Mathematics (School)

Algebra (Secondary) Arithmetic (Primary) Geometry Statistics & Probability Trigonometry

Q. If point A is at (3, 5) and point B is at (9, 1), what is the coordinates of point P that divides AB in the ratio 2:3?

A. (5.4, 3.6)
B. (6, 4)
C. (5, 3)
D. (4, 2)

Solution

Using the section formula: P = ((2*9 + 3*3)/(2+3), (2*1 + 3*5)/(2+3)) = (5.4, 3.6).

Correct Answer: A — (5.4, 3.6)

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Q. If point A(2, 3) and point B(8, 7) are endpoints of a line segment, what is the midpoint M?

A. (5, 5)
B. (4, 5)
C. (6, 5)
D. (5, 4)

Solution

Midpoint M = ((x1 + x2)/2, (y1 + y2)/2) = ((2 + 8)/2, (3 + 7)/2) = (5, 5).

Correct Answer: A — (5, 5)

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Q. If point A(2, 3) and point B(8, 7) are endpoints of a line segment, what is the midpoint M of AB?

A. (5, 5)
B. (4, 5)
C. (6, 5)
D. (5, 4)

Solution

Midpoint M = ((x1 + x2)/2, (y1 + y2)/2) = ((2 + 8)/2, (3 + 7)/2) = (5, 5).

Correct Answer: A — (5, 5)

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Q. If point A(2, 3) and point B(8, 7) are given, what is the midpoint M of segment AB?

A. (5, 5)
B. (4, 5)
C. (6, 5)
D. (5, 4)

Solution

Using the midpoint formula: M = ((x1 + x2)/2, (y1 + y2)/2) = ((2 + 8)/2, (3 + 7)/2) = (5, 5).

Correct Answer: A — (5, 5)

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Q. If point A(2, 3) and point B(8, 7) are the endpoints of a line segment, what is the midpoint M of AB?

A. (5, 5)
B. (4, 5)
C. (6, 5)
D. (5, 4)

Solution

Midpoint M = ((x1 + x2)/2, (y1 + y2)/2) = ((2 + 8)/2, (3 + 7)/2) = (5, 5).

Correct Answer: A — (5, 5)

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Q. If point C divides the segment AB in the ratio 2:3, where A(1, 2) and B(6, 8), what are the coordinates of point C?

A. (3.0, 4.0)
B. (4.0, 5.0)
C. (2.0, 3.0)
D. (5.0, 6.0)

Solution

Using the section formula: C = ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n)) = ((2*6 + 3*1)/(2+3), (2*8 + 3*2)/(2+3)) = (4.0, 5.0).

Correct Answer: B — (4.0, 5.0)

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Q. If point C is at (0, 0) and point D is at (6, 8), what is the length of segment CD?

A. 8
B. 10
C. 6
D. 12

Solution

Using the distance formula: d = √((6 - 0)² + (8 - 0)²) = √(36 + 64) = √100 = 10.

Correct Answer: B — 10

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Q. If point C is at (2, 3) and point D is at (10, 7), what is the coordinates of point E that divides CD in the ratio 1:3?

A. (5, 4)
B. (4, 5)
C. (6, 5)
D. (7, 6)

Solution

Using the section formula: E = ((1*10 + 3*2)/(1+3), (1*7 + 3*3)/(1+3)) = (5, 4).

Correct Answer: A — (5, 4)

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Q. If point C is at (3, 5) and point D is at (9, 1), what is the coordinates of the point that divides CD in the ratio 2:3?

A. (5.4, 3.2)
B. (6, 3)
C. (4.5, 4)
D. (5, 4)

Solution

Using the section formula: P = ((2*9 + 3*3)/(2+3), (2*1 + 3*5)/(2+3)) = (5.4, 3.2).

Correct Answer: A — (5.4, 3.2)

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Q. If point C is at (4, 5) and point D is at (10, 15), what is the distance between C and D?

A. 10
B. 11.18
C. 12
D. 9.22

Solution

Using the distance formula: d = √((10 - 4)² + (15 - 5)²) = √(36 + 100) = √136 ≈ 11.18.

Correct Answer: B — 11.18

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Q. If point C is at (5, 5) and point D is at (5, 10), what is the distance between points C and D?

A. 5
B. 10
C. 15
D. 20

Solution

Distance = |y2 - y1| = |10 - 5| = 5.

Correct Answer: A — 5

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Q. If point C(0, 0) and point D(6, 8) are endpoints of a line segment, what is the length of CD?

A. 10
B. 8
C. 6
D. 12

Solution

Using the distance formula: d = √((6 - 0)² + (8 - 0)²) = √(36 + 64) = √100 = 10.

Correct Answer: A — 10

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Q. If point C(0, 0) and point D(6, 8) are the endpoints of a line segment, what is the length of CD?

A. 8.0
B. 10.0
C. 6.0
D. 7.0

Solution

Using the distance formula: d = √((6 - 0)² + (8 - 0)²) = √(36 + 64) = √100 = 10.0.

Correct Answer: B — 10.0

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Q. If point C(1, 1) divides the line segment joining points A(0, 0) and B(4, 4) in the ratio k:1, what is the value of k?

A. 1
B. 2
C. 3
D. 4

Solution

Using the section formula, C = ((kx2 + x1)/(k+1), (ky2 + y1)/(k+1)). Setting C(1, 1) gives k = 1.

Correct Answer: B — 2

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Q. If point C(1, 2) and point D(7, 8) are given, what is the coordinates of the point that divides CD in the ratio 3:2?

