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
91
Solve the quadratic equation 2x^2 - 5x + 2 = 0.
Answer:
1/2 and 2
To factor, multiply a and c (2*2=4). We need numbers that multiply to 4 and add to -5, which are -4 and -1. Rewriting: 2x^2 - 4x - x + 2 = 0 -> 2x(x-2) - 1(x-2) = 0 -> (2x-1)(x-2) = 0. Thus, x = 1/2 and x = 2.
92
Find the roots of x^2 - x - 20 = 0.
Answer:
-4 and 5
The factors of -20 that add up to -1 are -5 and 4. The factored form is (x-5)(x+4) = 0. Setting each factor to zero gives the roots x=5 and x=-4.
93
Solve the equation x^2 - 7x + 12 = 0.
Answer:
3 and 4
Factoring the equation involves finding numbers that multiply to 12 and add to -7. These are -3 and -4. So, (x-3)(x-4) = 0, which means the roots are x=3 and x=4.
94
What are the roots of x^2 + 5x + 6 = 0?
Answer:
-2 and -3
We need factors of 6 that add up to 5. The factors are 2 and 3. The equation becomes (x+2)(x+3) = 0. Solving for x gives x = -2 and x = -3.
95
Find the roots of the quadratic equation x^2 - 5x + 6 = 0.
Answer:
2 and 3
Factoring the quadratic equation, we look for two numbers that multiply to 6 and add to -5. These numbers are -2 and -3. So, (x-2)(x-3) = 0, yielding roots x=2 and x=3.
96
What are the roots of the equation x^2 - 9 = 0?
Answer:
3 and -3
Adding 9 to both sides gives x^2 = 9. Taking the square root of both sides gives x = 3 and x = -3. Hence, the roots are +3 and -3.
97
What is the product of the roots of 5x^2 - 10x - 15 = 0?
Answer:
-3
The product of the roots is c/a. Here, a=5 and c=-15. The product is -15 / 5 = -3. You can also divide the entire equation by 5 first to get x^2 - 2x - 3 = 0, where the constant is directly -3.
98
What is the sum of the roots of the equation 3x^2 + 8x + 2 = 0?
Answer:
-8/3
The sum of the roots is calculated as -b/a. In this equation, a=3 and b=8. Therefore, the sum of the roots is -8/3.
99
If the roots of a quadratic equation are reciprocals of each other, then which condition holds?
Answer:
a = c
If the roots are alpha and 1/alpha, their product is 1. The product of roots formula is c/a. Setting c/a = 1 gives c = a. Thus, the leading coefficient and the constant term must be equal.
100
If the sum of the roots of ax^2 + bx + c = 0 is zero, what must be true?
Answer:
b = 0
The sum of the roots is -b/a. For the sum to be zero, the numerator must be zero. Therefore, b must equal 0. This occurs in equations like x^2 - 9 = 0, where roots are +3 and -3.