analytic review

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TOPIC: ANALYTIC/SOLID GEOMETRY, PROBABILITY AND STATISTICS DATE\TIME: LECTURER: Engr. Jeffrey P. Landicho Lines General EQ. Ax + By + C = 0 Point-Slope Form y - = m(x - ) Slope intercept Form y = mx + b Two- Point Form y - = (x - ) Intercept Form Distance between two points d = Distance from a point to a line d = Angle between two lines General Equation of Conic Section A + Bxy + C + Dx + Ey + F = 0 Circle Parabola ( e = 1) upward\ downward right\ left Latus rectum = 4a Ellipse ( e < 1 ) ; Horizontal Major Axis ; Vertical Major Axis Latus rectum = 2 / a Eccentricity = c/a Distance between directrices = 2a/e Distance between vertices = 2a Distance between foci = 2c Hyperbola ( e > 1 ) ; Horizontal Trasverse Axis ; Vertical Transverse Axis Latus rectum = 2 / a Eccentricity = c/a

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Page 1: Analytic Review

TOPIC: ANALYTIC/SOLID GEOMETRY, PROBABILITY AND STATISTICSDATE\TIME: LECTURER: Engr. Jeffrey P. Landicho

Lines General EQ. Ax + By + C = 0

Point-Slope Form y - = m(x - )

Slope intercept Form y = mx + b

Two- Point Form y - = (x -

)

Intercept Form

Distance between two points

d =

Distance from a point to a line

d =

Angle between two lines

General Equation of Conic Section

A + Bxy + C + Dx + Ey + F = 0

Circle

Parabola ( e = 1)

upward\

downward

right\left

Latus rectum = 4aEllipse ( e < 1 )

; Horizontal Major

Axis

; Vertical Major Axis

Latus rectum = 2 / a

Eccentricity = c/a

Distance between directrices = 2a/e Distance between vertices = 2a Distance between foci = 2c

Hyperbola ( e > 1 )

; Horizontal

Trasverse Axis

; Vertical Transverse

Axis

Latus rectum = 2 / a

Eccentricity = c/a

Page 2: Analytic Review

Solid GeometryCones

V = h

Lateral Area =

Frustum of a Cone

V =

Lateral Area =

Pyramids

V = h)/3

Frustum of Pyramids

V =

Cube

V =

Lateral Area = 6

Sphere

V = =

Surface Area = 4

Area of Zone

A =2 r h

Spherical Segment

V = ; one base

V =

Prismatoid

V = L/6 (

Probability, Combination and PermutationProbability

Is the ratio of successful event to the total number of events.

p(E) =

p(E1 E2) = p(E1) + p(E2) - p(E1 E2) ;

Mutually Exclusive Events

p(E1 E2) = p(E1) x p(E2) ; Independent

Events

p(E1 E2) = p(E1) x p(E1 / E2) ;

Dependent Events nCr (pr)(1 – p)n-r ; Repeated Trials

Combination arrangement of things called element

with definite order

nCr = ; combination of n elements

taken r at a time C = 2n – 1 ; combination taken 1,2,3, …n

at a timePermutation

nPr =

nPr = ; Permutation of like

elements P = (n – 1)! ; Cyclic permutation

Page 3: Analytic Review

SAMPLE PROBLEMS:

1. Given three points A(-1,1), B(5,-1), C(4,3), find the angle made by the segment AB and AC.a.40.236° b.30.57° c. 63.409° d. 57.3°

2. A point P(x,3) is equidistant from the point A(1,5), B(-1,2). Find x.a.2/5 b.3/4 c.-3/4 d.-2/5

3. Find the locus of a points, P(x,y), such that the distance from P to (3,0) is twice its distance to (1,0). a. 3x2 + 3y2 +2x + 5 = 0 b. 3x2 + 3y2 - 2x – 5 = 0 c. -3x2 – 3y2 - 2x +5 = 0 d. 3x2 – 3y2 - 2x – 5 = 0

