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Main » 2013 » October » 5 » Compiled Mathematics Problems Part 2
5:17 PM Compiled Mathematics Problems Part 2 |
COMPILATION OF MATH PROBLEMS ANALYTIC GEOMETRY 1. Find the equation of the directrix of the parabola y^2=16x. 2. The midpoint of the line segment between P1 and P2 (-2, 4) is P (2, -1). Find P1 3. Given an ellipse: x^2/36+y^2/32=1. Determine the distance between foci. 4. Convert the θ = π/3 to the Cartesian equation. 5. Find the coordinates of the point P(2, 4) with respect to the translated axis with origin at (1, 3). 6. The segment from (-1, 4) to (2, -2) is extended three times its own length. Find the terminal point. 7. Determine B such that 3x + 2y – 7 = 0 is perpendicular to 2x – By + 2 = 0. 8. Find the value of k for which the equation x^2+y^2+4x-2y-k=0 represents a point circle. 9. What is the diameter of a circle described by 9x^2+9y^2=16? 10. The major axis of the elliptical path in which the earth moves around the sun is approximately 186,000,000 miles and the eccentricity of the ellipse is 1/60. Determine the apogee of the earth. 11. Point P(x, y) moves with a distance from point (0, 1) one-half of its distance from the line y = 4. What is the equation of the locus? 12. A line passes through point (2, 2). Find the equation of the line if the length of the line segment intercepted by the coordinate axis is √5. 13. Find the area of the triangle which the line 2x – 3y + 6 = 0 forms with the coordinate axes. 14. Determine the coordinates of the point which is three-fifths of the way from the point (2, -5) to the point (-3, 5). 15. Two vertices of a triangle are (2, 4) and (-2, 3) and the area is 2 square units. Find the locus of the third vertex. 16. If the points (-2, 3), (x, y), and (-3, 5) lie on a straight line, find the equation of the line. 17. Find the inclination of the line passing through (-5, 3) and (10, 7). 18. Find the distance of the directrix from the center of an ellipse if its major axis is 10 and its minor axis is 8. 19. A point moves so that its distance from the point (2, -1) is equal to its distance from the x-axis. What is the equation of the locus? 20. The parabolic antenna has an equation of y^2+ 8x=0. Determine the length of the latus rectum. 21. Find the equation of the ellipse with center at (4, 2), major axis horizontal and of length 8 and with minor axis of length 6. 22. Find the area of the hexagon ABCDEF formed by joining the points A(1, 4), B(0, -3), C(2, 3), D(-1, 2), E(-2, -1) and F(3, 0). 23. The directrix of a parabola is the line y = 5 and its focus is at the point (4, -3). What is the length of its latus rectum? 24. Find the eccentricity of an ellipse when the length of the latus rectum is 2/3 of the length of the major axis. 25. Find the equation of the parabola whose axis is parallel to the x-axis and passes through the points (3, 1), (0, 0) and (8, -4). 26. A point P (x, 2) is equidistant from the points (-2, 9) and (4, -7). What is the value of x? 27. Find the equation of a line whose x-intercept is 2 and y-intercept is -2. 28. If the length of the latus rectum of an ellipse is three-fourth of the length of its minor axis, find its eccentricity. CALCULUS 1 1. What is the allowable error in measuring the edge of the cube that is intended to hold 8 cu.m. if the error of the computed volume is not to exceed 0.02 cu.m? 2. Find the altitude of a cylinder of maximum volume which can be inscribed in a right circular cone of radius r and height h. 3. Find the approximate change in the volume of a cube of side "x” inches caused by increasing its side by 1%. 4. Three sides of a trapezoid are each 8-cm long. How long is the fourth side when the area of the trapezoid has the greatest value? 5. A statue 3 m high is standing on a base of 4 m high. If an observer’s eye is 1.5 m above the ground, how far should he stand from the base in order that the angle subtended by the statue is a maximum. 6. A balloon is rising vertically over a point A on the ground at the rate of 15 ft/sec. A point B on the ground level with and 30 ft from A. When the balloon is 40 ft from A, at what rate is its distance from B changing? 7. Find the point in the parabola y^2=4x at which the rate of change of the ordinate and abscissa are equal. 8. Find the slope of x^2 y=8 at the point (2, 2). 9. Find the equation of the normal to x^2+y^2=1 at the point (2, 1). 10. The depth of water in a cylindrical tank 4 m in diameter is increasing at the rate of 0.70 m/min. Find the rate at which the water is flowing into the tank. 11. Find the coordinates of the vertex of the parabola 〖y=x〗^2-4x+1 if the slope of the tangent at the vertex is zero. 12. Find the minimum distance from the point (4, 2) to the parabola y^2=8x. 13. Two posts, one 8 m and the other 12 m high are 15 cm apart. If the posts are supported by a cable running from the top of the first post to a stake on the ground and then back to the top of the second post, find the distance from the lower post to the stake to use minimum amount of wire? 14. What is the area of the largest rectangle that can be inscribed in a semi-circle of radius 10? 15. Find the approximate increase in the volume of the sphere if the radius increases from 2 to 2.05 in one second. 16. Differentiate: ln(ln y) + ln y = ln x 17. The volume of the sphere is increasing at the rate of 6 cm3/hr. At what rate is its surface area increasing when the radius is 50 cm? 18. If the sum of two numbers is 4, find the minimum value of the sum of their cubes. 