Arts (English Medium) कक्षा १२ - CBSE Important Questions

If f(x) = `{{:((kx)/|x|"," if x < 0),(  3","   if x ≥ 0):}` is continuous at x = 0, then the value of k 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 equation of the normal at a point on the curve x² = 4y which passes through the point (1, 2). Also find the equation of the corresponding tangent.

Find the equation of the normal at a point on the curve x² = 4y which passes through the point (1, 2). Also find the equation of the corresponding tangent.

The equation of tangent at (2, 3) on the curve y² = ax³ + b is y = 4x – 5. Find the values of a and b.

Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is `6sqrt3` r.

Find the equation of tangents to the curve y= x³ + 2x – 4, which are perpendicular to line x + 14y + 3 = 0.

If the function f(x)=2x³−9mx²+12m²x+1, where m>0 attains its maximum and minimum at p and q respectively such that p²=q, then find the value of m.

Prove that `y=(4sintheta)/(2+costheta)-theta `

Show that the equation of normal at any point t on the curve x = 3 cos t – cos³t and y = 3 sin t – sin³t is 4 (y cos³t – sin³t) = 3 sin 4t

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

If the sum of the lengths of the hypotenuse and a side of a right triangle is given, show that the area of the triangle is maximum, when the angle between them is 60º.

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 points on the curve 9y² = x³, where the normal to the curve makes equal intercepts with the axes are

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