4. Find the inclination of the line 2x + 5y = 10a.140.236° b.130.57° c. 158.2° d. 157.3°

5. Find the equation of a a line that passes thru (6,8) and parallel to the line 2x – 7y – 5 = 0 a. 4x – 6y – 56 = 0 b. 2x – 7y – 68 = 0 c. 2x + 6y + 49 + 0 d. 7y – 2 = 2x

6. Find the equation of a line passing through (1,-2) and perpendicular to the line through (2,-1) and (-3,2).a. 5x – 3y – 11 = 0 b. 2x – 7y – 11 = 0 c. 3x + 5y + 11 = 0 d. 3y – 2 = 5x

7. Find the distance from the point (3,-1) to the line 3x – 4y – 3 = 0.a.2 b.3 c.-3 d.-2

8. Find the distance between parallel lines 8x + 15y + 18 = 0 and 8x + 15y + 1 = 0a.2 b.3 c.4 d.1

9. Find the center and radius of the circle whose equation is x2 + y2 – 4x – 6y – 12 = 0.a. C(2,-3) r = 4 b. C(2,-3) r = 5 c. C(2,3) r = 5 d. C(-2,-3) r = 4

10. Given the end points of the diameter of the circle (5,2),(-1,2). Find its equation. a. x2 + y2 +2x + 5 = 0 b. x2 + y2 - 4x – 4y – 1 = 0 c. x2 + y2 - 2x +5y +3 = 0 d. x2 + y2 + 4x – 4y – 1 = 0

11. Find the vertex of the parabola x2 – 2x – 8y – 15 = 0.a. (2,-3) b. (2,-3) c. (2,1) d. (1,-2)

12. Given the ellipse 16x2 + 25y2 – 64x – 50y – 311 = 0, determine its center.a. (1,2) b. (2,-1) c. (2,1) d. (1,-2)

13. Given the ellipse 16x2 + 25y2 – 64x – 50y – 311 = 0, determine its latus rectum.a.32/5 b.13/5 c.46/5 d.23/5

14. An ellipse has the equation 16x2 + 9y2 – 32x – 128 = 0. Its eccentricity is:a.0.88 b.0.77 c.0.66 d.0..55

15. Find the equation of the hyperbola, center (2,0) focus (2,3) eccentricity =

a. 2y2 - x2 +4x – 10 = 0 b. 2x2 - y2 - 4x – 1 = 0 c. x2 - y2 +5y +3 = 0 d. x2 - y2 + 4x - 15 = 0

16. The semi-conjugate axis of the hyperbola

a.2 b.3 c.4 d.617. A hyperbola with major axis 8 and minor axis 6. Find the eccentricity.

a.4/3 b. 5/4 c. 5/3 d. none of these18. An elliptical plot of garden has semi major axis of 6 meters and semi minor axis of 4.8 meters. If these are

increased by 0.15 meter each, find the increase in the area of the garden in square meters.a.4.62 b. 5.16 c. 6.24 d. 3.62

19. The point (3,2) is the midpoint of the segment defined by the points (h,1) and (5,k) , h and k are respectively:

a.1 and 3 b. 3 and 1 c. -2 and 1 d. 2 and -120. Find the area of the triangle which the line 2x – 3y + 6 = 0 forms with the coordinates axes.

a.3 b. 4 c. 5 d. 223. Find the latus rectum of the curve 4x2+9y2+8x -32 = 0

a.1.23 b. 2.67 c. 1.33 d. 0.624. Find the equation of the directrix of the parabola y2 = 16x

a.x=-4 b. x=-8 c. x=8 d. x=425. The semi-major axis of an ellipse is 4 and its semi-minor axis is 3. The distance from the center to the directrix is:

a.6.222 b. 6.532 c. 6.047 d. 0.661

Page 4: Analytic Review

SUPPLEMENTARY PROBLEMS:

