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
91
If the angle of elevation of the sun is 45°, what is the length of the shadow of a 10m tall tree?
Answer:
10 m
Let the shadow length be x. Using the tangent function: tan(45°) = height / shadow = 10 / x. Since tan(45°) = 1, we have 1 = 10 / x, which means x = 10 meters.
92
Simplify the expression: √(1 - cos²θ), where θ is an acute angle.
Answer:
sin(θ)
From the Pythagorean identity sin²θ + cos²θ = 1, we can derive sin²θ = 1 - cos²θ. Taking the square root of both sides gives √(1 - cos²θ) = |sin(θ)|. Since θ is an acute angle, sin(θ) is positive, so the result is simply sin(θ).
93
Which of the following equals (sec²θ - 1)?
Answer:
tan²θ
Using the standard Pythagorean identity 1 + tan²θ = sec²θ, we can isolate the tangent term by subtracting 1 from both sides. This gives tan²θ = sec²θ - 1.
94
Find the value of arcsin(1/2) in degrees.
Answer:
30°
The expression arcsin(1/2) asks for the angle within the principal range [-90°, 90°] whose sine is exactly 1/2. We know from standard triangles that sin(30°) = 1/2, so the answer is 30°.
95
What is the principal value range for the inverse sine function, arcsin(x)?
Answer:
[-π/2, π/2]
To make the sine function invertible, its domain must be restricted to a range where it passes the horizontal line test. By standard convention, the principal value range of arcsin(x) is restricted to [-π/2, π/2].
96
The angle of elevation of a ladder leaning against a wall is 60° and the foot of the ladder is 4.6 m away from the wall. The length of the ladder is:
Answer:
9.2 m
Let the length of the ladder be L. The base of the triangle is 4.6 m. Using cosine: cos(60°) = adjacent / hypotenuse = 4.6 / L. Since cos(60°) = 1/2, we have 1/2 = 4.6 / L, yielding L = 9.2 m.
97
Which of these is the correct expansion of sin(A + B)?
Answer:
sin(A)cos(B) + cos(A)sin(B)
The compound angle formula for the sine of a sum dictates that sin(A + B) distributes as the product of the sine and cosine of the alternating angles, added together: sin(A)cos(B) + cos(A)sin(B).
98
If sin(A) = cos(A), then what is the value of 2*tan(A) + cos²(A)?
Answer:
2.5
Since sin(A) = cos(A), dividing by cos(A) gives tan(A) = 1, meaning A = 45°. Evaluating the expression: 2*tan(45°) + cos²(45°) = 2*(1) + (1/√2)² = 2 + 1/2 = 2.5.
99
What is the value of tan²(60°) + sin²(45°)?
Answer:
3.5
We know tan(60°) = √3, so tan²(60°) = 3. We also know sin(45°) = 1/√2, so sin²(45°) = 1/2. Adding them together gives 3 + 1/2 = 3.5.
100
Simplify: sin(A) * csc(A) + cos(A) * sec(A)
Answer:
2
The cosecant function is the reciprocal of sine, so sin(A) * csc(A) = 1. Similarly, secant is the reciprocal of cosine, so cos(A) * sec(A) = 1. Adding these together yields 1 + 1 = 2.