Science (English Medium) कक्षा १२ - CBSE Important Questions for Mathematics

If `"x" = "e"^(cos2"t")  "and"  "y" = "e"^(sin2"t")`, prove that `(d"y")/(d"x") = - ("y"log"x")/("x"log"y")`.

If y = (log x)^x + x^log x, find `"dy"/"dx".`

The value of ‘k’ for which the function f(x) = `{{:((1 - cos4x)/(8x^2)",",  if x ≠ 0),(k",",  if x = 0):}` is continuous at x = 0 is ______.

If `ysqrt(1 - x^2) + xsqrt(1 - y^2)` = 1, then prove that `(dy)/(dx) = - sqrt((1 - y^2)/(1 - x^2))`

The function f(x) = x | x |, x ∈ R is differentiable ______.

If x = a cos t and y = b sin t, then find `(d^2y)/(dx^2)`.

If f(x) = | cos x |, then `f((3π)/4)` is ______.

The set of all points where the function f(x) = x + |x| is differentiable, is ______.

Read the following passage and answer the questions given below:

1. Find the rate of growth of the plant with respect to the number of days exposed to the sunlight.
2. Does the rate of growth of the plant increase or decrease in the first three days? What will be the height of the plant after 2 days?

The amount of pollution content added in air in a city due to x-diesel vehicles is given by P(x) = 0.005x³ + 0.02x² + 30x. Find the marginal increase in pollution content when 3 diesel vehicles are added and write which value is indicated in the above question.

Find the absolute maximum and absolute minimum values of the function f given by f(x)=sin²x-cosx,x ∈ (0,π)

The Volume of cube is increasing at the rate of 9 cm 3/s. How fast is its surfacee area increasing when the length of an edge is 10 cm?

The total cost C(x) in rupees associated with the production of x units of an item is given by C(x) = 0.007x³ – 0.003x² + 15x + 4000. Find the marginal cost when 17 units are produced

A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening

The volume of a sphere is increasing at the rate of 8 cm³/s. Find the rate at which its surface area is increasing when the radius of the sphere is 12 cm.

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 per cm2 and the material for the sides costs Rs 2.50 per cm2. Find the least cost of the box

Verify Rolle's theorem for the following function on the indicated interval f (x) = log (x² + 2) − log 3 on [−1, 1] ?

Show that the cone of the greatest volume which can be inscribed in a given sphere has an altitude equal to \[ \frac{2}{3} \] of the diameter of the sphere.

A given quantity of metal is to be cast into a half cylinder with a rectangular base and semicircular ends. Show that in order that the total surface area may be minimum the ratio of the length of the cylinder to the diameter of its semi-circular ends is \[\pi : (\pi + 2)\].

Find the intervals in which function f given by f(x)  = 4x³ - 6x² - 72x + 30 is (a) strictly increasing, (b) strictly decresing .