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Find the vector and Cartesian equations of the plane passing through the points 2,2,-1 , 3,4,2 and 7,0, www. Example 3. The road way is 5 mts below the lowest point of the cable. If an extra support is provided across the cable 30 mts above the ground level find the length of the support if the height of the pillars are 55 mts.
Find the equation of the path traced by him if the distance between the poles is 6m. The shortest distance that the satellite gets to the earth is kms. Find the longest distance that the satellite gets from the earth. The semi-major axis is of length 36 million miles and the eccentricity of the orbit is 0.
Find i how close the mercury gets to sun? How high is the arch, 9ft from the right or left of the centre. Example 4. Find the axis, vertex, focus, directrix, equation of the latus rectum, length of the latus rectumfor the following parabolas and hence draw their graphs.
The girder of a railway bridge is in the parabolic form with span ft. Find the height of the bridge at 10ft, to the left or right from the midpoint of the bridge. On lighting a rocket cracker it gets projected in a parabolic path and reaches a maximum height of 4mts when it is 6 mts away from the point of projection.
Finally it reaches the ground 12 mts away from the starting point. Find the angle of projection. Assume that water issuing from the end of a horizontal pipe, 7.
The vertex of the parabolic path is at the end of the pipe. At a position 2. How far beyond this vertical line will the water strike the ground?
A comet is moving in a parabolic orbit around the sun which is at the focus of a parabola. Take the orbit as open rightward. A cable of a suspension bridge hangs in the form of a parabola when the load is uniformly distributed horizontally. The distance between two towers if ft, the points of support of the cable on the towers are ft above the road way and the lowest point on the circle is 70ft above the roadway.
Find the vertical distance to the cable from a pole whose height is ft. Find the eccentricity, centre, foci, vertices of the following ellipses: An arch is in the form of a semi-ellipse whose span is 48 feet wide. The height of the arch is 20 feet. How wide is the arch at a height of 10 feet above the base?
The ceiling in a hallway 20ft wide is in the shape of a semi ellipse and 18ft high at the centre. Find the height of the ceiling 4 feet from either wall if the height of the side walls is 12ft. A ladder of length 15m moves with its ends always touching the vertical wall and the horizontal floor.
Determine the equation of the locus of a point P on the ladder, which is 6m from the end of the ladder in contact with the floor. Find the eccentricity, centre, foci, and vertices of the following hyperbola and draw the diagram: A boy, who is standing on a pole of height It moves in a vertical line slightly away from the pole and falls on the ground. Its equation of motion in meters and seconds is 2 9. A ladder 10m long rests against a vertical wall.
Example 5. Both are headed for the intersection of the two roads. At what rate are the cars approaching each other when car A is 0. A water tank has the shape of an inverted circular cone with base radius 2 metres and height 4 metres. Find i the initial velocity of the missile, ii the time when the height of the missile is a maximum iii the maximum height reached and iv the velocity with which the missile strikes the ground. At what rate is the base of the triangle changing when the altitude is 10 cm and the area is cm2.
How fast is the distance between the ships changing at 4.
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How fast is the height of the pile increasing when the pile is 10 ft high? Prove that the slope at Q is four times the slope at P.
A farmer has feet of fencing and want to fence of a rectangular field that borders a straight river. He needs no fence along the river. What ar the dimensions of the field that has the largest area? Find the area of the largest rectangle that can be inscribed in a semi circle of radius r.
The top and bottom margins of a poster are each 6 cms and the side margins are each 4cms. If the area of the printed material on the poster is fixed at cms2 , find the dimension of the poster with the smallest area. Show that the volume of the largest right circular cone that can be inscribed in a sphere of radius a is 27 8 volume of the sphere.
A closed cuboid box with a square base is to have a volume of c. The material for the top and bottom of the box is to cost Rs. If the cost of the materials is to be the least, find the dimensions of the box.
A man is at a point P on a bank of a straight river, 3 km wide, and wants to reach point Q, 8 km downstream on the opposite bank, as quickly as possible. He could row his boat directly across the river to point R and then run to Q, or he could row directly to Q, or he could row to some point to between Q and R and then run to Q.