1. Find the midpoint between two points (2,4) and (8,12)a.2,4 b. 7,3 c. 4,2 d. 5,8

2. Find the point on the y-axis that is equidistant from (5,1) and (-3,-1)a.0,4 b. 2,1 c. 0,6 d. 2,4

3. What is the center of an ellipse 64x2 + 25y2 = 1600?a.2,3 b. 1,1 c. 4,2 d. 0,0

4. Find the minor axes of the ellipse x2 + 9y2 = 9?a.1 b. 2 c. 3 d. 4

5. What are the focus of ellipse with an equation of 4x2 + y2 = 100? a.7.90 b. 5.76 c. 8.66 d. 4.44

6. What is the major axes of an ellipse with an equation of 4x2 + y2 + 16x – 10y – 23 =0?a.8 b. 9 c. 10 d. 11

7. What is the eccentricity of an ellipse 9x2+y2+18x-12y-180=0? a.0.789 b.0.897 c. 0.943 d. 0.576

8. Find the distance between center and directrix of an ellipse x2/36 + y2/16 = 1 a.5 b. 6 c. 7 d. 8

9. A hyperbola has an equation of 4x2-y2 = 256. Find the eccentricity. a.2.23 b. 3.42 c.4.35 d. 1.65

10. Find the latus rectum if the equation of parabola y2 + 20x -10 = 0 a.20 b. 15 c. 25 d. 30

11. Find the equation of straight line through point (3,2) and is parallel to line y = 3x-2 a.y=3x-19 b. y=4x+16 c. y=3x-7 d. y=4x-8

12. Two vertices of a triangle are (2,4) and (-2,3) and the area is 2sq. units, the locus of the third vertex is : a.4x-y=14 b. 4x+4y=14 c. x+4y= d. x-4y=-18

13. Compute the y-intercept of a line passing through the point (5,3) and a slope of ¾.a.-3/4 b. -5/4 c. -3/5 d. -4/5

14. The distance between points (4,7,z) and (5,1,6) is 7.28.Find z. a.2 b. 6 c. 4 d. 8

15. Find the equation of perpendicular bisector of the line passing through points (2,-5) to the point (-3,5) a.5x-9y=2 b. 3x-4y=1 c. x+y=4 d.3x+7y=3

16. Determine the coordinates of the point which is three-fifths of the way from the point (2,-5) to the point (-3,5) a.(-1,1) b. (-1,-2) c. (-2,-1) d. (1,-1)

17. Find the area of the triangle which the line 2x – 3y + 6 = 0 forms with the coordinates axes. a.3 b. 4 c. 5 d. 2

18. The linear distance between -4 and 17 on the number line is a.13 b. 21 c. -17 d. -13

19. The midpoint of the line segment between P1 (x,y) and P2 (-2,4) is Pm (2,-1). Find the coordinate of P1. a.(6,-5) b. (5,-6) c. (6,-6) d. (-6,6)

20. Find the equation of the normal to x2 + y2 = 1 at the point (2,1) a.y=2x b. x=2y c. 2x+3y=3 d. x+y=1

21. Given the equation of the curve 9x2+25y2-144x +200y + 751=0. Find the distance between the foci. a.6units b. 10units c. 8units d. 12units

22. .Find the equation of the directrix of the parabola y2 = 16x a.x=-4 b. x=-8 c. x=8 d. x=4

23. Determine the equation of the directrix of the curve 3x=2y2-4y+7 a.x=30/20 b. x=30/24 c. x=31/20 d. x=31/24

24. A hyperbola with major axis 8 and minor axis 6. Find the eccentricity. a.4/3 b. 5/4 c. 5/3 d. none of these

25. The semi-major axis of an ellipse is 4 and its semi-minor axis is 3. The distance from the center to the directrix is:

