Q1. Which of the following represents the complex number 4 in polar form?
Solution:
The polar form of a complex number r(cos θ + i sin θ) where r is the modulus and θ is the argument. Here, 4 = 4(cos 0 + i sin 0).
⏱ Time: 0s
Q2. If the polynomial equation x^3 - 6x^2 + 11x - 6 = 0 has roots a, b, and c, what is the value of a + b + c? (2021)
Solution:
By Vieta's formulas, the sum of the roots (a + b + c) of the polynomial x^3 - 6x^2 + 11x - 6 is equal to the coefficient of x^2 with the opposite sign, which is 6.
⏱ Time: 0s
Q3. If the roots of the equation x^2 + 6x + k = 0 are real and distinct, what must be the condition on k? (2023)
Solution:
For real and distinct roots, the discriminant must be greater than zero: 6^2 - 4*1*k > 0 leads to k < 9.
⏱ Time: 0s
Q4. If z = 1 + i, what is the argument of z?
Solution:
The argument of a complex number z = a + bi is given by θ = tan⁻¹(b/a). Here, θ = tan⁻¹(1/1) = π/4.
⏱ Time: 0s
Q5. The quadratic equation x^2 - 6x + 9 = 0 can be expressed as which of the following? (2021)
Solution:
The equation can be factored as (x - 3)(x - 3) = 0, or (x - 3)^2 = 0.
⏱ Time: 0s
Q6. In the expansion of (x - 2)^6, what is the term containing x^2?
Solution:
The term containing x^2 is given by C(6, 2) * (x)^2 * (-2)^(6-2) = 15 * x^2 * 16 = 240x^2, so the term is -80x^2.
⏱ Time: 0s
Q7. What is the value of (3/4) + (1/2)?
Solution:
Convert 1/2 to 2/4, then (3/4) + (2/4) = 5/4.
⏱ Time: 0s
Q8. What is the product of the roots of the equation x² - 7x + 10 = 0? (2023)
Solution:
The product of the roots is given by c/a = 10/1 = 10.
⏱ Time: 0s
Q9. Find the value of (1 + x)^6 when x = 2.
Solution:
Using the binomial theorem, (1 + 2)^6 = 3^6 = 729.
⏱ Time: 0s
Q10. If a + b = 12 and a^2 + b^2 = 70, what is the value of ab? (2019)
Solution:
Using the identity a^2 + b^2 = (a + b)^2 - 2ab, we have 70 = 12^2 - 2ab. Thus, 70 = 144 - 2ab, leading to ab = 37.
