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
31
If the equation 3x^2 - 12x + m = 0 has real and equal roots, what is the value of m?
Answer:
12
For real and equal roots, the discriminant D = 0. D = b^2 - 4ac = (-12)^2 - 4(3)(m) = 144 - 12m. Setting this to 0 yields 144 = 12m, so m = 12.
32
What is the sum of the roots of the equation 4x^2 = 12x?
Answer:
3
First, write the equation in standard form: 4x^2 - 12x = 0. The sum of the roots is -b/a. Here, a=4 and b=-12. The sum is -(-12)/4 = 12/4 = 3. (The roots are 0 and 3, which sum to 3).
33
Find the value of c for which x^2 - 10x + c is a perfect square trinomial.
Answer:
25
To complete the square for x^2 + bx + c, the constant term c must equal (b/2)^2. Here, b = -10. So c = (-10/2)^2 = (-5)^2 = 25. The expression becomes (x - 5)^2.
34
What are the roots of the equation x^2 = x?
Answer:
0 and 1
To solve, bring all terms to one side: x^2 - x = 0. Factor out an x to get x(x - 1) = 0. This means x = 0 or x - 1 = 0, giving the roots 0 and 1.
35
If the product of the roots of x^2 - 3x + k = 0 is 10, what is the value of k?
Answer:
10
The product of the roots in the equation ax^2 + bx + c = 0 is c/a. In this equation, a=1 and c=k. Therefore, the product of the roots is k/1 = k. We are given the product is 10, so k = 10.
36
If x^2 - 7x + a = 0 has 3 as one of its roots, what is the other root?
Answer:
4
The sum of the roots is given by -b/a. Here, the sum of the roots must be -(-7)/1 = 7. If one root is 3, let the other be y. Then 3 + y = 7, which gives the other root y = 4.
37
Find the x-coordinate of the vertex of the parabola y = x^2 + 8x + 15.
Answer:
-4
The x-coordinate of the vertex is given by x = -b / (2a). For the given equation, a = 1 and b = 8. Plugging these into the formula gives x = -8 / (2*1) = -4.
38
If a parabola intersects the x-axis at two distinct points, what must be true about its discriminant?
Answer:
It is positive
Intersecting the x-axis at two different points means the quadratic equation has two distinct real roots. The mathematical condition for this is a positive discriminant (D > 0).
39
If the graph of a quadratic equation does not intersect the x-axis, what can be said about its roots?
Answer:
They are imaginary
The x-intercepts of a quadratic graph represent its real roots. If the graph never intersects the x-axis, it has no real roots, which implies both roots are complex or imaginary.
40
If a parabola touches the x-axis at exactly one point, what is true about its discriminant?
Answer:
It is zero
When a parabola touches the x-axis at exactly one point, it means the related quadratic equation has exactly one repeated real root. This only happens when the discriminant (b^2 - 4ac) is exactly zero.