2 - ANALYTIC GEOMETRY

Given a triangle of sides 10 cm and 15 cm and an included angle of 60 deg. Determine the area of the triangle.

A. 70
B. 72
C. 80
D. 65

D. 65

Find the distance between (4, -3) and (-2, 5).

A. 10
B. 8
C. 9
D. 11

A. 10

The segment from (-1, 4) to (2, 2) is extended three times its own length. Find the terminal point.

A. (11,-24)
B. (-11,-20)
C. (11,-18)
D. (11,-20)

D. (11,-20)

Given three vertices of a triangle whose coordinates are A(1,1), B(3,-3) and C(5,-3). Find the area of the triangle.

A. 3
B. 4
C. 5
D. 6

B. 4

What is the equation of the line that passes through (4,0) and is parallel to the line x - y - 2 = 0?

A. y + x + 4 = 0
B. x - y - 4 = 0
C. y - x - 4 = 0
D. x + y - 4 = 0

B. x - y - 4 = 0

Determine B such that 4x + 2y - 7 = 0 is perpendicular to 2x - By + 2 = 0. (Note: Assuming typo in source 3x vs 4x based on options).

A. 5
B. 4
C. 6
D. 3

B. 4

Find the distance from the line 4x - 3y + 5 = 0 to the point (2,1).

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

B. 2

Find the distance between the lines, 3x + y - 12 = 0 and 3x + y - 4 = 0.

A. 16/sqrt(10)
B. 12/sqrt(10)
C. 4/sqrt(10)
D. 8/sqrt(10)

D. 8/sqrt(10)

What is the x-intercept of the line passing through (1,4) and (4,1)?

A. 4.5
B. 5
C. 6
D. 4

B. 5

Find the equation of the circle whose center is at (3,-5) and whose radius is 4.

A. x^2 + y^2 - 6x + 10y + 18 = 0
B. x^2 + y^2 + 6x + 10y + 18 = 0
C. x^2 + y^2 - 6x - 10y + 18 = 0
D. x^2 + y^2 + 6x - 10y + 18 = 0

A. x^2 + y^2 - 6x + 10y + 18 = 0

What conic section is represented by x^2 + 4xy + 4y^2 + 2x = 10?

A. circle
B. parabola
C. ellipse
D. hyperbola

B. parabola

Compute the focal length and the length of latus rectum of parabola y^2 + 8x - 6y + 25 = 0.

A. 2, 8
B. 4, 16
C. 16, 64
D. 1, 4

A. 2, 8

An arch 18 m high has the form of parabola with a vertical axis. The length of a horizontal beam placed across the arch 8 m from the top is 64 m. Find the width of the arch at the bottom.

A. 86 m
B. 96 m
C. 106 m
D. 76 m

B. 96 m

Find the center of the ellipse 9x^2 + 25y^2 + 18x - 100y = 116.

A. (1,2)
B. (-1,2)
C. (-1,-2)
D. (1,-2)

B. (-1,2)

Find the eccentricity of the curve 9x^2 - 4y^2 - 36x + 8y = 4.

A. 1.8
B. 1.7
C. 1.9
D. 1.6

A. 1.8

Find the equation of the hyperbola whose asymptotes are y = 2x and y = -2x and which passes through (5/2, 3).

A. 4x^2 - y^2 - 16 = 0
B. 2x^2 - y^2 - 4 = 0
C. 3x^2 - y^2 - 9 = 0
D. 5x^2 - y^2 - 25 = 0

A. 4x^2 - y^2 - 16 = 0

Find the area of a regular octagon inscribed in a circle of radius 10 cm.

A. 186.48
B. 148.91
C. 282.24
D. 166.24

C. 282.24

A regular pentagon has sides of 20 cm. An inner pentagon with sides of 10 cm is inside and concentric to the larger pentagon. Determine the area inside and concentric to the larger pentagon but outside of the smaller pentagon.

A. 430.70
B. 573.26
C. 473.77
D. 516.14

D. 516.14

In a circle with a diameter of 10 m, a regular five pointed star touching its circumference is inscribed. What is the area of the part not covered by the star?

A. 61
B. 51
C. 57
D. 67

B. 51

A goat is tied to a corner of a 30 ft by 35 ft building. If the rope is 40 ft long and the goat can reach 1 ft farther than the rope length (Radius 41), what is the maximum area the goat can cover?

A. 4840
B. 4804
C. 8044
D. 4084

D. 4084

A cubical container that measures 2 in on a side is tightly packed with 8 marbles and is filled with water. All the 8 marbles are in contact with the walls of the container and the adjacent marbles. All the marbles are the same in size. What is the volume of water in the container?

A. 0.38
B. 2.5
C. 3.8
D. 4.2

C. 3.8

The base of a right prism is a hexagon with one of each side equal to 6 cm. The bases are 12 cm apart. What is the volume of the right prism?

A. 1212
B. 2212
C. 1213
D. 1122

D. 1122

A pyramid with a square base has an altitude of 25 cm. If the edge of the base is 15 cm. Calculate the volume of the pyramid.

A. 1785
B. 1875
C. 5178
D. 5871

B. 1875

The angle of a sector is 30 deg and the radius is 15 cm. What is the area of a sector?

A. 59.8
B. 58.9
C. 89.5
D. 85.9

B. 58.9

A frustum of a regular pyramid has an upper base of 8 m x 80 m and a lower base of 10 m x 100 m and an altitude of 5 m. Find the volume of the pyramid.

A. 4066.67
B. 5066.67
C. 6066.67
D. 7066.67

A. 4066.67

A circular cone having an altitude of 9 m is divided into two segments having the same vertex. If the smaller altitude is 6 m. Find the ratio of the volume of small cone to the big cone.

A. 0.296
B. 0.386
C. 0.186
D. 0.486

A. 0.296

The volume of water in a hemisphere having a radius of 2 m is 2.05 cu.m. Find the height of the water.

A. 0.602
B. 0.498
C. 0.782
D. 0.865

A. 0.602

The volume of water in a spherical tank having a diameter of 4 m is 5.236 cu. m. Determine the depth of the water in the tank.

A. 1.6
B. 1.4
C. 1.2
D. 1