Questions: Complex Numbers and Quadratic Equations

Q1. Express (2 + 3i) + (4 - 5i) and (2 + 3i) - (4 - 5i) in the form a + ib.

Answer: Add part by part for the sum: 2 + 4 gives 6 and 3i - 5i gives -2i, so the sum is 6 - 2i. For the difference, distribute the minus sign to get 2 + 3i - 4 + 5i, which is -2 + 8i. Keep real and imaginary parts on separate lines so a sign slip does not hide.

Hinglish: Sum ke liye real aur imaginary parts alag jodo: 6 - 2i. Difference ke liye minus sign distribute karo: -2 + 8i. Parts alag lines me likho.

Q2. Find the value of i to the power 47.

Answer: Divide 47 by 4. The quotient is 11 and the remainder is 3, so i to the 47 equals i to the 44 times i cubed. The 44th power is 1 because 44 is a multiple of 4, and i cubed is minus i. The answer is minus i.

Hinglish: 47 ko 4 se divide karo: remainder 3. Isliye i^47 = i^44 x i^3 = 1 x (-i) = -i. Powers char ke cycle me chalti hain.

Q3. Express (1 + 2i)(3 - i) in the form a + ib.

Answer: Multiply term by term: 1 times 3 is 3, 1 times minus i is minus i, 2i times 3 is 6i, and 2i times minus i is minus 2 i squared. Since i squared is minus 1, the last term is plus 2. Adding the parts gives 5 + 5i.

Hinglish: Term by term multiply karo: 3 - i + 6i - 2i squared. i squared = -1 se last term +2 ban jaata hai. Total 5 + 5i.

Q4. Write the conjugate and the modulus of z = 3 - 4i.

Answer: The conjugate flips the sign of the imaginary part, so it is 3 + 4i. The modulus is the square root of 3 squared plus 4 squared, which is the square root of 25, that is 5. Checking, the product of z and its conjugate is 3 squared plus 4 squared equal to 25, which equals the modulus squared.

Hinglish: Conjugate me imaginary part ka sign badalta hai: 3 + 4i. Modulus = root of 9 + 16 = 5. z times conjugate = 25, jo modulus squared hai.

Q5. Express (4 + 3i) / (2 - i) in the form a + ib.

Answer: Multiply the numerator and denominator by the conjugate of the denominator, 2 + i. The denominator becomes 4 + 1 equal to 5. The numerator becomes 8 + 4i + 6i + 3 i squared, that is 5 + 10i after replacing i squared by minus 1. Dividing each part by 5 gives 1 + 2i.

Hinglish: Denominator ke conjugate 2 + i se upar neeche multiply karo. Denominator 5 ban jaata hai, numerator 5 + 10i. Divide karke 1 + 2i milta hai.

Q6. If (x + iy)(2 - 3i) = 4 - i, find x and y.

Answer: Expand the left side: 2x - 3xi + 2yi - 3i squared y, which becomes (2x + 3y) + i(2y - 3x). Compare with the right side 4 - i. Then 2x + 3y = 4 and 2y - 3x = -1. Solving the pair gives x = 11 by 13 and y = 10 by 13.

Hinglish: Left side expand karo: (2x + 3y) + i(2y - 3x). Right side se compare karo: 2x + 3y = 4 aur 2y - 3x = -1. Solve karne par x = 11/13, y = 10/13.

Q7. Find the multiplicative inverse of 4 - 3i.

Answer: The inverse is the conjugate divided by the modulus squared. The conjugate is 4 + 3i and the modulus squared is 16 + 9, that is 25. So the inverse is (4 + 3i) by 25, that is 4 by 25 plus 3 by 25 i. Multiplying by the original gives 1, which confirms the answer.

Hinglish: Inverse = conjugate / modulus squared. Conjugate 4 + 3i, modulus squared 25, to inverse (4 + 3i)/25. Original se multiply karke check karo, product 1 aata hai.

Q8. Solve x squared + 2x + 5 = 0 in the complex number system.

Answer: The discriminant is 4 minus 20, that is -16. The square root of -16 is 4i. Applying the quadratic formula gives x = (-2 plus or minus 4i) by 2, so the roots are -1 + 2i and -1 - 2i. They form a conjugate pair, and their sum is -2 while their product is 5, matching the coefficients.

Hinglish: Discriminant = 4 - 20 = -16, jiska root 4i hai. Quadratic formula se roots = (-2 plus/minus 4i)/2, yaani -1 + 2i aur -1 - 2i. Sum -2 aur product 5 check karo.

Q9. Find the modulus of (1 + i) / (1 - i).

Answer: Simplify the quotient first. Multiply top and bottom by 1 + i to get (2i) by 2, which is i. The modulus of i is 1. Alternatively, the modulus of the quotient equals the quotient of the moduli, and both numerator and denominator have modulus root 2, so the answer is 1 without simplifying.

Hinglish: Pehle simplify karo: (1 + i)/(1 - i) = i, jiska modulus 1 hai. Ya modulus property se: root 2 / root 2 = 1. Dono tarah answer 1.

Q10. Plot 2 + 3i and its conjugate on the Argand plane, and state their distance from the origin.

Answer: The point for 2 + 3i is (2, 3), two units right and three units up. The conjugate 2 - 3i is (2, -3), its mirror image in the real axis. Both points have modulus the square root of 4 plus 9, that is the square root of 13, so both lie on the circle of radius root 13 centred at the origin.

Hinglish: 2 + 3i ka point (2, 3) hai, aur conjugate 2 - 3i ka point (2, -3) real axis par mirror image. Dono ka modulus root 13 hai, isliye dono origin se same distance par hain.

Common mistakes on this chapter

The first slip is writing the imaginary part as i b instead of b. The second is forgetting that i squared is minus 1 while expanding a product, which flips the sign of the last term. The third is dividing by a complex number without using the conjugate, which leaves an imaginary denominator. The fourth is treating the rule for roots of products as valid when both factors are negative. A useful last check is to multiply a quotient answer by the denominator and confirm the numerator comes back. Check each final answer for these errors, and keep the answer in standard form a plus ib, because an unsimplified fraction or a stray i in the denominator is treated as incomplete.

Quiz

Quiz below: five multiple choice questions on powers of i, algebra, modulus, conjugate and complex roots. Attempt them after the notes and questions above, then move to the previous year page.