HSC Science (Computer Science) इयत्ता १२ वी - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

Differentiate `tan^-1((sqrt(1 + x^2) - 1)/x)` w.r.t. `cos^-1(sqrt((1 + sqrt(1 + x^2))/(2sqrt(1 + x^2))))`

If y² = a²cos²x + b²sin²x, show that `y + (d^2y)/(dx^2) = (a^2b^2)/y^3`

If log y = log (sin x) – x², show that `(d^2y)/(dx^2) + 4x "dy"/"dx" + (4x^2 + 3)y` = 0.

If x= a cos θ, y = b sin θ, show that `a^2[y(d^2y)/(dx^2) + (dy/dx)^2] + b^2` = 0.

If y = Ae^mx + Be^nx, show that y₂ – (m + n)y₁ + mny = 0.

The lines `(x - 2)/(1) = (y - 3)/(1) = (z - 4)/(-k) and (x - 1)/k = (y - 4)/(2) = (z - 5)/(1)` are coplnar if ______.

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.