QUESTION BANK- MATHS

MAXIMUM AND MINIMUM 

PART-

1) Find two numbers whose sum is 24 and whose product is as large as possible.  12,12

2) Find two positive numbers x and y such that x+ y= 60 and xy³ is maximum.    15,45

3) Find two positive numbers x and y such that the sum is 35 and the product x²y⁵ is maximum.      10,25

4) Amongst all pairs of positive numbers with product 256, find whose sum is the least.         16,16

5) Find two positive numbers whose sum is 14 and the sum of whose squares is minimum .     7,7

6) Find the maximum value of ax+ by, where xy= c² and a,b,c are positive.

7) Show that all the rectangles with a given perimeter, the square has the largest area.

8) Show that of all rectangles of given area, the square has the smallest perimeter.

9) Show that of all the rectangles inscribed in a given circle, the square has the maximum area.

10) Show that the rectangle of maximum perimeter which can be inscribed of radius a is a square of side √2 a.

11) AB is a diameter of a circle and C is any point on the circle. Show that the area of  ∆ ABC is maximum, when it is isosceles.

12) If the sum of the length of the hypotenuse and a side of a right angled triangle is  given, show that the area of the triangle is maximum when the angle between is π/3.

13) Show that the area of right angled triangle of given hypotenuse is maximum when the triangle is isosceles.

14) Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is cube.

15) An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of the material will be least when depth of the tank is half of its width.

16) A metal box with a square base and vertical sides is to contain 1024 cm³ of water,  the material for the top and bottom costs Rs 5 per cm² and the material for the sides costs Rs 2.50 per cm². Find the least cost of the box.     8x8x16, Rs1920

17) An open box with a square base is to be made out of a given quantity of cardboard of area c¹ square units. Show that the maximum volume of the boxe is c³/6√3. cubic units.

18) The sum of the surface area of a rectangular parallelopiped with sides x, 2x and x/3 and a sphere is given to be constant. Prove that the sum of the volume is minimum, if x is equal to 3 times the radius of the sphere. also, find the minimum value of the sum of their volumes.     (2/3) x³(1+ 2π/27)

19) a wire of length 36 m is to be cut into two pieces . One of the pieces is to be made into a square and the other into a circle. What should be the lengths of the two pieces, so that the combined area of the square and the circle is minimum.

20) A figure consists of a semi-circle with a rectangle on its diameter. Given the perimeter of the figure, find its dimensions in order that the area may be maximum.   2P/(π+4), P/(π+4)

21) A square piece of tin of side 24cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. What should be the side of the square to be cut off so that the volume of the box is maximum? Also, find this maximum volume.     1024 cm³

22) A rectangular sheet of fix perimeter with sides having their length in the ratio 8:15 is converted into an open rectangular box by folding after removing squares of equal area from all 4 corners. if the total area of the removed square is 100 square unit, the resultant box has maximum volume. Find the length of the sides of the rectangular sheet .    24

23) Show that a cylinder of a given volume which is open at the top, has minimum total surface area, provided its height is equal to the radius of its base .  

24) Show that the height of the closed cylinder of given surface and maximum volume, is equal to the diameter of its base.

25) Show that the height of a cylinder, which is open at the top, having a given surface area and greatest volume, is equal to the radius of its base.

26) A telephone company in a town has 500 subscribers on its list and collects fixed charges of Rs 300 per subscriber. The company purposes to increase the annual subscription and it is believed that every increase of Rs 1 one subscriber will discontinue the service. Find what increase will bring maximum revenue.    100

27) Find the point on the curve y²= 4x which is nearest to the point (2,1). (1,2)

28) A jet age of an enemy is flying along the curve y= x²+ 2. A soldier is placed at the point (3,2). What is the shortest distance between the soldier and the jet ?  √5