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An open box with square base is to be made of a given quantity of cardboard of area c2. Show that the maximum volume of the box is `"c"^3/(6sqrt(3))` cubic units
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Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible, when revolved about one of its sides. Also, find the maximum volume.
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If the sum of the surface areas of cube and a sphere is constant, what is the ratio of an edge of the cube to the diameter of the sphere, when the sum of their volumes is minimum?
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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.
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A metal box with a square base and vertical sides is to contain 1024 cm3. The material for the top and bottom costs Rs 5/cm2 and the material for the sides costs Rs 2.50/cm2. Find the least cost of the box.
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The sum of the surface areas of a rectangular parallelopiped with sides x, 2x and `x/3` and a sphere is given to be constant. Prove that the sum of their volumes is minimum, if x is equal to three times the radius of the sphere. Also find the minimum value of the sum of their volumes.
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If x is real, the minimum value of x2 – 8x + 17 is ______.
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The smallest value of the polynomial x3 – 18x2 + 96x in [0, 9] is ______.
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The function f(x) = 2x3 – 3x2 – 12x + 4, has ______.
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The maximum value of sin x . cos x is ______.
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Maximum slope of the curve y = –x3 + 3x2 + 9x – 27 is ______.
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The maximum value of `(1/x)^x` is ______.
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The curves y = 4x2 + 2x – 8 and y = x3 – x + 13 touch each other at the point ______.
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If A, B be two square matrices such that |AB| = O, then ____________.
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A square matrix A is invertible if det A is equal to ____________.
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Which of the following is a second order differential equation?
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Integrating factor of the differential equation `(1 - x^2) ("d"y)/("d"x) - xy` = 1 is ______.
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The solution of `(1 + x^2) ("d"y)/("d"x) + 2xy - 4x^2` = 0 is ______.
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If A `= [(0,-1,2),(1,0,3),(-2,-3,0)],` then A + 2AT equals
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Find the adjoint of the matrix A `= [(1,2),(3,4)].`
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