HSC Commerce (English Medium) 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

The company makes concrete bricks made up of cement and sand. The weight of a concrete brick has to be at least 5 kg. Cement costs ₹ 20 per kg and sand costs of ₹ 6 per kg. Strength consideration dictates that a concrete brick should contain minimum 4 kg of cement and not more than 2 kg of sand. Form the L.P.P. for the cost to be minimum.

Solve the following L.P.P. by graphical method:

Minimize: z = 8x + 10y

Subject to: 2x + y ≥ 7, 2x + 3y ≥ 15, y ≥ 2, x ≥ 0, y ≥ 0.

Solve the following LPP:

Maximize z = 6x + 10y subject to 3x + 5y ≤ 10, 5x + 3y ≤ 15, x ≥ 0, y ≥ 0.

Solve the following LPP:

Minimize z = 4x + 2y

Subject to 3x + y ≥ 27, x + y ≥ 21, x + 2y ≥ 30, x ≥ 0, y ≥ 0

A firm manufactures two products A and B on which profit earned per unit are ₹ 3 and ₹ 4 respectively. Each product is processed on two machines M₁ and M₂. The product A requires one minute of processing time on M₁ and two minutes of processing time on M₂, B requires one minute of processing time on M₁ and one minute of processing time on M₂. Machine M₁ is available for use for 450 minutes while M₂ is available for 600 minutes during any working day. Find the number of units of products A and B to be manufactured to get the maximum profit.

The edge of a cube is decreasing at the rate of`( 0.6"cm")/sec`. Find the rate at which its volume is decreasing, when the edge of the cube is 2 cm.

Test whether the following functions are increasing or decreasing : f(x) = x³ – 6x² + 12x – 16, x ∈ R.

Test whether the following functions are increasing or decreasing : f(x) = 2 – 3x + 3x² – x³, x ∈ R.

Test whether the following functions are increasing or decreasing: f(x) = `x-(1)/x`, x ∈ R, x ≠ 0.

Find the values of x for which the following functions are strictly increasing : f(x) = 2x³ – 3x² – 12x + 6

Find the values of x for which the following functions are strictly increasing:

f(x) = 3 + 3x – 3x² + x³

Find the values of x for which the following func- tions are strictly increasing : f(x) = x³ – 6x² – 36x + 7

Find the values of x for which the following functions are strictly decreasing:

f(x) = 2x³ – 3x² – 12x + 6

Find the values of x for which the following functions are strictly decreasing : f(x) = `x + (25)/x`

Find the values of x for which the following functions are strictly decreasing : f(x) = x³ – 9x² + 24x + 12

Find the values of x for which the function f(x) = x³ – 12x² – 144x + 13 (a) increasing (b) decreasing

Find the values of x for which f(x) = `x/(x^2 + 1)` is (a) strictly increasing (b) decreasing.

show that f(x) = `3x + (1)/(3x)` is increasing in `(1/3, 1)` and decreasing in `(1/9, 1/3)`.

Show that f(x) = x – cos x is increasing for all x.

Show that y = `log (1 + x) – (2x)/(2 + x), x > - 1` is an increasing function on its domain.