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
Evaluate: sin(45°) * csc(45°)
Answer:
1
Because the cosecant function is defined strictly as the reciprocal of the sine function (csc = 1/sin), multiplying the sine of any valid angle by its cosecant always results in exactly 1.
32
Find the value of arctan(1) in radians.
Answer:
π/4
The arctangent function gives the principal angle whose tangent is 1. Since tan(45°) = 1, the angle in degrees is 45°. Converting 45 degrees to radians gives 45 * (π/180) = π/4.
33
Simplify: sec²(x) - tan²(x) + cos²(x) + sin²(x)
Answer:
2
We can break this into two known Pythagorean identities. First, sec²(x) - tan²(x) always equals 1. Second, cos²(x) + sin²(x) always equals 1. Adding these together yields exactly 1 + 1 = 2.
34
If sin(θ) = 0.8 and θ is acute, find tan(θ).
Answer:
1.33
Given sin(θ) = 0.8 = 4/5. Using a right triangle, opposite=4, hypotenuse=5. By Pythagoras, adjacent = √(5² - 4²) = 3. Tangent is opposite/adjacent = 4/3. Converting 4/3 to a decimal yields approximately 1.33.
35
Which trigonometric identity represents cos(A - B)?
Answer:
cos(A)cos(B) + sin(A)sin(B)
The compound angle formula for the cosine of a difference states that cos(A - B) evaluates to the product of cosines plus the product of sines: cos(A)cos(B) + sin(A)sin(B).
36
If cot(θ) = 1, what is the value of θ (acute)?
Answer:
45°
Cotangent is the reciprocal of tangent. If cot(θ) = 1, then tan(θ) must also equal 1. In a right triangle, tangent is 1 when the opposite and adjacent sides are equal, which occurs precisely at 45°.
37
What is the exact value of tan(60°)?
Answer:
√3
In a 30-60-90 triangle with sides 1, √3, and 2, the angle 60° sits opposite the side measuring √3 and adjacent to the side measuring 1. Thus, tan(60°) = opposite/adjacent = √3 / 1 = √3.
38
Which trigonometric function has no undefined points over the real numbers?
Answer:
sin(x)
The sine and cosine functions are defined for all real numbers and represent continuous waves without any vertical asymptotes. Tangent, secant, and cosecant involve division by zero at certain angles, making them undefined there.
39
What is the relation between degrees and radians?
Answer:
1 degree = π/180 radians
A full circle encompasses 360 degrees, which is equivalent to 2π radians. Therefore, 360° = 2π rad. Dividing both sides by 360 gives exactly 1 degree = 2π/360 = π/180 radians.
40
Calculate the area of a triangle having sides a=5, b=6, and the included angle C=30°.
Answer:
7.5
The trigonometric formula for the area of a triangle given two sides and the included angle is Area = (1/2)*a*b*sin(C). Substituting the values: Area = (1/2)*5*6*sin(30°) = 15 * (1/2) = 7.5 square units.