JEE Mathematics Complex Numbers MCQs Set 33

Answer: (b) \(32(x_1^2+y_1^2)\)

Explanation:
1. Expanding, the cross terms \(\mp2x_1y_1\,\mathrm{Re}(z_1\bar z_2)\) cancel.
2. The result is \((x_1^2+y_1^2)(|z_1|^2+|z_2|^2)=(x_1^2+y_1^2)(16+16)=32(x_1^2+y_1^2)\).

Teacher's Note:
1. Write complex numbers in polar form or use z times its conjugate for modulus problems.
2. Use geometry on the Argand plane (distances, arguments, rotations) to check the algebra.

Answer: (c) \(-\bar\omega\)

Explanation:
1. Let \(\omega=re^{i\phi}\). Then \(\operatorname{Arg}z=\pi-\phi\) and \(z=re^{i(\pi-\phi)}\).
2. \(re^{i(\pi-\phi)}=-re^{-i\phi}=-\bar\omega\).

Teacher's Note:
1. Write complex numbers in polar form or use z times its conjugate for modulus problems.
2. Use geometry on the Argand plane (distances, arguments, rotations) to check the algebra.

Answer: (d) \([1,9]\)

Explanation:
1. The region is a filled ellipse with foci \(1\) and \(-3\) and major axis \(8\): centre \(-1\), semi-major axis 4, so it meets the real axis from \(-5\) to \(3\).
2. The point 4 lies on the major axis, so the nearest point is \(3\) (distance 1) and the farthest is \(-5\) (distance 9).

Teacher's Note:
1. Write complex numbers in polar form or use z times its conjugate for modulus problems.
2. Use geometry on the Argand plane (distances, arguments, rotations) to check the algebra.

Answer: (a) for any positive integers n_1, n_2

Explanation:
1. \(i^3=-i,\ i^5=i,\ i^7=-i\), so the sum is \((1+i)^{n_1}+(1-i)^{n_1}+(1+i)^{n_2}+(1-i)^{n_2}\).
2. Each pair is a number plus its conjugate, which is real for every n_1 and n_2.

Teacher's Note:
1. Write complex numbers in polar form or use z times its conjugate for modulus problems.
2. Use geometry on the Argand plane (distances, arguments, rotations) to check the algebra.

Answer: (c) c

Explanation:
1. The midpoint of \(z_1,z_2\) lies on the line: \(b(\bar z_1+\bar z_2)+\bar b(z_1+z_2)=2c\).
2. The segment \(z_1z_2\) is perpendicular to the line, whose normal is \(b\), so \(\dfrac{z_1-z_2}{b}\) is real, giving \(\bar b(z_1-z_2)=b(\bar z_1-\bar z_2)\).
3. Subtracting the second relation from the first: \(2b\bar z_1+2\bar bz_2=2c\), so \(\bar bz_2+b\bar z_1=c\).

Teacher's Note:
1. Write complex numbers in polar form or use z times its conjugate for modulus problems.
2. Use geometry on the Argand plane (distances, arguments, rotations) to check the algebra.

Answer: (b) \(\dfrac{2z_1z_2}{z_1+z_2}\)

Explanation:
1. Take radius 1 (scaling does not matter) and \(\bar z=1/z\) on the circle. The tangent at \(z_1\): \(z+z_1^2\bar z=2z_1\).
2. Tangents at \(z_1,z_2\) meet where \(z+z_1^2\bar z=2z_1\) and \(z+z_2^2\bar z=2z_2\); subtracting gives \(\bar z=\dfrac{2}{z_1+z_2}\).
3. Substituting back: \(z=2z_1-\dfrac{2z_1^2}{z_1+z_2}=\dfrac{2z_1z_2}{z_1+z_2}\).

Teacher's Note:
1. Write complex numbers in polar form or use z times its conjugate for modulus problems.
2. Use geometry on the Argand plane (distances, arguments, rotations) to check the algebra.

Answer: (d) 26

Explanation:
1. \(z=e^{\pm i\pi/3}\), a primitive 6th root of unity. \(2017=6\cdot336+1\), so \(z^{2017}=z\) and \(a=1\).
2. For \(n\ge2\), \(2^{2^n}\) is a power of 2 with exponent a multiple of 4, so it ends in 6 and \(2^{2^n}-1\) ends in 5: b = 5.
3. \(a^2+b^2=1+25=26\).

Teacher's Note:
1. Write complex numbers in polar form or use z times its conjugate for modulus problems.
2. Use geometry on the Argand plane (distances, arguments, rotations) to check the algebra.