Page 5: Analytic Review

a.6.222 b. 6.532 c. 6.047 d. 0.66126. A car headlight reflector is cut by a plane along its axis. The section is a parabola having the light center at

the focus. If the distance of focus from vertex is 3/4cm and if the diameter of the reflector is 0cm, find its depth. a.23/3 b. 22/3 c. 27/3 d. 25/3

27. Given the ellipse x2/36 + y2/32=1. Determine the distance between foci. a.3 b. 4 c. 2 d. 8

28. The distance between the vertices of an ellipse is 10. The distance between the foci is 6. What is the distance between directrices. a.0.60 b. 7.33 c. 16.7 d. 8.33

29. Find the latus rectum of the curve 4x2+9y2+8x -32 = 0a.1.23 b. 2.67 c. 1.33 d. 0.6

30. The area of the ellipse is 62.83m2. The semi-minor axis is 0.8 times the semi-major axis. Find the perimeter of the ellipse.a.19.97 b. 28.45 c. 4.53 d. 4

31. Find the angle in degrees between the asymptote of the hyperbola 9x2-25y2+50y-250=0 and the x axis.a.30.96° b. 25.66° c.11.76° d. 15.4°

32. The equilateral hyperbola xy= 25 has the coordinate axes as asymptotes. Find the distance from its vertex to the origin.a.25 b. 5.07 c. 7.07 d. 2.07

33. The chords of the ellipse 4x2 + 9y2 = 144 having equal slopes of ¾ are bisected by its diameter. Find the equation of this diameter.a.8x + 8y=0 b. 27y – 18x = 0 c. 16x + 27y = 0 d. 8x + 18y = 0

34. A hotel swimming pool has for its boundary a curve given by the equation: x2+y2 = 2x-4y+20 where the units are all in meters. Find the volume of water required to fill the pool to a depth of 1.5m.a.117.8m3 b. 168.7m3 c. 121.5m3 d. 171.5m3

Page 6: Analytic Review

SAMPLE PROBLEMS:Solid Geometry

1. The volume of a sphere is increased by how much if its radius is increased by 20%?a.32.6% b. 33% c. 44% d. 72.8%

2. The volume of a sphere is 904.78 cu. m. find the volume of the spherical segment of height 4m.a.234.57 cu. m. b. 256.58 cu. m. c. 145.69 cu. m. d. 124.58 cu. m.

3. The diameters of two spheres are in the ratio 2:3. If the sum of their volumes is1260 cu. m, the volume of the larger sphere is:

a.972 cu. m b. 927 cu. m c. 856 cu. m d. 865 cu. m4. The corners of a cubical block touch the close spherical shell that encloses it. The volume of the box is

2744cc. What volume in cc, inside the shell is not occupied by the block?a. a.1356cc b. 4721cc c. 3423cc d. 7623cc

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

a. a.0.38 cu. in b. 2.5 cu. in c. 3.8 cu. in d. 4.2 cu. in6. A right circular cone, with vertical axis and with base uppermost is surmounted by a hemisphere. If the

volume of hemisphere is twice that of the cone, what angle is formed by a slant height and the axis of the cone?

a.25 b. 45 c. 90 d. 307. A piece of thin card board in the form of a sector of a circle of radius 36cm. is rolled into a cone. Find the

volume of the cone if the angle of the sector is 60. a.35.50cc b. 37.70cc c.1338.3cc d. 3355.6cc

8. The upper base of the frustum of a regular triangular pyramid is an equilateral triangle with an edge of 3m. The volume of the frustum is 84.44cu. m and its altitude is 5m. What is the lower base edge in meters?

a.8m b. 9m c. 6m d. 5m9. If a solid steel ball is immersed in an eight cm diameter, it displaced water to a depth of 2.25 cm. The

radius of the ball is:a.3 cm b.6cm c. 9cm d. 12cm

10. A circular piece of cardboard with a diameter of 1m will be made into a conical hot 40 cm high by cutting a sector off and joining the edges to form a cone. Determine the angle subtended by the sector removed?

