  • The value of a for which the sum of the squares of the roots of the equation x2−(a−2)x−a−1=0 assume the least value is :2005 · MCQ
  • In a triangle PQR,∠R=2π​.Iftan(2P​) and tan(2Q​) are the roots of ax2+bx+c=0,ae0 then2005 · MCQ
  • If both the roots of the quadratic equation x2−2kx+k2+k−5=0 are less than 5, then k lies in the interval2005 · MCQ
  • Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation2004 · MCQ
  • If (1−p) is a root of quadratic equation x2+px+(1−p)=0 then its root are2004 · MCQ
  • If one root of the equation x2+px+12=0 is 4, while the equation x2+px+q=0 has equal roots, then the value of ′q′ is2004 · MCQ
  • If the sum of the roots of the quadratic equation ax2+bx+c=0 is equal to the sum of the squares of their reciprocals, then ca​,ab​ and bc​ are in2003 · MCQ