Mathematics/General Ability MCQs
Topic Notes: Mathematics/General Ability
MCQs and preparation resources for competitive exams, covering important concepts, past papers, and detailed explanations.
Plato
- Biography: Ancient Greek philosopher (427–347 BCE), student of Socrates and teacher of Aristotle, founder of the Academy in Athens.
- Important Ideas:
- Theory of Forms
- Philosopher-King
- Ideal State
101
If the product of the roots of ax^2 + bx + c = 0 is zero, what must be true?
Answer:
c = 0
The product of the roots is given by c/a. For this fraction to be zero, the numerator c must be exactly zero (assuming a is non-zero, which it must be for a quadratic equation).
102
If one root of the equation x^2 - 5x + k = 0 is 2, what is the value of k?
Answer:
6
If 2 is a root, it must satisfy the equation. Substitute x = 2: (2)^2 - 5(2) + k = 0. This gives 4 - 10 + k = 0, which simplifies to -6 + k = 0, yielding k = 6.
103
Find the product of the roots of the equation 2x^2 - 7x + 3 = 0.
Answer:
3/2
The formula for the product of roots is c/a. In the equation 2x^2 - 7x + 3 = 0, c = 3 and a = 2. Therefore, the product of the roots is 3/2.
104
Find the sum of the roots of the equation x^2 - 5x + 6 = 0.
Answer:
5
Using the formula for the sum of the roots (-b/a), where a=1 and b=-5, the sum is -(-5)/1 = 5. (The actual roots are 2 and 3, which indeed sum to 5).
105
What is the product of the roots of the quadratic equation ax^2 + bx + c = 0?
Answer:
c/a
By Vieta's formulas, the product of the roots of a quadratic equation ax^2 + bx + c = 0 is defined as the constant term divided by the leading coefficient, which is c/a.
106
What is the sum of the roots of the quadratic equation ax^2 + bx + c = 0?
Answer:
-b/a
By Vieta's formulas, the sum of the roots of a quadratic equation ax^2 + bx + c = 0 is always given by the ratio -b/a.
107
For what value of p does the equation 4x^2 - px + 9 = 0 have equal roots?
Answer:
12 or -12
Equal roots require the discriminant D = 0. Here, D = (-p)^2 - 4(4)(9) = p^2 - 144. Setting this to 0 yields p^2 = 144, which implies p can be either 12 or -12.
108
Find the value of k if the equation x^2 + kx + 16 = 0 has real and equal roots.
Answer:
8 or -8
For real and equal roots, the discriminant must be zero. So, k^2 - 4(1)(16) = 0. This means k^2 - 64 = 0, leading to k^2 = 64. Taking the square root gives k = 8 or k = -8.
109
What is the nature of the roots for the equation x^2 + x + 1 = 0?
Answer:
Imaginary
For x^2 + x + 1 = 0, a=1, b=1, c=1. The discriminant D = (1)^2 - 4(1)(1) = 1 - 4 = -3. Because D < 0, the roots are imaginary (complex conjugates).
110
If the discriminant is less than zero, what kind of roots does the quadratic equation have?
Answer:
No real roots (imaginary)
When the discriminant (b^2 - 4ac) is negative, the square root part of the quadratic formula yields an imaginary number. Hence, the equation has no real roots, only complex or imaginary ones.