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
21
Find the roots of x² -11x +30 = 0.
Answer:
6 and 5
Step-by-step solution: 1. Factor the quadratic using sum and product of roots. 2. Sum of roots = -b = 11. 3. Product of roots = c = 30. 4. Roots are 6 and 5.
22
Find the roots of x² -12x +35 = 0.
Answer:
7 and 5
Step-by-step solution: 1. Factor the quadratic using sum and product of roots. 2. Sum of roots = -b = 12. 3. Product of roots = c = 35. 4. Roots are 7 and 5.
23
Find the roots of x² -13x +42 = 0.
Answer:
7 and 6
Step-by-step solution: 1. Factor the quadratic using sum and product of roots. 2. Sum of roots = -b = 13. 3. Product of roots = c = 42. 4. Roots are 7 and 6.
24
Find the roots of x² -14x +48 = 0.
Answer:
8 and 6
Step-by-step solution: 1. Factor the quadratic using sum and product of roots. 2. Sum of roots = -b = 14. 3. Product of roots = c = 48. 4. Roots are 8 and 6.
25
Find the roots of x² -9x +18 = 0.
Answer:
3 and 6
Step-by-step solution: 1. Factor the quadratic using sum and product of roots. 2. Sum of roots = -b = 9. 3. Product of roots = c = 18. 4. Roots are 3 and 6.
26
Find the roots of x² -10x +24 = 0.
Answer:
6 and 4
Step-by-step solution: 1. Factor the quadratic using sum and product of roots. 2. Sum of roots = -b = 10. 3. Product of roots = c = 24. 4. Roots are 6 and 4.
27
For the quadratic equation x^2 + px + q = 0, if the non-zero roots are p and q, what are the values of p and q?
Answer:
p = 1, q = -2
The sum of the roots is -p, so p + q = -p, leading to q = -2p. The product of the roots is q, so p*q = q. Since the roots are non-zero, q ≠ 0, allowing us to divide by q to get p = 1. Substituting p = 1 into q = -2p gives q = -2. The values are p=1, q=-2.
28
Which of the following equations has roots -1 and -2?
Answer:
x^2 + 3x + 2 = 0
If the roots are -1 and -2, the sum is -3 and the product is 2. The quadratic equation is x^2 - (sum)x + (product) = 0, which translates to x^2 - (-3)x + 2 = 0, simplifying to x^2 + 3x + 2 = 0.
29
Solve the equation for x: √x = x - 2
Answer:
4
Squaring both sides gives x = (x - 2)^2. Expanding gives x = x^2 - 4x + 4. Rearranging yields x^2 - 5x + 4 = 0, which factors to (x-4)(x-1) = 0. Potential roots are 4 and 1. Checking the original equation: if x=1, √1 = 1-2, which means 1 = -1 (False). If x=4, √4 = 4-2, which means 2 = 2 (True). The only valid root is 4.
30
If a quadratic equation has no linear term (b = 0), what can be said about its real roots (if they exist)?
Answer:
They are equal in magnitude and opposite in sign
When b=0, the equation is ax^2 + c = 0, so x^2 = -c/a. If real roots exist, taking the square root gives x = ±√(-c/a). These two values have the same absolute value (magnitude) but opposite signs.