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
51
What is the condition for a quadratic equation to have real roots?
Answer:
Discriminant >= 0
Real roots can be either distinct (if D > 0) or equal (if D = 0). Therefore, the overall condition for having real roots is that the discriminant must be greater than or equal to 0 (D >= 0).
52
If the roots of ax^2 + bx + c = 0 are equal in magnitude but opposite in sign, then what is true?
Answer:
b = 0
Let the roots be r and -r. Their sum is r + (-r) = 0. We also know the sum of the roots is -b/a. Setting -b/a = 0 implies that the coefficient b must be 0.
53
What is the value of the discriminant for a perfect square trinomial?
Answer:
Equal to 0
A perfect square trinomial can be written in the form (px + q)^2 = 0. This means it has two identical real roots. For roots to be real and equal, the discriminant must be exactly 0.
54
If the roots of x^2 - kx + 8 = 0 are in the ratio 1:2, find the positive value of k.
Answer:
6
Let the roots be r and 2r. Their product is 2r^2 = 8, which means r^2 = 4, so r = 2 (for positive k). The roots are 2 and 4. Their sum is 6, which equals k. So, k = 6.
55
If one root of ax^2 + bx + c = 0 is double the other, what is the relation between a, b, and c?
Answer:
2b^2 = 9ac
Let the roots be r and 2r. Sum = 3r = -b/a, so r = -b/(3a). Product = 2r^2 = c/a. Substituting r: 2(-b/(3a))^2 = c/a. This becomes 2b^2 / 9a^2 = c/a, which simplifies to 2b^2 = 9ac.
56
If x = 1 is a root of x^2 + kx + 3 = 0, what is the value of k?
Answer:
-4
If x = 1 is a root, it must satisfy the equation. Plugging in x = 1 gives (1)^2 + k(1) + 3 = 0. This simplifies to 1 + k + 3 = 0, which means k + 4 = 0, and therefore k = -4.
57
The altitude of a right triangle is 2 cm less than its base. If its area is 12 cm^2, what is the length of the base?
Answer:
6 cm
Let base be b. Altitude is b - 2. Area = (1/2)*b*(b-2) = 12. Multiplying by 2 gives b(b-2) = 24. This simplifies to b^2 - 2b - 24 = 0. Factoring yields (b-6)(b+4) = 0. The positive length for the base is 6 cm.
58
The perimeter of a rectangle is 20 and its area is 24. Find its length and width.
Answer:
6 and 4
Perimeter = 2(L + W) = 20, so L + W = 10. Area = L*W = 24. Substituting W = 10 - L into the area equation gives L(10 - L) = 24, or L^2 - 10L + 24 = 0. The roots are 6 and 4. The dimensions are 6 and 4.
59
The difference between two numbers is 3, and their product is 40. Find the larger positive number.
Answer:
8
Let the numbers be x and x-3. Their product is x(x-3) = 40, leading to x^2 - 3x - 40 = 0. Factoring gives (x-8)(x+5) = 0. The larger positive number is x = 8.
60
The sum of a number and its reciprocal is 5/2. What is the number?
Answer:
2 or 1/2
Let the number be x. The equation is x + 1/x = 5/2. Multiplying by 2x gives 2x^2 + 2 = 5x. Rearranging yields 2x^2 - 5x + 2 = 0. Factoring gives (2x-1)(x-2) = 0. The roots are 2 and 1/2.