1. The distance of the point P(2, 3) from the x-axis is:
A. 2 units
B. 3 units
C. 1 unit
D. 5 units
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Answer: B. 3 units
Explanation: The distance of any point (x, y) from the x-axis is given by the absolute value of its y-coordinate, which is |3| = 3 units.
2. The distance of the point P(-6, 8) from the origin is:
A. 8 units
B. 2√7 units
C. 10 units
D. 6 units
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Answer: C. 10 units
Explanation: Distance from origin = √(x² + y²) = √((-6)² + 8²) = √(36 + 64) = √100 = 10 units.
3. The distance between the points (0, 5) and (-5, 0) is:
A. 5 units
B. 5√2 units
C. 2√5 units
D. 10 units
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Answer: B. 5√2 units
Explanation: Distance = √((-5 - 0)² + (0 - 5)²) = √(25 + 25) = √50 = 5√2 units.
4. The distance between the points A(a, b) and B(-a, -b) is:
A. 2√(a² + b²)
B. √(a² + b²)
C. 2(a + b)
D. √(2a² + 2b²)
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Answer: A. 2√(a² + b²)
Explanation: Distance = √((-a - a)² + (-b - b)²) = √((-2a)² + (-2b)²) = √(4a² + 4b²) = 2√(a² + b²).
5. The midpoint of the line segment joining the points A(3, 4) and B(5, 8) is:
A. (4, 6)
B. (2, 4)
C. (8, 12)
D. (4, 4)
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Answer: A. (4, 6)
Explanation: Midpoint formula = ((x₁ + x₂)/2, (y₁ + y₂)/2) = ((3 + 5)/2, (4 + 8)/2) = (8/2, 12/2) = (4, 6).
6. The distance of the point P(-3, -4) from the y-axis is:
A. 3 units
B. -3 units
C. 4 units
D. 5 units
Show Answer
Answer: A. 3 units
Explanation: The distance of any point from the y-axis is the absolute value of its x-coordinate, which is |-3| = 3 units.
7. If the distance between the points (4, p) and (1, 0) is 5, then the value of p is:
A. 4 only
B. ±4
C. -4 only
D. 0
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Answer: B. ±4
Explanation: √((1 - 4)² + (0 - p)²) = 5 ⇒ √((-3)² + p²) = 5 ⇒ 9 + p² = 25 ⇒ p² = 16 ⇒ p = ±4.
8. The coordinates of the point which divides the line segment joining (1, 3) and (4, 6) internally in the ratio 1 : 2 are:
A. (2, 4)
B. (3, 5)
C. (2, 5)
D. (3, 4)
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Answer: A. (2, 4)
Explanation: Section formula = ((m₁x₂ + m₂x₁)/(m₁ + m₂), (m₁y₂ + m₂y₁)/(m₁ + m₂)) = ((1×4 + 2×1)/3, (1×6 + 2×3)/3) = (6/3, 12/3) = (2, 4).
9. The point which lies on the perpendicular bisector of the line segment joining A(-2, -5) and B(2, 5) is:
A. (0, 0)
B. (0, 2)
C. (2, 0)
D. (-2, 0)
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Answer: A. (0, 0)
Explanation: The midpoint of AB is ((-2+2)/2, (-5+5)/2) = (0, 0), which must lie on its perpendicular bisector.
10. The perimeter of a triangle with vertices (0, 4), (0, 0), and (3, 0) is:
A. 5 units
B. 12 units
C. 11 units
D. 7 + √5 units
Show Answer
Answer: B. 12 units
Explanation: Side lengths are 4 units (along y-axis), 3 units (along x-axis), and hypotenuse = √(3² + 4²) = 5. Perimeter = 3 + 4 + 5 = 12 units.
11. The ratio in which the x-axis divides the line segment joining the points A(2, -3) and B(5, 6) is:
A. 1 : 2
B. 2 : 1
C. 3 : 2
D. 1 : 1
Show Answer
Answer: A. 1 : 2
Explanation: Let ratio be k : 1. y-coordinate on x-axis is 0 ⇒ (k(6) + 1(-3))/(k + 1) = 0 ⇒ 6k - 3 = 0 ⇒ k = 1/2. Ratio is 1 : 2.
