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
71
The standard formula for the area of a triangle given its three vertex coordinates uses concepts from which mathematical structure?
Answer:
Determinant of a matrix
Step 1: The formula for the area of a triangle with vertices (x1,y1), (x2,y2), (x3,y3) is 0.5 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|. Step 2: This exact algebraic structure is derived from evaluating a 3x3 determinant composed of the points' coordinates and a column of ones. Step 3: Thus, it uses the determinant of a matrix.
72
Find the area of the triangle with vertices (-3, 0), (3, 0), and (0, 4).
Answer:
12
Step 1: The base of the triangle lies on the x-axis from x = -3 to x = 3. Length of base = 3 - (-3) = 6. Step 2: The third vertex is at (0,4), so the height (perpendicular distance to x-axis) is 4. Step 3: Area = (1/2) * base * height = (1/2) * 6 * 4 = 12.
73
If the calculated area of a triangle using coordinate geometry formulas is exactly zero, what can be concluded about the three points?
Answer:
They are collinear
Step 1: The area of a triangle represents the space enclosed by three non-collinear points. Step 2: If the area is exactly zero, it means no space is enclosed. Step 3: This implies that the three points lie on a single straight line, meaning they are collinear.
74
What is the area of a triangle with vertices (1, 1), (1, 4), and (5, 1)?
Answer:
6
Step 1: This is a right triangle since two points share the same x-coordinate (1) forming a vertical segment, and two share the same y-coordinate (1) forming a horizontal segment. Step 2: Height = |4 - 1| = 3. Base = |5 - 1| = 4. Step 3: Area = (1/2) * 4 * 3 = 6.
75
Calculate the area of the triangle whose vertices are (0, 0), (5, 0), and (0, 12).
Answer:
30
Step 1: The vertices form a right-angled triangle positioned at the origin. Step 2: The base lies along the x-axis with length 5, and the height lies along the y-axis with length 12. Step 3: Area = (1/2) * base * height = (1/2) * 5 * 12 = 30.
76
What is the x-intercept of a line that is parallel to the x-axis?
Answer:
No x-intercept
Step 1: A line parallel to the x-axis is perfectly horizontal. Step 2: Unless it is the x-axis itself, it will never cross or intersect the x-axis. Step 3: Thus, a parallel line has no x-intercept.
77
What is the length of the portion of the line 3x + 4y = 12 intercepted between the coordinate axes?
Answer:
5
Step 1: Find the intercepts. x-intercept (y=0) is 3x=12 => x=4. y-intercept (x=0) is 4y=12 => y=3. The points are (4,0) and (0,3). Step 2: Use the distance formula to find the length between these two points. Step 3: Length = √[(4-0)² + (0-3)²] = √[16 + 9] = √25 = 5.
78
What is the area of the triangle formed by the line x/3 + y/4 = 1 and the coordinate axes?
Answer:
6
Step 1: The equation x/3 + y/4 = 1 shows the x-intercept is 3 and the y-intercept is 4. Step 2: This forms a right-angled triangle with the axes, with base = 3 and height = 4. Step 3: Area = (1/2) * base * height = (1/2) * 3 * 4 = 6.
79
What is the x-intercept of the line y = 2x - 4?
Answer:
2
Step 1: To find the x-intercept, set the y-value to 0 in the given equation. Step 2: 0 = 2x - 4. Step 3: Solve for x: 2x = 4, which gives x = 2.
80
What is the y-intercept of the line y = 3x - 7?
Answer:
-7
Step 1: The equation is in the slope-intercept form y = mx + c. Step 2: In this format, 'c' directly represents the y-intercept. Step 3: Comparing y = 3x - 7 to the standard form, we see that c = -7.