Science (English Medium) इयत्ता १२ - CBSE Question Bank Solutions for Mathematics

`tan^-1 (("a"cosx - "b"sinx)/("b"cosx - "a"sinx)), - pi/2 < x < pi/2` and `"a"/"b" tan x > -1`

`sec^-1 (1/(4x^3 - 3x)), 0 < x < 1/sqrt(2)`

`tan^-1 ((3"a"^2x - x^3)/("a"^3 - 3"a"x^2)), (-1)/sqrt(3) < x/"a" < 1/sqrt(3)`

`tan^-1 ((sqrt(1 + x^2) + sqrt(1 - x^2))/(sqrt(1 + x^2) - sqrt(1 - x^2))), -1 < x < 1, x ≠ 0`

If x^m . yⁿ = (x + y)^m+n, prove that `("d"^2"y")/("dx"^2)` = 0

If y = `sqrt(sinx + y)`, then `"dy"/"dx"` is equal to ______.

For the curve `sqrt(x) + sqrt(y)` = 1, `"dy"/"dx"` at `(1/4, 1/4)` is ______.

Show that the function f(x) = 4x³ – 18x² + 27x – 7 has neither maxima nor minima.

Find all the points of local maxima and local minima of the function f(x) = `- 3/4 x^4 - 8x^3 - 45/2 x^2 + 105`

Let f have second derivative at c such that f′(c) = 0 and f"(c) > 0, then c is a point of ______.

If the sum of the lengths 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 them is `pi/3`

Find the points of local maxima, local minima and the points of inflection of the function f(x) = x⁵ – 5x⁴ + 5x³ – 1. Also find the corresponding local maximum and local minimum values.

A telephone company in a town has 500 subscribers on its list and collects fixed charges of Rs 300/- per subscriber per year. The company proposes to increase the annual subscription and it is believed that for every increase of Re 1/- one subscriber will discontinue the service. Find what increase will bring maximum profit?

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

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.

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?

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.

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.

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.

If x is real, the minimum value of x² – 8x + 17 is ______.