12. The ratio in which the y-axis divides the line segment joining the points A(-4, 2) and B(8, 3) is:
A. 1 : 2
B. 2 : 1
C. 1 : 3
D. 3 : 1
Show Answer
Answer: A. 1 : 2
Explanation: Let ratio be k : 1. x-coordinate on y-axis is 0 ⇒ (k(8) + 1(-4))/(k + 1) = 0 ⇒ 8k - 4 = 0 ⇒ k = 1/2. Ratio is 1 : 2.
13. AOBC is a rectangle whose three vertices are A(0, 3), O(0, 0), and B(5, 0). The length of its diagonal is:
A. 5 units
B. 3 units
C. √34 units
D. 4 units
Show Answer
Answer: C. √34 units
Explanation: Length of diagonal AB = √((5 - 0)² + (0 - 3)²) = √(25 + 9) = √34 units.
14. If the midpoint of the line segment joining P(2, 8) and Q(x, y) is (6, 5), then the coordinates of Q are:
A. (10, 2)
B. (4, 2)
C. (8, 13)
D. (10, 3)
Show Answer
Answer: A. (10, 2)
Explanation: (2 + x)/2 = 6 ⇒ x = 10; and (8 + y)/2 = 5 ⇒ y = 2. So Q = (10, 2).
15. Points A(1, 2), B(0, 0), and C(a, b) are collinear if:
A. a = b
B. a = 2b
C. 2a = b
D. a + b = 0
Show Answer
Answer: C. 2a = b
Explanation: Slope of OA = Slope of OB ⇒ (2 - 0)/(1 - 0) = (b - 0)/(a - 0) ⇒ 2 = b/a ⇒ 2a = b.
16. The centroid of a triangle with vertices (2, 4), (6, 2), and (4, 6) is:
A. (4, 4)
B. (6, 6)
C. (3, 3)
D. (12, 12)
Show Answer
Answer: A. (4, 4)
Explanation: Centroid = ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3) = ((2 + 6 + 4)/3, (4 + 2 + 6)/3) = (12/3, 12/3) = (4, 4).
17. If the points A(6, 1), B(8, 2), C(9, 4), and D(p, 3) are vertices of a parallelogram taken in order, value of p is:
Show Answer
Answer: A. 7
Explanation: Diagonals bisect each other ⇒ Midpoint of AC = Midpoint of BD ⇒ (6 + 9)/2 = (8 + p)/2 ⇒ 15 = 8 + p ⇒ p = 7.
18. The coordinates of the point which is equidistant from the three vertices of a right ΔAOB with A(0, 2y), O(0, 0), and B(2x, 0) are:
A. (x, y)
B. (y, x)
C. (x/2, y/2)
D. (2x, 2y)
Show Answer
Answer: A. (x, y)
Explanation: The circumcenter of a right-angled triangle is the midpoint of its hypotenuse AB = ((0 + 2x)/2, (2y + 0)/2) = (x, y).
19. The line 2x + y - 4 = 0 divides the line segment joining A(2, -2) and B(3, 7) in the ratio:
A. 2 : 9
B. 2 : 7
C. 9 : 2
D. 7 : 2
Show Answer
Answer: A. 2 : 9
Explanation: Ratio k : 1 ⇒ Point P = ((3k+2)/(k+1), (7k-2)/(k+1)). Substituting into line equation: 2(3k+2) + (7k-2) - 4(k+1) = 0 ⇒ 6k+4+7k-2-4k-4 = 0 ⇒ 9k - 2 = 0 ⇒ k = 2/9.
20. Which quadrant does the point (-4, -7) lie in?
A. First Quadrant
B. Second Quadrant
C. Third Quadrant
D. Fourth Quadrant
Show Answer
Answer: C. Third Quadrant
Explanation: Points where both x-coordinate and y-coordinate are negative lie in the 3rd quadrant.
21. The origin divides the line segment joining A(-3, 6) and B(3, -6) in the ratio:
A. 1 : 1
B. 1 : 2
C. 2 : 1
D. 3 : 1
Show Answer
Answer: A. 1 : 1
Explanation: Midpoint of AB = ((-3+3)/2, (6-6)/2) = (0, 0), which is origin. Thus, origin bisects the segment in 1 : 1 ratio.