Answer: (a) \(y_1=\sqrt3,\ y_2<\sqrt3\)

Explanation:
1. By the triangle inequality \(\big||z-\omega|-|z-\omega^2|\big|\le|\omega-\omega^2|=\sqrt3\), with equality when z lies on the line through \(\omega,\omega^2\) (the line \(x=-\tfrac12\)) outside the segment between them.
2. The circle \(|z|=2\) cuts that line at points outside the segment, so \(y_1=\sqrt3\).
3. The circle \(|z|=\tfrac12\) touches the line only at \(-\tfrac12\), which lies between \(\omega\) and \(\omega^2\), so equality is not reached; numerically \(y_2\approx0.866<\sqrt3\).

Teacher's Note:
1. Write complex numbers in polar form or use z times its conjugate for modulus problems.
2. Use geometry on the Argand plane (distances, arguments, rotations) to check the algebra.

Answer: (b) 1

Explanation:
1. \(|a+b\omega+c\omega^2|^2=a^2+b^2+c^2-ab-bc-ca=\tfrac12[(a-b)^2+(b-c)^2+(c-a)^2]\).
2. The numbers are integers and not all equal, so the three differences (which sum to zero) are integers, not all zero. At least two are non-zero, so the sum of their squares is at least 1 + 1 + 0 = 2 and the modulus squared is at least 1. It equals 1 for a = 1, b = c = 0.

Teacher's Note:
1. Write complex numbers in polar form or use z times its conjugate for modulus problems.
2. Use geometry on the Argand plane (distances, arguments, rotations) to check the algebra.

Answer: (c) 0

Explanation:
1. \(|f(i)|=\prod|i-r_k|=\prod\sqrt{1+r_k^2}\ge1\) for real roots \(r_k\).
2. Equality needs every \(r_k=0\), so \(f(x)=x^4\) and \(a=b=c=d=0\).

Teacher's Note:
1. Write complex numbers in polar form or use z times its conjugate for modulus problems.
2. Use geometry on the Argand plane (distances, arguments, rotations) to check the algebra.

Answer: (d) \(c\in(-\infty,-1-\sqrt2)\)

Explanation:
1. Put \(z=x+iy\). Imaginary part: \(-2x+2c=0\), so \(x=c\). Real part: \(x^2+y^2+2y+2c=0\).
2. Then \((y+1)^2=1-2c-c^2\), which has a real solution iff \(c^2+2c-1\le0\), i.e. \(-1-\sqrt2\le c\le-1+\sqrt2\).
3. So there is no solution for c outside this interval, for example for c < -1 - sqrt 2.

Teacher's Note:
1. Write complex numbers in polar form or use z times its conjugate for modulus problems.
2. Use geometry on the Argand plane (distances, arguments, rotations) to check the algebra.

Answer: (a) \(\mathrm{Im}(\omega_1\bar\omega_2)=0\)

Explanation:
1. The conditions say the rows \((a,b)\) and \((c,d)\) of the matrix \(\begin{pmatrix}a&b\\c&d\end{pmatrix}\) are orthonormal, so the matrix is orthogonal and its columns are orthonormal too.
2. Hence \(a^2+c^2=1\), \(b^2+d^2=1\) and \(ab+cd=0=\mathrm{Re}(\omega_1\bar\omega_2)\).
3. \(\mathrm{Im}(\omega_1\bar\omega_2)=cb-ad=\mp1\ne0\) (it is the determinant).

Teacher's Note:
1. Write complex numbers in polar form or use z times its conjugate for modulus problems.
2. Use geometry on the Argand plane (distances, arguments, rotations) to check the algebra.

Answer: (d) 1 and 2

Explanation:
1. Multiply by \(z_1z_2z_3\): \(z_1^3+z_2^3+z_3^3=-z_1z_2z_3\). With \(p=z_1+z_2+z_3\), \(e_3=z_1z_2z_3\) and \(e_2=e_3\bar p\) (as \(|z_k|=1\)): \(p^3-3pe_2+3e_3=-e_3\).
2. So \(p^3=e_3(3|p|^2-4)\) and \(r^3=|3r^2-4|\) for \(r=|p|\).
3. \(r^3-3r^2+4=(r+1)(r-2)^2=0\) gives \(r=2\); \(r^3+3r^2-4=(r-1)(r+2)^2=0\) gives \(r=1\).

Teacher's Note:
1. Write complex numbers in polar form or use z times its conjugate for modulus problems.
2. Use geometry on the Argand plane (distances, arguments, rotations) to check the algebra.