a. 1440 b. 1480 c. 1520 d. 1540

10. A trough, whose ends are isosceles right triangles with vertical axis which is 6.0 m long. If it contains 150 liters of red horse, how deep is the red horse?

a. 0.158 m b. 0.518 m c. 0,025 m d. 0.205 m11. A hole of 2 cm radius is to be made through a right circular cone. Find the volume removed.

a. 97.74 cm3 b. 79.94 cm3 c. 99.74 cm3 d. 74.99 cm3

12. Two vertical conical tanks are joined at the vertices by a pipe. Initially the bigger tank is full of SanMig. The pipe valve is opened to allow the SanMig to flow to the smaller tank until it is full. At this instant, how deep is the SanMig in the bigger tank? If the diameter and height of the bigger and smaller tanks are 6 m, 10 m, and 6 m, 6 m respectively.

a. 7.36 m b. 6.37 m c. 3.76 m d. 7.63 m13. A pyramid has a base whose sides are 8 cm, 15 cm, and 17 cm., respectively. If the altitude of the pyramid

is 20 cm., find the volume of the inscribed and circumscribed cones.a. 188.5, 1513 b. 188.5, 1531 c. 185.8, 1513 d. 185.8, 1531

14. An ice cream cone is filled with ice cream and more ice cream in the form of hemisphere is place on top. The diameter of the hemisphere is equal to the diameter of the cone. If the hemispherical surface is equal to the lateral surface of the cone, find the total volume of ice cream if the radius of the hemisphere is 25 mm.

a. 3.73 in3 b. 33.7 in3 c. 37.3 in3 d. 7.33 in3

Page 7: Analytic Review
Page 8: Analytic Review

SUPPLEMENTARY PROBLEMS:Solid Geometry

1. What is the surface area of a sphere whose volume is 36 cu. m?a.52.7m2 b. 48.7m2 c. 46.6m2 d. 54.6m2

2. The surface area of a sphere is 4 r2. Find the percentage increase in its diameter when the surface area

increased by 21%a.5% b. 10% c. 15% d. 20%

3. Find the percentage increase in volume of a sphere if its surface area is increased by 21%a.30.2% b. 33.1% c. 34.5% d. 30.9%

4. Given two spheres whose combined volume is known to be 819 cu. m. if their radii are in the ratio 3:4, what is the volume of the smaller sphere?

a.576 cu. m b. 243 cu. m c. 343 cu. m d. 476 cu. m5. How much will the surface area of a sphere be increased if its radius is increased by 5%?

.25% b. 15.5% c. 12.5% d. 10.25%6. A sphere of radius r just fits into a cylindrical container of radius r and altitude 2r. Find the empty space in

the cylinder.a.(8/9) r3 b. (20/27) r3 c. (4/5) r3 d. (2/3) r3

7. A hemispherical bowl of radius 10 cm is filled with water to such a depth that the water surface area is equal to 75 sq. cm. the volume of the water is:

a.625/3cm3 b. 625 /3cm3 c. 625 /2cm3 d. 625 cm3

8. Find the volume of a spherical sector of altitude 3cm and radius 5cm.a.75 cm3 b. 100 cm3 c. 50 cm3 d. 25 cm3

9. How far from the center of a sphere of a radius 10 cm should a plane be passed so that the ration of the areas of two zones is 3:7 .

a.3cm b. 4cm c. 5cm d. 6cm10. The volume of water in a spherical tank having a diameter of 4m is 5.236m3 . Determine the depth of the

water on the tank.a.1 b.1.4 c. 1.2 d. 1.6

11. A mixtre compound from equal parts of two liquids, one white and the other black was placed in a hemispherical bowl. The total depth of the liquids is 6 inches. After standing for a short time, the mixture separated the white liquid setting below the lack. If the thickness of the segment of the liquid is 2inches, find the radius of the bowl in the inches.

