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JEE Mathematics 2024 Question with Solution

Let z1z_1 and z2z_2 be two complex numbers such that z1+z2=5z_1 + z_2 = 5 and z13+z23=20+15iz_1^3 + z_2^3 = 20 + 15i. Then ∣z14+z24∣|z_1^4 + z_2^4| equals:

  • A

    30330\sqrt{3}

  • B

    7575

  • C

    151515\sqrt{15}

  • D

    25325\sqrt{3}

Answer

Correct answer:B

Step-by-step solution

Standard Method

Given: z1+z2=5z_1 + z_2 = 5 and z13+z23=20+15iz_1^3 + z_2^3 = 20 + 15i.

Find: ∣z14+z24∣|z_1^4 + z_2^4|.

Using the identity for sum of cubes,

z13+z23=(z1+z2)(z12−z1z2+z22)z_1^3 + z_2^3 = (z_1 + z_2)(z_1^2 - z_1z_2 + z_2^2)

So,

20+15i=5(z12−z1z2+z22)20 + 15i = 5(z_1^2 - z_1z_2 + z_2^2)

Hence,

z12−z1z2+z22=4+3iz_1^2 - z_1z_2 + z_2^2 = 4 + 3i

Also,

(z1+z2)2=z12+2z1z2+z22=25(z_1 + z_2)^2 = z_1^2 + 2z_1z_2 + z_2^2 = 25

Subtracting,

25−(4+3i)=3z1z225 - (4 + 3i) = 3z_1z_2

Therefore,

3z1z2=21−3i3z_1z_2 = 21 - 3i z1z2=7−iz_1z_2 = 7 - i

Now,

z12+z22=(z1+z2)2−2z1z2=25−2(7−i)=11+2iz_1^2 + z_2^2 = (z_1 + z_2)^2 - 2z_1z_2 = 25 - 2(7 - i) = 11 + 2i

Using

z14+z24=(z12+z22)2−2(z1z2)2z_1^4 + z_2^4 = (z_1^2 + z_2^2)^2 - 2(z_1z_2)^2

we get

(11+2i)2=117+44i(11 + 2i)^2 = 117 + 44i

and

(7−i)2=48−14i(7 - i)^2 = 48 - 14i

So,

z14+z24=(117+44i)−2(48−14i)z_1^4 + z_2^4 = (117 + 44i) - 2(48 - 14i) =117+44i−96+28i= 117 + 44i - 96 + 28i =21+72i= 21 + 72i

Therefore,

∣z14+z24∣=212+722=5625=75|z_1^4 + z_2^4| = \sqrt{21^2 + 72^2} = \sqrt{5625} = 75

So the correct option is B.

Note: The answer key says option DD, but the solution working on the page gives 7575, which matches option B.

Common mistakes

  • Using z13+z23=(z1+z2)3z_1^3 + z_2^3 = (z_1 + z_2)^3 is incorrect because the cubic expansion has extra mixed terms. Use the identity z13+z23=(z1+z2)(z12−z1z2+z22)z_1^3 + z_2^3 = (z_1 + z_2)(z_1^2 - z_1z_2 + z_2^2) instead.

  • Writing z12+z22=(z1+z2)2−z1z2z_1^2 + z_2^2 = (z_1 + z_2)^2 - z_1z_2 is wrong. The correct relation is z12+z22=(z1+z2)2−2z1z2z_1^2 + z_2^2 = (z_1 + z_2)^2 - 2z_1z_2.

  • Forgetting the factor 22 in z14+z24=(z12+z22)2−2(z1z2)2z_1^4 + z_2^4 = (z_1^2 + z_2^2)^2 - 2(z_1z_2)^2 leads to the wrong fourth-power sum. Expand carefully before substituting.

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