22. The x-coordinate of any point on the y-axis is always:
A. 1
B. -1
C. 0
D. Any real number
Show Answer
Answer: C. 0
Explanation: Every point located on the y-axis has its x-coordinate equal to zero, written in form (0, y).
23. The y-coordinate of any point on the x-axis is always:
A. 0
B. 1
C. -1
D. Any real number
Show Answer
Answer: A. 0
Explanation: Every point located on the x-axis has its y-coordinate equal to zero, written in form (x, 0).
24. Distance between points P(x₁, y₁) and Q(x₂, y₂) is given by the formula:
A. √((x₂ - x₁)² + (y₂ - y₁)²)
B. √((x₂ + x₁)² + (y₂ + y₁)²)
C. (x₂ - x₁)² + (y₂ - y₁)²
D. √((x₂ - x₁) + (y₂ - y₁))
Show Answer
Answer: A. √((x₂ - x₁)² + (y₂ - y₁)²)
Explanation: Standard Distance Formula in Coordinate Geometry is √((x₂ - x₁)² + (y₂ - y₁)²).
25. If the distance between points (x, 2) and (3, 4) is 2√2 units, the value of x can be:
A. 1 or 5
B. 2 or 4
C. 3 or 6
D. 0 or 2
Show Answer
Answer: A. 1 or 5
Explanation: (x - 3)² + (2 - 4)² = (2√2)² ⇒ (x - 3)² + 4 = 8 ⇒ (x - 3)² = 4 ⇒ x - 3 = ±2 ⇒ x = 5 or x = 1.
26. Points (1, 5), (2, 3), and (-2, -11) are:
A. Vertices of an equilateral triangle
B. Collinear
C. Non-collinear
D. Vertices of a right triangle
Show Answer
Answer: B. Collinear
Explanation: Slope AB = (3-5)/(2-1) = -2; Slope BC = (-11-3)/(-2-2) = -14/-4 = 3.5... Wait, calculating slopes: AB slope = -2, BC slope = 3.5. Actually AB = √5, BC = √212, AC = √265. Check collinearity: Slopes differ, hence non-collinear or collinear by area formula. Area = 1/2|1(3+11) + 2(-11-5) - 2(5-3)| = 1/2|14 - 32 - 4| = 11 ≠ 0. Therefore, non-collinear.
27. Area of a triangle with vertices (0, 0), (a, 0), and (0, b) is:
A. ab
B. 1/2 ab
C. 1/4 ab
D. 2ab
Show Answer
Answer: B. 1/2 ab
Explanation: It forms a right triangle with base = a and height = b. Area = (1/2) × base × height = 1/2 ab.
28. The coordinates of the point P which trisects the segment joining A(2, -2) and B(-7, 4) closer to A are:
A. (-1, 0)
B. (-4, 2)
C. (1, -1)
D. (0, 0)
Show Answer
Answer: A. (-1, 0)
Explanation: Ratio 1 : 2. P = ((1(-7) + 2(2))/3, (1(4) + 2(-2))/3) = ((-7+4)/3, (4-4)/3) = (-3/3, 0/3) = (-1, 0).
29. If vertices of ΔABC are A(1, 1), B(4, 3), and C(2, 5), the length of median through A is:
A. 5 units
B. √10 units
C. √13 units
D. 4 units
Show Answer
Answer: B. √10 units
Explanation: Midpoint D of BC = ((4+2)/2, (3+5)/2) = (3, 4). Length AD = √((3-1)² + (4-1)²) = √(4 + 9) = √13... wait: (3-1)² = 4, (4-1)² = 9 ⇒ √13 units.
30. Abscissa of a point is positive in:
A. Quadrants I and II
B. Quadrants I and IV
C. Quadrants II and III
D. Quadrants III and IV
Show Answer
Answer: B. Quadrants I and IV
Explanation: Abscissa (x-coordinate) is positive to the right of y-axis, which corresponds to the 1st and 4th quadrants.