a.7.53 b.7.33 c. 7.73 d. 7.9312. 20.5 cu. m of water is inside a spherical tank whose radius 2m. Find the height of the water surface above

the bottom of the tank.a.2.7 b. 2.5 c. 2.3 d. 2.1

13. A spherical segment of one base has an altitude of 3 meters and radius of 4m. Find the volume.a.84.28 b. 88.24 c. 82.84 d. 84.82

14. The face of a regular icosahedrons is a a. triangle b. square c. pentagon d. hexagon

15. A closed conical vessel has diameter of 2.4m across the top and a height of 4.8m. It contains water at a depth of m2.4m. If the vessel is inverted, how deep is the water inside?

a.0.56m b. 0.21m c. 0.92m d. 0.45m16. Two perpendicular elements of a right cone subtend a chord of 2 units at the base of the cone 1 unit

altitude. Find the volume of the cone.a.1.85 b. 1.705 c. 1.05 d. 1.259

17. A solid any section of which is an ellipsea .spheroid b. conoid c. torus d.prismatoid

18. .Regular polyhedrons are also known as solidsa.platonic b. archimedian c. euclidians d. neutonians

19. The solid formed by revolving the ellipse about its minor axis is called a a.spheroid b. oblate spheroid c. prolate spheroid d. ellipsoid

20. The solid formed by revolving the ellipse about its major axis is called a

Page 9: Analytic Review

a.spheroid b. oblate spheroid c. prolate spheroid d. ellipsoid21. A hole is drilled vertically at the horizontal base of a right circular cone at the axis. The cone has a base

diameter of 30cm. and an altitude of 45 cm. If the hole diameter is 12cm.,what volume is taken out?a.6871cc b. 3732cc c. 10,602.87cc d. 678.58cc

22. A spherical ball of radius 3cm was dropped into a conical vessel of depth 8cm and radius of base 6cm. What is the area of the portion of the sphere which lies above the circle of contact with the cone?

a.25.48cm2 b. 90.48cm2 c. 80.48cm2 d. 75.48cm2

23. What is the distance in cm. between two vertices of a cube which are farthest from each other, if an edge measures 8cm?

a.12.32cm b. 13.86cm c. 8.66cm d. 6.93cm24. In the story about the crow that wanted to drink water from a cylindrical can but could not reach the water,

it is said that the crow dropped a pebble which was a perfect sphere 3cm. in radius into the can. If the can was 6cm radius, what was the rise in water level inside the can after that pebble was dropped?

a.2 cm b. 1cm c. 3cm d. 2.5cm25. A solid has a circular base of radius 20cm. Find the volume of solid if every plane section perpendicular to

a particular fixed diameter is an equilateral triangle.a.12,453.57 b. 21,342.56 c. 18,475.21 d. 15,453.67

26. A cylindrical water tank which is 35ft in diameter and 105 ft in length is placed temporarily on an 18.5 deg. Slope. The filler is located flush with the top of the tank at midpoint. What is the maximum volume of water which can be placed in the tank?

a.90,303.09 b. 100,455.45 c. 89,232.56 d. 90,343.9227. If the LPG sphere of Shell company has an inside diameter of 15 meters and it could safely be filled to

75%of its volume, compute for the volume of LPG that could safely be contained in the sphere.a.1,235.56 b. 2,324.5 c. 1325.36 d. none of these

28. The diameters of two spheres are in the ratio 2:3; while the sum of their volumes is 1260cu. m. Find the volume of the larger sphere in cu. meter.

a.980 b. 972 c. 960 d. 95829. A circle with radius 6 has half its area removed by cutting off a border of uniform width. Find the width of

the border.a.1.76 b. 2.2 c. 1.35 d. 3.75

30. The sides of the right triangle are 8, 15 and 17 units. If each sides is doubled. How many square units will be the area of the new triangle?

a.240 b. 420 c. 300 d. 320