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

Read the following passage and answer the questions given below.

1. If the length and the breadth of the rectangular field be 2x and 2y respectively, then find the area function in terms of x.
2. Find the critical point of the function.
3. Use First derivative Test to find the length 2x and width 2y of the soccer field (in terms of a and b) that maximize its area.
   OR
   Use Second Derivative Test to find the length 2x and width 2y of the soccer field (in terms of a and b) that maximize its area.

If the circumference of circle is increasing at the constant rate, prove that rate of change of area of circle is directly proportional to its radius.

If equal sides of an isosceles triangle with fixed base 10 cm are increasing at the rate of 4 cm/sec, how fast is the area of triangle increasing at an instant when all sides become equal?

The interval in which the function f(x) = 2x³ + 9x² + 12x – 1 is decreasing is ______.

The median of an equilateral triangle is increasing at the ratio of `2sqrt(3)` cm/s. Find the rate at which its side is increasing.

Sum of two numbers is 5. If the sum of the cubes of these numbers is least, then find the sum of the squares of these numbers.

The function f(x) = x³ + 3x is increasing in interval ______.

Find the interval/s in which the function f : R `rightarrow` R defined by f(x) = xe^x, is increasing.

If f(x) = `1/(4x^2 + 2x + 1); x ∈ R`, then find the maximum value of f(x).

Find the maximum profit that a company can make, if the profit function is given by P(x) = 72 + 42x – x², where x is the number of units and P is the profit in rupees.

Check whether the function f : R `rightarrow` R defined by f(x) = x³ + x, has any critical point/s or not ? If yes, then find the point/s.

Evaluate : `int_0^3dx/(9+x^2)`

Evaluate :`int_(pi/6)^(pi/3) dx/(1+sqrtcotx)`

Evaluate: `int(5x-2)/(1+2x+3x^2)dx`

Evaluate : `int_0^4(|x|+|x-2|+|x-4|)dx`

find : `int(3x+1)sqrt(4-3x-2x^2)dx`

Find : `int x^2/(x^4+x^2-2) dx`

Evaluate :`int_0^(pi/2)(2^(sinx))/(2^(sinx)+2^(cosx))dx`

Find:

`int(x^3-1)/(x^3+x)dx`

Find : `int((2x-5)e^(2x))/(2x-3)^3dx`