Answer: (c) \(i\)

Explanation:
1. \(\sin t-i\cos t=-i(\cos t+i\sin t)=-ie^{it}\).
2. \(\sum_{k=1}^{10}e^{2\pi ik/11}=-1\) (sum of the 11th roots of unity is 0 and the k = 0 term is 1), so \(S=-i\cdot(-1)=i\).

Teacher's Note:
1. Write complex numbers in polar form or use z times its conjugate for modulus problems.
2. Use geometry on the Argand plane (distances, arguments, rotations) to check the algebra.

Answer: (a) 48

Explanation:
1. \(z=e^{i\pi/3}\), so \(z^k+z^{-k}=2\cos\tfrac{k\pi}3\) and the term is \(4\cos^2\tfrac{k\pi}3\).
2. Over one period k = 1..6 the values are 1, 1, 4, 1, 1, 4, which sum to 12.
3. 24 terms = 4 periods: 4 * 12 = 48.

Teacher's Note:
1. Write complex numbers in polar form or use z times its conjugate for modulus problems.
2. Use geometry on the Argand plane (distances, arguments, rotations) to check the algebra.

Answer: (b) 10

Explanation:
1. \(z=e^{-i\pi/6}\), \(z^{89}=e^{-89i\pi/6}=e^{-5i\pi/6}=-\tfrac{\sqrt3}2-\tfrac i2\) and \(i^{97}=i\).
2. \(z^{89}+i^{97}=-\tfrac{\sqrt3}2+\tfrac i2=e^{5i\pi/6}\), and the 94th power is \(e^{470i\pi/6}=e^{i\pi/3}\).
3. \(z^n=e^{-in\pi/6}=e^{i\pi/3}\) needs \(n\equiv-2\pmod{12}\), so the smallest positive n is 10.

Teacher's Note:
1. Write complex numbers in polar form or use z times its conjugate for modulus problems.
2. Use geometry on the Argand plane (distances, arguments, rotations) to check the algebra.

Answer: (b) |z| = |w| = 1

Explanation:
1. |z + iw| <= |z| + |w| <= 2, and equality forces |z| = |w| = 1.
2. Both conditions are consistent with this: z = iw and z = -i conj(w) give w purely imaginary, e.g. w = i, z = -1.

Teacher's Note:
1. Write complex numbers in polar form or use z times its conjugate for modulus problems.
2. Use geometry on the Argand plane (distances, arguments, rotations) to check the algebra.

Answer: (c) Im(z) = Re(w)

Explanation:
1. Equality in the triangle inequality gives z = iw (z has the same direction as iw, both of modulus 1) and z = -i conj(w).
2. So iw = -i conj(w), i.e. w = -conj(w): w is purely imaginary, Re(w) = 0. Then z = iw is real, so Im(z) = 0 = Re(w). (For w = i, z = -1: Re z = -1, Im w = 1, so Re z = Im w is false.)

Teacher's Note:
1. Write complex numbers in polar form or use z times its conjugate for modulus problems.
2. Use geometry on the Argand plane (distances, arguments, rotations) to check the algebra.

Answer: (d) \(a=b=2-\sqrt3\)

Explanation:
1. Rotating \(z_1\) about the origin by \(\pm60^\circ\) must give \(z_2\): \((a+i)\left(\tfrac12\pm\tfrac{\sqrt3}2i\right)=1+bi\).
2. With the + sign: real part gives \(a=2+\sqrt3>1\), rejected. With the - sign: \(\tfrac a2+\tfrac{\sqrt3}2=1\) gives \(a=2-\sqrt3\).
3. Imaginary part: \(b=\tfrac12-\tfrac{\sqrt3}2a=\tfrac12-\tfrac{\sqrt3(2-\sqrt3)}2=2-\sqrt3\).

Teacher's Note:
1. Write complex numbers in polar form or use z times its conjugate for modulus problems.
2. Use geometry on the Argand plane (distances, arguments, rotations) to check the algebra.

Answer: (a) \(3z_0^2\)

Explanation:
1. For an equilateral triangle \(z_1^2+z_2^2+z_3^2=z_1z_2+z_2z_3+z_3z_1\), so \((z_1+z_2+z_3)^2=3(z_1^2+z_2^2+z_3^2)\).
2. The centroid is \(z_0\): \(z_1+z_2+z_3=3z_0\), so the sum of squares is \(\dfrac{9z_0^2}3=3z_0^2\).

Teacher's Note:
1. Write complex numbers in polar form or use z times its conjugate for modulus problems.
2. Use geometry on the Argand plane (distances, arguments, rotations) to check the algebra.