Find the intervals of concavity and the points of inflection of the following functions: Example 7. Solve the following: Solve the following differential equations: In a certain chemical reaction the rate of conversion of a substance at time t is proportional to the quantity of the substance still untransformed at that instant.
At the end of one hour. How many grams of the first substance was there initially? Example 8. A bank pays interest by continuous compounding, that is by treating the interest rate as the instantaneous rate of change of principal. Calculate the percentage increase in such an account over one year.
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For a postmortem report, a doctor requires to know approximately the time of death of the deceased. He records the first temperature at After 2 hours he finds the temperature to be Assume normal temperature of a human body to be A drug is excreted in a patients urine. The urine is monitored continuously using a catheter.
The number of bacteria in a yeast culture grows at a rate which is proportional to the number present. If the population of a colony of yeast bacteria triples in 1 hour.
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Show that the number of bacteria at the end of five hours will be 5 3 times of the population at initial time. In how many years will the amount be twice the original principal? Find its temperature after a further interval of 5 minutes. If there were 1,30, people in the city in and 1,60, in what population may be anticipated in ?
When its mass is 10 mgm, the rate of disintegration is 0. How long will it take for the mass to be reduced from 10 mgm to 5 mgm. Example 9. Show that the nth roots of unity form an abelian group of finite order with usual multiplication. Example An urn contains 4 white and 3 red balls. Find the probability distribution of number of red balls in three draws one by one from the urn.
If the number of incoming buses per minute at a bus terminus is a random variable having a Poisson distribution with ,9.
If X is normally distributed with mean 6 and standard deviation 5 find. The mean score of students for an examination is 34 and S. The air pressure in a randomly selected tyre put on a certain model new car is normally distributed with mean value 31 psi and standard deviation 0.
Assuming the weights are normally distributed, find how many students weigh i between and pounds ii more than pounds.
Solve by matrix inversion method each of the following system of linear equations: Find the rank of the following matrices: If it is consistent then solve the same. Solve the following system of linear equations by determinant method.
Find the work done by the forces.
It is enough to prove any three vectors formed by these four points are coplanar. Find the equation of the plane. Justify your answer.
Also find the equation in Cartesian form. Find also the centre and radius of the sphere. Angle in a semi-circle is a right angle.
Prove by vector method. Diagonals of a rhombus are at right angles. Prove by vector methods. Find the vector and Cartesian equations of the straight line passing through the points -5,2,3 and 4,-3,6 Example 2.
Show that the points 3,-1,-1 , 1,0,-1 and 5,-2,-1 are collinear. The foot of perpendicular drawn from the origin to the plane is 4,-2,-5 , find the equation of the plane. Find the point of intersection of the line passing through the two points 1,1,-1 ; -1,0,1 and the xy -plane.
Take the orbit as open rightward. The distance between two towers if ft, the points of support of the cable on the towers are ft above the road way and the lowest point on the circle is 70ft above the roadway. Find the vertical distance to the cable from a pole whose height is ft.
The height of the arch is 20 feet. How wide is the arch at a height of 10 feet above the base? Find the height of the ceiling 4 feet from either wall if the height of the side walls is 12ft. Determine the equation of the locus of a point P on the ladder, which is 6m from the end of the ladder in contact with the floor. It moves in a vertical line slightly away from the pole and falls on the ground. Its equation of motion in meters and seconds is x 9.
Both are headed for the intersection of the two roads. At what rate are the cars approaching each other when car A is 0. A missile fired from ground level rises x metres vertically upwards in t seconds and x t 25 t Find i the initial velocity of the missile, ii the time when the height of the missile is a maximum iii the maximum height reached and iv the velocity with which the missile strikes the ground.Prove that the slope at Q is four times the slope at P.
The crisis of foundations was stimulated by a number of controversies at the time, including the controversy over Cantor's set theory and the Brouwer—Hilbert controversy.
Query Query. The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Revision notes are found at the start of each chapter, emphasising important concepts and formulae in the chapter. Applied Mathematics - I is one of the important subject in Amity University. The history of mathematics can be seen as an ever-increasing series of abstractions.