19. Water is running out a conical funnel at the rate of 1 cu.inch per second. If the radius of the base of the funnel is 4 inches and the altitude is 8 inches, find the rate at which the water level is dropping when it is 2 inches from the top. 20. Divide the number 120 into two parts such that the product of one and the square of the other is a maximum. 21. If y = 2x + sin 2x, find x if y’ = 0. 22. What is the equation of the tangent to the curve y=x+ 5/x at the point (1, 3)? 23. If y=arc tan〖(ln〖x)〗 〗, find y’ at x = 1/e. 24. Find the change in y = 2x – 3 if x changes from 3.3. to 3.5. 25. The radius of a sphere is r inches at time t seconds. Find the radius when the rates of increase of the surface area and the radius are numerically equal. 26. Using the two existing corner sides of an existing wall, what is the maximum rectangular area that can be fenced by a fencing material 30 ft. long? 27. A man on a wharf 3.6 m above sea level is pulling a rope tied to a raft at 0.60 m/sec. How fast is the raft approaching the wharf when there are 6 m of rope out? 28. The sum of four positive integers is 32. Find the greatest possible product of these four numbers. 29. Find the maximum point of y=x+ 1/x 30. Find the height of a right circular cylinder of maximum volume which can be inscribed in a sphere of radius 10 cm. CALCULUS 2 1. Find the area bounded by the curve defined by the equation x^2=8y and its latus rectum. 2. Evaluate: ∫((cos)^8 3a da) from 0 to π/6. 3. Find the area bounded by the parabolas x^2-2y=0 and x^2+2y-8=0. 4. Evaluate: ∫(√(1-cosx ) dx). 5. Find the area bounded by the curve y=9-x^2 and the x-axis. 6. Evaluate: ∫(3^x/e^x dx). 7. Evaluate: ∫((tan〗^2 xdx) from 0 to 1. CALCULUS 3 1. Find the value of (1 + i)^5. 2. Simplify the expression: i^1997+i^1999 3. What is the quotient when 4 + 8i is divided by i3? 4. If (x+yi)(2-4i)= 14-8i, find x. 5. Find the angle between the planes 3x – y + z – 5 = 0 and x +2y +2z + 2 = 0 6. Find the area of the geometric figure whose vertices are at (3, 0, 0), (3, 3, 0), (0, 0, 4) and (0, 3, 4). 7. Find the length of the vector (2, 4, 4) TERMINOLOGIES: 1. A function F(x) is called ________ of f(x) if F’(x) = f(x). 2. In an ellipse, a chord which contains a focus and is in a line perperdicular to the major axis is a ________. 3. If all y-terms have even exponents, the curve is symmetric with respect to the ______. 4. Convergent series is a sequence of decreasing numbers or when the succeeding term is ______ than the preceding term. 5. The graph of r = a + b cos θ is a ____________. 6. 4x^2-256=0 is the equation of __________. 7. Each of the faces of a regular hexahedron is a _________. 8. It is a sequence of numbers such that successive terms differ by a constant. 9. Equations relating x and y that cannot readily solved explicitly for y as a function of x or for x as a function of y. Such function may nonetheless determine y as a function of x or vice versa. Such a function is called ____________. 10. If the roots of an equation are zero, then they are classified as _______. 11. If a = b, then b = a. This illustrates what axiom in Algebra? 12. The integral of any quotient whose numerator is the differential of the denominator is the ________. 13. It is a polyhedron of which two faces are equal polygons in parallel planes and the other faces are parallelograms. 14. When f”(x)is negative, the curve of y = f(x) is concave ________. 15. In polar coordinate system, the length of the ray segment from a fixed origin is known as _________. 16. If the first derivative of the function is constant, then the function is _________. 17. A locus of a point which moves so that it is always equidistant from a fixed point to a fixed line is a __________. 18. A line which a curve approaches indefinitely near as its tracing point passes off to the infinity is called the _________. 19. __________ is the concept of finding the derivative of composite functions. 20. It is a part of a circle bounded by a chord and an arc. 21. In Algebra, the operation of root extraction is called as ________. 22. The chords of an ellipse, which pass through the center, are known as _______. 23. A horizontal line has a slope of _____. 24. The line passing through the focus and is perpendicular to the directrix of a parabola is called the _________. 25. At maximum point the value of y” is ________. 26. The altitude of the sides of a triangle intersect at the point known as ______. 27. A sequence of numbers where the succeeding term is greater than the preceding term is_______. 28. A line, which is perpendicular to the x-axis, has a slope equal to ________. 29. Convergent series is a sequence of decreasing numbers or when the succeeding term is _______ than the preceding term. 30. The area of the region bounded by two concentric circles is called ______. 31. Point of derivatives which do not exists (and so equals zero) are called ______. 32. It can be defined as the set of all points in the plane whose distances from two fixed points is a constant is called __________. 33. If the equation is unchanged by the substitution of y for x, it curve is symmetric with respect to the ________. 34. The apothem of a polygon is the _______ of its inscribed circle. |
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