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📐 Mathematics  ·  Complex Numbers and Quadratic Equations  ·  JEE

The quadratic f(x) = ax² + bx + c (a > 0) satisfies f(1) = 1, f(2) = 1, f(3) = 5. Find f(0).

Answer: 5.

  • A 1
  • B 2
  • C 3
  • D 5

Correct answer: D. 5

Explanation: Three equations: a+b+c=1, 4a+2b+c=1, 9a+3b+c=5. From first two: 3a+b=0. From last two: 5a+b=4. So 2a=4, a=2, b=-6. c=1-2+6=5. f(0)=c=5.

Argand Plane: z = a + ibReImz = a+iba (real part)b (imaginary part)θ = arg(z)|z| = length of the vector OZ = √(a²+b²); θ = angle OZ makes with the positive real axis

A complex number z = a+ib is plotted as a point (or vector from the origin) on the Argand plane, with the real part along the horizontal axis and the imaginary part along the vertical; its modulus |z| is the vector's length, and its argument θ is the angle from the positive real axis.

Concept context

Complex numbers in a+ib form, modulus, argument, polar form, cube roots of unity, and quadratic equations with complex roots

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