Arts (English Medium) इयत्ता १२ - CBSE Important Questions for Mathematics

Find the local maxima and local minima, of the function f(x) = sin x − cos x, 0 < x < 2π.

Find the value(s) of x for which y = [x(x − 2)]² is an increasing function.

Find the equations of the tangent and normal to the curve `x^2/a^2−y^2/b^2=1` at the point `(sqrt2a,b)` .

Show that the function f(x) = 4x³ - 18x² + 27x - 7 is always increasing on R.

The total cost C(x) associated with the production of x units of an item is given by C(x) = 0.005x³ – 0.02x² + 30x + 5000. Find the marginal cost when 3 units are produced, whereby marginal cost we mean the instantaneous rate of change of total cost at any level of output.

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.

A particle moves along the curve 3y = ax³ + 1 such that at a point with x-coordinate 1, y-coordinate is changing twice as fast at x-coordinate. Find the value of a.

Read the following passage:

Engine displacement is the measure of the cylinder volume swept by all the pistons of a piston engine. The piston moves inside the cylinder bore. One complete of a four-cylinder four-stroke engine. The volume displace is marked The cylinder bore in the form of circular cylinder open at the top is to be made from a metal sheet of area 75π cm2.

Based on the above information, answer the following questions:

1. If the radius of cylinder is r cm and height is h cm, then write the volume V of cylinder in terms of radius r. (1)
2. Find `(dV)/(dr)`. (1)
3. (a) Find the radius of cylinder when its volume is maximum. (2)
   OR
   (b) For maximum volume, h > r. State true or false and justify. (2)

Read the following passage:

Based on the above information, answer the following questions:

1. Can the above function be used to estimate number of vehicles in the year 2000? Justify. (2)
2. Prove that the function V(t) is an increasing function. (2)

Write the antiderivative of `(3sqrtx+1/sqrtx).`

Evaluate : `∫(sin^6x+cos^6x)/(sin^2x.cos^2x)dx`

Evaluate : `int(x-3)sqrt(x^2+3x-18)  dx`

Evaluate `int_0^(pi)e^2x.sin(pi/4+x)dx`

Find `intsqrtx/sqrt(a^3-x^3)dx`

Evaluate `int_(-1)^2|x^3-x|dx`

Integrate the following w.r.t. x `(x^3-3x+1)/sqrt(1-x^2)`

Evaluate :

`∫_(-pi)^pi (cos ax−sin bx)^2 dx`

Evaluate `int_1^3(e^(2-3x)+x^2+1)dx`  as a limit of sum.

If `f(x) =∫_0^xt sin t dt` , then write the value of f ' (x).

Evaluate :

`∫_0^π(4x sin x)/(1+cos^2 x) dx`