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

2002 · Shift 0 · Q104
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. /2002 · Shift 0 · Q104

Quadratic Equation and Inequalities question

2002 · Shift 0 · Q104

JEE MainMathematicsQuadratic Equation and InequalitiesMCQ+4 / −1
If a, b, ca,\,b,\,ca,b,c are distinct +ve+ ve+ve real numbers and a2+b2+c2=1{a^2} + {b^2} + {c^2} = 1a2+b2+c2=1 then ab+bc+caab + bc + caab+bc+ca is
  1. A
    less than 1
  2. B
    equal to 1
  3. C
    greater than 1
  4. D
    any real no.
View written solutionFree

Correct answer: A

  1. We are given

    \qquad a,b,c\text{ are distinct,} \qquad a^2+b^2+c^2=1.$$ We need to determine the value of $$ab+bc+ca.$$
  2. Use the identity (a+b+c)2=a2+b2+c2+2(ab+bc+ca).(a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ca).(a+b+c)2=a2+b2+c2+2(ab+bc+ca). Also, (a−b)2+(b−c)2+(c−a)2≥0.(a-b)^2+(b-c)^2+(c-a)^2 \ge 0.(a−b)2+(b−c)2+(c−a)2≥0.

  3. Expanding this,

    =2(a^2+b^2+c^2)-2(ab+bc+ca). $$ Hence, $$2(a^2+b^2+c^2)-2(ab+bc+ca)\ge 0,$$ so $$a^2+b^2+c^2 \ge ab+bc+ca.$$
  4. Since a2+b2+c2=1a^2+b^2+c^2=1a2+b2+c2=1, we get ab+bc+ca≤1.ab+bc+ca \le 1.ab+bc+ca≤1.

  5. But a,b,ca,b,ca,b,c are distinct, so equality cannot hold. Equality in a2+b2+c2≥ab+bc+caa^2+b^2+c^2 \ge ab+bc+caa2+b2+c2≥ab+bc+ca occurs only when a=b=c,a=b=c,a=b=c, which is impossible because they are distinct.

  6. Therefore, ab+bc+ca<1.ab+bc+ca<1.ab+bc+ca<1.

  7. Checking options:

    • A: less than 1 — correct
    • B: equal to 1 — false
    • C: greater than 1 — false
    • D: any real number — false

Therefore, the correct option is A.\boxed{\text{A}}.A​.

Previous

More from Quadratic Equation and Inequalities

  • Let Pn​=αn+βn,n∈N. If P10​=123,P9​=76,P8​=47 and P1​=1, then the quadratic equation having roots α1​…2025 · MCQ
  • If the set of all a∈R−{1}, for which the roots of the equation (1−a)x2+2(a−3)x+9=0 are positive is (−∞,−α]∪[β,γ), then 2α+β+γ is equal to .2025 · Numerical
  • Let α and β be the roots of x2+3​x−16=0, and γ and δ be the roots of x2+3x−1=0. If Pn​=αn+βn and Qn​=γn+o^n, then 2P23​P25​+3​P24​​+Q24​Q25​−Q23​​…2025 · MCQ
  • Let the equation x(x+2)(12−k)=2 have equal roots. Then the distance of the point (k,2k​) from the line 3x+4y+5=0 is2025 · MCQ
  • Consider the equation x2+4x−n=0, where n∈[20,100] is a natural number. Then the number of all distinct values of n, for which the given equation has integral roots, is equal to2025 · MCQ
  • Let the set of all values of p∈R, for which both the roots of the equation x2−(p+2)x+(2p+9)=0 are negative real numbers, be the interval (α,β]. Then β−2α is equal to2025 · MCQ
  • The number of real roots of the equation x∣x−2∣+3∣x−3∣+1=0 is :2025 · MCQ
  • The sum of the squares of the roots of ∣x−2∣2+∣x−2∣−2=0 and the squares of the roots of x2−2∣x−3∣−5=0, is2025 · MCQ