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
131
If an observer looks up at the top of a tower, the angle formed by the line of sight and the horizontal is called the:
Answer:
Angle of elevation
When an observer looks at an object that is higher than their horizontal eye level, they must raise their eyes. The angle formed between the horizontal line and the upward line of sight is defined as the angle of elevation.
132
What is the value of tan(90°)?
Answer:
Undefined
By definition, tan(θ) = sin(θ) / cos(θ). At 90 degrees, sin(90°) = 1 and cos(90°) = 0. Therefore, tan(90°) = 1 / 0, which involves division by zero and is mathematically undefined.
133
Evaluate: sin(90° - θ)
Answer:
cos(θ)
In a right triangle, the two acute angles are complementary (sum to 90°). The sine of one acute angle (opposite/hypotenuse) is exactly the cosine (adjacent/hypotenuse) of the other acute angle. Thus, sin(90° - θ) = cos(θ).
134
If tan(θ) = 1, what is the principal value of θ in degrees?
Answer:
45°
The tangent of an angle is 1 when the opposite side equals the adjacent side in a right-angled triangle. This forms an isosceles right triangle, meaning the two acute angles must both be exactly 45°.
135
Which of the following identities is correct?
Answer:
1 + tan²(θ) = sec²(θ)
Starting with the Pythagorean identity sin²(θ) + cos²(θ) = 1, if we divide the entire equation by cos²(θ), we get (sin²(θ)/cos²(θ)) + 1 = 1/cos²(θ). This simplifies to tan²(θ) + 1 = sec²(θ).
136
What is the exact value of cos(45°)?
Answer:
1/√2
In an isosceles right triangle with angles 45-45-90, the two legs are equal. If the legs are 1, the hypotenuse is √2 by the Pythagorean theorem. The cosine is adjacent/hypotenuse, which is 1/√2 (or √2/2 when rationalized).
137
If sin(θ) = 3/5, what is the value of csc(θ)?
Answer:
5/3
The cosecant function, csc(θ), is the reciprocal of the sine function. Therefore, csc(θ) = 1 / sin(θ). If sin(θ) = 3/5, then calculating the reciprocal gives csc(θ) = 5/3.
138
What is the value of sin²(θ) + cos²(θ)?
Answer:
1
This is the fundamental Pythagorean identity in trigonometry. For any angle θ, the sum of the square of its sine and the square of its cosine always equals 1, derived directly from the Pythagorean theorem (a² + b² = c²) divided by c².
139
Which of the following is equivalent to tan(θ)?
Answer:
sin(θ)/cos(θ)
The tangent of an angle is defined as the ratio of the opposite side to the adjacent side. Since sine is opposite/hypotenuse and cosine is adjacent/hypotenuse, dividing sine by cosine yields (opposite/hypotenuse) / (adjacent/hypotenuse) = opposite/adjacent = tan(θ).
140
What is the value of sin(30°)?
Answer:
1/2
The sine of an angle in a right triangle is the ratio of the opposite side to the hypotenuse. For a 30-degree angle in a standard 30-60-90 triangle, the opposite side is exactly half the length of the hypotenuse. Thus, sin(30°) = 1/2.