A. (4, 5)
B. (5, 6)
C. (3, 4)
D. (6, 7)

Solution

Using the section formula: P = ((3*7 + 2*1)/(3+2), (3*8 + 2*2)/(3+2)) = (4, 5).

Correct Answer: A — (4, 5)

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Q. If point C(4, 5) is the midpoint of segment AB, and A is at (2, 3), what are the coordinates of point B?

A. (6, 7)
B. (8, 9)
C. (4, 5)
D. (2, 3)

Solution

Using the midpoint formula: B = (2*C - A) = (2*4 - 2, 2*5 - 3) = (6, 7).

Correct Answer: A — (6, 7)

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Q. If point D divides the line segment joining points (2, 5) and (8, 9) in the ratio 3:1, what are the coordinates of point D?

A. (5, 7)
B. (6, 8)
C. (4, 6)
D. (7, 8)

Solution

Using the section formula: D = ((3*8 + 1*2)/(3+1), (3*9 + 1*5)/(3+1)) = (5, 7).

Correct Answer: A — (5, 7)

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Q. If point D divides the segment AB with A(2, 3) and B(8, 7) in the ratio 2:3, what are the coordinates of D?

A. (4.8, 5.2)
B. (5.2, 4.8)
C. (6, 5)
D. (5, 5)

Solution

Using the section formula: D = ((2*8 + 3*2)/(2+3), (2*7 + 3*3)/(2+3)) = (4.8, 5.2).

Correct Answer: A — (4.8, 5.2)

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Q. If point D is at (5, 5) and point E is at (1, 1), what is the distance DE?

A. 4
B. 5
C. 6
D. 7

Solution

Using the distance formula: d = √((5 - 1)² + (5 - 1)²) = √(16 + 16) = √32 = 4√2 ≈ 5.66.

Correct Answer: B — 5

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Q. If point D is at (6, 8) and point E is at (2, 4), what is the length of DE?

A. 4.47
B. 5.66
C. 6.32
D. 7.21

Solution

Using the distance formula: d = √((6 - 2)² + (8 - 4)²) = √(16 + 16) = √32 = 5.66.

Correct Answer: B — 5.66

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Q. If point D is at (6, 8) and point E is at (2, 4), what is the section formula ratio if point F divides DE in the ratio 1:3?

A. (3, 5)
B. (4, 6)
C. (2.5, 4.5)
D. (3.5, 5.5)

Solution

Using the section formula: F = ((1*2 + 3*6)/(1+3), (1*4 + 3*8)/(1+3)) = (5, 7).

Correct Answer: A — (3, 5)

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Q. If point D(3, 4) is the midpoint of segment AB where A(1, 2) and B(x, y), what are the coordinates of B?

A. (5, 6)
B. (7, 8)
C. (6, 8)
D. (4, 6)

Solution

Using the midpoint formula: D = ((x1 + x2)/2, (y1 + y2)/2). Solving gives B = (5, 6).

Correct Answer: A — (5, 6)

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Q. If point E is at (3, 4) and point F is at (6, 8), what is the distance between E and F?

A. 3
B. 5
C. 6
D. 7

Solution

Using the distance formula: d = √((6 - 3)² + (8 - 4)²) = √(9 + 16) = √25 = 5.

Correct Answer: B — 5

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Q. If point E(1, 1) and point F(4, 5) are the endpoints of a line segment, what is the length of EF?

A. 3.61
B. 4.24
C. 5.0
D. 6.0

Solution

Using the distance formula: d = √((4 - 1)² + (5 - 1)²) = √(9 + 16) = √25 = 5.0.

Correct Answer: B — 4.24

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Q. If point P divides the segment joining (2, 3) and (8, 7) in the ratio 3:1, what are the coordinates of P?

A. (5, 4)
B. (6, 5)
C. (7, 6)
D. (4, 5)

Solution

Using the section formula: P = ((3*8 + 1*2)/(3+1), (3*7 + 1*3)/(3+1)) = (6, 5).

Correct Answer: B — (6, 5)

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Q. If point P divides the segment joining A(1, 2) and B(5, 6) in the ratio 1:3, what are the coordinates of P?

A. (3, 4)
B. (2, 3)
C. (4, 5)
D. (1, 2)

Solution

Using the section formula: P = ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n)) = ((1*5 + 3*1)/(1+3), (1*6 + 3*2)/(1+3)) = (3, 4).

Correct Answer: A — (3, 4)

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Q. If point P is at (3, 5) and point Q is at (9, 1), what is the coordinates of point R that divides PQ in the ratio 2:3?

A. (5.4, 3.6)
B. (6, 4)
C. (5, 4)
D. (4, 3)

Solution

Using the section formula: R = ((2*9 + 3*3)/(2+3), (2*1 + 3*5)/(2+3)) = (5.4, 3.6).

Correct Answer: A — (5.4, 3.6)

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Q. If sin(x) = 0.5, what are the possible values of x in degrees?

A. 30°, 150°
B. 45°, 135°
C. 60°, 120°
D. 90°, 270°

Solution

sin(x) = 0.5 gives x = 30° and 150°.

Correct Answer: A — 30°, 150°

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Q. If sin(x) = 0.8, what is the value of csc(x)?

A. 0.8
B. 1.25
C. 1.2
D. 0.5

Solution

csc(x) = 1/sin(x) = 1/0.8 = 1.25.

Correct Answer: B — 1.25

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Q. If sin(θ) = 0.6, what is cos(θ) using the Pythagorean identity?

A. 0.8
B. 0.6
C. 0.4
D. 0.2

Solution

Using sin²θ + cos²θ = 1, cos²θ = 1 - 0.6² = 0.64, thus cos(θ) = 0.8.

Correct Answer: A — 0.8

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