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Quadratic Equation and Inequalities question

2021 · 25 Jul · Shift 2 · Q44

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Quadratic Equation and Inequalities
2021 · 25 Jul · Shift 2 · Q44

Quadratic Equation and Inequalities question

2021 · 25 Jul · Shift 2 · Q44

JEE MainMathematicsQuadratic Equation and InequalitiesNumerical+4 / −1

If a + b + c = 1, ab + bc + ca = 2 and abc = 3, then the value of a4 + b4 + c4 is equal to

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.

Numerical answer

View written solutionFree

Correct answer: 13

We are given

\qquad ab+bc+ca=2, \qquad abc=3.$$ We need to find $$a^4+b^4+c^4.$$ 2. Let $$p=a+b+c=1,\qquad q=ab+bc+ca=2,\qquad r=abc=3.$$ We will use symmetric identities. 3. First find $a^2+b^2+c^2$: $$a^2+b^2+c^2=(a+b+c)^2-2(ab+bc+ca)=p^2-2q.$$ Substituting $p=1, q=2$, $$a^2+b^2+c^2=1^2-2\cdot 2=1-4=-3.$$ 4. Now use $$a^4+b^4+c^4=(a^2+b^2+c^2)^2-2(a^2b^2+b^2c^2+c^2a^2).$$ So we need $a^2b^2+b^2c^2+c^2a^2$. But $$a^2b^2+b^2c^2+c^2a^2=(ab+bc+ca)^2-2abc(a+b+c).$$ That is, $$a^2b^2+b^2c^2+c^2a^2=q^2-2rp.$$ Substitute the given values: $$a^2b^2+b^2c^2+c^2a^2=2^2-2\cdot 3\cdot 1=4-6=-2.$$ 5. Therefore, $$a^4+b^4+c^4=(-3)^2-2(-2)=9+4=13.$$ 6. Final answer: $$\boxed{13}$$ 7. Comparison with stored correct answer: The stored correct answer is $13$, which matches our derived result.

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