JEE MainMathematicsQuadratic Equation and InequalitiesMCQ+4 / −1
If are distinct real numbers and then is
- Aless than 1
- Bequal to 1
- Cgreater than 1
- Dany real no.
View written solutionFree
Correct answer: A
-
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.$$ -
Use the identity Also,
-
Expanding this,
=2(a^2+b^2+c^2)-2(ab+bc+ca). $$ Hence, $$2(a^2+b^2+c^2)-2(ab+bc+ca)\ge 0,$$ so $$a^2+b^2+c^2 \ge ab+bc+ca.$$ -
Since , we get
-
But are distinct, so equality cannot hold. Equality in occurs only when which is impossible because they are distinct.
-
Therefore,
-
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
More from Quadratic Equation and Inequalities
- Let . If and , then the quadratic equation having roots …2025 · MCQ
- If the set of all , for which the roots of the equation are positive is , then is equal to .2025 · Numerical
- Let and be the roots of , and and be the roots of . If and , then …2025 · MCQ
- Let the equation have equal roots. Then the distance of the point from the line is2025 · MCQ
- Consider the equation , where is a natural number. Then the number of all distinct values of , for which the given equation has integral roots, is equal to2025 · MCQ
- Let the set of all values of , for which both the roots of the equation are negative real numbers, be the interval . Then is equal to2025 · MCQ
- The number of real roots of the equation is :2025 · MCQ
- The sum of the squares of the roots of and the squares of the roots of , is2025 · MCQ