AP EAMCET (EAPCET) Mathematic Multiple Choice Questions (MCQs)

31. Which of the following represents a complex number?

• a) 3 + 4i
• b) 5
• c) -2
• d) 7 + 2i

Answer: a) 3 + 4i

32. The modulus of a complex number \( z = a + ib \) is:

• a) \( |z| = \sqrt{a^2 + b^2} \)
• b) \( |z| = \sqrt{a - b^2} \)
• c) \( |z| = \sqrt{a^2 - b^2} \)
• d) \( |z| = a + b \)

Answer: a) \( |z| = \sqrt{a^2 + b^2} \)

33. The amplitude (argument) of a complex number \( z = a + ib \) is:

• a) \( \tan^{-1} \left( \frac{b}{a} \right) \)
• b) \( \tan^{-1} \left( \frac{a}{b} \right) \)
• c) \( \cos^{-1} \left( \frac{b}{a} \right) \)
• d) \( \cos^{-1} \left( \frac{a}{b} \right) \)

Answer: a) \( \tan^{-1} \left( \frac{b}{a} \right) \)

34. The complex number \( z = 3 + 4i \) is represented in the Argand plane as:

• a) A point at (3, 4)
• b) A point at (4, 3)
• c) A point at (0, 4)
• d) A point at (3, 0)

Answer: a) A point at (3, 4)

35. Which of the following represents the polar form of the complex number \( z = 1 + i \)?

• a) \( 1 + i \)
• b) \( \sqrt{2} (\cos 45^\circ + i \sin 45^\circ) \)
• c) \( \sqrt{2} (\cos 30^\circ + i \sin 30^\circ) \)
• d) \( \sqrt{2} (\cos 60^\circ + i \sin 60^\circ) \)

Answer: b) \( \sqrt{2} (\cos 45^\circ + i \sin 45^\circ) \)

36. In the Argand plane, the distance of the point representing the complex number \( z = -4 + 3i \) from the origin is:

• a) 5
• b) 7
• c) 3
• d) 4

Answer: a) 5

37. If \( z = 2 + 3i \) and \( w = 1 - 2i \), then \( z + w \) is:

• a) 3 + i
• b) 3 - i
• c) 3 + 5i
• d) 1 + i

Answer: a) 3 + i

38. The geometric representation of a complex number \( z = a + ib \) on the Argand diagram corresponds to:

• a) A point at coordinates (a, b)
• b) A point at coordinates (b, a)
• c) A point on the real axis
• d) A point on the imaginary axis

Answer: a) A point at coordinates (a, b)

39. If \( z = 3 + 4i \), the amplitude (argument) of \( z \) is:

• a) \( \tan^{-1} \left( \frac{4}{3} \right) \)
• b) \( \tan^{-1} \left( \frac{3}{4} \right) \)
• c) \( \tan^{-1} \left( \frac{3}{5} \right) \)
• d) \( \tan^{-1} \left( \frac{4}{5} \right) \)

Answer: a) \( \tan^{-1} \left( \frac{4}{3} \right) \)

40. Which of the following is the correct representation of the complex number \( z = 5 \text{cis} 60^\circ \)?

• a) \( 5 (\cos 60^\circ + i \sin 60^\circ) \)
• b) \( 5 (\cos 30^\circ + i \sin 30^\circ) \)
• c) \( 5 + 60i \)
• d) \( 5 (\cos 90^\circ + i \sin 90^\circ) \)

Answer: a) \( 5 (\cos 60^\circ + i \sin 60^\circ) \)